Nonlinear Fractional Schrödinger Equations in R^N (Frontiers in Mathematics) 3030602192, 9783030602192

Citation preview

Frontiers in Elliptic and Parabolic Problems

Vincenzo Ambrosio

Nonlinear Fractional Schrödinger Equations in

Frontiers in Mathematics

Frontiers in Elliptic and Parabolic Problems Series Editor Michel Chipot, Institute of Mathematics, Zürich, Switzerland

The goal of this series is to reflect the impressive and ongoing evolution of dealing with initial and boundary value problems in elliptic and parabolic PDEs. Recent developments include fully nonlinear elliptic equations, viscosity solutions, maximal regularity, and applications in finance, fluid mechanics, and biology, to name a few. Many very classical notions have been revisited, such as degree theory or Sobolev spaces. Books in this series present the state of the art keeping applications in mind wherever possible. The series is curated by the Series Editor.

More information about this subseries at http://www.springer.com/series/16595

Vincenzo Ambrosio

Nonlinear Fractional Schr¨odinger Equations in RN

Vincenzo Ambrosio Dipartimento di Ingegneria Industriale e Scienze Matematiche Universit`a Politecnica delle Marche Ancona, Italy

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISSN 2730-549X ISSN 2730-5503 (electronic) Frontiers in Elliptic and Parabolic Problems ISBN 978-3-030-60219-2 ISBN 978-3-030-60220-8 (eBook) https://doi.org/10.1007/978-3-030-60220-8 Mathematics Subject Classification: 35R11, 35A15, 35B33, 35S05, 35J60, 35B09, 35B65, 35B40, 47G20, 58E05 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com, by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The aim of this book is to collect a set of results concerning nonlinear Schrödinger equations in the whole space driven by fractional operators. The material presented here was mainly taken from some papers carried out by the author and some joint works with his collaborators in these last years. It concerns some existence, multiplicity, and regularity results and qualitative properties of solutions established by means of suitable variational and topological methods which take care of the nonlocal character of the involved fractional operator. We deal with fractional Schrödinger equations involving different types of potentials and nonlinearities satisfying certain growth assumptions. Moreover, fractional Kirchhoff problems, fractional Choquard equations, fractional Schrödinger– Poisson systems, and fractional Schrödinger equations with magnetic field are considered. The book is principally addressed to researchers interested in pure and applied mathematics, physics, mechanics, and engineering. It is also appropriate for graduate students in mathematical and applied sciences, who will find updated information and a systematic exposition of important parts of modern mathematics. In particular, the proofs are given in detail and some of them are presented in a more different and elegant way with respect to the original ones in order to make the exposition more accessible and attractive for the interested reader. The author would like to express his sincere gratitude to some dear friends and colleagues, including C. O. Alves, V. Coti Zelati, P. d’Avenia, G. M. Figueiredo, T. Isernia, G. Molica Bisci, H.-M. Nguyen, and E. Valdinoci, for their friendship, encouragement, and discussions on mathematics of common interest. Finally, the author would like to thank Prof. M. Chipot for giving him the opportunity to write this book. Ancona, Italy 31 July 2020

Vincenzo Ambrosio

v

Introduction

Recently, a great attention has been devoted to the study of nonlinear problems involving fractional elliptic operators, whose prototype is given by the fractional Laplacian operator (−)s defined for any u : RN → R sufficiently smooth by  (−) u(x) = C(N, s)P .V . s

RN

u(x) − u(y) dy |x − y|N+2s

where s ∈ (0, 1), N ∈ N, P .V . stands for the Cauchy principal value and C(N, s) is a positive constant depending only on N and s. Indeed, this operator has a great interest both for pure mathematical research, due to its intriguing structure, and in view of several concrete real-world applications, such as phase transition phenomena, crystal dislocation, population dynamics, flame propagation, chemical reactions of liquids, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, minimal surface, and game theory. Moreover, from a probabilistic point of view, the fractional Laplacian is the infinitesimal generator of a (rotationally) symmetric 2s-stable Lévy process. For a very nice introduction on fractional nonlocal operators and their applications, we refer to [106, 115, 168, 203, 273]. In this book we focus on the following nonlinear fractional Schrödinger equation: (−)s u + V (x)u = f (x, u) in RN ,

(1)

where V : RN → R is a continuous potential and f : RN × R → R is a suitable nonlinear term. Equation (1) arises in the study of the time-dependent fractional Schrödinger equation ı

∂ = (−)s  + V (x) − F (x, ) in RN × R, ∂t

(2)

vii

viii

Introduction

when we look for standing wave solutions, which is solutions of the form (x, t) = u(x)e−ıct , where c is a constant. We recall that (2) plays a fundamental role in the study of fractional quantum mechanics. It has been discovered by Laskin [245, 246] as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths; see also [158, 247] for more physical background. We stress that for N > 1, when s → 1− , (−)s u → −u for all u ∈ Cc∞ (RN ), and Eq. (1) boils down to the well-known nonlinear Schrödinger equation −u + V (x)u = f (x, u) in RN , which has been widely investigated in these last 30 years by many authors [1, 13, 28, 90, 146, 154, 165, 197, 284, 299, 330] due to its relevance in several physical problems arising in nonlinear optics, plasma physics, and condensed matter physics, only to cite a few. Motivated by the interest shared by the mathematical community in nonlocal fractional problems, the goal of this monograph is to present some recent results concerning the existence, multiplicity, and qualitative properties of different types of solutions to nonlinear fractional Schrödinger equations by applying suitable variational and topological methods. Indeed, due to the variational nature of (1), we will look for critical points of the functional      |u(x) − u(y)|2 1 C(N, s) 2 J (u) = dxdy + V (x)u (x) dx − F (x, u) dx N+2s 2 2 R2N |x − y| RN RN

t where F (x, t) = 0 f (x, τ ) dτ . For our purpose, the constant C(N,s) appearing in the 2 above definition does not play an essential role, so, along the book, we will omit it. The book consists of 17 chapters and is organized as follows. In Chap. 1, we recall some basic properties of fractional Sobolev spaces, we give different definitions of the fractional Laplacian, and we present some regularity theorems, maximum principles, technical lemmas, and a concentration-compactness principle which will be frequently used along the book. Chapter 2 addresses variational and topological principles and critical point theory, including minimax theorems, genus and category theory, and the recent approach developed by Szulkin and Weth [321,322] concerning the Nehari method for non-differentiable manifolds. In Chap. 3, we consider fractional scalar field equations of the type (−)s u = g(u) in RN , where g is a Berestycki-Lions type nonlinearity with subcritical or critical growth. We use the mountain pass theorem [29] to deduce the main existence and multiplicity results. We further analyze regularity, decay, and symmetry properties of ground state solutions.

Introduction

ix

In Chap. 4, we focus on the existence and qualitative properties of ground state solutions for a fractional Schrödinger equation driven by the pseudo-relativistic operator (−+m2)s used in the study of stability of relativistic matter; see [250, 251]. In Chap. 5, we introduce fractional Schrödinger equations with periodic potentials and bounded potentials, involving nonlinearities with subcritical or critical growth. A compactness lemma in this last case is proved. In Chap. 6, we study existence, multiplicity, and concentration of positive solutions for ∗

ε2s (−)s u + V (x)u = f (u) + γ |u|2s −2 u in RN ,

(3)

where ε > 0 is a small parameter, γ ∈ {0, 1}, f is a continuous nonlinearity with subcritical growth, and the potential V satisfies the following global condition due to Rabinowitz [299]: inf V (x) < lim inf V (x) ∈ (0, ∞]. |x|→∞

x∈RN

In Chap. 7, we construct a family of positive solutions to (3), which concentrates around a local minimum of the potential V that fulfills the following local condition introduced by del Pino and Felmer [165]: there exists a bounded open set  ⊂ RN such that inf V (x) < min V (x).

x∈

x∈∂

Here, we consider superlinear nonlinearities satisfying the Ambrosetti-Rabinowitz condition [29] (shortly (AR)), and the monotonicity assumption t → f (tt ) is increasing for t > 0, which plays a fundamental role in applying Nehari method. An existence result for a supercritical fractional Schrödinger equation is also discussed. In Chap. 8, we investigate the previous problem in the subcritical case with asymptotically linear nonlinearities as well as superlinear nonlinearities which do not satisfy (AR). We point out that no monotonicity assumption on the function f (tt ) is required. In Chap. 9, we study the multiplicity and concentration of positive solutions for the following fractional Choquard equation:  2s

ε (−) u + V (x)u = ε s

μ−N

 1 ∗ F (u) f (u) in RN , |x|μ

where 0 < μ < 2s and f is a continuous nonlinearity with subcritical growth. The main results will be obtained by applying suitable variational methods and LusternikSchnirelman theory.

x

Introduction

In Chap. 10, we examine the following fractional Kirchhoff equation [196]   M

R2N

 |u(x) − u(y)|2 dxdy (−)s u = g(u) in RN , |x − y|N+2s

where M(t) = p + qt and g fulfills Berestycki-Lions type assumptions. A multiplicity result is obtained provided that q > 0 is small enough. Chapter 11 focuses on the multiplicity and concentration of positive solutions for a fractional Kirchhoff problem in R3 under del Pino and Felmer assumptions on the potential. In Chap. 12, we show the existence of positive concentrating solutions for a fractional Kirchhoff equation with critical growth. Chapter 13 concerns the study of a fractional Schrödinger–Poisson system with critical growth 

∗

ε2s (−)s u + V (x)u + φu = f (u) + |u|2s −2 u in R3 , in R3 , ε2t (−)t φ = u2

where s ∈ ( 34 , 1), t ∈ (0, 1), V satisfies local conditions, and f is a subcritical nonlinearity. Applying variational and topological arguments, we relate the number of positive solutions with the topology of the set where the potential attains its minimum value. In Chap. 14, we obtain an existence result for the fractional Kirchhoff-Schrödinger– Poisson system   p + q R6 (−)t φ =

|u(x)−u(y)|2 |x−y|N+2s μu2

dxdy (−)s u + μφu = g(u) in R3 , in R3 ,

as long as μ > 0 is sufficiently small, by means of the Struwe-Jeanjean monotonicity trick. Chapter 15 is devoted to the multiplicity of positive solutions for the non-homogeneous fractional Schrödinger equation (−)s u + V (x)u = k(x)f (u) + h(x) in RN , where k(x) is a bounded positive function and h is a L2 -small perturbation. In Chap. 16, we use a minimization argument and a quantitative deformation lemma to deal with the existence of sign-changing solutions for a fractional Schrödinger equation with vanishing potentials. In Chap. 17 we consider the following fractional magnetic Schrödinger equation: ε2s (−)sA u + V (x)u = f (|u|2 )u in RN ,

Introduction

xi

where A : RN → RN is a smooth magnetic potential and  (−)sA u(x)

= C(N, s)P .V .

x+y

RN

u(x) − u(y)eıA( 2 |x − y|N+2s

)·(x−y)

dy

is the magnetic fractional Laplacian introduced in [157, 225]. We present a multiplicity result when the potential V satisfies the global Rabinowitz condition and an existence result under local del Pino–Felmer assumptions. We also prove a multiplicity result when s = 12 , N = 1 and f has exponential critical growth at infinity. A list of references and an index conclude the book.

Contents

1

Preliminaries . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.1.1 Fractional Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.1.2 Sobolev Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.2 The Fractional Laplacian Operator . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.2.1 Definition via Singular Integrals .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.2.2 Definition via Fourier Transform: Riesz and Bessel Potentials .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.2.3 Definition via Caffarelli-Silvestre Extension .. . . . .. . . . . . . . . . . . . . . 1.3 Regularity Results and Maximum Principles . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.4 Some Useful Lemmas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 1.5 A Concentration-Compactness Principle.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

1 1 1 4 5 5 7 11 14 22 28

2

Some Abstract Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.1 Critical Point Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.2 Minimax Methods .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.3 Genus and Category .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 2.4 The Method of Nehari Manifold.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

35 35 38 43 46

3

Fractional Scalar Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.2 The Subcritical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.2.1 Introduction of a Penalty Functional.. . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.2.2 Comparison between Functionals.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 3.2.3 An Auxiliary Functional on the Augmented Space s (RN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . R × Hrad 3.2.4 Mountain Pass Value Gives the Least Energy Level . . . . . . . . . . . . . 3.2.5 Regularity, Symmetry and Asymptotic Behavior of Ground States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

51 51 56 56 59 64 69 75

xiii

xiv

Contents

3.3 3.4

3.5

The Zero Mass Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 88 The Critical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 92 3.4.1 Ground State for the Critical Case . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 92 3.4.2 Mountain Pass Characterization of Least Energy Solutions .. . . . 100 The Pohozaev Identity for (−)s . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 102

4

Ground States for a Pseudo-Relativistic Schrödinger Equation . . . . . . . . . . . . . 4.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.2 Preliminaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.2.1 The Extension Method for (− + m2 )s . . . . . . . . . .. . . . . . . . . . . . . . . 4.2.2 Local Schauder Estimates and Maximum Principles . . . . . . . . . . . . 4.3 A Minimization Argument via the Extension Method . . . . .. . . . . . . . . . . . . . . 4.4 Regularity, Decay and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.5 Passage to the Limit as m → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 4.5.1 Final Comments on (− + m2 )s . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

107 107 109 109 112 117 123 138 142

5

Ground States for a Superlinear Fractional Schrödinger Equation with Potentials . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.1 Nonlinearities with Subcritical Growth . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.1.2 Preliminaries .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.1.3 Periodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.1.4 Bounded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.2 Nonlinearities with Critical Growth .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.2.2 Splitting Lemmas .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 5.2.3 A Compactness Lemma when the Potential is Constant .. . . . . . . . 5.2.4 Ground States when the Potential is Not Constant .. . . . . . . . . . . . . .

145 145 145 148 150 156 159 159 162 170 187

Fractional Schrödinger Equations with Rabinowitz Condition . . . . . . . . . . . . . . 6.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.2 Preliminaries.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.3 The Subcritical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.3.1 The Nehari Method for (6.1.1) .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.3.2 The Autonomous Subcritical Problem.. . . . . . . . . . . .. . . . . . . . . . . . . . . 6.3.3 An Existence Result for (6.1.1) . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.3.4 A Multiplicity Result for (6.1.1) .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.3.5 Concentration Phenomenon for (6.1.1) .. . . . . . . . . . .. . . . . . . . . . . . . . . 6.4 The Critical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.4.1 The Nehari Method for (6.1.3) .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 6.4.2 The Autonomous Critical Problem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

195 195 199 207 207 213 219 225 232 236 236 238

6

Contents

6.4.3 An Existence Result for the Critical Case . . . . . . . . .. . . . . . . . . . . . . . . 6.4.4 A Multiplicity Result for (6.1.3) .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . A Remark on the Ambrosetti-Rabinowitz Condition .. . . . . .. . . . . . . . . . . . . . . Further Generalizations: The Fractional p-Laplacian Operator.. . . . . . . . . .

246 248 251 253

Fractional Schrödinger Equations with del Pino-Felmer Assumptions . . . . . 7.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.2 The Subcritical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.2.1 The Modified Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.2.2 Proof of Theorem 7.1.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.3 The Critical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.3.1 The Modified Critical Problem and the Local (PS) Condition.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.3.2 Proof of Theorem 7.1.2.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.4 A Supercritical Fractional Schrödinger Equation .. . . . . . . . . .. . . . . . . . . . . . . . . 7.4.1 The Truncated Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.4.2 A Moser Type Iteration Argument .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 7.5 Some Extensions to Fractional Schrödinger Systems . . . . . .. . . . . . . . . . . . . . .

255 255 258 258 270 276

6.5 6.6 7

8

xv

276 283 286 286 288 290

Fractional Schrödinger Equations with Superlinear or Asymptotically Linear Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.2 Modification of the Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.3 Mountain Pass Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.4 Limit Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 8.5 ε-Dependent Concentration-Compactness Result . . . . . . . . . .. . . . . . . . . . . . . . . 8.6 Proof of Theorem 8.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

295 295 297 300 316 320 332

Multiplicity and Concentration Results for a Fractional Choquard Equation . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 9.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 9.2 Variational Framework .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 9.3 The Autonomous Choquard Problem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 9.4 Multiplicity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 9.5 Proof of Theorem 9.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

335 335 338 350 355 359

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General Nonlinearity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 10.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 10.2 The Truncated Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 10.3 A Multiplicity Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

363 363 365 376

9

xvi

Contents

11 Multiplicity and Concentration of Positive Solutions for a Fractional Kirchhoff Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.2 The Modified Kirchhoff Problem.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.3 The Autonomous Kirchhoff Problem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.4 Multiple Solutions for (11.2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 11.5 Proof of Theorem 11.1.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

379 379 381 398 405 412

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical Growth . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 12.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 12.2 The Modified Critical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 12.3 The Autonomous Critical Fractional Kirchhoff Problem . .. . . . . . . . . . . . . . . 12.4 Proof of Theorem 12.1.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

417 417 419 430 432

13 Multiplicity and Concentration Results for a Fractional Schrödinger-Poisson System with Critical Growth . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.2 Functional Setting .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 13.3 An Existence Result for the Modified Problem .. . . . . . . . . . . .. . . . . . . . . . . . . . . 13.4 The Autonomous Schrödinger-Poisson Equation .. . . . . . . . . .. . . . . . . . . . . . . . . 13.5 Barycenter Map and Multiplicity of Solutions to (13.2.2) .. . . . . . . . . . . . . . . 13.6 Proof of Theorem 13.1.1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

443 443 447 458 466 470 477

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 483 14.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 483 14.2 Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional . . . . . . . . 484 15 Multiple Positive Solutions for a Non-homogeneous Fractional Schrödinger Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 15.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 15.2 The Asymptotically Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 15.3 The Superlinear Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

497 497 502 514

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with Vanishing Potential .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.2 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.3 The Nehari Manifold Argument .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.4 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 16.5 Existence and Multiplicity Results . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

521 521 525 531 537 545

Contents

xvii

17 Fractional Schrödinger Equations with Magnetic Fields . . . . . .. . . . . . . . . . . . . . . 17.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.2 A Multiplicity Result Under the Rabinowitz Condition .. . .. . . . . . . . . . . . . . . 17.2.1 The Variational Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.2.2 A First Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.2.3 A Multiplicity Result via Lusternik–Schnirelman Category.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.3 An Existence Result Under del Pino-Felmer Conditions .. .. . . . . . . . . . . . . . . 17.3.1 The Modified Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.3.2 Proof of Theorem 17.3.1: A Kato’s Inequality for (−)sA . . . . . . 17.4 A Multiplicity Result for a Fractional Magnetic Schrödinger Equation with Exponential Critical Growth.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.4.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 17.4.2 Variational Framework and Modified Problem.. . .. . . . . . . . . . . . . . . 17.4.3 Multiple Solutions for the Modified Problem . . . . .. . . . . . . . . . . . . . . 17.4.4 Proof of Theorem 17.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

553 553 556 556 562 575 583 585 593 610 613 619 632 637

Bibliography . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 645 Index . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 661

1

Preliminaries

In this chapter we collect some useful facts about the fractional Sobolev spaces, define the fractional Laplacian operator and give some its interesting properties. For more details, we refer to [2, 78, 166, 209, 244, 258, 268, 315, 326] and the recent works [115, 168, 203, 273].

1.1

Function Spaces

1.1.1

Fractional Sobolev Spaces

Let be an open set in RN . For any s ∈ (0, 1) and p ∈ [1, ∞), we define the fractional Sobolev space W s,p ( ) as  W

s,p

( ) = u ∈ L ( ) : p

|u(x) − u(y)| |x − y|

N+sp p

 2

∈ L ( ) , p

endowed with the norm 



uW s,p ( ) =

|u| dx + p



2

|u(x) − u(y)|p dxdy |x − y|N+sp

1

p

,

where the term   [u]

W s,p ( )

=

2

|u(x) − u(y)|p dxdy |x − y|N+sp

1

p

is the so-called Gagliardo (semi)norm of u. Here we used the notation 2 = × . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_1

1

2

1 Preliminaries

Next we list some useful properties on W s,p spaces. Proposition 1.1.1 ([168]) Let p ∈ [1, ∞) and 0 < s ≤ s < 1. Let ⊂ RN be an open set and u : → R be a measurable function. Then uW s,p ( ) ≤ CuW s ,p ( ) for some

positive constant C = C(N, s, p), and W s ,p ( ) ⊂ W s,p ( ). Proposition 1.1.2 ([168]) Let p ∈ [1, ∞) and 0 < s ≤ s < 1. Let ⊂ RN be an open set of class C 0,1 with bounded boundary and u : → R be a measurable function. Then uW s,p ( ) ≤ CuW 1,p ( ) for some positive constant C = C(N, s, p), and W 1,p ( ) ⊂ W s,p ( ). Remark 1.1.3 As proved in [334], the previous result holds true if one assumes that has the W 1,p -extension property. Lemma 1.1.4 ([334]) Let p ∈ [1, ∞) and s ∈ (0, 1). Let u ∈ W s,p ( ) and ϕ ∈ C 0,1 ( ) ∩ L∞ ( ). Then ϕu ∈ W s,p ( ) and there exists a positive constant C = C(N, s, p, ϕL∞ ( ) ) such that ϕuW s,p ( ) ≤ CuW s,p ( ) . Lemma 1.1.5 ([334]) Let p ∈ [1, ∞) and s ∈ (0, 1). Then the following assertions hold: (i) If u ∈ W s,p ( ), then |u|, u+ , u− ∈ W s,p ( ) and |u|W s,p ( ) ≤ uW s,p ( ) , u+ W s,p ( ) ≤ uW s,p ( ) , u− W s,p ( ) ≤ uW s,p ( ) , where u+ = max{u, 0} and u− = max{−u, 0}. (ii) If k ∈ R, k ≥ 0 and u ∈ W s,p ( ) is nonnegative, then u ∧ k = min{u, k} ∈ W s,p ( ) and u ∧ kW s,p ( ) ≤ uW s,p ( ) . (iii) If u1 , u2 ∈ W s,p ( ) are non-negative, then u = u1 ∨ u2 = max{u1 , u2 } and v = u1 ∧ u2 = min{u1 , u2 } are in W s,p ( ) and p

p

p

uW s,p ( ) + uW s,p ( ) ≤ u1 W s,p ( ) + u2 2W s,p ( ) . (iv) If u ∈ W s,p ( ) and n ∈ N, then (u ∧ n) ∨ (−n) ∈ W s,p ( ). When s > 1 and s ∈ / N, we can write s = m + σ , where m ∈ N and σ ∈ (0, 1). Then W s,p ( ) consists of the equivalence classes of functions u ∈ W m,p ( ) whose distributional derivatives D α u, with |α| = m, belong to W σ,p ( ). When s = m ∈ N, the space W s,p ( ) coincides with the classical Sobolev space W m,p ( ). Corollary 1.1.6 ([168]) Let p ∈ [1, ∞) and t ≥ s > 1. Let be an open set in RN of class C 0,1 . Then, W t,p ( ) ⊂ W s,p ( ).

1.1 Function Spaces

3

We also have the following density result. Theorem 1.1.7 ([2, 166, 258]) For any s ≥ 0, the space Cc∞ (RN ) of smooth functions with compact support is dense in W s,p (RN ). In general, if = RN , the space Cc∞ ( ) is not dense in W s,p ( ). Hence, for s ≥ 0, s,p we denote by W0 ( ) the closure of Cc∞ ( ) with respect to the  · W s,p ( ) -norm. When −s,p

s < 0 and p ∈ (1, ∞), we can define W s,p ( ) as the dual space of W0 ( ), where p

is the conjugate exponent of p, i.e., p1 + p1 = 1. It follows by the reflexivity of W s,p ( ) spaces, with s ∈ R and p ∈ (1, ∞), that the dual of W s,p (RN ) is isometrically isomorphic

to W −s,p (RN ), with p the conjugate exponent of p. Now, we focus on the case = RN , s ∈ (0, 1) and p = 2. We denote by Ds,2 (RN ) the completion of Cc∞ (RN ) with respect to the Gagliardo seminorm   [u]s = [u]W s,2 (RN ) =

R2N

|u(x) − u(y)|2 dxdy |x − y|N+2s

 12 ,

or equivalently (see [164]), when N > 2s, ∗

Ds,2 (RN ) = {u ∈ L2s (RN ) : [u]s < ∞}, where 2∗s =

2N N − 2s

is the fractional critical Sobolev exponent. Occasionally, we also use the notation  u, vDs,2 (RN ) =

R2N

(u(x) − u(y))(v(x) − v(y)) dxdy |x − y|N+2s

for all u, v ∈ Ds,2 (RN ). We let H s (RN ) denote the space W s,2 (RN ) endowed with the natural norm

1 uH s (RN ) = [u]2s + u2L2 (RN ) 2 . Clearly, H s (RN ) is a Hilbert space with the inner product  u, vH s (RN ) = u, vDs,2 (RN ) + for any u, v ∈ H s (RN ).

RN

uv dx

4

1 Preliminaries

1.1.2

Sobolev Embeddings

We review the main embedding results for the fractional Sobolev space H s (RN ). Theorem 1.1.8 ([2, 166, 268]) Let s ∈ (0, 1) and N ≥ 1. • If N > 2s, then there exists a sharp constant S∗ = S(N, s) > 0 such that S∗ u2 2∗s

L (RN )

≤ [u]2s

(1.1.1)

for all u ∈ Ds,2 (RN ). Moreover, H s (RN ) is continuously embedded in Lq (RN ) for q every q ∈ [2, 2∗s ] and compactly embedded in Lloc (RN ) for any q ∈ [1, 2∗s ). • When N = 2s, then H s (RN ) is continuously embedded in Lq (RN ) for every q ∈ q [2, ∞) and compactly embedded in Lloc (RN ) for any q ∈ [1, ∞). N

• If N < 2s, then H s (RN ) is continuously embedded in C 0,s− 2 (RN ) and compactly 0,λ N embedded in Cloc (R ) for all 0 < λ < s − N2 . Remark 1.1.9 As proved in [155], (1.1.1) becomes an equality if and only if u(x) = c(μ2 + (x − x0 )2 )−

N−2s 2

, x ∈ RN ,

where c ∈ R, μ > 0 and x0 ∈ RN are fixed constants. Moreover, the value of the best constant S∗ is given by S∗−1

=2

− 2s N

π

− s(N+1) N

 2s   ( N−2s ) N +1 N 2

 . 2  N+2s 2

We also have the following fractional Hardy inequality: Theorem 1.1.10 ([200, 217]) Let s ∈ (0, 1) and N > 2s. Then for all u ∈ Ds,2 (RN ),  RN

|u(x)|2 dx ≤ HN,s [u]2s , |x|2s

with HN,s = 2π

N 2

N+2s  2 ( N+2s 4 ) ( 2 ) . |(−s)|  2 ( N−2s 4 )

(1.1.2)

1.2 The Fractional Laplacian Operator

5

Now we introduce the space of radial functions in H s (RN ), namely s Hrad (RN ) = {u ∈ H s (RN ) : u(x) = u(|x|)}.

We recall the following compactness result due to Lions [255]: s Theorem 1.1.11 ([255]) Let s ∈ (0, 1) and N ≥ 2. Then Hrad (RN ) is compactly q N ∗ embedded in L (R ) for every q ∈ (2, 2s ).

1.2

The Fractional Laplacian Operator

1.2.1

Definition via Singular Integrals

Consider the Schwartz space S(RN ) of rapidly decreasing functions in RN , that is, the set of functions φ ∈ C ∞ (RN ) such that φα,β = sup |x α D β φ(x)| < ∞, x∈RN

for all multi-indices α and β. The vector space S(RN ) with the natural topology given by the seminorms  · α,β is a Fréchet space, i.e., a complete metrizable locally convex space. As it is customary, if α = (α1 , . . . , αN ) is a multi-index (that is an N-tuple of nonnegative integers), then |α| = α1 + · · · + αN , the monomial x α (with x = (x1 , . . . , xN ) ∈ RN ) is |α| αN , and D α denotes the differential operator α1∂ αN , with the defined by x α = x1α1 · · · xN ∂x1 ...∂xN

convention that D (0,...,0) is the identity operator. Let S (RN ) be the space of tempered distributions, that is, the topological dual of S(RN ). For any φ ∈ S(RN ), we denote by F φ(ξ ) =



1 N

RN

(2π) 2

e−ıξ ·x φ(x) dx

the Fourier transform of φ. We recall that F is an isomorphism from S(RN ) onto itself, with inverse given by F −1 φ(x) =



1 (2π)

N 2

RN

eıx·ξ φ(ξ ) dξ,

which is called the inverse Fourier transform of φ ∈ S(RN ). Moreover, one can extend the Fourier transformation to S (RN ) so that F is an isomorphism from S (RN ) onto itself, with inverse F −1 .

6

1 Preliminaries

Fix s ∈ (0, 1) and u ∈ S(RN ). Then we define the fractional Laplace operator by  (−) u(x) = C(N, s) P.V. s



RN

u(x) − u(y) dy |x − y|N+2s

= C(N, s) lim

r→0 RN \Br (x)

(1.2.1)

u(x) − u(y) dy |x − y|N+2s

where P.V. stands for the Cauchy principal value, and C(N, s) is a positive dimensional constant that depends on N and s, precisely given by (see [124, 317])  C(N, s) =

RN

1 − cos(ξ1 ) dξ |ξ |N+2s

−1

= π − 2 22s

( N+2s 2 ) −(−s)

= π − 2 22s

( N+2s 2 ) s (1 − s)

= π − 2 22s

( N+2s 2 ) s(1 − s). (2 − s)

N

N

N

We recall the following useful integral representation formula for the fractional Laplacian. Lemma 1.2.1 ([168]) For all u ∈ S(RN ), 1 (−)s u(x) = − C(N, s) 2

 RN

u(x + y) + u(x − y) − 2u(x) dy |y|N+2s

∀x ∈ RN .

For u ∈ S(RN ) it is not true in general that (−)s u ∈ S(RN ). However, one can verify that (−)s u ∈ C ∞ (RN ) and decays at infinity according to the following result. Lemma 1.2.2 ([203]) Let u ∈ S(RN ). Then, (−)s u ∈ C ∞ (RN ) ∩ L1 (RN ). Moreover, for all x ∈ RN such that |x| > 1, we have |(−)s u(x)| ≤ Cu,N,s |x|−(N+2s), where

Cu,N,s = CN,s uN+2,S + uN,S + uL1 (RN ) , CN,s is a positive constant depending only on N and s, and k

uk,S = sup sup (1 + |x|2 ) 2 |D α u(x)| |α|≤k x∈RN

for k ∈ N ∪ {0}.

1.2 The Fractional Laplacian Operator

7

One can also define the fractional Laplacian acting on spaces of functions with weaker regularity. Indeed, following [244, 313], one considers the space S¯s (RN ) of C ∞ functions u such that, for every k ∈ N ∪ {0}, the quantity (1 + |x|N+2s )D k u is bounded. The class S¯s (RN ) is endowed with the topology induced by the countable family of seminorms ρk (u) = sup |(1 + |x|N+2s )D k u|. x∈RN

Denote by S¯s (RN ) the dual space of Ss (RN ). One can then verify that (−)s maps S(RN ) into S¯s (RN ). The symmetry of (−)s allows us to extend its definition to S¯s (RN ) by duality; i.e., if u ∈ S¯s (RN ), (−)s u, f  = u, (−)s f 

for every f ∈ S(RN ).

This definition coincides with the previous ones in the case where u ∈ S(RN ), and (−)s is a continuous operator from S¯s (RN ) to S (RN ). Anyway, it is more convenient to define (−)s for functions in the space    |u(x)| N N dx < ∞ = L1loc (RN ) ∩ S¯s (RN ). Ls (R ) = u : R → R measurable : N+2s RN 1 + |x| Then we have the following result: Proposition 1.2.3 ([313]) Let be an open subset of RN and let u ∈ Ls (RN ), s ∈ (0, 1). If u ∈ C 0,2s+ε ( ) (or u ∈ C 1,2s+ε −1 ( ) if s ∈ [ 12 , 1)) for some ε > 0, then (−)s u is a continuous function in and (−)s u(x) is given by the integral in (1.2.1), for every x ∈ .

1.2.2

Definition via Fourier Transform: Riesz and Bessel Potentials

The fractional Laplacian (−)s can be also defined via the Fourier transform and viewed as a pseudo-differential operator of symbol |ξ |2s (see [220, 221]). More precisely, for any u ∈ S(RN ), we have F ((−)s u)(ξ ) = |ξ |2s F u(ξ ). This last identity is motivated by the fact that, for u ∈ S(RN ), F (−u)(ξ ) = |ξ |2 F u(ξ ), and thus one can define (−)s as the operator given by multiplication with the function |ξ |2s on the Fourier transform. Using Plancherel’s formula we see that [u]2s

= 2C(N, s)

−1

 RN

|ξ |2s |F u(ξ )|2 dξ

for all u ∈ S(RN ),

(1.2.2)

8

1 Preliminaries

which combined with the definition of (−)s via the Fourier transform implies that [u]2s = 2C(N, s)−1 (−) 2 u2L2 (RN ) s

for all u ∈ S(RN ).

In the light of these relations, we can see that for s ∈ (0, 1), H s (RN ) is equivalent to the following space:   s (RN ) = u ∈ L2 (RN ) : H

RN

 (1 + |ξ |2 )s |F u(ξ )|2 dξ < ∞ .

We note that the above space, unlike the spaces defined via the Gagliardo seminorm, is defined for any real s ≥ 0, whereas if s < 0 we set   s N

N  H (R ) = u ∈ S (R ) :

 (1 + |ξ | ) |F u(ξ )| dξ < ∞ . 2 s

RN

2

s (RN ), and H −s (RN ) coincides with the Moreover, for any s ∈ R, Cc∞ (RN ) is dense in H s N  dual of H (R ); see [2,166,258]. From now on, whenever s ∈ (0, 1), we will use the same s (RN ), and will denote by H −s (RN ) the dual of H s (RN ). notation for H s (RN ) and H Now, we note that when N > 2s, the operator (−)−s is defined as the inverse of (−)s and is given by (−)−s u(x) = c(N, −s)

 RN

u(y) dy, |x − y|N−2s

for all u ∈ S(RN ),

which is a Riesz potential of order 2s. More generally, we have the following definition. Definition 1.2.4 Let N ∈ N and 0 < s < N. The Riesz potential of order s is the operator whose action on a function u ∈ S(RN ) is given by Is u(x) = (−)

− 2s



where Is (x) =

 π

N 2

u(x) = (Is ∗ u)(x) =

N−s 2

 π

N 2

N−s  2 2s  2s

RN

u(y) dy, |x − y|N−s

(1.2.3)



2s  ( 2s )

|x|s−N is called the Riesz kernel.

The following identities exhibit essential properties of the Riesz operators Is (see [315]): Is (It u) = Is+t u,

for all u ∈ S(RN ), s, t > 0, s + t < N,

(Is u) = Is (u) = −Is−2 u,

for all u ∈ S(RN ), N ≥ 3, 2 ≤ s ≤ N.

1.2 The Fractional Laplacian Operator

9

The importance of the Riesz potentials lies in the fact that they are indeed smoothing operators. This is the essence of the Hardy–Littlewood–Sobolev theorem on fractional integration. Theorem 1.2.5 ([315]) Let s ∈ (0, N) and let 1 ≤ p < q < ∞ satisfy 1 1 s = − . N p q • If u ∈ Lp (RN ), then the integral defined by (1.2.3) is absolutely convergent for a.e. x ∈ RN . • If p > 1, then there exists a positive constant C = C(N, p, s) such that for all u ∈ Lp (RN ), Is uLq (RN ) ≤ CuLp (RN ) . • If p = 1, then for some positive constant C = C(N, s) and for all u ∈ L1 (RN ), N    Cu 1 N  N−s   L (R ) {x ∈ RN : |Is u(x)| > λ} ≤ λ

for all λ > 0. While the behavior of the kernels |x|s−N as |x| → 0 is well suited to their smoothing properties, their decay as |x| → ∞ gets worse as s increases. A way out of this dilemma is to slightly adjust the Riesz potentials so that we maintain their essential behavior near zero, but achieve exponential decay at infinity. The simplest way to accomplish this goal is to replace the “nonnegative” operator − by the “strictly positive” operator 1 − . This fact justifies the introduction of Bessel potentials (see [78, 128, 209, 315]). Definition 1.2.6 Let s > 0. The Bessel potential of order s of u ∈ S(RN ) is defined as s

Js u(x) = (1 − )− 2 u(x) = (Gs ∗ u)(x) =

 RN

Gs (x − y)u(y) dy,

where Gs is called the Bessel kernel and its Fourier transform satisfies N

s

F Gs (ξ ) = (2π)− 2 (1 + |ξ |2 )− 2 .

10

1 Preliminaries

Remark 1.2.7 If s ∈ R (or s ∈ C), then we may define the Bessel potential of a temperate distribution u ∈ S (RN ) (see [128]) by setting F Js u(ξ ) = (2π)− 2 (1 + |ξ |2 )− 2 F u(ξ ). N

s

It is possible to prove (see [78]) that Gs (x) =

1 2

N+s−2 2

π

N 2

( 2s )

K N−s (|x|)|x|

s−N 2

2

,

where Kν denotes the modified Bessel function of the third kind (or Macdonald function) and order ν (see [178,336]), and satisfies the following asymptotic formulas for ν ∈ R and r > 0: (ν) r −ν as r → 0, for ν > 0, 2 2  π − 1 −r Kν (r) ∼ as r → ∞, for ν ∈ R. r 2e 2 Kν (r) ∼

(1.2.4) (1.2.5)

Thus Gs (x) is a positive, decreasing function of |x|, analytic except at x = 0, and for x ∈ RN \ {0}, Gs (x) is an entire function of s. Moreover, Gs (x) ∼ Gs (x) ∼

 π

N 2

N−s s−N 2 s |x| s 2 2

1 2

N+s−1 2

π

N−1 2

( 2s )

|x|

as |x| → 0, if 0 < s < N, s−N−1 2

e−|x|

as |x| → ∞,

 Gs ∈ L1 (RN ) for all s > 0, and RN Gs (x) dx = 1. We also have the composition formula Gs+t = Gs ∗ Gt for all s, t > 0. One the most interesting facts concerning Bessel potentials is they can be employed to define the Bessel potential spaces; see [2, 78, 128, 315]. For p ∈ [1, ∞] and α ∈ R we define the Banach space Lαp = Gα (Lp (RN )) = {u : u = Gα ∗ f,

f ∈ Lp (RN )}

endowed with the norm uLαp = f Lp (RN ) p

if u = Gα ∗ f.

Thus Lα is a subspace of Lp (RN ) for all α ≥ 0. We also have the following useful result:

1.2 The Fractional Laplacian Operator

11

Theorem 1.2.8 ([2, 128]) p

(i) If α ≥ 0 and 1 ≤ p < ∞, then D(RN ) is dense in Lα . p (ii) If 1 < p < ∞ and p its conjugate exponent, then the dual of Lα is isometrically

p isomorphic to L−α . p p (iii) If β < α, then Lα is continuously embedded in Lβ . Np N (iv) If β ≤ α and if either 1 < p ≤ q ≤ N−(α−β)p < ∞, or p = 1 and 1 ≤ q < N−α+β , p q then Lα is continuously embedded in Lβ . p (v) If 0 < μ ≤ α − Np < 1, then Lα is continuously embedded in C 0,μ (RN ). p (vi) Lk = W k,p (RN ) for all k ∈ N and 1 < p < ∞, Lα2 = W α,2 (RN ) for any α. (vii) If 1 < p < ∞ and ε > 0, then for every α we have the following continuous embeddings: p

p

Lα+ε ⊂ W α,p (RN ) ⊂ Lα−ε .

1.2.3

Definition via Caffarelli-Silvestre Extension

Caffarelli and Silvestre [127] showed that (−)s can be characterized as the Dirichletto-Neumann operator for a suitable local degenerate elliptic problem posed on the upper = {(x, y) ∈ RN+1 : y > 0}. half-space RN+1 + Let s ∈ (0, 1) be given. For a measurable function u : RN → R, we first formally by setting define its s-harmonic extension to RN+1 +  Ext(u)(x, y) = (Ps (·, y) ∗ u)(x) =

RN

Ps (x − z, y)u(z) dz,

where the Poisson kernel Ps (x, y) is given by Ps (x, y) = pN,s

y 2s (|x|2 + y 2 )

and the constant pN,s is such that

 RN

N+2s 2

,

pN,s = π − 2

N

( N+2s 2 ) , (s)

Ps (x, y) dx = 1 for all y > 0.

N+1 If u ∈ S(RN ), then U = Ext(u) ∈ C ∞ (RN+1 + ) ∩ C(R+ ), U (·, y) → 0 as y → ∞, and U solves the following problem



− div(y 1−2s ∇U ) = 0 in RN+1 + , ∼ U (·, 0) = u on ∂RN+1 = RN . +

(1.2.6)

12

1 Preliminaries

Moreover, for any x ∈ RN , ∂U ∂U (x, y) = κs (−)s u(x) (x, 0) = − lim y 1−2s 1−2s y→0 ∂ν ∂y

(1.2.7)

where κs =

(1 − s) . 22s−1(s)

Then (1.2.7) expresses the fact that (−)s can be regarded as the Dirichlet-to-Neumann map for the extension problem (1.2.6) with the weight y 1−2s . Qualitatively, the result in [127] states that one can localize the fractional Laplacian by adding the new variable y. We now consider a more general setting. Let us define the spaces H˙ s (RN ) and N+1 s ∞ N ∞ X0 (RN+1 + ) as the completion of Cc (R ) and Cc (R+ ), respectively, under the norms  uH˙ s (RN ) =

1

RN

|ξ |2s |F u(ξ )|2 dξ

2

  U Xs (RN+1 ) = +

0

, 1 2

RN+1 +

y

1−2s

|∇U | dxdy 2

.

If u ∈ H˙ s (RN ), then Ext(u) ∈ X0s (RN+1 + ) and Ext(u)2 s

X0 (RN+1 + )

= κs u2H˙ s (RN ) .

Furthermore, the function U = Ext(u) is a weak solution to div(y 1−2s ∇U ) = 0 in RN+1 + , and, as ε → 0, we have U (·, ε) → u in H˙ s (RN )

− ε1−2s

and

∂U (·, ε) → κs (−)s u in H˙ −s (RN ), ∂y

where H˙ −s (RN ) is the dual of H˙ s (RN ). Now, we recall (see [109, 198, 199]) that there ˙s N exists a linear trace operator Tr : X0s (RN+1 + ) → H (R ) such that √

κs Tr(v)H˙ s (RN ) ≤ vXs (RN+1 ) 0

+

for any v ∈ X0s (RN+1 + ),

(1.2.8)

and the equality is attained if and only if v = Ext(u) for some u ∈ H˙ s (RN ). Combining Theorem 1.1.8 with (1.2.8), we derive the following Sobolev inequality: Tr(v)L2∗s (RN ) ≤ CvXs (RN+1 ) 0

+

(1.2.9)

1.2 The Fractional Laplacian Operator

13

N+1 s for any v ∈ X0s (RN+1 + ). In what follows, for v ∈ X0 (R+ ), we denote Tr(v) by v(·, 0). The extension technique is commonly used in the recent literature since it allows to transform a given nonlocal problem into a local one and use classical PDE tools and ideas that are available for these kind of problems. Further results about the extension method can be found in [31, 109, 124, 126, 131, 181, 198, 317].

Remark 1.2.9 As shown in [127], if u ∈ S(RN ), a way to prove (1.2.7) is to use a partial Fourier transform with respect to x ∈ RN . Indeed, it suffices to verify that U 2 s N+1 = X0 (R+

)

κs u2H˙ s (RN ) . First, one observes that F U (ξ, y) = F u(ξ )θ (|ξ |y), where θ (y) solves the problem ⎧ 1−2s



⎪ ⎨ θ + y θ − θ = 0 in (0, ∞), θ (0) = 1, ⎪ ⎩ limy→∞ θ (y) = 0.

(1.2.10)

We note that (see [131, 181]) θ (y) =

2 y s Ks (y) (s) 2

and that Ks (y) solves (see [78, 178, 336]) the following differential equation of second order: y 2 Ks

+ yKs − (y 2 + s 2 )Ks = 0. Then we obtain   y 1−2s |∇U |2 dxdy =







+1 RN +

RN

=

RN

 =

∞

|ξ |2 |F U (ξ, y)|2 + |F Uy (ξ, y)|2 y 1−2s dξ dy

0 ∞



|ξ |2|F u(ξ )|2 |θ (|ξ |y)|2 + |θ (|ξ |y)|2 y 1−2s dξ dy

0 ∞



y 1−2s |θ (y)|2 + |θ (y)|2 dy

0

  RN

 |ξ |2s |F u(ξ )|2 dξ .

Now, we observe that (see [78, 178, 336]) (y s Ks (y)) = −y s Ks−1 (y)

and Ks−1 (y) = K1−s (y)

14

1 Preliminaries

y s−1 yield (y s Ks (y)) = −y s K1−s (y). Moreover, by (1.2.4), K1−s (y) ∼ (1−s) as 2 2 y → 0. Recalling the expression of θ (y), θ (0) = 1, and the precise value of κs , it follows from (y 1−2s θ (y)) = y 1−2s θ (y) (by (1.2.10)) and an integration by parts that 

∞ 0

1.3



y 1−2s |θ (y)|2 + |θ (y)|2 dy = − lim y 1−2s θ (y) = κs . y→0

(1.2.11)

Regularity Results and Maximum Principles

The following propositions obtained in [313] recall how the fractional Laplace operators interact with the Hölder norms. For more details see also [316]. Proposition 1.3.1 ([313]) Let w = (−)s u. Assume w ∈ C 0,α (RN ) and u ∈ L∞ (RN ) for α ∈ (0, 1] and s > 0. • If α + 2s ≤ 1, then u ∈ C 0,α+2s (RN ). Moreover,

uC 0,α+2s (RN ) ≤ C uL∞ (RN ) + wC 0,α (RN ) for a constant C that depends only on n, α, and s. • If α + 2s > 1, then u ∈ C 1,α+2s−1(RN ). Moreover,

uC 1,α+2s−1 (RN ) ≤ C uL∞ (RN ) + wC 0,α (RN ) for a constant C that depends only on n, α, and s. Proposition 1.3.2 ([313]) Let w = (−)s u. Assume w ∈ L∞ (RN ) and u ∈ L∞ (RN ) for s > 0. • If 2s ≤ 1, then u ∈ C 0,α (RN ) for any α < 2s. Moreover,

uC 0,α (RN ) ≤ C uL∞ (RN ) + wL∞ (RN ) for a constant C depending only on n, α, and s. • If 2s > 1, then u ∈ C 1,α (RN ) for any α < 2s − 1. Moreover,

uC 1,α (RN ) ≤ C uL∞ (RN ) + wL∞ (RN ) for a constant C depending only on n, α, and s.

1.3 Regularity Results and Maximum Principles

15

Next, we collect some maximum principles for (−)s . For more details we refer to [106, 163, 222, 228, 282, 313]. We start by recalling the following maximum principle for distributional supersolutions. Proposition 1.3.3 ([313]) Let  RN be an open set, and let u ∈ Ls (RN ) be a lower¯ such that (−)s u ≥ 0 in and u ≥ 0 in RN \ . Then semicontinuous function in

u ≥ 0 in RN . Moreover, if u(x) = 0 for one point x inside , then u ≡ 0 in all RN . Next we introduce several definitions. Let be an open set in RN . Set s ( ) = {u ∈ W s,2 ( ) : u˜ ∈ W s,2 (RN )}, H −s ( ) the dual space of where u˜ is the extension of u by zero outside . Denote by H s s  ( ) and H −s ( ). Let Hs ( )  ( ), and denote by ·, · the duality pairing between H H 2 N be the space of functions u ∈ Lloc (R ) such that for any bounded

⊆ there is an open set U 

so that u ∈ W s,2 (U ), and  RN

|u(x)| dx < ∞. (1 + |x|)N+2s

When is bounded, then u ∈ Hs ( ) if and only if there is an open set U  such that u ∈ W s,2 (U ) and  RN

|u(x)| dx < ∞. (1 + |x|)N+2s

−s ( ), we say that f ≥ (≤)0 if for any φ ∈ H s ( ), φ ≥ 0, it holds that Given f ∈ H f, φ ≥ (≤)0. When is bounded, we say that u ∈ Hs ( ) is a weak supersolution (subsolution) of (−)s u = f in if u, φDs,2 (RN ) ≥ (≤)f, φ s ( ), φ ≥ 0. for all φ ∈ H When is unbounded, we say that u ∈ Hs ( ) is a weak supersolution (subsolution) of (−)s u = f in if u is a weak supersolution (subsolution) of (−)s u = f in

for every bounded set

⊂ . In both cases, u is a weak solution of (−)s u = f in if u is a supersolution and subsolution of (−)s u = f in . Given c ∈ L1loc ( ), we say that u ∈ Hs ( ) is a weak supersolution (subsolution) −s ( ) and u is a weak supersolution of (−)s u = c(x)u in if f = c(x)u ∈ H s (subsolution) of (−) u = f in . Finally, u ∈ Hs ( ) is a weak solution of (−)s u = c(x)u in if u is both a supersolution and a subsolution of (−)s u = c(x)u in .

16

1 Preliminaries

Theorem 1.3.4 ([163] Strong Maximum Principle (Variant 1)) Let c ∈ L1loc ( ) be a non-positive function and u ∈ Hs ( ) be a weak supersolution of (−)s u = c(x)u in .

(1.3.1)

• If is bounded and u ≥ 0 a.e. in RN \ , then either u > 0 a.e. in , or u = 0 a.e. in RN . • If u ≥ 0 a.e. in RN , then either u > 0 a.e. in , or u = 0 a.e. in RN . Theorem 1.3.5 ([163] Strong Maximum Principle (Variant 2)) Let c ∈ C( ) be a nonpositive function and u ∈ Hs ( ) ∩ C( ) be a weak supersolution of (1.3.1). • If is bounded and u ≥ 0 a.e. in RN \ , then either u > 0 in , or u = 0 a.e. in RN . • If u ≥ 0 a.e. in RN , then either u > 0 in , or u = 0 a.e. in RN . Remark 1.3.6 In the case where u ≥ 0 a.e. in RN , the non-positivity assumption on the function c(x) in the previous theorems is not necessary. Theorem 1.3.7 ([163] Hopf Lemma) Let satisfy the interior ball condition in x0 ∈ ∂ , namely there exists a ball BR ⊆ such that x0 ∈ ∂BR . Let c ∈ C( ) and u ∈ Hs ( ) ∩ C( ) be a weak supersolution of (1.3.1). • If is bounded, c(x) ≤ 0 in and u ≥ 0 a.e. in RN \ , then either u = 0 a.e. in RN , or lim inf

x∈BR ,x→x0

u(x) >0 δR (x)s

(1.3.2)

where BR ⊆ and x0 ∈ ∂BR and δR (x) = dist(x, RN \ BR ). • If u ≥ 0 a.e. in RN , then either u = 0 a.e. in RN , or (1.3.2) holds. Lemma 1.3.8 (Comparison Principle [164]) Let ⊂ RN be an open set, μ ∈ R+ , u1 , u2 ∈ H s (RN ) such that u1 ≤ u2 in RN \ and (−)s u1 + μu1 ≤ (−)s u2 + μu2

in ,

that is 

 (u1 (x) − u1 (y))(φ(x) − φ(y)) dxdy + μu1 φ dx |x − y|N+2s R2N RN   (u2 (x) − u2 (y))(φ(x) − φ(y)) ≤ dxdy + μu2 φ dx |x − y|N+2s R2N RN

1.3 Regularity Results and Maximum Principles

17

s ( ), φ ≥ 0. Then u1 ≤ u2 in . for all φ ∈ H Due to the strong connection between the fractional Laplacian and degenerate elliptic problems with Neumann boundary conditions, we also recall some useful regularity results and maximum principle established in [124,237] (see also [180]). First we introduce some N 2 N notations. We let |x| = i=1 xi denote the euclidean norm of x = (x1 , . . . , xN ) ∈ R  and let |(x, y)| = |x|2 + y 2 denote the euclidean norm of (x, y) ∈ RN+1 + . Let D ⊂ N+1 R be a bounded domain, that is, a bounded connected open set, with boundary ∂D. in RN , and we set ∂

D = ∂D \ ∂ D. For We denote by ∂ D the interior of D ∩ ∂RN+1 + R > 0, we set : |(x, y)| < R}, BR+ = {(x, y) ∈ RN+1 + R0 = ∂ BR+ = {(x, 0) ∈ ∂RN+1 : |x| < R}, + R+ = ∂

BR+ = {(x, y) ∈ RN+1 : y ≥ 0, |(x, y)| = R}. Now, we introduce the weighted Lebesgue space (see [180, 237, 328] for more details). Let D ⊂ RN+1 be an open set, s ∈ (0, 1) and r ∈ [1, ∞). Denote by Lr (D, y 1−2s ) the + weighted Lebesgue space of all measurable functions v : D → R such that   vLr (D,y 1−2s ) =

1 y 1−2s |v|r dxdy

r

< ∞.

D

We say that v ∈ H 1 (D, y 1−2s ) if v ∈ L2 (D, y 1−2s ) and its weak derivatives, collectively denoted by ∇v, exist and belong to L2 (D, y 1−2s ). The norm of v in H 1 (D, y 1−2s ) is given by   vH 1 (D,y 1−2s ) =

1 y

1−2s

(|∇v| + v ) dxdy 2

2

2

< ∞.

D

It is clear that H 1 (D, y 1−2s ) is a Hilbert space with the inner product  y 1−2s (∇v · ∇w + vw) dxdy. D

We note that when D = RN+1 then we have the following weighted Sobolev embeddings. +

18

1 Preliminaries

Lemma 1.3.9 ([170, 181]) (i) There exists a constant Sˆ > 0 such that  

1

2γ

RN+1 +

y

1−2s

|u|

2γ

dxdy

≤ Sˆ

 

1 2

RN+1 +

y

1−2s

2

|∇u| dxdy

2 . N − 2s N+1 s (ii) Let R > 0 and let T be a subset of X0 (R+ ) such that for all u ∈ X0s (RN+1 + ), where γ = 1 +  sup

u∈T

N+1 R+

y 1−2s |∇u|2 dxdy < ∞.

Then T is pre-compact in L2 (BR+ , y 1−2s ). Now we give the following definitions. Definition 1.3.10 Let D ⊂ RN+1 be a bounded domain, with ∂ D = ∅. Let a ∈ + 2N

N+2s Lloc (∂ D) and b ∈ L1loc (∂ D). Consider the problem

⎧ ⎨ −div(y 1−2s ∇v) = 0 in D, ∂v ⎩ = a(x)v + b(x) on ∂ D. ∂ν 1−2s

(1.3.3)

We say that v ∈ H 1 (D, y 1−2s ) is a weak supersolution (resp. subsolution) to (1.3.3) in D if for any nonnegative ϕ ∈ Cc∞ (D ∪ ∂ D),   y 1−2s ∇v(x, y) · ∇φ(x, y) dxdy ≥ (≤) [a(x)v(x, 0) + b(x)]ϕ(x, 0) dx. ∂ D

D

We say that v ∈ H 1 (D, y 1−2s ) is a weak solution to (1.3.3) in D if it is both a weak supersolution and a weak subsolution. N N Set QR = BR ×(0, R) ⊂ RN+1 + , where BR ⊂ R is the ball in R of radius R centered at 0. We recall the following De Giorgi–Nash–Moser type theorems.

Proposition 1.3.11 ([237]) Let a, b ∈ Lq (B1 ) for some q >

N 2s .

(i) Let v ∈ H 1 (Q1 , y 1−2s ) be a weak subsolution to (1.3.3) in Q1 . Then for all ν > 0,

sup v + ≤ C v + Lν (Q1 ,y 1−2s ) + bLq (B1 ) ,

Q1/2

1.3 Regularity Results and Maximum Principles

19

where C > 0 depends only on N, s, q, ν and a + Lq (B1 ) . (ii) (Weak Harnack inequality) Let v ∈ H 1 (Q1 , y 1−2s ) be a nonnegative weak supersolution to (1.3.3) in Q1 . Then for any 0 < μ < τ < 1 and 0 < ν ≤ N+1 N , inf v + b− Lq (B1 ) ≥ CvLν (Qτ ,y 1−2s ) ,

Qμ

where C > 0 depends only on N, s, q, ν, μ, τ and a− Lq (B1 ) . (iii) Let v ∈ H 1 (Q1 , y 1−2s ) be a non-negative weak solution to (1.3.3) in Q1 . Then  sup v ≤ C Q1/2

 inf v + bLq (B1 ) ,

Q1/2

where C > 0 depends only on N, s, q, aLq (B1 ) . Consequently, there exists α ∈ (0, 1) depending only on N, s, q, aLq (B1 ) such that any weak solution v of (1.3.3) is of class C 0,α (Q1/2 ). Moreover,

vC 0,α (Q1/2 ) ≤ C vL∞ (Q1 ) + bLq (B1 ) , where C > 0 depends only on N, s, q, aLq (B1 ) . The next results are very useful for obtaining local Schauder estimates for non-negative solutions of fractional Laplace equations. Proposition 1.3.12 ([237]) Let a, b ∈ C k (B1 ) for some positive integer k ≥ 1. Let v ∈ H 1 (Q1 , y 1−2s ) be a weak solution to (1.3.3) in Q1 . Then we have k 



∇xi vL∞ (Q1/2 ) ≤ C vL2 (Q1 ,y 1−2s ) + bC k (B1 ) ,

i=0

where C > 0 depends only on N, s, k, aC k (B1 ) . Lemma 1.3.13 ([237] (Schauder Estimate)) Let v ∈ H 1 (Q1 , y 1−2s ) be a weak solution / N. If α + 2s is not an integer, then to (1.3.3), where a, b ∈ C α (B1 ) for some 0 < α ∈ v(·, 0) ∈ C α+2s (B1/2 ). Moreover, we have v(·, 0)C α+2s (B1/2 ) ≤ C(vL∞ (Q1 ) + bC α (B1 ) ), where C > 0 depends only on N, s, α, aC α (B1 ) . Here, for simplicity, we used the symbol C α ( ) to denote the standard Hölder space C 0,α ( ) over a bounded domain whenever / N. α ∈ (0, 1), and the space C [α],α−[α] ( ) when 1 < α ∈

20

1 Preliminaries 2N

N+2s Let be a bounded domain in RN , a ∈ Lloc ( ) and b ∈ L1loc ( ). We say u ∈ s,2 N D (R ) is a weak solution of

(−)s u = a(x)u + b(x) in ,

(1.3.4)

if for every non-negative v ∈ C ∞ (RN ) supported in ,  u, vDs,2 (RN ) =

[a(x)u(x) + b(x)]v(x) dx.

Then u ∈ Ds,2 (RN ) is a weak solution of (−)s u =

1 (a(x)u + b(x)) in B1 , κs

if and only if the s-harmonic extension U (x, y) = Ps (x, y) ∗ u(x) of u is a weak solution of (1.3.3) in Q1 . / N. Let u ∈ Ds,2 (RN ) and Theorem 1.3.14 ([237]) Let a, b ∈ C α (B1 ) with 0 < α ∈ N u ≥ 0 in R be a weak solution of (−)s u = a(x)u + b(x) in B1 . Suppose that 2s + α is not an integer. Then u ∈ C 2s+α (B1/2 ). Moreover, 

 uC 2s+α (B1/2 ) ≤ C

inf u + bC α (B3/4 ) ,

B3/4

where C > 0 depends only on N, s, α, aC α (B3/4 ) . Finally, we collect some maximum principles for (1.3.3). The following (weak) maximum principle holds true. Lemma 1.3.15 ((Weak Maximum Principle)[124]) Let v ∈ H 1 (BR+ , y 1−2s ) be a weak supersolution to (1.3.3) in BR+ with a ≡ b ≡ 0. If v ≥ 0 on R+ in the trace sense, then v ≥ 0 in BR+ . Remark 1.3.16 In addition, one has the strong maximum principle: either v ≡ 0, or v > 0 in BR+ ∪ R0 . That v cannot vanish at an interior point follows from the classical strong maximum principle for strictly elliptic operators. That v cannot vanish at a point in R0 follows from the Hopf principle below or by the strong maximum principle of [180]. Note

1.3 Regularity Results and Maximum Principles

21

that the same weak and strong maximum principles hold in other bounded domains of different than BR+ . It also holds for the Dirichlet problem in BR+ . RN+1 + The next lemma provides estimates for solutions of the Neumann problem in a half-ball. 0 Lemma 1.3.17 ([124]) Let g ∈ C 0,γ (2R ) for some γ + + H 1 (B2R , y 1−2s ) ∩ L∞ (B2R ) be a weak solution to

∈

(0, 1). Let v

∈

⎧ + ⎨ −div(y 1−2s ∇v) = 0 in B2R , ∂v 0 . ⎩ =g on 2R ∂ν 1−2s Then there exists α ∈ (0, 1) depending only on N, s, γ such that v ∈ C 0,α (BR+ ) and

0,α (B + ). Moreover, there exists positive constants C 1 and C 2 depending y 1−2s ∂v R R R ∂y ∈ C only on N, s, R, uL∞ (B + ) and also on gL∞ ( 0 ) (for CR1 ) and gC 0,γ ( 0 ) (for CR2 ) 2R 2R 2R such that

vC 0,α (B + ) ≤ CR1 R

and    1−2s ∂v   y ≤ CR2 .  ∂y C 0,α (B + ) R

The following proposition provides a Hopf boundary lemma. Proposition 1.3.18 ([124] Hopf Boundary Lemma) Let CR,1 = BR × (0, 1) ⊂ RN+1 + and let v ∈ H 1 (CR,1 , y 1−2s ) ∩ C(CR,1 ) be a weak solution to ⎧ 1−2s ∇v) ≥ 0 in C ⎪ R,1 , ⎨ −div(y v>0 in CR,1 , ⎪ ⎩ v(0, 0) = 0. Then, lim sup −y 1−2s y→0

v(0, y) < 0. y

In addition, if y 1−2s ∂v ∂y ∈ C(CR,1 ), then ∂v (0, 0) < 0. ∂ν 1−2s

22

1 Preliminaries

Corollary 1.3.19 ([124]) Let d be a Hölder continuous function in R0 and v ∈ L∞ (BR+ ) ∩ H 1 (BR+ , y 1−2s ) be a weak solution to ⎧ ⎪ −div(y 1−2s ∇v) = 0 in BR+ , ⎪ ⎨ v≥0 on BR+ , ⎪ ⎪ ⎩ ∂v + d(x)v = 0 on  0 . R ∂ν 1−2s Then, v > 0 in BR+ ∪ R0 unless v ≡ 0 in BR+ .

1.4

Some Useful Lemmas

We recall some useful technical lemmas which will be frequently used later. We start with two well-known results that can be found in [100]. Lemma 1.4.1 ([100] Radial Lemma) Let u ∈ Lt (RN ), 1 ≤ t < ∞ be a non-negative radially decreasing function (that is, 0 ≤ u(x) ≤ u(y) if |x| ≥ |y|). Then  |u(x)| ≤

1

N

t

ωN−1

N

|x|− t uLt (RN ) ,

for any x ∈ RN \ {0},

(1.4.1)

where ωN−1 is the surface area of the unit sphere SN−1 = {x ∈ RN : |x| = 1}. Lemma 1.4.2 ([83,100,318] The Compactness Lemma of Strauss) Let P and Q : R → R be two continuous functions satisfying P (t) = 0. |t |→∞ Q(t) lim

Let (vn ), v and w be real measurable functions on RN , with w bounded, such that  sup

n∈N RN

|Q(vn (x))w(x)| dx < ∞,

P (vn (x)) → v(x)

a.e. x ∈ RN , as n → ∞.

Then (P (vn ) − v)wL1 (B) → 0 as n → ∞, for any bounded Borel set B ⊂ RN . Moreover, if we also have lim

t →0

P (t) = 0, Q(t)

1.4 Some Useful Lemmas

23

and lim sup |vn (x)| = 0,

|x|→∞ n∈N

then (P (vn ) − v)wL1 (RN ) → 0 as n → ∞. The next result will be useful for obtaining compactness when the property of uniform decay at infinity, such as in the Strauss type radial lemma, is not clear. Lemma 1.4.3 ([139]) Let (X,  · ) be a Banach space such that X is embedded continuously and compactly in Lq (RN ) for q ∈ [q1, q2 ] and q ∈ (q1 , q2 ), respectively, where q1 , q2 ∈ (0, ∞). Let (un ) ⊂ X, let u : RN → R be a measurable function and let P ∈ C(R, R). Assume that P (t) = 0, |t|q1 P (t) (ii) lim = 0, |t |→∞ |t|q2 (i) lim

|t |→0

(iii) sup un  < ∞, n∈N

(iv) lim P (un (x)) = u(x) for a.e. x ∈ RN . n→∞

Then, up to a subsequence, lim P (un ) − uL1 (RN ) = 0.

n→∞

We recall the following version of vanishing Lions-type result (see [256]) for H s (RN ). Lemma 1.4.4 Let N > 2s and r ∈ [2, 2∗s ). If (un ) is a bounded sequence in H s (RN ) and if  |un |r dx = 0

lim sup

n→∞

y∈RN

(1.4.2)

BR (y)

for some R > 0, then un → 0 in Lτ (RN ) for all τ ∈ (r, 2∗s ). Proof Let τ ∈ (r, 2∗s ). Given y ∈ RN , the Hölder inequality shows that α un Lτ (BR (y)) ≤ un 1−α Lr (BR (y)) un  2∗s

L (BR (y))

for all n ∈ N,

24

1 Preliminaries

where 1−α α 1 = + ∗. t r 2s Now, covering RN by balls of radius R, in such a way that each point of RN is contained in at most N + 1 balls, and applying Theorem 1.1.8, we find that (1−α)τ

 un τLτ (RN ) ≤ (N + 1) sup

|un |r dx

L (RN )

(1−α)τ

 ≤ C sup y∈RN

un ατ2∗s

BR (y)

y∈RN

|un | dx

for all n ∈ N.

r

BR (y)

Using (1.4.2), we conclude that un → 0 in Lτ (RN ).

 

In what follows we give some technical results whose proofs can be obtained arguing as in [40, 47, 196, 344]. However, we present different proofs. Lemma 1.4.5 Let (un ) ⊂ Ds,2 (RN ) be a bounded sequence in Ds,2 (RN ), and let η ∈ C ∞ (RN ) be a function such that 0 ≤ η ≤ 1 in RN , η = 0 in B1 and η = 1 in RN \ B2 . Set ηR (x) = η( Rx ) for all R > 0. Then  lim lim sup

R→∞ n→∞

R2N

|un (x)|2

|ηR (x) − ηR (y)|2 dxdy = 0. |x − y|N+2s

Proof Fix R > 0. We start by proving that  lim sup n→∞

R2N

|un (x)|2

|ηR (x) − ηR (y)|2 dxdy ≤ |x − y|N+2s

 R2N

|u(x)|2

|ηR (x) − ηR (y)|2 dxdy. |x − y|N+2s (1.4.3)

Set ψR = 1 − ηR and note that ψR ∈ Cc∞ (RN ), 0 ≤ ψR ≤ 1, ψR = 1 in BR and ψR = 0 in RN \ B2R . Define  WR (x) =

RN

|ηR (x) − ηR (y)|2 dy = |x − y|N+2s

 RN

|ψR (x) − ψR (y)|2 dy. |x − y|N+2s

1.4 Some Useful Lemmas

25

We observe that WR ∈ L∞ (RN ), because 

 |ψR (x) − ψR (y)|2 |ψR (x) − ψR (y)|2 dy + dy N+2s |x − y| |x − y|N+2s |y−x|≤R |y−x|>R   1 1 dy + 4 dy ≤ R −2 ∇η2L∞ (RN ) N+2s−2 N+2s |y−x|≤R |x − y| |y−x|>R |x − y|  R  ∞ 1 1 dr + 4ω dr = R −2 ∇η2L∞ (RN ) ωN−1 N−1 2s−1 2s+1 r r 0 R

WR (x) =

≤

C1 . R 2s

(1.4.4)

On the other hand, for all |x| > 4R, we have  WR (x) =

|y|≤2R

|ψR (y)|2 2N+2s |B2R | C2 R N dy ≤ = , |x − y|N+2s |x|N+2s |x|N+2s

(1.4.5)

where we used that ψR = 0 in RN \ B2R , 0 ≤ ψR ≤ 1, and |x − y| ≥ |x| − |y| ≥ |x| − 2R ≥

|x| 2

for |x| > 4R, |y| ≤ 2R.

We note that the constants C1 and C2 in (1.4.4) and (1.4.5) are independent of R. Take ∗ M > 4R. Using (1.4.5) and the boundedness of (un ) in L2s (RN ), and applying the Hölder inequality, we see that 

 RN

|un (x)|2 WR (x) dx =

 |x|≤M

|un (x)|2 WR (x) dx +

|x|≤M

|un (x)|2 WR (x) dx + un 2 2∗s

L (RN )

|x|>M



 ≤

|un (x)|2 WR (x) dx



 ≤

|x|>M

|x|≤M

|un (x)|2 WR (x) dx + C3 R N

|WR (x)|

1 N2

|x| 2s +N

|x|>M

2∗ s 2∗ s −2

2s

dx



 n→∞ |x|≤M

|un (x)|2WR (x) dx =

|x|≤M

dx

 2∗s −2 ∗

Since un → u in L2loc (RN ) and WR ∈ L∞ (RN ), we have lim

∗

 2s2−2 ∗

|u(x)|2 WR (x) dx,

.

s

26

1 Preliminaries

and consequently  lim sup n→∞

|un (x)|2 WR (x) dx

RN



 ≤

|x|≤M

|u(x)|2WR (x) dx + C3 R N

 2∗s −2 ∗ 2s

1

|x|>M

|x|

dx

N2 2s +N

∀M > 4R.

Letting here M → ∞, we obtain  lim sup

R2N

n→∞

|un (x)|2

|ηR (x) − ηR (y)|2 dxdy ≤ |x − y|N+2s

 RN

|u(x)|2 WR (x) dx,

that is, (1.4.3) holds true. Since (1.4.4) and (1.4.5) imply that WR (x) ≤ C1 R −2s for all |x| ≤ 4R,

WR (x) ≤ C2 (R|x|−1 )N |x|−2s for all |x| > 4R,

we deduce that, for some C¯ > 0 independent of R, ¯ −2s 0 ≤ WR (x) ≤ C|x|

for all x = 0.

This fact combined with the fractional Hardy–Sobolev inequality (1.1.2) yields |u(x)|2WR (x) ≤ C¯

|u(x)|2 ∈ L1 (RN ). |x|2s

On the other hand, (1.4.4) implies feat WR → 0 as R → ∞. Then, by the dominated convergence theorem,  lim sup lim sup R→∞

n→∞

R2N

|un (x)|2

|ηR (x) − ηR (y)|2 dxdy ≤ lim R→∞ |x − y|N+2s

 RN

|u(x)|2 WR (x) dx = 0.

  Remark 1.4.6 If (un ) ⊂ H s (RN ), then the proof of Lemma 1.4.5 can be simplified. Indeed, using the boundedness of (un ) in L2 (RN ) and polar coordinates, we have  R2N

 =

RN

|ηR (x) − ηR (y)|2 dxdy |x − y|N+2s    |ηR (x) − ηR (y)|2 |ηR (x) − ηR (y)|2 2 |un (x)| dy + dy dx |x − y|N+2s |x − y|N+2s |y−x|>R |y−x|≤R

|un (x)|2

1.4 Some Useful Lemmas



 ≤

27

RN

|un (x)|

4η2L∞ (RN )

2



 ≤C

|un (x)| dx 2

RN

|x − y|N+2s

|y−x|>R ∞ R

1

dy + R

dρ + R

ρ 2s+1

−2



−2

R 0





∇η2L∞ (RN ) |x − y|N+2s−2 

|y−x|≤R

1 dρ ρ 2s−1

dy

dx

C ≤ 2s , R and letting first n → ∞ and then R → ∞ we get the thesis. Lemma 1.4.7 Let (un ) ⊂ Ds,2 (RN ) be a bounded sequence in Ds,2 (RN ) and ψ ∈ Cc∞ (RN ) be such that 0 ≤ ψ ≤ 1 in RN , ψ = 1 in B1 , ψ = 0 in RN \ B2 and N 0 ∇ψL∞ (RN ) ≤ 2. Set ψρ (x) = ψ( x−x ρ ) for all ρ > 0, where x0 ∈ R is a fixed point. Then  lim lim sup

ρ→0 n→∞

R2N

|un (x)|2

|ψρ (x) − ψρ (y)|2 dxdy = 0. |x − y|N+2s

Proof We modify the proof of Lemma 1.4.5 in a convenient way. Without loss of generality we may assume that x0 = 0. Fix ρ > 0 and set  Wρ (x) =

RN

|ψρ (x) − ψρ (y)|2 dy. |x − y|N+2s

It is easy to verify that Wρ (x) ≤ C1 ρ 2s

for all x ∈ RN ,

Wρ (x) ≤ C2 ρ −N |x|−(N+2s)

for all |x| >

4 . ρ

Then, arguing as in the proof of Lemma 1.4.5, we get  lim sup n→∞

R2N

|un (x)|2

|ψρ (x) − ψρ (y)|2 dxdy ≤ |x − y|N+2s

 |u(x)|2Wρ (x) dx.

RN

¯ −2s and (1.1.2) imply that |u(x)|2 Wρ (x) ≤ C¯ |u(x)| Since 0 ≤ Wρ (x) ≤ C|x| ∈ L1 (RN ), |x|2s and Wρ → 0 as ρ → 0, we can apply the dominated convergence theorem to deduce that 2

 lim sup lim sup ρ→0

n→∞

R2N

|un (x)|2

|ψρ (x) − ψρ (y)|2 dxdy ≤ lim ρ→0 |x − y|N+2s

 RN

|u(x)|2 Wρ (x) dx = 0.

 

28

1 Preliminaries

The next result was established in [288] and provides a way to manipulate smooth truncations for the fractional Laplacian. Here we give an alternative proof. Lemma 1.4.8 ([288]) Let u ∈ Ds,2 (RN ) and φ ∈ Cc∞ (RN ) be such that 0 ≤ φ ≤ 1 in RN , φ = 1 in B1 and φ = 0 in RN \ B2 . Set φR (x) = φ( Rx ) for R > 0. Then lim [uφR − u]s = 0.

R→∞

Proof We have that   [uφR − u]2s ≤ 2

R2N

|u(x)|2

|φR (x) − φR (y)|2 dxdy + |x − y|N +2s



|φR (x) − 1|2 |u(x) − u(y)|2 dxdy |x − y|N +2s R2N



= 2[AR + BR ].

Since |φR (x) − 1| ≤ 2, |φR (x) − 1| → 0 a.e. in RN and u ∈ Ds,2 (RN ), the dominated convergence theorem implies that BR → 0

as R → ∞.

On the other hand, setting  WR (x) =

RN

|φR (x) − φR (y)|2 dy, |x − y|N+2s

we can argue as in the second part of the proof of Lemma 1.4.5 to deduce that AR → 0

as R → ∞.  

This ends the proof of lemma.

1.5

A Concentration-Compactness Principle

Here we present a concentration-compactness lemma in the spirit of Lions [256, 257] and Chabrowski [135] (see also [94, 104, 340]). For more details and comments on this topic we refer to [40, 61, 170, 288, 344]. Lemma 1.5.1 ([61]) Let (un ) be a sequence that converges weakly to u in Ds,2 (RN ) and such that s

|(−) 2 un |2  μ and

∗

|un |2s  ν,

(1.5.1)

1.5 A Concentration-Compactness Principle

29

in the sense of measures, where μ and ν are two bounded nonnegative measures on RN . Then, there exist an at most countable index set I and three families of distinct points, (xi )i∈I in RN and (μi )i∈I and (νi )i∈I in (0, ∞), such that ∗

ν = |u|2s +



(1.5.2)

νi δxi ,

i∈I s

μ ≥ |(−) 2 u|2 +



(1.5.3)

μi δ x i ,

i∈I 2 2∗

μi ≥ S∗ νi s

∀i ∈ I,

(1.5.4)

where δx is the Dirac mass at x ∈ RN . Moreover, if we define  μ∞ = lim lim sup R→∞ n→∞

s

|x|>R

|(−) 2 un |2 dx,

(1.5.5)

and  ν∞ = lim lim sup R→∞ n→∞

∗

|x|>R

|un |2s dx,

(1.5.6)

then  lim sup n→∞



lim sup

RN

RN



∗

|un |2s dx =

RN

n→∞



s

|(−) 2 un |2 dx =

RN

dμ + μ∞ ,

(1.5.7)

dν + ν∞ ,

(1.5.8)

2 2∗

μ∞ ≥ S∗ ν∞s .

(1.5.9)

Proof In order to prove (1.5.2), we aim to pass to the limit in the following relation which holds in view of the Brezis-Lieb lemma [113]:  RN

∗

∗



|ψ|2s |un |2s dx =

∗

RN



∗

|ψ|2s |u|2s dx +

∗

RN

∗

|ψ|2s |un − u|2s dx + on (1), (1.5.10)

where ψ ∈ Cc∞ (RN ). Set u˜ n = un − u. Then, by Theorem 1.1.8, u˜ n → 0 in L2loc (RN ) and a.e. in RN . Up to subsequence, we may assume that s

|(−) 2 u˜ n |2  μ˜

and

∗

|u˜ n |2s  ν˜

(1.5.11)

30

1 Preliminaries

in the sense of measures. Fix ψ ∈ Cc∞ (RN ). Using the fractional Sobolev inequality (1.1.1) in Theorem 1.1.8 we have 

∗

RN

∗

|ψ|2s |un − u|2s dx

 1∗



2s

=



∗

RN

≤

−1 S∗ 2

=

−1 S∗ 2

|ψ u˜ n |2s dx



1 2∗ s

1

s 2

RN

|(−) (ψ u˜ n )| dx

  R2N

2

2

|(ψ u˜ n )(x) − (ψ u˜ n )(y)|2 dxdy |x − y|N+2s

 12 . (1.5.12)

Next, by the Minkowski inequality,   R2N

|ψ(x)u˜ n (x) − ψ(y)u˜ n (y)|2 dxdy |x − y|N+2s

  ≤

R2N

|u˜ n (x) − u˜ n (y)| |ψ(y)|2 |x − y|N+2s

 =

s

RN

1

|ψ|2 |(−) 2 u˜ n |2 dx

2

 12  12

2

dxdy

  +

 +

R2N s

RN

|u˜ n (x)|

|u˜ n |2 |(−) 2 ψ|2 dx

2 |ψ(x) −

ψ(y)|2 dxdy |x − y|N+2s

 12

1 2

,

where in the last equality we used a simple change of variable. Now we prove that 

s

RN



|u˜ n |2 |(−) 2 ψ|2 dx =

R2N

|u˜ n (x)|2

|ψ(x) − ψ(y)|2 dxdy = on (1). |x − y|N+2s

(1.5.13)

Assume for simplicity that supp(ψ) = B¯ 1 and set  W (x) =

RN

|ψ(x) − ψ(y)|2 dy. |x − y|N+2s

Note that W ∈ L∞ (RN ), because 

 |ψ(x) − ψ(y)|2 |ψ(x) − ψ(y)|2 dy + dy N+2s N+2s |y−x|≤1 |x − y| |y−x|>1 |x − y|   1 1 2 ≤ ∇ψ2L∞ (RN ) dy + 4ψ dy L∞ (RN ) N+2s−2 N+2s |x − y| |x − y| |y−x|≤1 |y−x|>1

W (x) =

≤ C.

(1.5.14)

1.5 A Concentration-Compactness Principle

31

Moreover, for |x| > 2, we get  W (x) =

|y|≤1

2N+2s ωN ψ2L∞ (RN ) C |ψ(y)|2 dy ≤ = , |x − y|N+2s |x|N+2s |x|N+2s

(1.5.15)

where we used the fact that, for all |x| > 2 and |y| ≤ 1, |x − y| ≥ |x| − |y| ≥ |x| − 1 ≥

|x| . 2

Fix R > 2. In view of (1.5.14), (1.5.15) and using the Hölder inequality we see that 

 RN

|u˜ n (x)|2 W (x) dx =

 |x|≤R

|u˜ n (x)|2 W (x) dx +

|x|>R



 ≤ W L∞ (RN )

|x|≤R

|u˜ n (x)|2 dx + u˜ n 2 2∗s

L (RN )



 ≤C

|x|≤R

|u˜ n (x)|2 W (x) dx

|u˜ n (x)|2 dx + C

N2

|x| 2s

∗

 2s2−2 ∗ dx

s

 2∗s −2 ∗ 2s

1 |x|>R

|x|>R

|W (x)|

2∗ s 2∗ s −2

+N

dx

,

and recalling that u˜ n → 0 in L2 (BR ) we have 

 lim sup n→∞

RN

|u˜ n (x)|2 W (x) dx ≤ C

 2∗s −2 ∗ 2s

1 N2

|x| 2s +N

|x|>R

dx

∀R > 2.

Taking here the limit as R → ∞, we obtain (1.5.13). On the other hand, by the first limit relation in (1.5.11), we have 



s

RN

|ψ|2 |(−) 2 u˜ n |2 dx →

RN

|ψ|2 d μ. ˜

Since the second limit relation in (1.5.11) implies that 

∗

RN

∗

|ψ|2s |un − u|2s dx →



∗

RN

|ψ|2s d ν˜ ,

from (1.5.12) we obtain that 

∗

RN

|ψ|2s d ν˜



1 2∗ s

− 12

≤ S∗

 RN

1 2 |ψ|2 d μ˜ , for all ψ ∈ Cc∞ (RN ).

32

1 Preliminaries

Then, using Lemma 1.2 in [257], there exist an at most a countable set I , families (xi )i∈I of distinct points in RN , (νi )i∈I in (0, ∞) such that ν˜ =



μ˜ ≥ S∗

νi δxi ,

i∈I



2 2∗

νi s δxi .

(1.5.16)

i∈I ∗

In view of (1.5.10), we deduce that ν = |u|2s + ν˜ , which together with (1.5.16) implies that ∗

ν = |u|2s +



νi δxi ,

i∈I

that is, (1.5.2) is satisfied. i Now, we prove that (1.5.4) holds true. Fix i ∈ I and take ψρ (x) = ψ( x−x ρ ), where ψ ∈ Cc∞ (RN ), 0 ≤ ψ ≤ 1 in RN , ψ(0) = 1 and supp(ψ) = B¯ 1 . As before, we obtain 1

S∗2

 RN

∗



∗

|ψρ |2s |un |2s dx

1 2∗ s

 

 12 |ψρ (x)un (x) − ψρ (y)un (y)|2 dxdy |x − y|N +2s R2N   1  1 2 2 s s ≤ |un |2 |(−) 2 ψρ |2 dx + |ψρ |2 |(−) 2 un |2 dx . ≤

RN

RN

(1.5.17) Now, taking into account (1.5.1) and (1.5.2), we have 

∗

lim

n→∞ RN

∗



∗

|ψρ |2s |un |2s dx =

∗

|ψρ |2s |u|2s dx + νi . Bρ (xi )

Since 0 ≤ ψρ ≤ 1 implies      ∗ ∗ ∗   2s 2s |ψρ | |u| dx  ≤ C |u|2s dx → 0 as ρ → 0,    Bρ (xi ) Bρ (xi ) we deduce that  lim lim

ρ→0 n→∞ RN

∗

∗

|ψρ |2s |un |2s dx = νi .

(1.5.18)

On the other hand, (1.5.1) gives  lim

n→∞ R2N

s

|ψρ |2 |(−) 2 un |2 dxdy =

 RN

|ψρ |2 dμ,

(1.5.19)

1.5 A Concentration-Compactness Principle

33

and using Lemma 1.4.7 we see that  lim lim sup

ρ→0 n→∞

s

RN

|un |2 |(−) 2 ψρ |2 dx = 0.

(1.5.20)

Combining (1.5.17), (1.5.18), (1.5.19) and (1.5.20), we get 2 2∗

S∗ νi s ≤ lim μ(Bρ (xi )). ρ→0

Setting μi = lim μ(Bρ (xi )), ρ→0

we deduce that (1.5.4) holds true. Now, we note that μ≥



μi δ x i ,

i∈I s

and that the weak lower semi-continuity implies that μ ≥ |(−) 2 u|2 . Then, since  s |(−) 2 u|2 is orthogonal to i∈I μi δxi , we infer that (1.5.3) is satisfied. Finally, we show the validity of (1.5.7)–(1.5.9). Let ηR be defined as in Lemma 1.4.5. Then we have 



s

RN

|(−) 2 un |2 dx =



s

RN

2 |(−) 2 un |2 ηR dx +

s

RN

2 |(−) 2 un |2 (1 − ηR ) dx.

(1.5.21) Since  |x|>2R



s

|(−) 2 un |2 dx ≤



s

RN

2 |(−) 2 un |2 ηR dx ≤

s

|x|>R

|(−) 2 un |2 dx,

we obtain that  μ∞ = lim lim sup R→∞ n→∞

s

RN

2 |(−) 2 un |2 ηR dx.

(1.5.22)

2 is smooth and compactly supported, we can Now, using the fact that μ is finite and 1 − ηR apply the dominated convergence theorem to get

 lim lim sup

R→∞ n→∞

RN

s





2 |(−) 2 un |2 (1 − ηR ) dx = lim

R→∞ RN

2 (1 − ηR ) dμ =

RN

dμ. (1.5.23)

34

1 Preliminaries

Combining (1.5.21), (1.5.22) and (1.5.23), we obtain (1.5.7). In a similar fashion, we can prove that  ν∞ = lim lim sup R→∞ n→∞

∗

RN

2∗

|un |2s ηRs dx,

(1.5.24)

and arguing as before we deduce that (1.5.8) is verified. To show that (1.5.9) is satisfied, we observe that 1

S∗2



∗

RN

|ηR un |2s dx



1 2∗ s

 ≤

RN

 ≤

RN

1

s

2

|(−) 2 (ηR un )|2 dx s 2 ηR |(−) 2 un |2 dx

1 2

 +

RN

s 2

1

|un | |(−) ηR | dx 2

2

2

.

(1.5.25) By Lemma 1.4.5,  lim lim sup

R→∞ n→∞

s

RN

|un |2 |(−) 2 ηR |2 dx = 0,

which together with (1.5.22), (1.5.24), and (1.5.25) yields (1.5.9). This ends the proof of lemma.   Remark 1.5.2 In the case = RN , the concentration-compactness principle established in [288] does not provide any information about a possible loss of mass at infinity. Then Lemma 1.5.1 expresses this fact in quantitative terms.

2

Some Abstract Results

In this section we review some useful results on critical point theory, minimax methods and Nehari manifold arguments. For more details we refer to [27, 137, 267, 298, 319, 321, 322, 340].

2.1

Critical Point Theory

We start with the definitions of derivatives in Banach spaces. Let X be a Banach space equipped with the norm  · , and denote by X∗ its topological dual, that is, the space of all continuous linear functionals on X. The (dual) norm on X∗ is defined by f X∗ =

|f (u)| =

sup u∈X,u≤1

sup

f (u).

u∈X,u≤1

We denote by ·, · the duality pairing between X and X∗ , and by L(X, X∗ ) the vector space of all continuous linear operators from X into X∗ . Definition 2.1.1 Let U ⊂ X be an open set of X and ϕ : U → R be a functional. We say that ϕ has a Gateaux derivative f ∈ X∗ at u ∈ X if, for every h ∈ X, lim

t →0

ϕ(u + th) − ϕ(u) − f, th = 0. t

The Gateaux derivative at u is denoted by ϕ (u) and it holds that ϕ (u), h = lim

t →0

ϕ(u + th) − ϕ(u) . t

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_2

35

36

2 Some Abstract Results

The functional ϕ has a Fréchet derivative f ∈ X∗ at u ∈ U if lim

h→0

ϕ(u + h) − ϕ(u) − f, h = 0. h

Evidently, any Fréchet derivative is a Gateaux derivative. The functional ϕ belongs to C 1 (U, R) if the Fréchet derivative of ϕ exists and is continuous on U . Using the mean value theorem, it is easy to see that if ϕ has a continuous Gateaux derivative on U , then ϕ ∈ C 1 (U, R). If X is a Hilbert space with inner product (·, ·) and ϕ has a Gateaux derivative at u ∈ U , the gradient ∇ϕ(u) of ϕ at u is defined by (∇ϕ(u), h) = ϕ (u), h. A point u ∈ U is called a critical point of ϕ if ϕ (u) = 0; otherwise u is called a regular point. A number c ∈ R is a critical value of ϕ if there exists a critical point u of ϕ with ϕ(u) = c. Otherwise, c is called regular. Definition 2.1.2 Let ϕ ∈ C 1 (U, R). The functional ϕ has a second Gateaux derivative L ∈ L(X, X∗ ) at u ∈ U if, for every h, v ∈ X, ϕ (u + th) − ϕ (u) − Lth, v = 0. t →0 t lim

The second Gateaux derivative at u is denoted by ϕ

(u). The functional ϕ has a second Fréchet derivative L ∈ L(X, X∗ ) at u ∈ U if ϕ (u + h) − ϕ (u) − Lh = 0. h→0 h lim

The functional ϕ belongs to C 2 (X, R) if the second Fréchet derivative of ϕ exists and is continuous on U . The second Gateaux derivative is given by ϕ (u + th) − ϕ (u), k . t →0 t

ϕ

(u)h, k = lim

Clearly, any second Fréchet derivative of ϕ is a second Gateaux derivative. Using the mean value theorem, if ϕ has a continuous second Gateaux derivative on U , then ϕ ∈ C 2 (U, R). Let f (x, t) be a function on × R, where is either bounded or unbounded. We say that f is a Carathéodory function if f (x, t) is continuous in t for a.e. x ∈ and measurable in x for every t ∈ R.

2.1 Critical Point Theory

37

Lemma 2.1.3 ([351]) Assume p, q ≥ 1. Let f (x, t) be a Carathéodory function on ×R satisfying p

|f (x, t)| ≤ a + b|t| q ,

∀(x, t) ∈ × R,

where a, b > 0 and is either bounded or unbounded. Define a Carathéodory operator by setting for all u ∈ Lp ( ) Bu = f (x, u(x)). Let (un ) ⊂ Lp ( ) be a sequence. If un → u in Lp ( ) as n → ∞, then Bun → Bu in Lq ( ) as n → ∞. In particular, if is bounded, then B is a continuous and bounded mapping from Lp ( ) to Lq ( ) and the same conclusion holds true if is unbounded and a = 0. Lemma 2.1.4 ([351]) Assume p1 , p2 , q1 , q2 ≥ 1. Let f (x, t) be a Carathéodory function on × R satisfying p1

p2

|f (x, t)| ≤ a|t| q1 + b|t| q2 ,

∀(x, t) ∈ × R,

where a, b ≥ 0 and is bounded or unbounded. Define a Carathéodory operator by setting for every u ∈ D = Lp1 ( ) ∩ Lp2 ( ) Bu = f (x, u(x)). Define the space E = Lq1 ( ) + Lq2 ( ), equipped with the norm uE = inf{vLq1 ( ) + wLq2 ( ) : u = v + w ∈ E, v ∈ Lq1 ( ), w ∈ Lq2 ( )}. Then B = B1 + B2 , where Bi is a bounded and continuous mapping from Lpi ( ) to Lqi ( ), i = 1, 2. In particular, B is a bounded continuous mapping from D to E. Theorem 2.1.5 ([351]) Assume σ, p ≥ 0. Let f (x, t) be a Carathéodory function on

× R satisfying |f (x, t)| ≤ a|t|σ + b|t|p ,

∀(x, t) ∈ × R,

38

2 Some Abstract Results

where a, b > 0 and is either bounded or unbounded. Define a functional I by 

 I (u) =

where

F (x, u) dx,

t

F (x, t) =

f (x, τ ) dτ. 0



Assume that (X,  · ) is a Sobolev Banach space such that E is continuously embedded in Lp+1 ( ) and Lσ +1 ( ). Then, I ∈ C 1 (X, R) and



I (u), h =

f (x, u)h dx

∀h ∈ X.



Moreover, if E is compactly embedded in Lp+1 ( ) and Lσ +1 ( ), then I : X → X∗ is compact.

2.2

Minimax Methods

This subsection is devoted to some classical minimax theorems of the critical point theory. We start by recalling Ekeland’s variational principle [176]: Theorem 2.2.1 ([176] Ekeland’s Variational Principle) Let (X, d) be a a complete metric space with metric d, and let ϕ : X → R ∪ {∞} be a lower semi-continuous function, bounded from below and ϕ ≡ ∞. Let ε > 0 and λ > 0 be given and u ∈ X be such that ϕ(u) ≤ infX ϕ + ε. Then there exists v ∈ X such that • ϕ(v) ≤ ϕ(u), • d(u, v) ≤ λ, • ϕ(w) > ϕ(v) − λε d(v, w) for all w = v. Let us introduce some useful definitions (see [111, 267, 287, 319] for more details). Definition 2.2.2 Let X be a Banach space and ϕ ∈ C 1 (X, R). We say that ϕ satisfies the Palais–Smale condition (shortly, (PS) condition) if any sequence (un ) ⊂ X such that (ϕ(un )) is bounded and ϕ (un ) → 0 in X∗

(2.2.1)

admits a strongly convergent subsequence. Any sequence satisfying (2.2.1) is called a Palais–Smale sequence.

2.2 Minimax Methods

39

Let c ∈ R. The functional ϕ is said to satisfy the Palais–Smale condition at the level c ∈ R (shortly, (PS)c condition), if every sequence (un ) ⊂ X such that ϕ(un ) → c and ϕ (un ) → 0 in X∗ admits a strongly convergent subsequence. Now we give the notion of pseudogradient defined in [286]. Definition 2.2.3 Let E be a metric space, X a normed space and h : E → X∗ \ {0} a continuous mapping. A pseudogradient vector field for h on E is a locally Lipschitz continuous vector field g : E → X such that, for every u ∈ E, g(u) ≤ 2h(u)X∗ , h(u), g(u) ≥ h(u)2X∗ . Lemma 2.2.4 Under the assumptions of the preceding definition, there exists a pseudogradient vector field for h on E. Before stating the next result, we introduce some notations. Let ϕ : X → R be a functional and S ⊂ X. Then, for all d ∈ R, we set ϕ d = {v ∈ X : ϕ(v) ≤ d}, and for all δ > 0 Sδ = {u ∈ X : dist(u, S) ≤ δ}. At this point we present the following quantitative deformation lemma for continuously differentiable functions defined on a Banach space. Lemma 2.2.5 ([340]) Let X be a Banach space, ϕ ∈ C 1 (X, R), S ⊂ X, c ∈ R, ε, δ > 0 such that ϕ (u) ≥

8ε δ

∀u ∈ ϕ −1 ([c − 2 ε, c + 2 ε]) ∩ S2δ .

Then there exists η ∈ C([0, 1] × X, X) such that (1) η(t, u) = u, if t = 0 or if u ∈ ϕ −1 ([c − 2 ε, c + 2 ε]) ∩ S2δ , (2) η(1, ϕ c+ε ∩ S) ⊂ ϕ c−ε , (3) η(t, ·) is an homeomorphism of X, ∀t ∈ [0, 1],

40

2 Some Abstract Results

(4) η(t, u) − u ≤ δ, ∀u ∈ X, ∀t ∈ [0, 1], (5) ϕ(η(·, u)) is nonincreasing, ∀u ∈ X, (6) ϕ(η(t, u)) < c, ∀u ∈ ϕ c ∩ Sδ , ∀t ∈ (0, 1]. Remark 2.2.6 As pointed out in [340], a simple application of Lemma 2.2.5 gives a version of Ekeland variational principle for Banach spaces. A very important application of Lemma 2.2.5 is the following general minimax principle. Theorem 2.2.7 ([340]) Let X be a Banach space. Let M0 be a closed subspace of the metric space M and let 0 ⊂ C(M0 , X). Define  = {γ ∈ C(M, X) : γ |M0 ∈ 0 }. If ϕ ∈ C 1 (X, R) satisfies ∞ > c = inf sup ϕ(γ (u)) > a = sup sup ϕ(γ0 (u)) γ ∈ u∈M

γ0 ∈0 u∈M0

(2.2.2)

then, for every ε ∈ (0, c−a 2 ), δ > 0 and γ ∈  such that sup ϕ ◦ γ ≤ c + ε, M

there exists u ∈ X such that (a) c − 2 ε ≤ ϕ(u) ≤ c + 2 ε, (b) dist(u, γ (M)) ≤ 2δ, (c) ϕ (u)X∗ ≤ 8δε . As a consequence of Theorem 2.2.7 we have: Theorem 2.2.8 ([340]) Under assumption (2.2.2), there exists a sequence (un ) ⊂ X satisfying ϕ(un ) → c

and

ϕ (un ) → 0 in X∗ .

In particular, if ϕ satisfies the (PS)c condition, then c is a critical value of J . With the aid of Theorem 2.2.8 we obtain the next renowned minimax theorems.

2.2 Minimax Methods

41

Theorem 2.2.9 (Mountain Pass Theorem [29, 298, 340]) Let X be a Banach space, ϕ ∈ C 1 (X, R), e ∈ X and r > 0 be such that e > r and b = inf ϕ(u) > ϕ(0) ≥ ϕ(e). u=r

If ϕ satisfies the (PS)c condition with c = inf max ϕ(γ (t)), γ ∈ t ∈[0,1]

 = {γ ∈ C([0, 1], X) : γ (0) = 0, γ (1) = e}, then c is a critical value of ϕ. Remark 2.2.10 If ϕ satisfies the geometric assumptions of the mountain pass theorem, then Theorem 2.2.8 guarantees the existence of a Palais–Smale sequence at the mountain pass level c. In the literature, this result is sometimes referred to as a variant of the mountain pass theorem without the Palais–Smale condition (see [111, 114, 177]). More precisely, Brezis and Nirenberg [114] observed that if X is a Banach space, ϕ ∈ C 1 (X, R), and there exist a neighborhood U of 0 in X and a constant ρ such that ϕ(u) ≥ ρ for every u in the boundary of U , ϕ(0) < ρ and ϕ(v) < ρ for some v ∈ U , then there exists a Palais–Smale sequence at the level c = inf max ϕ(w) ≥ ρ, P ∈P w∈P

where P denotes the class of paths joining 0 to v. Theorem 2.2.11 (Saddle-Point Theorem [298,340]) Let X = Y ⊕ Z be a Banach space with dim Y < ∞. Define, for ρ > 0, M = {u ∈ Y : u ≤ ρ},

M0 = {u ∈ Y : u = ρ}.

Let ϕ ∈ C 1 (X, R) be such that b = inf ϕ > a = max ϕ. Z

M0

If ϕ satisfies the (PS)c condition with c = inf max ϕ(γ (u)), γ ∈ u∈M

 = {γ ∈ C(M, X) : γ = id on M0 }, then c is a critical value of ϕ.

42

2 Some Abstract Results

Theorem 2.2.12 (Linking Theorem [298, 340]) Let X = Y ⊕ Z be a Banach space with dimY < ∞. Let ρ > r > 0 and let z ∈ Z be such that z = r. Define M = {u = y + λz : u ≤ ρ, λ ≥ 0, y ∈ Y }, M0 = {u = y + λz : y ∈ Y, u = ρ and λ ≥ 0 or u ≤ ρ and λ = 0}, N = {u ∈ Z : u = r}. Let ϕ ∈ C 1 (X, R) be such that b = inf ϕ > a = max ϕ. N

M0

If ϕ satisfies the (PS)c condition with c = inf max ϕ(γ (u)), γ ∈ u∈M

 = {γ ∈ C(M, X) : γ = id on M0 }, then c is a critical value of ϕ. We also recall the following compactness-type condition proposed in [134]. Definition 2.2.13 Let X be a Banach space and ϕ ∈ C 1 (X, R). We say that ϕ satisfies the Cerami condition (shortly, (C) condition) if any sequence (un ) ⊂ X such that (ϕ(un )) is bounded and (1 + un )ϕ (un ) → 0 in X∗

(2.2.3)

admits a strongly convergent subsequence. Any sequence satisfying (2.2.3) is called a Cerami sequence. Let c ∈ R. The functional ϕ is said to satisfy the Cerami condition at the level c ∈ R (shortly, (C)c condition), if every sequence (un ) ⊂ X such that ϕ(un ) → c and (1 + un )ϕ (un ) → 0 in X∗ admits a strongly convergent subsequence. Remark 2.2.14 The Cerami condition is weaker than the Palais–Smale condition. When a C 1 -functional possesses a mountain pass geometry, we can use Ekeland’s variational principle to obtain the following variant of the mountain pass lemma with the Cerami condition.

2.3 Genus and Category

43

Theorem 2.2.15 ([177]) Let X be a Banach space and suppose that ϕ ∈ C 1 (X, R) satisfies max{ϕ(0), ϕ(e)} ≤ μ < α ≤ inf ϕ(u), u=ρ

for some μ < α, ρ > 0 and e ∈ X with e > ρ. Let c ≥ α be characterized by c = inf max ϕ(γ (t)), γ ∈ t ∈[0,1]

where  = {γ ∈ C([0, 1], X) : γ (0) = 0, γ (1) = e}. Then there exists a Cerami sequence (un ) ⊂ X at the level c.

2.3

Genus and Category

Topological tools play a central role in the study of variational problems. In this subsection we recall the notions of genus and category as well as some of their basic properties. Definition 2.3.1 Let X be a real Banach space and denote (X) = {A ⊂ X \ {0}| A is closed and symmetric with respect to 0}. We say that the positive integer k is the Krasnoselski genus of A ∈ (X), if there exists an odd map ϕ : A → Rk \ {0} and k is the smallest integer with this property. The genus of the set A is denoted by γ (A) = k. When there is no finite such k, set γ (A) = ∞. Finally, set γ (∅) = 0. Remark 2.3.2 The notion of genus is due to Krasnoselskii [242]; here we use the equivalent definition due to Coffman [149]. The genus has the following well-known properties: Proposition 2.3.3 ([29, 298]) Let A, B ∈ (X). Then: (i) (ii) (iii) (iv)

if there exists an odd map f ∈ C(A, B), then γ (A) ≤ γ (B); if A ⊂ B, then γ (A) ≤ γ (B); if there exists an odd homeomorphism between A and B, then γ (A) = γ (B); γ (A ∪ B) ≤ γ (A) + γ (B);

44

2 Some Abstract Results

(v) if γ (B) < ∞, then γ (A \ B) ≥ γ (A) − γ (B); (vi) if A is compact, then γ (A) < ∞, and there exists δ > 0 such that γ (Nδ (A)) = γ (A), where Nδ (A) = {x ∈ X : dist(x, A) ≤ δ}; (vii) if A is homeomorphic by an odd homeomorphism to the boundary of a symmetric bounded open neighborhood of 0 in Rm , then γ (A) = m; (viii) let A ∈ (X), V be a k-dimensional subspace of X, and V ⊥ an algebraically and topologically complementary subspace. If γ (A) > k, then A ∩ V ⊥ = ∅. The genus can be used to get infinitely many distinct pairs of critical points for even functionals: Theorem 2.3.4 ([298]) Let X be an infinite-dimensional Hilbert space and let ϕ ∈ C 1 (X, R) be even. Suppose r > 0, ϕ|∂Br satisfies the (PS) condition, and ϕ|∂Br is bounded from below. Then ϕ|∂Br possesses infinitely many distinct pairs of critical points. We also recall the following symmetric mountain pass theorem due to Ambrosetti and Rabinowitz [29]. Theorem 2.3.5 (Symmetric Mountain Pass Theorem [29]) Let X be an infinitedimensional Banach space and let ϕ ∈ C 1 (X, R) be even, satisfy the (PS) condition, and ϕ(0) = 0. If X = V ⊕ W , where V is finite-dimensional, and ϕ satisfies (i) there are constants ρ, α > 0 such that ϕ(u) ≥ α for all u ∈ W such that u = ρ, and (ii) for each finite-dimensional subspace E ⊂ X, there exists an R = R(E) such that ϕ(u) ≤ 0 for any u ∈ E such that u ≥ R, then ϕ possesses an unbounded sequence of critical values. Next we present the notion of category introduced by Lusternik and Schnirelman [265]. Definition 2.3.6 Let M be a topological space and A ⊂ M a subset. The continuous map h : [0, 1] × A → M is called a deformation of A in M if h(0, u) = 0 for all u ∈ A. The set A is said be contractible in M if there exist a deformation h : [0, 1] × A → M and p ∈ M such that h(1, u) = {p} for all u ∈ A. Definition 2.3.7 Let M be a topological space. A set A ⊂ M is said to be of LusternikSchnirelman category k in M (denoted catM (A)) if it can be covered by k but not by k − 1 closed sets that are contractible to a point in M. If such k does not exist, then catM (A) = ∞.

2.3 Genus and Category

45

The Lusternik-Schnirelman category has the following basic properties. Proposition 2.3.8 ([137, 298, 319, 340]) Let A, B ⊂ M. catM (A) = 0 ⇐⇒ A = ∅; A ⊂ B $⇒ catM (A) ≤ catM (B); catM (A ∪ B) ≤ catM (A) + catM (B); if A is closed and η : [0, 1] × A → M is a deformation of A in M, then catM (A) ≤ catM (η(1, A)); (v) if M is a Banach-Finsler manifold of class C 1 and A ⊂ M, then there is a closed neighborhood N of A in M such that catM (A) = catM (N).

(i) (ii) (iii) (iv)

Let M be a Banach-Finsler manifold of class C 1 (see [137, 286, 319] for the definition) and ϕ ∈ C 1 (M, R). For any k ≤ cat(M) we define ck = inf max ϕ(u), A∈Ak u∈A

Ak = {A ⊂ M : A is compact, catM (A) ≥ k}. With the aid of the category we can state the following celebrated multiplicity result (see [137] for a proof). Theorem 2.3.9 (Lusternik-Schnirelman Theorem [265]) Let M be a complete BanachFinsler manifold of class C 1 , and let ϕ : M → R be a functional of class C 1 which is bounded from below on M. If ϕ satisfies the (PS) condition, then ϕ has at least catM (M) critical points. In what follows, we assume that (A) X is a real Banach space, ψ ∈ C 2 (X, R), V = {u ∈ X : ψ(v) = 1} = ∅, ψ (v) = 0 for every v ∈ V. The set V is a differentiable manifold of class C 2 . The norm on X induces a metric on V and so V becomes a metric manifold, i.e., a metric space and a manifold. We denote by Tv V the tangent space of V at v, i.e., Tv V = {w ∈ X : ψ (v), w = 0}.

46

2 Some Abstract Results

Let ϕ ∈ C 1 (X, R) and v ∈ V. The norm of the derivative of the restriction of ϕ to V is defined by ϕ (v)∗ =

sup

ψ (v), w.

w∈Tv V , w=1

The point v is a critical point of the restriction of ϕ to v if the restriction of ϕ (v) to Tv V is equal to 0. Proposition 2.3.10 ([340]) If ϕ ∈ C 1 (X, R) and u ∈ V, then ϕ (u)∗ = min ϕ (u) − λψ (u). λ∈R

In particular, u is a critical point of ϕ|V if and only if there exists λ ∈ R such that ϕ (u) = λψ (u). Definition 2.3.11 The functional ϕ|V satisfies the (PS)c condition if any sequence (un ) ⊂ V such that ϕ(un ) → c and ϕ (un )∗ → 0 has a convergent subsequence. Theorem 2.3.12 ([340]) If ϕ|V is bounded from below, d ≥ infV ϕ and ϕ satisfies the (PS)c condition for any c ∈ [infV ϕ, d], then ϕ|V has a minimum and ϕ d contains at least catϕ d (ϕ d ) critical points of ϕ|V .

2.4

The Method of Nehari Manifold

Let X be a Banach space and let ϕ ∈ C 1 (X, R) be a functional such that ϕ (0) = 0. Suppose that u ∈ X is a nontrivial critical point of ϕ. Then necessarily u is contained in the set N = {u ∈ X \ {0} : ϕ (u), u = 0} which is a natural constraint for the problem of finding nontrivial critical points of ϕ. N is called the Nehari manifold, though in general it may not be a manifold. Set c = inf ϕ(u). u∈N

2.4 The Method of Nehari Manifold

47

Under suitable conditions on ϕ one can prove that c is attained at some u0 ∈ N and that u0 is a critical point. For instance, if ϕ is bounded below on N and u ∈ N satisfies ϕ(u) = c, then u is a critical point of ϕ different from 0. We recall the following result: Proposition 2.4.1 ([280]) Let X be a Banach space and ϕ ∈ C 2 (X, R). Assume that N = ∅, ϕ

(u)u, u = 0 for all u ∈ N , and there exists r > 0 such that Br ∩ N = ∅. Then N is a complete C 1 -Banach submanifold of X of codimension 1, and it is a natural constraint of ϕ. In what follows, we present the approach developed in [321, 322] to the method of Nehari manifold. It does not need to make customary assumptions which imply that ϕ ∈ C 2 (X, R) and ϕ

(u)u, u = 0 on N . Let X be a uniformly convex real Banach space, ϕ ∈ C 1 (X, R) and ϕ(0) = 0. Let S = S1 (0) = {x ∈ X : x = 1} be the unit sphere in X. A function ν ∈ C(R+ , R+ ) is said to be a normalization function if ν(0) = 0, ν is increasing and ν(t) → ∞ as t → ∞. We also introduce the following assumptions: (A1 ) There exists a normalization function ν such that 

u

u → ψ(u) =

ν(t) dt ∈ C 1 (X \ {0}, R),

0

J = ψ is bounded on bounded sets and J (w), w = 1 for all w ∈ S.

(s) > 0 (A2 ) For each w ∈ X \ {0} there exists sw such that if αw (s) = ϕ(sw), then αw

for 0 < s < sw and αw (s) < 0 for s > sw . (A3 ) There exists δ > 0 such that sw ≥ δ for all w ∈ S and for each compact subset W ⊂ S there exists a constant CW such that sw ≤ CW for all w ∈ W. The map J in (A1 ) is called the duality mapping corresponding to ϕ. It follows from (A1 ) that S is a C 1 -submanifold of X. Let us note that (A2 ) implies that sw ∈ N if and only if s = sw . Moreover, by the first part of (A3 ), N is closed in X and bounded away from 0. Define the mappings m ˆ : X \ {0} → N and m : S → N by setting m(w) ˆ = sw w

and

m = m| ˆ S.

Proposition 2.4.2 ([322]) Suppose ν satisfies (A2 ) and (A3 ). Then: (a) The mapping m ˆ is continuous. (b) The mapping m is a homeomorphism between S and N , and the inverse of m is given by m−1 (u) = u/u.

48

2 Some Abstract Results

ˆ : X \ {0} → R and  : S → R defined by Let us consider the functionals  ˆ (w) = ϕ(m(w)) ˆ

ˆ S. and  = |

Proposition 2.4.3 ([322]) Suppose X is a Banach space satisfying (A1 ). If ϕ satisfies ˆ ∈ C 1 (X \ {0}, R) and (A2 ) and (A3 ), then  ˆ (w), z = 

m(w) ˆ ϕ (m(w)), ˆ z w

for all w, z ∈ X, w = 0.

Corollary 2.4.4 ([322]) Suppose X is a Banach space satisfying (A1 ). If ϕ satisfies (A2 ) and (A3 ), then: (a)  ∈ C 1 (S, R) and  (w), z = m(w)ϕ (m(w)), z

for all z ∈ Tw S.

(b) If (wn ) is a Palais–Smale sequence for , then (m(wn )) is a Palais–Smale sequence for ϕ. If (un ) ⊂ N is a bounded Palais–Smale sequence for ϕ, then (m−1 (un )) is a Palais–Smale sequence for . (c) w is a critical point of  if and only if m(w) is a nontrivial critical point of ϕ. Moreover, the corresponding values of  and ϕ coincide and infS  = infN ϕ. (d) If ϕ is even, then so is . We note that the infimum of ϕ over N has the following minimax characterization: c = inf ϕ(u) = u∈N

inf

max ϕ(sw) = inf max ϕ(sw).

w∈X\{0} s>0

w∈S s>0

(2.4.1)

Now, we introduce the subset S+ = {u ∈ X : u = 1, u+ = 0} of the unit sphere S in X, and consider the following assumptions: (I ) For each w ∈ X with w+ = 0 there exists sw such that if αw (s) = ϕ(sw), then

(s) > 0 for 0 < s < s and α (s) < 0 for s > s . αw w w w (II) There exists δ > 0 such that sw ≥ δ for all w ∈ S+ and for each compact subset W ⊂ S+ there exists a constant CW such that sw ≤ CW for all w ∈ W. (III) The map m : S+ → N defined by m(u) = su u is a homeomorphism between S+ and N , and the inverse of m is given by m−1 (u) = u/u. As before, N = {u ∈ X \ {0} : ϕ (u), u = 0} is the Nehari manifold. Let  : S+ → R be defined by (w) = ϕ(m(w)). ˆ Then  is a C 1 -functional on the open subset S+ of S

2.4 The Method of Nehari Manifold

49

and it holds  (w), z = m(w)ϕ (m(w)), z for all w ∈ S+ and z ∈ Tw S+ . Hence, the nontrivial critical points of  are in a one-to-one correspondence with the nontrivial critical points of ϕ. Moreover, the following result holds true: Lemma 2.4.5 ([322]) (i) Let (un ) ⊂ S+ be a sequence such that dist(un , ∂S+ ) → 0 as n → ∞ (where the distance is taken with respect to the norm  · ). Then (un ) → ∞. (ii)  satisfies the Palais–Smale condition in S+ . Similarly as before, we see that c = inf ϕ(u) = u∈N

inf

max ϕ(sw) = inf max ϕ(sw) = inf (w).

w∈X\{0} s>0

w∈S+ s>0

w∈S+

By standard deformation arguments with respect to the flow of a pseudogradient vector field of  on S+ , we have an abstract multiplicity result for critical points of  in terms of the Lusternik-Schnirelman category with respect to sublevel sets. d such that Theorem 2.4.6 ([322]) If there exist d ≥ c and a compact set K ⊂ S+ d d catS d (K) ≥ k for some k ∈ N, where S+ = {u ∈ S+ : (u) ≤ d}, then S+ contains + at least k critical points of . If furthermore k ≥ 2 and there exists e > d such that K is e , then there exists another critical point of  in S e \ S d . contractible in S+ + +

Using the definitions of  and m and property (I I I ) above, we obtain the following corollary. d such that Corollary 2.4.7 ([322]) If there exist d ≥ c and a compact set K ⊂ S+ d catN d (K) ≥ k for some k ∈ N, where N = {u ∈ N : ϕ(u) ≤ d}, then N d contains at least k critical points of ϕ. If furthermore k ≥ 2 and there exists e > d such that K is contractible in N e , then there exists another critical point of ϕ in N e \ N d .

In the next sections we will see that the approach presented above is fundamental for obtaining multiplicity results for some nonlinear fractional elliptic problems for which the corresponding Nehari manifolds are not differentiable.

3

Fractional Scalar Field Equations

3.1

Introduction

In the fundamental papers [100, 101], Berestycki and Lions studied the existence and the multiplicity of nontrivial solutions to the equation − u = g(u) in RN ,

(3.1.1)

where N ≥ 3 and g : R → R is an odd continuous function such that g(t) g(t) ≤ lim sup = −m < 0; t t t →0 g(t) 2N (BL2) −∞ < lim sup 2∗ −1 ≤ 0, where 2∗ = N−2 is the critical exponent; t →∞ t  (BL1) −∞ < lim inf t →0

t

(BL3) there exists ξ0 > 0 such that G(ξ0 ) > 0, where G(t) =

g(τ ) dτ . 0

To obtain the existence of a positive solution to (3.1.1), the authors in [100] developed a subtle Lagrange multiplier procedure which ultimately relies on the Pohozaev identity [293] for (3.1.1), namely N −2 2



 |∇u| dx = N 2

RN

RN

G(u) dx.

The lack of compactness due to the translational invariance of (3.1.1) is regained by 1 (RN ) of H 1 (RN ) of radially symmetric functions. They working in the subspace Hrad also proved the existence of a positive solution to (3.1.1) when m = 0, the so called zero mass case. In [101] the authors showed that (3.1.1) possesses infinitely many distinct

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_3

51

52

3 Fractional Scalar Field Equations

 bound states by applying the genus theory to the even functional V (u) = RN G(u) dx 1 (RN ) : ∇u defined on the symmetric manifold M = {u ∈ Hrad L2 (RN ) = 1}. In [234] Jeanjean and Tanaka studied a mountain pass characterization of least energy solutions of (3.1.1). Indeed, without the assumption of monotonicity of t → g(tt ) , they showed that the mountain pass value gives the least energy level. Subsequently, the existence and multiplicity of solutions of closely related problems to (3.1.1) have been extensively studied by many authors; see for instance [23, 83, 99, 218, 233]. Later, Zhang and Zou [345] extended the existence result in [100] under the following critical growth assumptions on g: (ZZ1) g ∈ C(R, R) and g is odd; g(t) = −a < 0; (ZZ2) lim t →0 t g(t) (ZZ3) lim 2∗ −1 = b > 0; t →∞ t  (ZZ4) there exist C > 0 and max 2, g(t) − bt 2

4 N −2

∗ −1



< q < 2∗ such that

+ at ≥ Ct q−1 for all t > 0.

In the spirit of [234], they also proved a mountain pass characterization of least energy solutions of (3.1.1) in the critical case. In this chapter we deal with the following fractional nonlinear scalar field equation with fractional diffusion (−)s u = g(u) in RN ,

u ∈ H s (RN ),

(3.1.2)

with s ∈ (0, 1) and N ≥ 2. A naturally arising question is whether or not the above classical existence and multiplicity results for (3.1.1) still hold in the nonlocal framework of (3.1.2). In [139] the authors obtained the existence of a positive solution to (3.1.2) when g is a subcritical nonlinearity, by using the Struwe–Jeanjean monotonicity trick as in [307], and by proving the following Pohozaev identity for the fractional Laplacian (see also [198, 302, 307]): N − 2s 2



s

RN

|(−) 2 u|2 dx = N

 RN

G(u) dx.

(3.1.3)

The first aim of this chapter is to answer the question concerning the existence of infinitely many solutions for (3.1.2) when g is a general nonlinearity with subcritical growth. We also provide a mountain pass characterization of least energy solutions to (3.1.2) as in [234]. In this way, we are able to complement and improve the result proved in [139].

3.1 Introduction

53

Now we state our main assumptions. We will assume that g : R → R is an odd function of class C 1,α , for some α > max{0, 1 − 2s}, with the following properties: g(t) g(t) ≤ lim sup = −m < 0; t t t →0 g(t) (g2) −∞ < lim sup 2∗ −1 ≤ 0; t →∞ t s  ξ0 (g3) there exists ξ0 > 0 such that G(ξ0 ) = g(τ ) dτ > 0.

(g1) −∞ < lim inf t →0

0

We emphasize that the regularity of g is higher than in [100,101], and this seems to be due to the more demanding assumptions for elliptic regularity in the framework of fractional operators; see [124, 313]. Under assumptions (g1)–(g3), problem (3.1.2) admits a variational formulation, so its weak solutions can be found as critical points of the energy functional I : H s (RN ) → R defined as 1 I (u) = 2

 R2N

|u(x) − u(y)|2 dxdy − |x − y|N+2s

 RN

G(u) dx.

We recall that by weak solution of problem (3.1.2) we mean a function u ∈ H s (RN ) such that   (u(x) − u(y))(ϕ(x) − ϕ(y)) dxdy = g(u)ϕ dx |x − y|N+2s R2N RN for all ϕ ∈ H s (RN ), and we say that u is a least energy solution (or ground state solution) to (3.1.2) if I (u) = m

and

 m = inf I (u) : u ∈ H s (RN ) \ {0} is a solution of (3.1.2) .

Our first main result is the following: Theorem 3.1.1 ([39]) Let s ∈ (0, 1) and N ≥ 2 and let g ∈ C 1,α (R, R), with α > max{0, 1 − 2s}, be an odd function satisfying (g1)–(g3). Then (3.1.2) possesses a positive least energy solution and infinitely many (possibly sign-changing) radiallysymmetric solutions (un ) such that I (un ) → ∞ as n → ∞. Moreover, these solutions are of class C 2,β (RN ), for some β ∈ (0, 1), and they are characterized by a mountain pass and a symmetric mountain pass arguments, respectively. In order to prove our main result, we borrow some arguments developed in [120,218,234]. Clearly, due to the nonlocal nature of the fractional Laplacian, some additional difficulties arise in the study of (3.1.2). We would like to point out that our results are in clear

54

3 Fractional Scalar Field Equations

agreement with those for the classical local counterpart. For this reason, Theorem 3.1.1 can be seen as the fractional version of the existence and multiplicity results given in [100, 101, 218]. Secondly, we use the mountain pass approach in [218, 234] to prove the existence of a positive solution to (3.1.2) in the null mass case, that is, when g satisfies the following properties: g(t) ≤ 0; 2∗ −1 t →0 t s g(t) (h2 ) lim 2∗ −1 = 0; t →∞ t s

(h1 ) lim sup



ξ0

(h3 ) there exists ξ0 > 0 such that G(ξ0 ) =

g(τ ) dτ > 0. 0

We point out that the main difficulty related to the zero mass case is due to the fact that the energy of solutions of (3.1.2) can be infinite. In the local setting, i.e., when s = 1, several results for zero mass problems were established in [24,26,84,97,98,100,102,319]. We also recall that from a physical point of view, this type of problem is related to the Yang–Mills equations; see for instance [205,206]. However, differently from the classic literature, there is only one work [38] dealing with zero mass problems in nonlocal setting. Motivated by this fact, here we study (3.1.2) when g is a general nonlinearity such that g (0) = 0. Our main second result can be stated as follows: Theorem 3.1.2 ([39]) Let s ∈ (0, 1) and N ≥ 2 and let g ∈ C 1,α (R, R), with α > max{0, 1 − 2s}, be an odd function satisfying (h1 )–(h3 ). Then (3.1.2) possesses a positive radially decreasing solution. The proof of Theorem 3.1.2 is obtained by combining the mountain pass approach and an approximation argument. Indeed, we show that a solution of (3.1.2) can be approximated by a sequence of positive radially-symmetric solutions (uε ) in H s (RN ), each of which solves an approximate “positive mass” problem (−)s u = g(u) − εu in RN . Taking into account the mountain pass characterization of least energy solution of (3.1.2) given in Theorem 3.1.1, we are able to obtain lower and upper bounds for the ε , which can be estimated independently of ε, when ε is mountain pass critical levels bmp sufficiently small. This allows us to pass to the limit in (−)s uε = g(uε ) − εuε in RN as ε → 0, and to find a nontrivial solution u to (3.1.2). We emphasize that our proof is different from the one given in [100], and that it works even when s = 1. In the second part of this chapter, we consider (3.1.2) with a general critical nonlinearity. More precisely, we assume that g : R → R satisfies the following conditions: (g1 ) g ∈ C 1,β (R, R) for some β > max{0, 1 − 2s}, and g is odd; g(t) = −a < 0; (g2 ) lim t →0 t

3.1 Introduction

(g3 ) lim

g(t)

∗ t →∞ t 2s −1

55

= b > 0;

! " (g4 ) there exist C > 0 and max 2∗s − 2, 2 < q < 2∗s such that ∗

g(t) − bt 2s −1 + at ≥ Ct q−1 for all t > 0. Then we are able to prove the following result: Theorem 3.1.3 ([36]) Let s ∈ (0, 1) and N ≥ 2. Assume that g satisfies (g1 )–(g4 ). s (RN ). Then (3.1.2) possesses a radial positive least energy solution ω ∈ Hrad ∗

Remark 3.1.4 Let us observe that if g(t) = b|t|2s −2 t − at, then g satisfies (g1 )–(g3 ), but not (g4 ). Moreover, in view of the Pohozaev identity, it is not difficult to prove that (3.1.2) does not admit solution. Hence, without (g4 ), assumptions (g1 )–(g3 ) are not sufficient to guarantee the existence of a ground state to (3.1.2). To establish the existence of a solution for (3.1.2), we set 1 I(u) = 2

 R2N

|u(x) − u(y)|2 dxdy − |x − y|N+2s

 RN

G(u) dx = T (u) − V(u),

and we consider the constrained minimization problem  M = inf T (u) : V(u) = 1, u ∈ H s (RN ) . Motivated by Zhang and Zou [345], we use some crucial estimates obtained in [310] and prove that M satisfies the following bounds: 0 0 for all t ≥ ξ0 , we simply extend g to the negative axis:  g(t) ˜ =

g(t), for t ≥ 0, −g(−t), for t < 0.

(ii) If there exists t0 > ξ0 such that g(t0 ) = 0, we put ⎧ ⎪ for t ∈ [0, t0 ], ⎨ g(t), g(t) ˜ = 0, for t > t0 , ⎪ ⎩ −g(−t), ˜ for t < 0.

3.2 The Subcritical Case

57

Then g˜ satisfies (g1), (g3) and lim

t →∞

g(t) ˜ = 0. ∗ |t|2s −1

(g2 )

Clearly, any solution to (−)s u = g(u) ˜ in RN is also a solution to (3.1.2). Indeed, in the s ˜ in RN satisfies −t0 ≤ u ≤ t0 in RN . case (ii) above, any solution to (−) u = g(u) + s N To prove this, we use (u − t0 ) ∈ H (R ) as test function in the weak formulation of (−)s u = g(u) ˜ in RN , and we get 

 (u(x) − u(y)) + + ((u(x) − t0 ) − (u(y) − t0 ) ) dxdy = g(u)(u(x) ˜ − t0 )+ dx. N+2s R2N |x − y| RN

 ˜ − t0 )+ dx = 0. Then, recalling that By the definition of g, ˜ it follows that RN g(u)(u(x) + + + + 2 (x − y)(x − y ) ≥ |x − y | for all x, y ∈ R, we see that  R2N

|(u(x) − t0 )+ − (u(y) − t0 )+ |2 dxdy ≤ 0, |x − y|N+2s

which implies that [(u − t0 )+ ]2s ≤ 0. Since u ∈ H s (RN ), we deduce that u ≤ t0 in RN . The other inequality is obtained in a similar way by using (u + t0 )− as test function. We recall that x + = max{x, 0} and x − = min{x, 0}. Hereafter, we tacitly write g instead of g, ˜ and we will assume that g satisfies (g1), (g2 ) and (g3). Now we introduce a penalty function to construct an auxiliary functional. For t ≥ 0 we define   1 f (t) = max 0, mt + g(t) , 2 and h(t) = t p sup

0 0 such that h = 0 = H on [−β, β]. (h3) For all t ∈ R 1 2 mt + g(t)t ≤ h(t)t 2

and

1 2 mt + G(t) ≤ H (t). 4

h(t) = 0. ∗ |t |→∞ |t|2s −1 (h5) h satisfies the following Ambrosetti-Rabinowitz condition: (h4) lim

0 ≤ (p + 1)H (t) ≤ h(t)t for all t ∈ R. s (RN ), then (h6) If (uk ) is a bounded sequence in Hrad



 lim

k→∞ RN

h(uk )uk dx =

RN

h(u)u dx.

Proof Clearly, (h1), (h2), and (h3) follow from (g1) and the definitions of f , h and H . Concerning (h4), we first note that ∗

f (τ ) f (τ ) τ 2s −1−p h(t) −(2∗s −1−p) = t sup = sup . ∗ p 2∗ −1 t 2∗s −1−p t 2s −1 0 0 such that    f (τ )   ∗  < ε for all τ ≥ tε .  τ 2s −1     f (τ )   Set Cε = sup  2∗ −1 . Hence we obtain 0 2, we can find δ, ρ > 0 such that J (u) ≥ δ for u = ρ, and J (u) ≥ 0 if u ≤ ρ. Now (i) implies (ii). (iii) Arguing as in the proof of Theorem 10 in [101], for every n ∈ N there exists an odd 1 (RN ) such that continuous map πn : Sn−1 → Hrad  0∈ / πn (Sn−1 ),

RN

G(πn (σ )) dx ≥ 1

for all σ ∈ Sn−1 .

Since H 1 (RN ) ⊂ H s (RN ) and πn (Sn−1 ) is compact, there exists M > 0 such that πn (σ )H s (RN ) ≤ M

for any σ ∈ Sn−1 .

3.2 The Subcritical Case

61

For t > 0, define ψnt (σ )(x) = πn (σ )( xt ). Hence, t N−2s [πn (σ )]2H s (RN ) − t N 2   M ≤ t N−2s − t 2s → −∞ 2

I (ψnt (σ )) =

 RN

G(πn (σ )) dx

as t → ∞,

and thus we can find t > 0 sufficiently large such that I (ψnt (σ )) < 0 for all σ ∈ Sn−1 . By setting γn (σ )(x) = ψnt (σ )(x), we infer that γn satisfies the required properties for I .   In view of (i), we deduce that γn has the desired properties for J . Differently from I (u), the comparison functional J (u) enjoys the following compactness property: Theorem 3.2.3 The functional J satisfies the Palais–Smale condition. s (RN ) be a sequence such that Proof Let c ∈ R and let (uk ) ⊂ Hrad

J (uk ) → c

and

s J (uk ) → 0 in (Hrad (RN ))∗ .

(3.2.1)

Using (h5) of Lemma 3.2.1, we get 1 J (uk ), uk  p+1      1 1 1 2 = − uk  − 2 h(uk )uk dx H (uk ) − 2 p+1 p+1 RN   1 1 ≥ − uk 2 2 p+1

C(1 + uk ) ≥ J (uk ) −

s (RN ). Then, in light of Theorem 1.1.11, up to a so we deduce that (uk ) is bounded in Hrad subsequence, we may assume that s uk  u in Hrad (RN ),

uk → u in Lq (RN ) uk → u a.e. RN .

∀q ∈ (2, 2∗s ),

(3.2.2)

62

3 Fractional Scalar Field Equations

Using (h1) and (h4) in Lemma 3.2.1 and (3.2.2), we can apply the first part of ∗ Lemma 1.4.2 with P (t) = h(t) and Q(t) = |t|2s −1 to infer that 

 lim

k→∞ RN

h(uk )ϕ dx =

RN

(3.2.3)

h(u)ϕ dx

for all ϕ ∈ Cc∞ (RN ). Putting together (3.2.1), (3.2.2), and (3.2.3) we obtain J (u), ϕ = J (uk ), ϕ − uk − u, ϕDs,2 (RN ) −  +2

RN

m 2

 RN

(uk − u)ϕ dx

(h(uk ) − h(u))ϕ dx → 0

s (RN ), it follows that for all ϕ ∈ Cc∞ (RN ). Since Cc∞ (RN ) is dense in Hrad

J (u), ϕ = 0

s for all ϕ ∈ Hrad (RN ).

 Moreover, J (u), u = 0, which implies that u2 = 2 RN h(u)u dx. On the other hand, by (h6) of Lemma 3.2.1, we know that 

 lim

k→∞ RN

h(uk )uk dx =

RN

h(u)u dx.

(3.2.4)

Taking into account the boundedness of (uk ) and (3.2.1), we get J (uk ), uk  → 0, which together with (3.2.4) implies that uk  → u as k → ∞. Therefore, uk → u in s (RN ) as k → ∞.   Hrad Now we define minimax values of I and J by using the maps (γn ) in Lemma 3.2.2. For any n ∈ N, we set bn = inf max I (γ (σ )), γ ∈n σ ∈Dn

cn = inf max J (γ (σ )), γ ∈n σ ∈Dn

where Dn = {σ ∈ Rn : |σ | ≤ 1} and  s n = γ ∈ C(Dn , Hrad (RN )) : γ is odd and γ = γn on Sn−1 . The values bn and cn have the following properties.

3.2 The Subcritical Case

63

Lemma 3.2.4 (i) n = ∅ for any n ∈ N. (ii) 0 < δ ≤ cn ≤ bn for any n ∈ N, where δ appears in Lemma 3.2.2. Proof (i) Let us define  γ˜n (σ ) =

|σ |γn ( |σσ | ), for σ ∈ Dn \ {0}, 0,

for σ = 0.

Clearly, γ˜n ∈ n . (ii) By (i) of Lemma 3.2.2, cn ≤ bn for any n ∈ N. The property δ ≤ cn follows from the fact that s (RN ) : u = ρ} ∩ γ (Dn ) = ∅ for all γ ∈ n . {u ∈ Hrad

 

Lemma 3.2.5 (i) The value cn is a critical value of J . (ii) cn → ∞ as n → ∞. Proof (i) Follows from Theorem 3.2.3. (ii) Set ! " n = h ∈ C(Dm \ Y ) : h ∈ m , m ≥ n, Y ∈ Em and genus(Y ) ≤ m − n where Em is the family of closed sets A ⊂ Rm \ {0} such that −A = A and genus(A) is the Krasnoselskii genus of A. Now we define another sequence of minimax values by setting dn = inf max J (u). A∈n u∈A

Then dn ≤ dn+1 for all n ∈ N, and dn ≤ cn for all n ∈ N. Since J satisfies the Palais–Smale condition, we can proceed as in the proof of Proposition 9.33 in [298] to infer that dn → ∞ as n → ∞. Consequently, cn → ∞ as n → ∞.  

64

3 Fractional Scalar Field Equations

We observe that Lemmas 3.2.4-(ii) and 3.2.5-(ii) yield bn > 0 for all n ∈ N

lim bn = ∞.

and

n→∞

(3.2.5)

In the next section we will prove that bn are critical values of I (u).

3.2.3

s (RN ) An Auxiliary Functional on the Augmented Space R × Hrad

Let us introduce the following functional 1 I˜(θ, u) = e(N−2s)θ [u]2s − eNθ 2

 RN

G(u) dx

s (RN ). We endow R × H s (RN ) with the norm for (θ, u) ∈ R × Hrad rad

(θ, u)R×H s (RN ) =



|θ |2 + u2 .

s (RN ), R), I˜(0, u) = I (u) and that Let us point out that I˜ ∈ C 1 (R × Hrad



|u(e−θ x) − u(e−θ y)|2 dxdy = e(N−2s)θ [u]2s |x − y|N+2s R2N   G(u(e−θ x)) dx = eNθ G(u(x)) dx. RN

RN

In particular, we have I˜(θ, u(x)) = I (u(e−θ x))

s for all θ ∈ R, u ∈ Hrad (RN ).

We also define minimax values b˜n for I˜(θ, u) by b˜n = inf max I˜(γ˜ (σ )), γ˜ ∈˜ n σ ∈Dn

where s (RN )) : γ˜ (σ ) = (θ (σ ), η(σ )) satisfies ˜ n = {γ˜ ∈ C(Dn , R × Hrad

(θ (−σ ), η(−σ )) = (θ (σ ), −η(σ )) for all σ ∈ Dn , (θ (σ ), η(σ )) = (0, γn (σ )) for all σ ∈ Sn−1 }.

(3.2.6)

3.2 The Subcritical Case

65

Through the modified functional I˜(θ, u) we aim to show that bn are critical values for I (u). We begin by proving that Lemma 3.2.6 b˜n = bn for all n ∈ N. Proof Observe that (0, γ (σ )) ∈ ˜ n for any γ ∈ n , and so n ⊂ ˜ n . Since I˜(0, u) = I (u), by the definitions of bn and b˜n , it follows that b˜n ≤ bn for all n ∈ N. Now, take γ˜ (σ ) = (θ (σ ), η(σ )) ∈ ˜ n and put γ (σ ) = η(σ )(e−θ(σ ) x). Then it is easily verified that γ ∈ n and, by using (3.2.6), I (γ (σ )) = I˜(γ˜ (σ )) for all σ ∈ Dn .   Consequently, we have b˜n ≥ bn for all n ∈ N. Next we show that the functional I˜(θ, u) admits a Palais–Smale sequence in R × s (RN ) with a property related to the Pohozaev identity (3.1.3). First, we give a version Hrad of Ekeland’s variational principle [176]. Lemma 3.2.7 Let n ∈ N and ε > 0. Assume that γ˜n ∈ ˜ n satisfies max I˜(γ˜ (σ )) ≤ b˜n + ε.

σ ∈Dn

s (RN ) such that: Then there exists (θ, u) ⊂ R × Hrad

√ (i) distR×H s (RN ) ((θ, u), γ˜ (Dn )) ≤ 2 ε; rad (ii) I˜(θ, u) ∈ [bn − ε, bn + ε]; √ (iii) ∇ I˜(θ, u)R×(H s (RN ))∗ ≤ 2 ε. rad

Here we used the notations distR×H s

N ((θ, u), A) rad (R )

= inf

(t,v)∈A



|θ − t|2 + u − v2

s for A ⊂ R × Hrad (RN ), and

∇ I˜(θ, u) =



 ∂ ˜

˜ I (θ, u), I (θ, u) . ∂θ

s (RN ), we deduce that ˜ n is stable Proof Since I˜(θ, −u) = I˜(θ, u) for all (θ, u) ∈ R×Hrad under the pseudo-deformation flow generated by I˜(θ, u). Since b˜n = bn > 0, I (0) = 0   and maxσ ∈Dn I˜(0, γn (σ )) < 0, the proof of Lemma 3.2.7 goes as in [230].

66

3 Fractional Scalar Field Equations

Then we can deduce the following result: s (RN ) such Theorem 3.2.8 For every n ∈ N there exists a sequence ((θk , uk )) ⊂ R × Hrad that

(i) (ii) (iii) (iv)

θk → 0; I˜(θk , uk ) → bn ; s (RN ))∗ ; I˜ (θk , uk ) → 0 strongly in (Hrad ∂ ˜ ∂θ I (θk , uk ) → 0.

Proof For every k ∈ N there exists γk ∈ n such that 1 max I (γk (σ )) ≤ bn + . σ ∈Dn k Since γ˜k (σ ) = (0, γk (σ )) ∈ ˜ n and b˜n = bn , we have 1 max I˜(γ˜k (σ )) ≤ b˜ n + . k

σ ∈Dn

s It follows from Lemma 3.2.7 that there exists ((θk , uk )) ⊂ R × Hrad (RN ) such that

2 ((θk , uk ), γ˜k (Dn )) ≤ √ , k   1 1 I˜(θk , uk ) ∈ bn − , bn + , k k distR×H s

rad (R

N)

∇ I˜(θk , uk )R×(H s

rad (R

N ))∗

2 ≤ √ . k

(3.2.7) (3.2.8) (3.2.9)

s (RN ) yield (i). Clearly, (ii) follows from (3.2.8), Then, (3.2.7) and γ˜k (Dn ) ⊂ {0} × Hrad while (iii) and (iv) are consequences of (3.2.9).  

Next, we investigate the boundedness and the compactness properties of the sequence s (θk , uk ) ⊂ R × Hrad (RN ) obtained in Theorem 3.2.8. More precisely, we are able to prove s Theorem 3.2.9 Let ((θk , uk )) ⊂ R × Hrad (RN ) be a sequence satisfying (i)–(iv) of Theorem 3.2.8. Then we have s (RN ). (a) (uk ) is bounded in Hrad s (b) (θk , uk ) has a strongly convergent subsequence in R × Hrad (RN ).

3.2 The Subcritical Case

67

Proof (a) Using (ii) and (iv) of Theorem 3.2.8, we can see that 1 (N−2s)θk e [uk ]2s − eNθk 2

 RN

G(uk ) dx → bn

and 

N − 2s (N−2s)θk e [uk ]2s − NeNθk 2

RN

G(uk ) dx → 0

as k → ∞. Consequently, [uk ]2s →

Nbn s

(3.2.10)

and  RN

G(uk ) dx →

N − 2s bn 2s

as k → ∞. Thanks to Lemma 3.2.1, there exists C > 0 such that ∗

|h(t)| ≤ C|t|2s −1 for all t ∈ R.

(3.2.11)

Set εk = I˜ (θk , uk )(H s

rad (R

N ))∗

.

By item (iii) of Theorem 3.2.8, we deduce that εk → 0 as k → ∞, so we get |I˜ (θk , uk ), uk | ≤ εk uk .

(3.2.12)

Taking into account (3.2.10), (3.2.11), and (3.2.12), (h3) of Lemma 3.2.1 and invoking Theorem 1.1.8, we have e

(N−2s)θk

[uk ]2s

m + eNθk uk 2L2 (RN ) ≤ eNθk 2 ≤ eNθk

 

RN

RN

m 2 u + g(uk )uk dx + εk uk  2 k h(uk )uk dx + εk uk  2∗s

≤ CeNθk uk 

∗

L2s (RN )

+ εk uk 

68

3 Fractional Scalar Field Equations 2∗

≤ CC∗ eNθk [uk ]s s + εk uk  ≤ CC∗ C eNθk + εk uk , which implies the boundedness of uk L2 (RN ) . Now using (3.2.10), we infer that (uk ) s (RN ). is bounded in Hrad (b) In view of (a), passing to a subsequence if necessary, we may assume that s (RN ), uk  u in Hrad

uk → u in Lq (RN )

∀q ∈ (2, 2∗s ),

(3.2.13)

uk → u a.e. RN . By item (iii) of Theorem 3.2.8,  e

(N−2s)θk R2N

(uk (x) − uk (y))(ϕ(x) − ϕ(y)) dxdy − eNθk |x − y|N+2s

 RN

g(uk )ϕ dx → 0 (3.2.14)

for all ϕ ∈ Cc∞ (RN ). Then I (u), ϕ = 0 for any ϕ ∈ Cc∞ (RN ), and by the denseness s (RN ), we get that I (u), ϕ = 0 for any ϕ ∈ H s (RN ). In of Cc∞ (RN ) in Hrad rad particular,  [u]2s

=

RN

g(u)u dx.

(3.2.15)

s Now, we observe that (3.2.14) holds for all ϕ ∈ Hrad (RN ) and (uk ) is bounded in s N H (R ). Therefore, taking ϕ = uk in (3.2.14), we have

 e(N−2s)θk [uk ]2s − eNθk

RN

g(uk )uk dx = o(1) as k → ∞.

Consequently, m e(N−2s)θk [uk ]2s + eNθk uk 2L2 (RN ) 2  % & m 2 uk + g(uk )uk dx + o(1) = eNθk RN 2   % & m = eNθk h(uk )uk dx − eNθk h(uk )uk − u2k − g(uk )uk dx + o(1) 2 RN RN = eNθk Ak − eNθk Bk + o(1).

(3.2.16)

3.2 The Subcritical Case

69

Now, item (h6) of Lemma 3.2.1 implies that  lim Ak =

k→∞

RN

h(u)u dx,

(3.2.17)

and by (h3) of Lemma 3.2.1 and Fatou’s lemma we get  lim inf Bk ≥ k→∞

RN

% & m h(u)u − u2 − g(u)u dx. 2

(3.2.18)

Finally, using (3.2.13), (3.2.15), (3.2.16), (3.2.17), and (3.2.18), we infer that % & m u2 ≤ lim sup uk 2 = lim sup e(N−2s)θk [uk ]2s + eNθk uk 2L2 (RN ) 2 k→∞ k→∞  % & m 2 u + g(u)u dx = u2 ≤ RN 2 s which gives uk → u in Hrad (RN ) as k → ∞.

 

Combining Theorem 3.2.9, Lemma 3.2.6 and (3.2.5), we can deduce the following multiplicity result. Theorem 3.2.10 Under the assumptions of Theorem 3.1.1, there exist infinitely many radially symmetric solutions to (3.1.2). s Proof Fix n ∈ N, and let ((θk , uk )) ⊂ R × Hrad (RN ) be a sequence satisfying (i)–(iv) of s (RN ) such that Theorem 3.2.8. Using Theorem 3.2.9, we know that there exists un ∈ Hrad s N uk → un in Hrad (R ) as k → ∞. Then un satisfies

I (un ) = I˜(0, un ) = bn and I (un ) = I˜ (0, un ) = 0. Since bn → ∞ as n → ∞, we conclude that (3.1.2) admits infinitely many distinct s (RN ).   solutions in Hrad

3.2.4

Mountain Pass Value Gives the Least Energy Level

In this section we prove the existence of a positive solution to (3.1.2) by using the mountain pass approach developed in the previous section. We recall that the existence of a positive ground state solution to (3.1.2) was obtained in [139]. Here, we give an alternative proof of this fact and, in addition, we show that the least energy solution of (3.1.2) coincides with the mountain pass value. This is in clear agreement with the result obtained in the

70

3 Fractional Scalar Field Equations

classic framework by Jeanjean and Tanaka in [234]. Moreover, we investigate regularity, decay and symmetry properties of solutions to (3.1.2). The main result of this section can be stated as follows. Theorem 3.2.11 Under assumptions (g1)–(g3), there exists a classical positive solution u of (3.1.2). Moreover, u is radially decreasing, has a polynomial decay at infinity, and s (RN ). can be characterized by a mountain pass argument in Hrad s s (RN ) → R and I˜ : R × Hrad (RN ) → R Let us consider the functionals I : Hrad introduced in the previous subsections:

1 I (u) = [u]2s − 2

 RN

G(u) dx

and 1 I˜(θ, u) = e(N−2s)θ [u]2s − eNθ 2

 RN

G(u) dx.

We define the following minimax values: bmp,r = inf max I (γ (t)), γ ∈r t ∈[0,1]

bmp = inf max I (γ (t)), γ ∈ t ∈[0,1]

b˜mp,r = inf max I˜(γ˜ (t)), γ˜ ∈˜ r t ∈[0,1]

where s r = {γ ∈ C([0, 1], Hrad (RN )) : γ (0) = 0 and I (γ (1)) < 0},

 = {γ ∈ C([0, 1], H s (RN )) : γ (0) = 0 and I (γ (1)) < 0}, s (RN )) : γ ∈ r and θ (0) = 0 = θ (1)}. ˜ r = {γ˜ = (θ, γ ) ∈ C([0, 1], R × Hrad

Then, we show that the following result holds: Lemma 3.2.12 bmp = bmp,r = b1 = b˜mp,r . Proof Arguing as in the proof of Lemma 3.2.6, we have that bmp,r = b˜mp,r . Furthermore, it follows from the above definitions that bmp ≤ bmp,r ≤ b1 . To prove the lemma, it is

3.2 The Subcritical Case

71

enough to show that b1 ≤ bmp . Firstly, we show that ¯ bmp = b,

(3.2.19)

where b¯ = inf max I (γ (t)) γ ∈¯ t ∈[0,1]

and  ¯ = γ ∈ C([0, 1], H s (RN )) : γ (0) = 0 and γ (1) = γ1 (1) . Here γ1 is the path appearing in Lemma 3.2.2 (with n = 1). As observed in [100, 101], we may assume that γ1 satisfies the following properties: γ (1)(x) ≥ 0 for all x ∈ RN , γ1 (|x|) = γ (1)(x) and r → γ1 (1)(r) is piecewise linear and nonincreasing. ¯ Now, we show that Since I (γ1 (1)) < 0, it is clear that ¯ ⊂ , which gives bmp ≤ b. ¯ bmp ≥ b. For this purpose, it is suffices to prove that B = {u ∈ H s (RN ) : I (u) < 0} is path connected. We proceed as in [120]. Let u1 , u2 ∈ B. By Theorem 1.1.7, we may assume that u1 , u2 ∈ Cc∞ (RN ). Set uti (x) = ui ( xt ) for t > 0 and i = 1, 2. Then I (uti ) =

t N−2s [ui ]2s − t N 2

 RN

G(ui ) dx,

and so    d t N−1 −2s 1 2 I (ui ) = Nt [ui ]s − G(ui ) dx − st N−2s−1 [ui ]2s . t dt 2 RN In particular, for t ≥ 1, d I (uti ) ≤ Nt N−1 I (ui ) − st N−2s−1 [ui ]2s < 0. dt  Since ui ∈ B, we deduce that RN G(ui ) dx > 0, so we see that I (uti ) → −∞ as t → ∞. Hence, we can choose t0 > 0 such that  I (uti0 )

≤ −2 max

 max I (tu1 ), max I (tu2 ) < 0,

t ∈[0,1]

t ∈[0,1]

i = 1, 2.

(3.2.20)

72

3 Fractional Scalar Field Equations

Noting that uti0 , ui ∈ Cc∞ (RN ) and ut10 , tu2 (· − Re1 )Ds,2 (RN ) +



 =t  RN

RN

G(ut10 (x) +

ut10 tu2 (· − Re1 ) dx

RN

(|ξ |2s + 1)(t0N F u1 (t0 ξ ))(F u2 (ξ )e−ıξ1 R ) dξ → 0, 

tu2 (x − Re1 )) dx →

RN

G(ut10 (x)) dx

 +

RN

G(tu2 (x)) dx

uniformly with respect to t ∈ [0, 1] as R → ∞, where e1 = (1, 0, . . . , 0), it follows from the choice of t0 that as R → ∞ max I (ut10 + tu2 (· − Re1 )) → I (ut10 ) + max I (tu2 ) < − max I (tut20 ) < 0.

t ∈[0,1]

t ∈[0,1]

t ∈[0,1]

Hence, choosing R0 > 0 sufficiently large, we have supp(ut1 ) ∩ supp(u2 (· − R0 e1 )) = ∅

for t ∈ [1, t0 ],

max I (ut10 + tu2 (· − R0 e1 )) < 0,

t ∈[0,1]

I (ut1 + u2 (· − R0 e1 )) = I (ut1 ) + I (u2 ) < 0

for t ∈ [1, t0 ].

Then, considering the following paths γ1 (t) = ut1 for t ∈ [1, t0 ], t0 γ2 (t) = u1 + tu2 (· − R0 e1 ) for t ∈ [0, 1], γ3 (t) = u1t0 −t + u2 (· − R0 e1 ) for t ∈ [0, t0 − 1], we can connect u1 and u1 + u2 (· − R0 e1 ) in B. In a similar fashion, we can find a path ¯ This between u2 and u1 + u2 (· − R0 e1 ) in B. Therefore, B is path connected and bmp ≥ b. implies that (3.2.19) holds. s (RN ) and γ (−t) = −γ (t) for any Now, observing that I (u) = I (−u), γ1 (1) ∈ Hrad γ ∈ 1 , we aim to show that b¯ = inf max I (γ (t))(= b1 ) γ ∈¯ r t ∈[0,1]

where s (RN )) : γ (0) = 0 and γ (1) = γ1 (1)}. ¯ r = {γ ∈ C([0, 1], Hrad

(3.2.21)

3.2 The Subcritical Case

73

By definition, it is clear that b¯ ≤ b1 . In what follows, we show that b¯ ≥ b1 . Take η ∈ ¯ and we set γ (t) = |η(t)|. Obviously, γ ∈ C([0, 1], H s (RN )). Moreover, recalling that G is even, and using the fact that [|u|]s ≤ [u]s for any u ∈ H s (RN ) (this inequality follows from ||x| − |y|| ≤ |x − y| for any x, y ∈ R), we can see that for any t ∈ [0, 1], 1 I (γ (t)) = [|η(t)|]2s − 2

 RN

G(|η(t)|) dx

 1 [|η(t)|]2s − G(η(t)) dx 2 RN  1 ≤ [η(t)]2s − G(η(t)) dx = I (η(t)). 2 RN

=

(3.2.22)

¯ Since γ1 (1) ≥ 0, we have that γ (1) = |η(1)| = |γ1 (1)| = γ1 (1), and thus γ ∈ . ∗ Let γ (t) denote the Schwartz symmetrization (symmetric-decreasing rearrangement) of s (RN )) and there holds the fractional γ (t). By Almgren and Lieb [3], γ ∗ ∈ C([0, 1], Hrad Polya-Szegö inequality (see also [200]) [u∗ ]s ≤ [u]s

for any u ∈ Ds,2 (RN ),

(3.2.23)

which yields [γ ∗ (t)]s ≤ [γ (t)]s for any t ∈ [0, 1]. On the other hand, observing that G ∈ C(R) is even, G(0) = 0 and G(u) ∈ L1 (RN ) for all u ∈ H s (RN ), we deduce that (see [100, 249]) 

∗

RN



G(u ) dx =

RN

G(u) dx

for all u ∈ H s (RN ),

and thus  RN

G(γ ∗ (t)) dx =

 RN

G(γ (t)) dx

for all t ∈ [0, 1].

Consequently, I (γ ∗ (t)) ≤ I (γ (t)) for all t ∈ [0, 1]. By using the properties of γ1 , we see that (γ1 (1))∗ = γ1 (1), and consequently γ ∗ ∈ ¯ r . Therefore, in view of (3.2.22), we have b1 ≤ max I (γ ∗ (t)) ≤ max I (γ (t)) ≤ max I (η(t)), t ∈[0,1]

t ∈[0,1]

t ∈[0,1]

¯ This shows that (3.2.21) is satisfied. Combining (3.2.19) which implies that b1 ≤ b. and (3.2.21), we infer that bmp = b¯ = b1 . This ends the proof of lemma.  

74

3 Fractional Scalar Field Equations

Then we are able to prove the following result. Theorem 3.2.13 There exists a positive solution to (3.1.2) such that I (u) = bmp,r . Moreover, for any non-trivial solution v to (3.1.2), we have bmp,r ≤ I (v). This means that u is the least energy solution to (3.1.2) and that bmp,r is the least energy level. Proof Let (γk ) ⊂ r be a sequence such that 1 max I (γk (t)) ≤ bmp,r + . t ∈[0,1] k

(3.2.24)

s (RN ), we may assume that γ satisfies Since I (|u|) ≤ I (u) for all u ∈ Hrad k

γk (t)(x) ≥ 0

for all t ∈ [0, 1] and x ∈ RN .

In view of Lemma 3.2.12 and the equality I˜(0, u) = I (u), we deduce that 1 1 max I˜(0, γk (t)) ≤ b˜mp,r + = b1 + . k k

t ∈[0,1]

(3.2.25)

Therefore, applying Lemma 3.2.7 to I˜ and (0, γk ), we can find a sequence ((θk , uk )) ⊂ s R × Hrad (RN ) such that as k → ∞ (i) distR×H s (RN ) ((θk , uk ), {0} × γk ([0, 1])) → 0; rad (ii) I˜(θk , uk ) → b1 ; (iii) ∇ I˜(θk , uk )R×(H s (RN ))∗ → 0. rad

Taking into account item (i), we see that θk → 0 as k → ∞, and u− k H s (RN ) ≤ distR×H s

rad (R

N)

((θk , uk ), {0} × γk ([0, 1])) → 0.

Proceeding as in the proof of Theorem 3.2.9, we can show that uk → u in H s (RN ), s I (u) = 0 and I (u) = b1 , for some u ∈ Hrad (RN ) such that u ≥ 0, u ≡ 0. By Proposition 3.2.14 below and Proposition 1.3.2, we deduce that u ∈ C 0,β (RN ), and then applying Proposition 1.3.11-(ii), we conclude that u > 0 in RN . This ends the proof of the first statement of theorem. Now, let v ∈ H s (RN ) be a nontrivial solution to (3.1.2) and consider the curve γ : [0, ∞) → H s (RN ) defined by  γ (t)(x) =

v( xt ), for t > 0, 0, for t = 0.

3.2 The Subcritical Case

75

Then, (1) γ (t)2H s (RN ) = t N−2s [v]2s + t N v2L2 (RN ) ,    t N−2s 2 t N−2s 2 N − 2s [v]s − t N [v]s − t N (2) I (γ (t)) = G(v) dx = [v]2s , 2 2 2N RN  where in (2) we used the Pohozaev identity (3.1.3) to deduce that RN G(v) dx = N−2s 2 s N 2N [v]s > 0. By (1), we have that γ ∈ C([0, ∞), H (R )). Using (2), we have d N − 2s 2 N−2s−1 I (γ (t)) = [v]s t (1 − t 2s ), dt 2 whence d I (γ (t)) > 0 for t ∈ (0, 1) dt

d I (γ (t)) < 0 for t > 1. dt

and

Therefore, maxt ≥0 I (γ (t)) = I (γ (1)) = I (v). On the other hand, by (2) and N ≥ 2 > 2s, we obtain that tN 2 [v]s I (γ (t)) = 2



1 t 2s

N − 2s − N

 → −∞

as t → ∞,

so there exists μ > 0 such that I (γ (μ)) < 0. Then we consider the rescaled curve γ˜ ∈ C([0, 1], H s (RN )) defined by γ˜ (t)(x) = γ (μt)(x). Then, γ˜ ∈ r and satisfies v ∈ γ˜ ([0, 1])

and

max I (γ˜ (t)) = I (v).

t ∈[0,1]

By using Lemma 3.2.12, we deduce that bmp,r ≤ I (v) and this completes the proof.

3.2.5

 

Regularity, Symmetry and Asymptotic Behavior of Ground States

We start by proving some interesting results for fractional elliptic PDEs in RN . Proposition 3.2.14 (Regularity of Solutions) Let s ∈ (0, 1) and N > 2s. Let u ∈ H s (RN ) be a weak solution to (−)s u = g(x, u) in RN , where g : RN × R → R is a Carathéodory function such that |g(x, t)| ≤ C(|t| + |t|p )

for a.e. x ∈ RN , for all t ∈ R,

for some 1 ≤ p ≤ 2∗s − 1 and C > 0. Then, u ∈ Lq (RN ) for all q ∈ [2, ∞].

76

3 Fractional Scalar Field Equations

Proof We combine a Brezis-Kato type argument [112] with a Moser iteration scheme [278]. For simplicity, we assume that u ≥ 0 in RN . The general case can be dealt with in a similar way by considering u+ and u− . For any L > 0 and β > 0, we consider 2β the Lipschitz function γ (t) = ttL for t ≥ 0, where tL = min{t, L}. Since γ is a nondecreasing function, (a − b)(γ (a) − γ (b)) ≥ 0

for any a, b ≥ 0.

Define the function 

t

(t) =

(γ (τ )) 2 dτ. 1

0

Fix a, b ≥ 0 such that a > b. Then, the Schwartz inequality yields,  (a − b)(γ (a) − γ (b)) = (a − b)

a

γ (t) dt

b

 = (a − b)

a

( (t))2 dt

b



a

≥



2

 (t) dt

= ((a) − (b))2 .

b

In a similar fashion, we can prove that this inequality is valid for any a ≤ b. Thus we can infer that (a − b)(γ (a) − γ (b)) ≥ |(a) − (b)|2

for any a, b ≥ 0.

(3.2.26)

In particular, this implies that 2β

2β

|(u(x)) − (u(y))|2 ≤ (u(x) − u(y))((uuL )(x) − (uuL )(y)).

(3.2.27)

2β

Taking γ (u) = uuL ∈ H s (RN ) as a test function in (3.1.2), in view of (3.2.27) we have  [(u)]2s

≤  =

R2N

(u(x) − u(y)) 2β 2β ((uuL )(x) − (uuL )(y)) dxdy |x − y|N+2s 2β

RN

(3.2.28)

g(x, u)uuL dx.

Since (t) ≥

1 β tt β +1 L

for any t ≥ 0,

3.2 The Subcritical Case

77

the Sobolev inequality (1.1.1) implies that  [(u)]2s ≥ S∗ (u)2 2∗s

L (RN )

≥

1 β +1

2

β

S∗ uuL 2 2∗s

L (RN )

(3.2.29)

.

In view of (3.2.29) and the growth assumption on g, we see that (3.2.28) yields  wL 2 2∗s

L (RN )

≤ C(β + 1)2

2β

RN

2β

(u2 uL + up+1 uL ) dx,

(3.2.30)

β

where wL = uuL . We claim that there exist a constant c > 0 and a function h ∈ LN/2s (RN ), h ≥ 0, and independent of L and β, such that 2β

2β

2β

u2 uL + up+1 uL ≤ (c + h)u2 uL

on RN .

(3.2.31)

To show this, notice first that 2β

2β

2β

2β

u2 uL + up+1 uL = u2 uL + up−1 u2 uL

on RN .

Moreover, up−1 ≤ 1 + h

on RN ,

for some h ∈ LN/2s (RN ). Indeed, up−1 = χ{0≤u≤1} up−1 + χ{u>1} up−1 ≤ 1 + χ{u>1} up−1

on RN ,

N and if (p − 1) 2s < 2, then





N

RN

χ{u>1} u 2s (p−1) dx ≤

N we deduce that while if 2 ≤ (p − 1) 2s and (3.2.31) we obtain that

 RN

χ{u>1} u2 dx ≤

N 2s (p

u2 dx < ∞,

− 1) ∈ [2, 2∗s ]. Taking into account (3.2.30) 

wL 2 2∗s N L (R )

RN

≤ C(β + 1)

2β

2 RN

(c + h)u2 uL dx,

and by the monotone convergence theorem (uL is nondecreasing with respect to L) we have as L → ∞   uβ+1 2 2∗s N ≤ Cc(β + 1)2 u2(β+1) dx + C(β + 1)2 hu2(β+1) dx. (3.2.32) L (R )

RN

RN

78

3 Fractional Scalar Field Equations

Fix M > 0 and let A1 = {h ≤ M} and A2 = {h > M}. Then,  RN

hu2(β+1) dx ≤ Muβ+1 2L2 (RN ) + ε(M)uβ+1 2 2∗s

(3.2.33)

L (RN )

where 

 2s

ε(M) =

N/2s

h

N

dx

→0

as M → ∞.

A2

In view of (3.2.32) and (3.2.33) we get uβ+1 2 2∗s

L (RN )

≤ (β + 1)2 (Cc + M)uβ+1 2L2 (RN ) + C(β + 1)2 ε(M)uβ+1 2 2∗s

L (RN )

.

(3.2.34) Choosing M > 0 sufficiently large so that ε(M)C(β + 1)2
0,  RN

hu2(β+1) dx ≤ h

L

N s

(RN )

uβ+1 L2 (RN ) uβ+1 L2∗s (RN )

  1 β+1 2 β+1 2 ≤ h Ns N λu L2 (RN ) + u  2∗s N . L (R ) L (R ) λ

3.2 The Subcritical Case

79

Then, using (3.2.32), we deduce that uβ+1 2 2∗s

L (RN )

≤ (β + 1)2 (Cc + h

L

+

C(β + 1)2 h

L

N s

N s

(RN )

(RN )

λ)uβ+1 2L2 (RN )

uβ+1 2 2∗s

L (RN )

λ

.

(3.2.36)

Taking λ > 0 such that C(β + 1)2 h

L

N s

(RN )

λ

=

1 2

obtain uβ+1 2 2∗s

L (RN )

≤ 2(β + 1)2 (Cc + h

L

N s

(RN )

λ)uβ+1 2L2 (RN ) = Mβ uβ+1 2L2 (RN ) ,

and the advantage with respect to (3.2.35) is that now we control the dependence of the constant Mβ on β. Indeed, for some M0 > 0 independent of β, Mβ ≤ C(1 + β)4 ≤ M02 e2

√ β+1

,

which implies that 1

uL2∗s (β+1) (RN ) ≤ M0β+1 e

√1 β+1

uL2(β+1) (RN ) .

Iterating this last relation and choosing β0 = 0 and 2(βn+1 + 1) = 2∗s (βn + 1), we get n

uL2∗s (βn +1) (RN ) ≤ M0

1 i=0 βi +1

n

e

i=0

√1

βi +1

uL2(β0 +1) (RN ) .

N Since 1 + βn = ( N−2s )n , we have that ∞  i=0

1 0, then u ∈ L∞ (RN ). Next we prove a symmetry result for classical positive solutions to (3.1.2) by using the method of moving planes [206, 207] for the fractional Laplacian; see for instance [126, 138, 171, 183, 185]. We recall that given g ∈ C(RN ), we say that a function u ∈ C(RN ) is a classical solution of (−)s u + u = g

in RN

(3.2.37)

if (−)s u is defined at any point of RN , according the definition given in (1.2.1), and Eq. (3.2.37) is satisfied pointwise in all RN . Theorem 3.2.16 (Symmetry of Solutions) Let s ∈ (0, 1) and N ≥ 2. Let u be a positive classical solution of 

(−)s u = g(u) in RN , u > 0 in RN , lim|x|→∞ u(x) = 0.

If g(t) is a locally Lipschitz function in [0, ∞), nonincreasing for positive and small values of t, then u is radially symmetric and radially decreasing with respect to a point of RN . Proof Here we modify in a convenient way some arguments in [185]. We point out that, compared with [185], we do not require any condition on the decay of u at infinity. We write points x ∈ RN as x = (x1 , x2 , . . . , xN ) = (x1 , x ) and, for λ ∈ R, we set λ = {(x1 , x ) ∈ RN : x1 > λ}, Tλ = {(x1 , x ) ∈ RN : x1 = λ}, xλ = (2λ − x1 , x ) and uλ (x) = u(xλ ). We divide the proof into three steps. Step 1 We show that λ0 = sup{λ : uλ ≤ u in λ } is finite. Let us define  (uλ − u)+ (x) if x ∈ λ , w(x) = (uλ − u)− (x) if x ∈ λc .

3.2 The Subcritical Case

81

Note that for x ∈ λ w(xλ ) = min{(uλ − u)(xλ ), 0} = min{(u − uλ )(x), 0} = −(uλ − u)+ (x) = −w(x),

and similarly w(x) = −w(xλ ) for x ∈ λc . Thus w(x) = −w(xλ ) for all x ∈ RN , and this implies that 



2∗s

RN



2∗s

|w| dx =

|w| dx + λ

λc

2∗s



|w| dx = 2

∗

|w|2s dx.

(3.2.38)

λ

We also see that for every x ∈ λ ∩ supp(w) we have that w(x) = (uλ − u)(x) and (−)s w(x) ≤ (−)s (uλ − u)(x),

∀x ∈ λ ∩ supp(w),

(3.2.39)

since (−)s w(x) − (−)s (uλ − u)(x)  (uλ − u)(y) − w(y) dy = N |x − y|N+2s R   (uλ − u)(y) (uλ − u)(y) = dy + dy N+2s N+2s λ ∩(supp(w))c |x − y| λc ∩(supp(w))c |x − y|    1 1 (uλ (y) − u(y)) − = dy ≤ 0, |x − y|N+2s |x − yλ |N+2s λ ∩(supp(w))c where we used that uλ − u ≤ 0 in λ ∩ (supp(w))c ,

and |x − y| ≤ |x − yλ | for x, y ∈ λ .

Choosing (uλ − u)+ as test function in the equations for u and uλ in λ , respectively, we obtain 

+



(g(uλ ) − g(u))(uλ − u)+ dx.

(−) (uλ − u)(uλ − u) dx = s

λ

λ

Since g(t) is nonincreasing for positive and small values of t (say 0 < t < ε0 ), and u(x) → 0 as |x| → ∞, there exists R > 0 such that for all λ < −R we have uλ < ε0 on λ , and hence g(uλ ) − g(u) ≤ 0 on λ ∩ {uλ > u}. Therefore, for λ < −R, 

(−)s (uλ − u)(uλ − u)+ dx ≤ 0. λ

82

3 Fractional Scalar Field Equations

Using (3.2.39), we deduce that 



(−)s (uλ − u)(uλ − u)+ dx ≤ 0.

(−)s w w dx ≤ λ

λ

Then, by (3.2.38) and Theorem 1.1.8, we see that  0≥

(−)s w w dx =

1 2

=

1 2

λ

 

RN

(−)s w w dx s

RN

|(−) 2 w|2 dx

 ≥C

2∗s

RN

|w| dx

  =C 2

2∗s

 2∗

|w| dx

2s

 2∗ 2s

,

λ

which shows that w = 0 on λ for all λ < −R. Thus uλ ≤ u in λ for all λ < −R, and this implies that λ0 ≥ −R. Since u decays at infinity, there exists λ1 such that u(x) < uλ1 (x) for some x ∈ λ1 . Hence λ0 is finite. Step 2 We prove that u ≡ uλ0 in λ0 . By continuity, u ≥ uλ0 in λ0 . Assume by contradiction that u ≡ uλ0 in λ0 . Note that if x0 ∈ λ0 is such that uλ0 (x0 ) = u(x0 ), then (−)s uλ0 (x0 ) − (−)s u(x0 ) = g(uλ0 (x0 )) − g(u(x0 )) = 0, while since |x0 − y| < |x0 − yλ0 | for all y ∈ λ0 , u ≥ uλ0 and u ≡ uλ0 in λ0 , we have (−)s uλ0 (x0 ) − (−)s u(x0 )  uλ0 (y) − u(y) dy =− N+2s N R |x0 − y|    1 1 =− dy > 0. (uλ0 (y) − u(y)) − |x0 − y|N+2s |x0 − yλ0 |N+2s λ0 Therefore, u > uλ0 in λ0 . In order to complete Step 2, we only need to prove that the inequality u ≥ uλ in λ continues to hold when λ > λ0 is close to λ0 , contradicting the definition of λ0 . Fix δ > 0 whose value will be chosen later, and take λ ∈ (λ0 , λ0 + δ). Let P = (λ, 0) and consider B˜ = λ ∩ B(P , R), where B(P , R) is the ball centered at P and with radius R > 0 such that u(x) < ε0 for x

3.2 The Subcritical Case

83

outside B(P , R). Choosing (uλ − u)+ as test function in the equation for u and uλ in λ , we have 

(−)s (uλ − u)(uλ − u)+ dx λ



(g(uλ ) − g(u))(uλ − u)+ dx

=  =  ≤

λ

B˜

B˜

(g(uλ ) − g(u))(uλ − u)+ dx +

 λ \B˜

(g(uλ ) − g(u))(uλ − u)+ dx

(g(uλ ) − g(u))(uλ − u)+ dx

(3.2.40)

where we used that g(uλ ) − g(u) ≤ 0 on λ \ B˜ (note that u < uλ < ε0 on that ˜ we deduce that set). Since g is Lipschitz on the compact set B, 

(−)s (uλ − u)(uλ − u)+ dx ≤ L



λ

B˜

|(uλ − u)+ |2 dx.

On the other hand, arguing as in Step 1, we get 

+

  (−) w w dx ≥ C 2



(−) (uλ − u)(uλ − u) dx ≥ s

λ

s

λ

2∗s

|w| dx

 2∗ 2s

.

λ

Consequently, w2 2∗s

L (λ )



≤ C

B˜

|(uλ − u)+ |2 dx.

(3.2.41)

Using Hölder’s inequality, we have that C





+ 2

B˜

|(uλ − u) | dx ≤ C



 B˜

≤ C

|B˜ ∩ supp(uλ − u)+ | N

2s

|w|2 χsupp(uλ −u)+ dx 

∗

B˜



|w|2s dx

2 2∗ s

.

(3.2.42)

Taking into account that u > uλ0 in λ0 and the continuity of u, we see that u > uλ on any compact K ⊂ λ , for λ close to λ0 . This means that supp(uλ − u)+ is small in B˜ for λ > λ0 , λ close to λ0 , so we can choose δ > 0 such that, for all λ ∈ (λ0 , λ0 + δ), C|B˜ ∩ supp(uλ − u)+ | N < 2s

1 . 2

(3.2.43)

84

3 Fractional Scalar Field Equations

Combining (3.2.41), (3.2.42), and (3.2.43), we deduce that w = 0 in λ , which gives a contradiction. This completes the proof of Step 2. Step 3 Conclusion. By translation, we may say that λ0 = 0. The symmetry (and monotonicity) in the x1 -direction follows by repeating the argument in the (−x1 )-direction. The radial symmetry result (and the monotonicity of the solution) follows as well by applying the moving plane method in any direction ν ∈ SN−1 .   Finally, we recall the following useful lemma proved in [183], which allows us to deduce some asymptotic estimates of solutions for |x| large. Lemma 3.2.17 ([183]) Let N ≥ 2 and s ∈ (0, 1). There exists a continuous function w in RN such that 1 (−)s w(x) + w(x) = 0 for |x| > 1 2 in the classical sense, and 0 < w(x) ≤ C/|x|N+2s for |x| > 1, for some C > 0. Remark 3.2.18 This result holds true for all N ≥ 1 and s ∈ (0, 1). Indeed, in [199] it is proved that if Gs,λ ∈ S (RN ), with λ > 0, denotes the fundamental solution of (−)s Gs,λ + λGs,λ = δ0 in RN , then Gs,λ is radial, positive, strictly decreasing in |x|, smooth for |x| = 0, Gs,λ ∈ Lr (R) for all r ∈ [1, ∞] with 1 − 1r < 2s N , and |x|N+2s Gs,λ (x) → λ−2 γN,s > 0 as |x| → ∞. Take w˜ = Gs,1 ∗ χBa , where χBa is the 1 ˜ satisfies characteristic function of the ball Ba with radius a = 2− 2s . Then w(x) = w(ax) all conclusions in Lemma 3.2.17. Now we are ready to give the proofs of Theorems 3.2.11 and 3.1.1. Proof of Theorem 3.2.11 Using Theorem 3.2.13 we know that there exists a nontrivial non-negative solution u to (3.1.2). By Proposition 3.2.14, u ∈ L∞ (RN ). Now we argue as in the proof of Lemma 4.4 in [124]. Since g(u) is bounded, we can apply Proposition 1.3.2 to deduce that • If s ≤ 12 , then for any α < 2s, u ∈ C 0,α (RN ). • If s > 12 , then for any α < 2s − 1, u ∈ C 1,α (RN ). This implies in particular that g(u) ∈ C α (RN ). Applying now Proposition 1.3.1 we have • If α + 2s ≤ 1, then u ∈ C 0,α+2s (RN ). • If α + 2s > 1, then u ∈ C 1,α+2s−1 (RN ).

3.2 The Subcritical Case

85

Therefore, iterating the procedure a finite number of times, one gets that u ∈ C 1,σ (RN ) for some σ ∈ (0, 1) that depends only on s. Indeed, if α + 2s > 1, then one can take σ = α+2s −1. On the other hand, if α+2s ≤ 1, then g(u) ∈ C 0,α+2s (RN ). Consequently, u ∈ C 0,α+4s (RN ), and so iterating a finite number of times, we end up with α + 2sk > 1 for some integer k. This gives the C 1,σ regularity. We now differentiate the equation to obtain (−)s uxi = g (u)uxi in RN , for i = 1, . . . , N, with uxi and g (u) belonging to C 0,σ (RN ) provided we take σ < γ . Therefore, applying Proposition 1.3.1 we get uxi ∈ C 0,σ +2s (RN ). We iterate this procedure a finite number of times (as long as the Hölder exponent is smaller than γ ). Now, since γ + 2s > 1 by assumption, we finally conclude that uxi ∈ C 1,δ (RN ) for all i = 1, . . . , N, and thus u ∈ C 2,δ (RN ) for some δ > 0 depending only on s and γ . By Proposition 1.3.11-(ii), we deduce that u > 0 in RN . Note that, by Proposition 1.2.3, u is a classical solution of (3.1.2). ∗ Observing that u ∈ C 0,δ (RN ) ∩ L2s (RN ), we have that u(x) → 0 as |x| → ∞. Let us show that 0 < u(x) ≤

C for |x| % 1. |x|N+2s

For t ≥ 0 we define g1 (t) = (g(t) + mt)+ and g2 (t) = g1 (t) − g(t). Extend g1 and g2 as odd functions for t ≤ 0. Then g = g1 − g2 , g1 , g2 ≥ 0 in [0, ∞), g1 (t) = o(t) as t → 0, limt →∞ g21∗s(t−1) = 0, g2 (t) ≥ mt for all t ≥ 0. t

Hence, (−)s u + g2 (u) = g1 (u) in RN . Since u(x) → 0 as |x| → ∞, there exists R > 0 such that g1 (u(x)) ≤ m2 u(x) for all |x| > R. Consequently, (−)s u + m2 u ≤ 0 for |x| > R. By Lemma 3.2.17, there exist a positive continuous function w and a constant C > 0 such that for large |x| > R (taking R larger if necessary), it holds that w(x) ≤ C|x|−(N+2s) and (−)s w + m2 w = 0. In view of the continuity of u and w, max|x|≤R u there exists C1 = min|x|≤R w > 0 such that η(x) = u(x) − C1 w(x) ≤ 0 for |x| ≤ R. m s Moreover, (−) η + η ≤ 0 in RN \ B¯ R . Applying Lemma 1.3.8 with = RN \ B¯ R , 2

μ = m2 , u1 = η ∈ H s (RN ) and u2 = 0, we conclude that η(x) ≤ 0 for |x| > R, that is 0 < u(x) ≤ C|x|−(N+2s) for all |x| > R. Finally, in view of (g2), we can use Theorem 3.2.16 to deduce that u is radially   symmetric and radially decreasing with respect to a point of RN . Remark 3.2.19 The positivity of u can be also deduced by using the integral representation of (−)s given in Lemma 1.2.1, which holds for functions in H 2s (RN ) ∩ C τ (RN ) with τ > 2s; see Lemma 2.4 in [122]. Hence, recalling that u ≥ 0 in RN and u ≡ 0, if by

86

3 Fractional Scalar Field Equations

contradiction there exists x0 ∈ RN such that u(x0 ) = 0, then 1 0 = g(u(x0 )) = (−) u(x0 ) = − C(N, s) 2



s

RN

u(x0 + y) + u(x0 − y) dy ≤ 0 |y|N+2s

that is u ≡ 0, which is absurd. Therefore, u > 0 in RN . Proof of Theorem 3.1.1 The result is a consequence of Theorems 3.2.10 and 3.2.11.

 

We conclude this subsection by proving that if L is the set of least energy positive solutions to (3.1.2) satisfying u(0) = maxRN u(x), then the following compactness result holds true. Proposition 3.2.20 L is compact in H s (RN ). Moreover, there exists C > 0 independent of u ∈ L, such that 0 < u(x) ≤

C 1 + |x|N+2s

for all x ∈ RN .

Proof The proof is inspired by Byeon and Jeanjean [121]. From (3.1.3), we see that I (u) =

s [u]2 N s

for all u ∈ L, thus {[u]2s : u ∈ L} is bounded in R. Note that for any u ∈ L  [u]2s

=

RN

 g(u)u dx =

RN

[g1 (u) − g2 (u)]u dx,

where g1 and g2 are defined as in the proof of Theorem 3.2.11. Observe that for all ε > 0 ∗ there exists cε > 0 such that g1 (t) ≤ ε g2 (t) + cε t 2s −1 for all t ≥ 0, and g2 (t) ≥ mt for all t ≥ 0. Hence, fixing ε ∈ (0, 1), we have  [u]2s +

 RN

g2 (u)u dx =

RN

 g1 (u)u dx ≤ ε

2∗s

RN

g2 (u)u dx + cε u

∗

L2s (RN )

and thus (1 − ε)

m u2L2 (RN ) ≤ [u]2s + (1 − ε) 2

 RN

2∗s

g2 (u)u dx ≤ cε u

∗

L2s (RN )

.

Using the Sobolev inequality (1.1.1) and the boundedness of {[u]2s : u ∈ L}, we deduce that {u2L2 (RN ) : u ∈ L} is bounded in R. Consequently, L is bounded in H s (RN ). Using Propositions 3.2.14 and 1.3.2, we deduce that L is bounded in C 0,α (RN ).

3.2 The Subcritical Case

87

Let m be the least energy level corresponding to (3.1.2). Then, for any u ∈ L, we have m = I (u) = Ns [u]2s . Moreover, from the above calculations (with ε = 12 ), and recalling that uL2 (RN ) ≤ C¯ for all u ∈ L, we see that N m m ≤ [u]2s + u2L2 (RN ) ≤ c 1 2 s 4 ≤



∗

RN

u2s dx

2∗ −2 c 1 uLs∞ (RN ) 2

 RN

u2 dx

2∗ −2

≤ c 1 C¯ 2 uLs∞ (RN ) 2

which implies that there exists σ > 0 such that u(0) = uL∞ (RN ) ≥ σ > 0 for all u ∈ L, i.e., L is bounded away from zero in L∞ (RN ). We now claim that lim|x|→∞ u(x) = 0 uniformly for u ∈ L. Suppose, by contradiction, that there exist sequences (uk ) ⊂ L and (xk ) ⊂ RN such that lim infk→∞ |xk | = ∞ and lim infk→∞ uk (xk ) > 0. Define vk (x) = uk (x + xk ). Then we may assume that uk  u and vk  v in H s (RN ) and uniformly on compact sets of RN , for some u, v ∈ H s (RN ). Since uk (0) ≥ σ and vk (0) = uk (xk ), u and v are nontrivial solutions to (3.1.2). Consequently, I (u), I (v) ≥ I (w)

for all w ∈ L.

Take R > 0. Thus, for k large enough, we have |xk | ≥ 2R. For these values of k, I (uk ) =

s s [uk ]2s ≥ [uk ]2W s,2 (B ) + R N N s ≥ [uk ]2W s,2 (B ) + R N s = [uk ]2W s,2 (B ) + R N

s [uk ]2W s,2 (B c ) R N s [uk ]2W s,2 (B (x )) R k N s [vk ]2W s,2 (B ) R N

whence lim inf I (uk ) ≥ k→∞

s s [u]2 s,2 + [v]2 s,2 . N W (BR ) N W (BR )

Letting R → ∞ we find that lim inf I (uk ) ≥ k→∞

s s [u]2s + [v]2s ≥ 2I (w) N N

for all w ∈ L,

and this is in contrast with I (uk ) = I (w) = m > 0 for all w ∈ L. Therefore, lim|x|→∞ u(x) = 0 uniformly for u ∈ L, and using the fact that L is bounded in C 0,α (RN ), we can argue as in the last part of the proof of Theorem 3.2.11 (in this case we take M C1 = min|x|≤R w , where M > 0 is such that uL∞ (RN ) ≤ M for all u ∈ L) to deduce, by

88

3 Fractional Scalar Field Equations

comparison, that there exists C > 0 such that for any u ∈ L and x ∈ RN , C . 1 + |x|N+2s

u(x) ≤

(3.2.44)

Let now (uk ) ⊂ L. Taking a subsequence if necessary, we can assume that uk  u in p H s (RN ), uk → u in Lloc (RN ) for all p ∈ [1, 2∗s ), and uk → u a.e. in RN . Clearly, u is a weak solution of (3.1.2). Moreover, by (3.2.44), uk (x) ≤

C 1 + |x|N+2s

for all x ∈ RN , k ∈ N, ∗

and consequently uk → u in L2 (RN ) ∩ L2s (RN ). Thus, using the growth assumptions on g, it follows by the dominated convergence theorem that 

 RN

g(uk )uk dx →

RN

g(u)u dx

as k → ∞.

  Since [uk ]2s = RN g(uk )uk dx and [u]2s = RN g(u)u dx, we deduce that [uk ]s → [u]2s as k → ∞, which combined with the strong convergence in L2 (RN ) shows that uk → u in   H s (RN ). This completes the proof of the proposition.

3.3

The Zero Mass Case

This last section is devoted to the proof of the existence of a positive solution of (3.1.2) in the zero mass case. 0) > 0 and define gε (t) = g(t) − εt with ε ∈ (0, ε0 ]. Then gε satisfies Let ε0 = G(ξ ξ2 0

the assumption of Theorem 3.2.11, so we know that for every ε ∈ (0, ε0 ] there exists s uε ∈ Hrad (RN ) positive and radially decreasing in r = |x|, such that Iε (uε ) = bε and Iε (uε ) = 0, where the mountain pass minimax value bε is defined by bε = inf max Iε (γ (t)), γ ∈ε t ∈[0,1]

with s ε = {γ ∈ C([0, 1], Hrad (RN )) : γ (0) = 0, Iε (γ (1)) < 0}.

Since uε satisfies the Pohozaev identity (3.1.3), we deduce that Ns [uε ]2s = bε . Now we prove that it is possible to estimate bε from above independently of ε:

3.3 The Zero Mass Case

89

Lemma 3.3.1 There exists b0 > 0 such that 0 < bε ≤ b0 for all ε ∈ (0, ε0 ]. Proof For R > 1, we define ⎧ ⎪ for |x| ≤ R, ⎨ ξ0 , wR (x) = ξ0 (R + 1 − |x|), for |x| ∈ [R, R + 1], ⎪ ⎩ 0, for |x| ≥ R + 1. It is clear that wR ∈ H 1 (RN ) ⊂ H s (RN ). By the definitions of wR and gε , we obtain that for all ε ∈ (0, ε0 ] 

 RN

Gε (wR ) dx ≥

RN

Gε0 (wR ) dx 

= Gε0 (ξ0 )|BR | +

{R≤|x|≤R+1}

Gε0 (ξ0 (R + 1 − |x|)) dx

≥ Gε0 (ξ0 )|BR | − |BR+1 − BR | max |Gε0 (t)| t ∈[0,ξ0 ]

= ≥

π ( N2 π ( N2

N 2

+ 1)

[Gε0 (ξ0 )R N − max |Gε0 (t)|((R + 1)N − R N )] t ∈[0,ξ0 ]

#

 $  1 N Gε0 (ξ0 ) − max |Gε0 (t)| 1 + − 1 RN t ∈[0,ξ0 ] R + 1)

N 2

 so there exists R¯ > 0 (independent of ε) such that RN Gε (wR ) dx > 0 for all R ≥ R¯ and ε ∈ (0, ε0 ]. Then, upon setting wt (x) = wR¯ ( xt ), we have that for every ε ∈ (0, ε0 ], Iε (wt ) =

t N−2s [wR¯ ]2s − t N 2

 RN

Gε (wR¯ ) dx → −∞

(3.3.1)

as t → ∞. Hence, there exists τ > 0 such that Iε (wτ ) < 0 for any ε ∈ (0, ε0 ]. We put e(x) ¯ = wτ (x). Therefore, Iε (uε ) = bε ≤ sup

max Iε (t e) ¯ = b0

ε∈(0,ε0 ] t ∈[0,1]

for any ε ∈ (0, ε0 ].

 

In view of Lemma 3.3.1, we infer that [uε ]2s =

N N bε ≤ b0 s s

for all ε ∈ (0, ε0 ],

(3.3.2)

90

3 Fractional Scalar Field Equations

and consequently, we may assume that as ε → 0 s,2 uε  u in Drad (RN ),

for any q ∈ [1, 2∗s ),

q

uε → u in Lloc (RN ),

(3.3.3)

uε → u a.e. in RN . Since uε is a weak solution to (3.1.2), we know that  uε , ϕDs,2 (RN ) =

RN

(g(uε ) − εuε )ϕ dx,

(3.3.4)

for all ϕ ∈ Cc∞ (RN ). Taking into account (3.3.2), (3.3.3), (h1)–(h2), and the fact that ∗ Ds,2 (RN ) ⊂ L2s (RN ) we apply the first part of Lemma 1.4.2 with P (t) = g(t), Q(t) = ∗ |t|2s −1 , vε = uε , v = u and w = ϕ to pass to the limit in (3.3.4) as ε → 0. Then we obtain  u, ϕDs,2 (RN ) =

RN

g(u)ϕ dx,

for all ϕ ∈ Cc∞ (RN ). This means that u is a weak nonnegative solution to (−)s u = g(u) in RN . Now, we only need to prove that u ≡ 0. We recall that uε > 0 and satisfies the Pohozaev identity (3.1.3) with G replaced by Gε , that is N − 2s [uε ]2s = N 2

 RN

ε

G(uε ) − u2ε dx. 2

(3.3.5)

Now, using (h1 )–(h2 ), there exists c > 0 such that ∗

G(t) ≤ c|t|2s

for all t ∈ R.

(3.3.6)

Then, in view of (3.3.5) and (3.3.6), we get 2N ε 2N uε 2L2 (RN ) + [uε ]2s = N − 2s 2 N − 2s  2Nc ∗ 2∗ ≤ |uε |2s dx ≤ C[uε ]s s N − 2s RN

[uε ]2s ≤

 RN

G(uε ) dx

which implies that [uε ]s ≥ c > 0, for some c independent of ε.

(3.3.7)

3.3 The Zero Mass Case

91

Next, we argue by contradiction, and we assume that u = 0. We begin by proving that  RN

G+ (uε ) dx → 0

as ε → 0.

(3.3.8)

Using Lemma 1.4.1 with t = 2∗s , Theorem 1.1.8 and (3.3.2), we can see that, for any x ∈ RN \ {0} and ε ∈ (0, ε0 ],  |uε (x)| ≤



1 2∗ s

ωN−1 

≤

N



1 2∗ s

ωN−1 

≤

N

N



1 2∗ s

ωN−1

|x| |x| |x|

− 2N∗ s

− 2N∗ s

− 2N∗ s

uε L2∗s (RN ) S∗ [uε ]s  S∗

N − N∗ b0 = C|x| 2s , s

(3.3.9)

where C is independent of ε. By (3.3.9), for any δ > 0 there exists R > 0 such that |uε (x)| ≤ δ for |x| ≥ R. Moreover, by assumption (h1 ), for every η > 0 there exists δ > 0 such that ∗

G+ (t) ≤ η|t|2s

for |t| ≤ δ.

Hence, for large R, we obtain the estimate 

+

RN \B



G (uε ) dx ≤ η R

2∗

∗

RN \B

|uε |2s dx ≤ Cη[uε ]s s ≤ Cη, R

uniformly in ε ∈ (0, ε0 ]. On the other hand, by assumption (h2 ), for every η > 0 there exists a constant Cη such that ∗

G+ (t) ≤ η|t|2s + Cη

for all t ∈ R.

Now, we fix ⊂ RN with sufficiently small measure | | < 

G+ (uε ) dx ≤ η



η Cη .

Then, for any ε ∈ (0, ε0 ]

∗

|uε |2s dx + Cη | | ≤ Cη.

Applying Vitali’s convergence theorem, we deduce that (3.3.8) is satisfied.

92

3 Fractional Scalar Field Equations

Therefore, using (3.3.5) and the facts G = G+ + G− and G− ≤ 0, we have [uε ]2s

  ε 2 2N 2N − u dx ≤ − G (uε ) dx + N − 2s RN N − 2s RN 2 ε  2N = G+ (uε ) dx, N − 2s RN [uε ]2s

which together with (3.3.8) shows that [uε ]s → 0 as ε → 0. This gives a contradiction in s,2 N (R ) is a nontrivial non-negative weak solution to (3.1.2). view of (3.3.7). Then, u ∈ Drad By combining Remark 3.2.15 and Proposition 1.3.2, we deduce that u ∈ C 0,β (RN ). From Proposition 1.3.11-(ii), we conclude that u > 0 in RN .

3.4

The Critical Case

3.4.1

Ground State for the Critical Case

This section is devoted to the proof of Theorem 3.1.3. Without loss of generality, we will assume that b = 1 in (g3 ). Let us consider the functional I : H s (RN ) → R given by I(u) = T (u) − V(u) for any u ∈ H s (RN ), where T (u) =

1 2 [u] 2 s

 and

V(u) =

RN

G(u) dx.

By Theorem 1.1.8 and the assumptions on g, it is clear that I is well defined on H s (RN ) and that I ∈ C 1 (H s (RN ), R) (to prove that V(u) is of class C 1 in H s (RN ) one can argue as in [100]; see also Theorem 2.1.5). We begin by proving the following result: Lemma 3.4.1 Consider the constrained minimization problem  M = inf T (u) : V(u) = 1, u ∈ H s (RN ) . Then, 0 < M < 12 (2∗s )

N−2s N

S∗ .

(3.4.1)

3.4 The Critical Case

93

Proof First, we show that {u ∈ H s (RN ) : V(u) = 1} is not empty. We know (see Remark 1.1.9) that S∗ is achieved by the extremal functions Uε (x) = 

κ ε−   μ2 + 

N−2s 2

x 1 ε S∗2s

2  

 N−2s , 2

for any ε > 0, where κ ∈ R, μ > 0 are fixed constants. Let η ∈ Cc∞ (RN ) be a cut-off function with support in B2 and such that 0 ≤ η ≤ 1, and η = 1 on B1 . For ε > 0, we define ψε (x) = η(x)Uε (x), and we set vε (x) =

ψε . ψε L2∗s (RN )

By performing similar calculations to those in Proposition 21 and 22 in [310] we can see that 2∗s

ψε 

∗ L2s (RN )

N

N

= S∗2s + O(εN )

and

[ψε ]2s ≤ S∗2s + O(εN−2s ),

(3.4.2)

so, in particular, we deduce that [vε ]2s ≤ S∗ + O(εN−2s ). By assumption (g4 ), we get V(v ) ≥ ε =

(3.4.3)

1 + ε , where 2∗s

C a q vε Lq (RN ) − vε 2L2 (RN ) . q 2

Our aim is to prove that lim

ε→0

ε = ∞. εN−2s

(3.4.4)

Since {2∗s − 2, 2} < q < 2∗s , we know that (N − 2s)q > N. Then we can find two positive constants C1 (N, s) and C2 (N, s) such that  1 q vε Lq (RN ) ≥ |Uε (x)|q dx q ψε  2∗s N B1 L (R )

≥ C1 (N, s) ε

N− (N−2s)q 2

1



1/(ε S∗2s ) 0

= O(εN−

(N−2s)q 2

)

r N−1 (μ2 + r 2 )

(N−2s)q 2

dr

94

3 Fractional Scalar Field Equations

and 

1

vε 2L2 (RN ) ≤



1

2/(ε S∗2s )

|Uε (x)|2 dx ≤ C2 (N, s) ε2s

ψε 2 2∗s N B2 L (R ) ⎧ 2s ⎪ if N > 4s, ⎨ O(ε ), = O(ε 2s log( 1ε )), if N = 4s, ⎪ ⎩ O(ε N−2s ), if N < 4s.

0

r N−1 dr (μ2 + r 2 )N−2s

(N−2s)q

Thus we deduce that ε ≥ O(εN− 2 ). Using the fact that {2∗s − 2, 2} < q < 2∗s , it is easy to deduce that (3.4.4) holds true. Consequently, there exists ε0 > 0 such that V(vε ) ≥ Set ω(x) = vε

x

1 2∗s

for all 0 < ε < ε0 .

, where σ = (V(vε ))− N . Then, V(ω) = 1 and we have that {u ∈ 1

σ H s (RN ) : V(u) = 1} is not empty. N−2s Now we show that 0 < M < 12 (2∗s ) N S∗ . Clearly, 0 ≤ M < ∞. Since T (vε ) = σ N−2s T (ω) and V(ω) = 1, we deduce that T (vε )

M ≤ T (ω) =

2 ∗

.

(V(vε )) 2s

Then, by (3.4.3), we can see that for every ε such that 0 < ε < ε0 0 ≤ M ≤

1 2 2 [vε ]s 1 2∗s

1 ∗ 22∗ 1 + O(εN−2s ) . 2

2∗ ≤ 2 (2s ) s S∗ 2s ∗  ) 2∗s (1 + 2 + ε s ε

(3.4.5)

Noting that for p ≥ 1 (1 + y)p ≤ 1 + p(1 + y)p+1 y

for all y ≥ −1,

by (3.4.4) we deduce that for all ε > 0 sufficiently small (1 + O(εN−2s ))

2∗ s 2

−1≤

2∗ 2∗s s (1 + O(εN−2s ))1+ 2 O(εN−2s ) < 2∗s ε , 2

that is, (1 + O(εN−2s ))

2∗ s 2

< 1 + 2∗s ε .

(3.4.6)

3.4 The Critical Case

95

Hence, in view of (3.4.5) and (3.4.6), M
0. If, by contradiction, M = 0, then we can find a sequence (un ) ⊂ H s (RN ) such that V(un ) = 1 and T (un ) → 0 as n → ∞. By (1.1.1), we see that un L2∗s (RN ) → 0 as n → ∞. On the other hand, using (g2 ) and (g3 ), we know that ∗

there is a constant K > 0 such that G(t) ≤ K|t|2s for all t ∈ R, which implies that 2∗ 1 = V(un ) ≤ Kun  s2∗s N → 0 as n → ∞, a contradiction.   L (R )

Using Lemma 3.4.1, we show that: Lemma 3.4.2 Under the assumptions of Theorem 3.1.3, there exists a solution u ∈ H s (RN ) of the minimization problem (3.4.1). Proof By Lemma 3.4.1, {T (u) : V(u) = 1, u ∈ H s (RN )} is not empty. Now, we show the existence of a radial minimizing sequence. Let (un ) ⊂ H s (RN ) be a sequence such that V(un ) = 1 for all n ∈ N, and T (un ) → M as n → ∞. Denote by u∗n the Schwarz spherical rearrangement of |un |. Using the fractional Polya-Szeg˝o inequality (3.2.23), we have [u∗n ]s ≤ [un ]s

for all n ∈ N.

On the other hand, observing that G ∈ C(R) is even, G(0) = 0, and G(u) ∈ L1 (RN ) for all u ∈ H s (RN ), we deduce that  RN

G(u∗n ) dx =

 RN

G(un ) dx

for all n ∈ N.

s Hence, u∗n ∈ Hrad (RN ), M ≤ T (u∗n ) ≤ T (un ) and V(u∗n ) = 1 for all n ∈ N, that is, (u∗n ) is also a minimizing sequence. Therefore, we can assume that for every n ∈ N, un is non∗ negative and radially symmetric. Using (1.1.1), we see that (un ) is bounded in L2s (RN ). By (g2 ) and (g3 ), there exists K > 0 such that ∗

G(t) ≤ K|t|2 −

a 2 t 4

for all t ∈ R.

(3.4.7)

Then, combining the equality V(un ) = 1, (3.4.7) and the fact that (un ) is bounded in ∗ L2s (RN ), we obtain that (un ) is bounded in L2 (RN ). Consequently, (un ) is bounded in H s (RN ), so un  u in H s (RN ), and by Theorem 1.1.11 we have un → u in Lq (RN ) and a.e. in RN .

96

3 Fractional Scalar Field Equations

Let vn = un − u. Then, the weak convergence in H s (RN ) implies that T (un ) = T (vn ) + T (u) + o(1),

(3.4.8)

as n → ∞, while the Brezis-Lieb lemma [113] yields 2∗s

un 

∗ L2s (RN )

2∗s

= vn 

∗ L2s (RN )

2∗s

+ u

∗

L2s (RN )

+ o(1)

(3.4.9)

and un 2L2 (RN ) = vn 2L2 (RN ) + u2L2 (RN ) + o(1), ∗

as n → ∞. Let f (t) = g(t) − (t + )2s −1 + at and F (t) =



(3.4.10)

t

f (ξ ) dξ . By (g2 ) and (g3 ), 0

lim

|t |→0

F (t) =0 t2

and

lim

|t |→∞

F (t) ∗ = 0, |t|2s

and by the boundedness of (un ) in H s (RN ), Lemma 1.4.3 implies that 

 RN

F (vn ) dx = o(1)

and

 RN

F (un ) dx =

RN

F (u) dx + o(1)

(3.4.11)

as n → ∞. Combining (3.4.9), (3.4.10), and (3.4.11) we get V(un ) = V(vn ) + V(u) + o(1)

as n → ∞.

(3.4.12)

Set τn = T (vn ), τ = T (u), νn = V(vn ) and ν = V(u). Then, with these notations, (3.4.8) and (3.4.12) read τn = M − τ + o(1)

and

νn = 1 − ν + o(1).

(3.4.13)

In order to prove the existence of a minimizer of (3.4.1), it is enough to prove that ν = 1. Firstly, we note that, if uσ (x) = u( σx ), then T (uσ ) = σ N−2s T (u) and V(uσ ) = σ N V(u), so we have T (u) ≥ M(V(u))

N−2s N

,

(3.4.14)

for all u ∈ H s (RN ) and V(u) ≥ 0. N−2s If, by contradiction, ν > 1, then τ ≥ Mν N > M, which is in contrast with the fact that τ ≤ M. Therefore, ν ≤ 1. If ν < 0, then νn > 1 − ν2 > 1 for n sufficiently large.

3.4 The Critical Case

97

Using (3.4.14) we see that N−2s

τn ≥ Mνn N

ν N−2s N >M 1− 2

for n sufficiently large. This is impossible, since τn ≤ M + o(1) by (3.4.13). Therefore, ν ∈ [0, 1]. Let us show that ν = 1. If, by contradiction, ν ∈ [0, 1), then νn > 0 for all n big enough. By (3.4.14) we have N−2s

τn ≥ Mνn N

τ ≥ Mν

and

N−2s N

.

This yields M = lim (τ + τn ) n→∞   N−2s N−2s N ≥ lim M ν N + νn n→∞

N−2s N−2s = M ν N + (1 − ν) N ≥ M(ν + 1 − ν) = M. N−2s

N−2s

Consequently, ν N + (1 − ν) N = 1, and since ν ∈ [0, 1), we get ν = 0. Then u = 0 and τn → M as n → ∞. Moreover, by (3.4.13), νn → 1 as n → ∞, and by the definition of F and (3.4.11) we can infer that 2∗s

vn 

∗ L2s (RN )

= 2∗s +

2∗s avn 2L2 (RN ) + o(1), 2

which implies that lim sup vn 2 2∗s n→∞

Therefore, recalling that S∗ =

2 ∗

L (RN )

≥ (2∗s ) 2s .

[u]2s 2 u∈H s (RN )\{0} u 2∗ s inf

, we get

L (RN )

M=

2 1 [vn ]2s 1 ∗ lim [vn ]2s ≥ (2∗s ) 2s lim inf n→∞ vn 2 ∗ 2 n→∞ 2 2

≥

L s (RN )

which contradicts Lemma 3.4.1. Consequently, ν s (RN ). u ∈ Hrad

=

1 ∗ 22∗ (2 ) s S∗ , 2 s

1 and M is achieved by  

98

3 Fractional Scalar Field Equations

Before giving the proof of Theorem 3.1.3, we recall that any solution u of (3.1.2) satisfies the following Pohozaev identity (see [36]): N − 2s 2 [u]s = N 2

 RN

(3.4.15)

G(u) dx.

Now we are able to prove the main result of this section: Proof of Theorem 3.1.3 Let us show that there exists a least energy solution ω(x)   N−2s 2 N s N − 2s of (3.1.2) and that m = (2M) 2s , where m is the least energy level of N 2N (3.1.2). We introduce the sets  S = v ∈ H s (RN ) : V(v) = 1 and  P = v ∈ H s (RN ) \ {0} : J (v) = 0 , where J (v) =

1 2 N [v]s − 2 N − 2s

 RN

G(v) dx ∈ C 1 (H s (RN ), R).

Then we have a one-to-one correspondence  : S → P between S and P, defined by  ((v))(x) = v

x τu



 ,

where τu =

N − 2s 2N

1

2s

1

[v]ss .

Let us notice that for any v ∈ S I((v)) = τuN−2s T (v) − τuN V(v) =

s N



N − 2s 2N

 N−2s 2s

N

[v]ss ,

and thus s inf I(v) = inf I((v)) = N v∈P v∈S



N − 2s 2N

 N−2s 2s

Using Lemma 3.4.2, we know that there exists u ∈ S such that inf [v]2s = [u]2s = 2M.

v∈S

N

inf [v]ss .

v∈S

3.4 The Critical Case

99

Consequently, 

s inf I(v) = I((u)) = N v∈P

N − 2s 2N

 N−2s 2s

N

(2M) 2s .

Let ω = (u). By the theorem on Lagrange multipliers, there exists λ ∈ R such that I (ω), ϕ = λJ (ω), ϕ

for any ϕ ∈ H s (RN ).

This means that ω is a weak solution to  (1 − λ)(−)s ω = 1 − λ

N N − 2s

 g(ω)

in RN ,

and, in view of (3.4.15), ω satisfies the Pohozaev identity (1 − λ)

  (N − 2s) 2 N [ω]s = N 1 − λ G(ω) dx. 2 N − 2s RN

(3.4.16)

Since ω = (u) ∈ P, we also know that N − 2s [ω]2s = N 2

 RN

G(ω) dx.

(3.4.17)

Putting together (3.4.16) and (3.4.17) we deduce that  λ 1−

N N − 2s

 RN

G(ω) dx = 0,

so we have λ = 0 and I (ω), ϕ = 0 for all ϕ ∈ H s (RN ). Then, ω is a least energy solution to (3.1.2) and m=

s N



N − 2s 2N

 N−2s 2

N

(2M) 2s .

Arguing as in the proof of Theorem 3.2.11, we conclude that ω ∈ C 2,β (RN ), for some   β > 0, and ω > 0 in RN . Remark 3.4.3 Since ω(x) → 0 as |x| → ∞, there exists R > 0 such that g(ω(x)) ≤ − a2 ω(x) for all |x| > R. Consequently, (−)s ω + a2 ω ≤ 0 for |x| > R. Using

100

3 Fractional Scalar Field Equations

Lemma 3.2.17 and arguing as in the proof of Theorem 3.2.11, we deduce that 0 < ω(x) ≤ C|x|−(N+2s) for |x| > R. Moreover, we can see that ω is radially decreasing with respect to some point of RN .

3.4.2

Mountain Pass Characterization of Least Energy Solutions

In this section we give the proof of Theorem 3.1.5. First we prove Lemma 3.4.4 I has a mountain pass geometry, that is: (i) I(0) = 0; (ii) there exist ρ > 0, and η > 0 such that I(u) ≥ η for all uH s (RN ) = ρ; (iii) there exists u0 ∈ H s (RN ) such that u0 H s (RN ) > ρ and I(u0 ) < 0. Then c = inf max I(γ (t)) γ ∈ t ∈[0,1]

where   = γ ∈ C([0, 1], H s (RN )) : γ (0) = 0 and I(γ (1)) < 0 , is well defined. Proof Clearly, I(0) = 0. By assumptions (g2 ) and (g3 ), there exists a positive constant Ca such that a ∗ G(t) ≤ − t 2 + Ca |t|2s for all t ∈ R. 4 Then, by (1.1.1) and (3.4.18), we get 1 2 a 2∗ [u]s + u2L2 (RN ) − Ca u s2∗s N L (R ) 2 4   1 a 2∗ , ≥ min u2H s (RN ) − Ca u s2∗s N L (R ) 2 4   2∗ 1 a − s 2∗ , ≥ min u2H s (RN ) − Ca S∗ 2 uHs s (RN ) , 2 4

I(u) ≥

(3.4.18)

3.4 The Critical Case

101

which implies that there exist ρ > 0 and η > 0 such that I(u) ≥ η for all uH s (RN ) = ρ. Using again (g2 ) and (g3 ), there exists Ca > 0 such that G(t) ≥ −

Ca 2 1 ∗ t + ∗ |t|2s 2 22s

for all t ∈ R.

Then, for any u ∈ H s (RN ) \ {0} and t > 0, ∗

I(tu) ≤

t 2s t 2 2 Ca t 2 2∗ [u]s + u2L2 (RN ) − ∗ u s2∗s N → −∞ L (R ) 2 2 22s

as t → ∞,

so we can find u0 ∈ H s (RN ) such that u0 H s (RN ) > ρ and I(u0 ) < 0.

 

Proof of Theorem 3.1.5 Let ω be a least energy solution to (3.1.2). We begin by proving that there exists γ ∈ C([0, 1], H s (RN )) such that γ (0) = 0, I(γ (1)) < 0, ω ∈ γ ([0, 1]) and max I(γ (t)) = m.

t ∈[0,1]

Let  γ (t)(x) =

ω( xt ), for t > 0, 0, for t = 0.

Then, γ (t)2H s (RN ) = t N−2s [ω]2s + t N ω2L2 (RN ) ,  t N−2s [ω]2s − t N G(ω) dx. I(γ (t)) = 2 RN

(3.4.19)

Hence, γ ∈ C([0, ∞), H s (RN )). By the Pohozaev identity (3.4.15),  RN

G(ω) dx =

N − 2s [ω]2s > 0. 2N

Combining (3.4.19) and (3.4.20) we have N − 2s d I(γ (t)) = t N−2s−1 (1 − t 2s ) [ω]2s , dt 2

(3.4.20)

102

3 Fractional Scalar Field Equations

and, in particular, d I(γ (t)) > 0 dt

for t ∈ (0, 1)

and

d I(γ (t)) < 0 dt

for t > 1.

Thus, for L > 1 sufficiently large, there exists a path γ : [0, L] → H s (RN ) such that γ (0) = 0, I(γ (L)) < 0, ω ∈ γ ([0, L]) and max I(γ (t)) = m.

t ∈[0,L]

After a suitable scale change in t, we get the desired path γ ∈ . Hence, c ≤ m. From the proof of Theorem 3.1.3, we deduce that m = infv∈P I(v). Now, let  N − 2s 2 [u]s − N G(u) dx = NI(u) − s[u]2s . H(u) = 2 RN Modifying slightly the arguments of the proof of Lemma 3.4.4, one can show that there exists ρ0 > 0 such that H(u) > 0 for all 0 < uH s (RN ) ≤ ρ0 . For any γ ∈ , we have γ (0) = 0 and H(γ (1)) ≤ NI(γ (1)) < 0. Consequently, there exists t0 ∈ [0, 1] such that γ (t0 )H s (RN ) > ρ0

and

H(γ (t0 )) = 0.

Since γ (t0 ) ∈ γ ([0, 1]) ∩ P, we have γ ([0, 1]) ∩ P = ∅ and thus c ≥ m. Therefore, c = m.  

3.5

The Pohozaev Identity for (−)s

In this section we give a proof of the Pohozaev identity for the fractional Laplacian in RN . For more details we refer to [36, 122, 139, 302, 307]. Theorem 3.5.1 Let g satisfy (g1 ) and either (g1)–(g3) or (g2 )–(g4 ). If u ∈ H s (RN ) is a weak solution to (3.1.2), then u satisfies the Pohozaev identity   s N − 2s 2 2 |(−) u| dx = N G(u) dx. 2 RN RN Proof Transforming (3.1.2) into a local problem via the extension method, we see that the s-harmonic extension v(x, y) = Ps (x, y) ∗ u(x) of u solves ⎧ 1−2s ∇v) = 0 in RN+1 , ⎪ ⎨ − div(y + v(·, 0) = u on RN , ⎪ ⎩ ∂v = κ g(u) on RN , s ∂ν 1−2s

(3.5.1)

3.5 The Pohozaev Identity for (−)s

103

and satisfies (see Sect. 1.2.3) 

 RN+1 +

y 1−2s |∇v|2 dxdy = κs

s

RN

|(−) 2 u|2 dx = κs

C(N, s) 2 [u]s < ∞. 2

(3.5.2)

Using Proposition 3.2.14 and the bootstrap argument in Theorem 3.2.11, we infer that u ∈ C 2,β (RN ) and thus v ∈ C 2 (RN+1 + ). For any R > 0 and δ ∈ (0, R), denote + = {(x, y) ∈ RN × [δ, ∞) : |(x, y)| ≤ R}, DR,δ

and 1 ∂DR,δ = {(x, y) ∈ RN × {y = δ} : |x|2 ≤ R 2 − δ 2 }, 2 ∂DR,δ = {(x, y) ∈ RN × [δ, ∞) : |(x, y)| = R}. + . Then, Denote by n the unit outward normal vector on ∂DR,δ

 n=

1 , (0, . . . , 0, −1), on ∂DR,δ y 2 . ( Rx , R ), on ∂DR,δ

Note that   |∇v|2 div(y 1−2s ∇v)((x, y) · ∇v) = div y 1−2s ∇v((x, y) · ∇v) − y 1−2s (x, y) 2 +

N − 2s 1−2s y |∇v|2 . 2

+ and applying the divergence theorem Multiplying (3.5.1) by (x, y)·∇v, integrating on DR,δ we get

 0= 

+ DR,δ

div(y 1−2s ∇v)((x, y) · ∇v) dxdy

% & y y 1−2s ((x, y) · ∇v)(−vy ) + |∇v|2 dσ 1 2 ∂DR,δ     N − 2s R 1−2s 1 2 2 |(x, y) · ∇v| − |∇v| dσ + + y y 1−2s |∇v|2 dxdy + 2 R 2 2 ∂DR,δ DR,δ =

= A1 (R, δ) + A2 (R, δ) + A3 (R, δ).

(3.5.3)

104

3 Fractional Scalar Field Equations

Standard computations imply the following identity: g(u)(x · ∇u) = div(x G(u)) − NG(u) on RN . Then, using (3.5.1) and the divergence theorem, we see that  lim

δ→0 ∂D 1 R,δ

 y 1−2s ((x, y) · ∇v)(−vy ) dσ = κs

BR

g(u)(x · ∇u) dx



= Rκs



∂ BR

G(u) dσ − Nκs

BR

G(u) dx, (3.5.4)

where BR = {(x, 0) ∈ ∂RN+1 : |x|2 ≤ R 2 }. Clearly, +  lim

δ→0 ∂D 1 R,δ

y 1−2s y|∇v|2 dσ = 0.

(3.5.5)

Taking into account that G(u) ∈ L1 (RN ) and (3.5.2), we can find a sequence (Rn ) with Rn → ∞ as n → ∞, such that  lim Rn

n→∞

∂ BRn

G(u) dσ = 0,

lim A2 (Rn , δ) = 0 for all δ > 0.

n→∞

(3.5.6) (3.5.7)

Indeed, suppose, by contradiction, that there exist τ, R1 > 0 such that, for all R ≥ R1 ,  ∂ BR1

|G(u)| dσ ≥

τ . R

Then 

 RN

|G(u)| dx ≥  ≥  ≥

∞ R1

∂ BR1

R1

∂ BR1

∞

∞ R1

|G(u)| dσ dr |G(u)| dσ dr

τ dr = ∞ r

3.5 The Pohozaev Identity for (−)s

105

that is absurd. The same argument works for proving (3.5.7) and this shows that the claim holds true. Combining (3.5.4)–(3.5.7), we deduce that  lim lim A1 (Rn , δ) = −N

n→∞ δ→0

RN

G(u) dx, (3.5.8)

lim lim A2 (Rn , δ) = 0.

n→∞ δ→0

Then, using (3.5.8) and the fact that lim lim A3 (Rn , δ) =

n→∞ δ→0

N − 2s 2

 RN+1 +

y 1−2s |∇v|2 dxdy,

we can see that (3.5.3) yields N − 2s 2



 RN+1 +

y 1−2s |∇v|2 dxdy = κs

Now (3.5.2) and (3.5.9) yield the desired result.

RN

G(v(x, 0)) dx.

(3.5.9)  

Remark 3.5.2 As observed in [122], the regularity assumptions on g needed to obtain the Pohozaev identity can be weakened (in fact, it is enough to require the C 1 regularity for weak solutions of (3.1.2) to prove Theorem 3.5.1).

4

Ground States for a Pseudo-Relativistic Schrödinger Equation

4.1

Introduction

In this chapter we consider the nonlinear fractional equation [(− + m2 )s − m2s ]u + μu = |u|p−2 u

in RN ,

(4.1.1)

where s ∈ (0, 1), N ≥ 2, p ∈ (2, 2∗s ), m ≥ 0 and μ > 0. The fractional operator (− + m2 )s

(4.1.2)

which appears in (4.1.1) is defined in the Fourier space by setting F (− + m2 )s u(ξ ) = (|ξ |2 + m2 )s F u(ξ ), or via the Bessel potential, that is, for every u ∈ Cc∞ (RN ) 2 s

(− + m ) u(x) = cN,s m for every x ∈

RN ;

N+2s 2

 P.V.

u(x) − u(y) RN

|x − y|

N+2s 2

K N+2s (m|x − y|) dy + m2s u(x) 2

(4.1.3)

see [181, 249]. Here P.V. stands for the Cauchy principal value, cN,s = 2−

N+2s 2 +1

N

π − 2 22s

s(1 − s) (2 − s)

and Kν denotes the modified Bessel function of the third kind and order ν; see [78, 178, 336] for more details.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_4

107

108

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

When s = 1/2 the operator (4.1.2) has a clear meaning in quantum mechanics: it corresponds to the √ free Hamiltonian of a free relativistic particle of mass m. We stress that the study of − + m2 has been strongly influenced by the study of Lieb and Yau [250, 251] on the stability of relativistic matter. For more details on this topic one can consult [152, 202, 217, 249, 337, 338]. On the other hand, the operator (4.1.2) has a deep connection with the theory of stochastic processes: in this context, −[(− + m2 )s − m2s ] is the infinitesimal generator of a Lévy process called the 2s-stable relativistic process: see [76, 119, 132, 304] for more details. We remark that the hardest issue in dealing with this operator is the lack of scaling properties: there is no standard group action under which (− + m2 )s behaves as a local differential operator. Indeed, the most striking difference between (− + m2 )s and (−)s is that the latter enjoys some scaling properties that the former does not have. As should be clear from (4.1.3), the operator (− + m2 )s is not compatible with the semigroup R+ acting on functions as t ∗ u : x → u(t −1 x) for t > 0. In simpler words, the operator (− + m2 )s does not scale. For this reason, some interesting papers studied fractional nonlocal problems involving (4.1.2); see for instance [31, 34, 143, 152, 181, 184, 308] and the references therein. In the present chapter we are interested in the study of the ground states of (4.1.1). Such problems are motivated in particular by the search of standing wave solutions, namely ψ(x, t) = u(x)e−ıωt , where ω is a constant, for the fractional Schrödinger-Klein-Gordon equation ı

∂ψ = [(− + m2 )s − m2s ]ψ + F (x, ψ) ∂t

in RN ,

which describes dynamics of boson systems. Our first result can be stated as follows: Theorem 4.1.1 ([34]) Let m > 0 and μ > 0. Then there exists a ground state solution u ∈ Hms (RN ) to (4.1.1) which is positive, radially symmetric and decreasing. The main difficulty in the study of (4.1.1) is related to the nonlocal character of the operator (4.1.2). To overcome this difficulty, we use a variant of the Caffarelli-Silvestre extension method [127], which consists in writing a given nonlocal problem in a local form via the Dirichlet-Neumann map. This allow us to apply known variational techniques to this kind of problems; see for instance [31, 152, 181, 317]. More precisely, for any u ∈ 1−2s ) that solves in the weak sense Hms (RN ) there exists a unique v ∈ Hm1 (RN+1 + ,y 

−div(y 1−2s ∇v) + m2 y 1−2s v = 0 in RN+1 + , v(·, 0) = u on ∂RN+1 + ,

4.2 Preliminaries

109

and such that ∂v ∂v (x, 0) = − lim y 1−2s (x, y) = (− + m2 )s u(x) 1−2s y→0 ∂ν ∂y

in Hm−s (RN ).

We exploit this fact and we study the existence, regularity and qualitative properties of solutions to the problem ⎧ ⎨ −div(y 1−2s ∇v) + m2 y 1−2s v = 0 in RN+1 + , ∂v ⎩ = κs [m2s v − μv + |v|p−2 v] on ∂RN+1 + . ∂ν 1−2s Finally, we are able to pass in (4.1.1) to the limit as m → 0 and find a nontrivial solution to (−)s u + μu = |u|p−2 u

in RN .

(4.1.4)

In this way, we rediscover the existence result contained in Theorem 3.1.1 of Chap. 3 with g(u) = −μu + |u|p−2 u. More precisely, we obtain the following result. Theorem 4.1.2 ([34]) There exists a solution u ∈ H s (RN ) to (4.1.4) which is positive and radially symmetric. In Sect. 4.2 we give some useful results needed to deal with (4.1.2). In Sect. 4.3 we obtain the existence of a ground state solution to (4.1.1). In Sect. 4.4 we investigate the qualitative properties of solutions to (4.1.1). Finally, we give the proof of Theorem 4.1.2.

4.2

Preliminaries

4.2.1

The Extension Method for (− + m2 )s

Let s ∈ (0, 1) and m > 0. We define Hms (RN ) as the completion of Cc∞ (RN ) with respect to the norm  |u|Hms (RN ) =

1 (|ξ | + m ) |F u(ξ )| dξ 2

RN

2 s

2

2

< ∞,

where F u(k) is the Fourier transform of u. We note that 

s

RN

|(− + m2 ) 2 u|2 dx =

 RN

(|ξ |2 + m2 )s |F u(ξ )|2 dξ

110

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

for all u ∈ Hms (RN ). When m = 1, we use the notation H s (RN ) = H1s (RN ) and | · |H s (RN ) = | · |H s (RN ) . In [181], it is also proved that 1



s

RN

[|(− + m2 ) 2 u|2 − m2s u2 ] dx cN,s N+2s = m 2 2

 R2N

|u(x) − u(y)|2 K N+2s (m|x − y|) dxdy 2 |x − y|N+2s

(4.2.1)

1−2s ) the completion of C ∞ (RN+1 ) with for all u ∈ Hms (RN ). We denote by Hm1 (RN+1 + ,y + c respect to the norm

  vH 1 (RN+1 ,y 1−2s ) = m

+

1 2

RN+1 +

y

1−2s

2

2 2

(|∇v| + m v ) dxdy

< ∞.

1−2s ) = H 1 (RN+1 , y 1−2s ) and  ·  When m = 1, we set H 1 (RN+1 1−2s ) = + ,y + 1 H 1 (RN+1 + ,y  · H 1 (RN+1 ,y 1−2s ) . 1

+

1−2s ) We recall that there exists a trace operator which relates the spaces Hm1 (RN+1 + ,y s N and Hm (R ): 1−2s ) → H s (RN ) Theorem 4.2.1 ([181]) There exists a trace operator Tr : Hm1 (RN+1 m + ,y such that:

(i) Tr(v) = v(·, 0) for every v ∈ Cc∞ (RN+1 + ). 1−2s ), where κ = (ii) κs |Tr(v)|2H s (RN ) ≤ v2 1 N+1 1−2s for every v ∈ Hm1 (RN+1 s + ,y m

21−2s (1−s) (s) .

Hm (R+

,y

)

1−2s ) if and only if v is a weak Equality holds in (ii) for a function v ∈ Hm1 (RN+1 + ,y solution to

−div(y 1−2s ∇v) + m2 y 1−2s v = 0

in RN+1 + .

Remark 4.2.2 By item (ii) in Theorem 4.2.1 we deduce that  m2s Tr(v)2L2 (RN ) ≤

N+1 R+

y 1−2s (|∇v|2 + m2 v 2 ) dxdy

(4.2.2)

4.2 Preliminaries

111

1−2s ). Since H s (RN ) ⊂ Lq (RN ) for any 2 ≤ q ≤ 2∗ , we deduce for all v ∈ Hm1 (RN+1 + ,y m s ' ( N+1 1−2s 1 ) and for any q ∈ 2, 2∗s that for any v ∈ Hm (R+ , y

 Cq,s,N u2Lq (RN )

≤ κs  ≤

RN

(|ξ |2 + m2 )s |F u(ξ )|2 dξ

RN+1 +

y 1−2s (|∇v|2 + m2 v 2 ) dxdy,

(4.2.3)

where u(x) = v(x, 0) is the trace of v on ∂RN+1 + . Remark 4.2.3 In view of (1.2.8) and (1.2.2), κs

C(N, s) [Tr(v)]2s ≤ ∇v2 2 N+1 1−2s L (R+ ,y ) 2

for all v ∈ X0s (RN+1 + ). The following result holds true (see also [317]): Theorem 4.2.4 ([181]) Let u ∈ Hms (RN ). Then there exists a unique v 1−2s ) which solves the problem Hm1 (RN+1 + ,y 

−div(y 1−2s ∇v) + m2 y 1−2s v = 0 in RN+1 + , v(·, 0) = u on ∂RN+1 + .

∈

(4.2.4)

In addition, − lim y 1−2s y→0

∂v (x, y) = κs (− + m2 )s u(x) ∂y

in Hm−s (RN ),

where Hm−s (RN ) denotes the dual of Hms (RN ). Remark 4.2.5 As proved in [181], the extension v = Extm (u) of u ∈ Cc∞ (RN ) can be defined via the Fourier transform with respect to the variable x as  F v(ξ, y) = F u(ξ )θ (y |ξ |2 + m2 ), where θ is the Bessel function defined as in Remark 1.2.9. More precisely, v(x, y) = (Pm (·, y) ∗ u)(x) =

 RN

m (x − z, y)u(z) dz, P

112

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

 m (x, y) is the Fourier transform of ξ → θ (y |ξ |2 + m2 ), namely and P

y 2s m Pm (x, y) = CN,s

N+2s 2

(|x|2 + y 2 )−

for some constant CN,s > 0. Furthermore,

 RN

N+2s 4

 K N+2s (m |x|2 + y 2 ), 2

(4.2.5)

Pm (x, y) dx = θ (my) for all y > 0.

In the light of Theorem 4.2.4, we will look for positive solutions of the following problem: 

−div(y 1−2s ∇v) + m2 y 1−2s v = 0 in RN+1 + , ∂v 2s v − μv + |v|p−2 v] on ∂RN+1 , = κ [m s + ∂ν 1−2s

(4.2.6)

where ∂v ∂v (x, 0) = − lim y 1−2s (x, y). y→0 ∂ν 1−2s ∂y For simplicity we will assume that κs = 1. Finally, by using Theorem 4.2.1 and Theorem 1.1.11, we deduce the following result: Theorem 4.2.6 Let  m 1−2s Xrad = v ∈ Hm1 (RN+1 ) : v is radially symmetric with respect to x . + ,y m Then, Xrad is compactly embedded in Lq (RN ) for any q ∈ (2, 2∗s ).

4.2.2

Local Schauder Estimates and Maximum Principles

Here we collect some results about local Schauder estimates and maximum principles for problems involving the operator −div(y 1−2s ∇v) + m2 y 1−2s v. We begin with the following definition: Definition 4.2.7 Let R > 0 and h ∈ L1 (R0 ). We say that v ∈ Hm1 (BR+ ) is a weak solution to  −div(y 1−2s ∇v) + m2 y 1−2s v = 0 in BR+ , ∂v =h on R0 , ∂ν 1−2s

4.2 Preliminaries

113

if 

 y

BR+

1−2s

[∇v · ∇φ + m vϕ] dxdy = 2

0 R

hϕ(·, 0) dx

for every ϕ ∈ C 1 (BR+ ) such that ϕ = 0 on R+ . Next we recall some regularity results whose proofs can be found in [181]. N Proposition 4.2.8 ([181]) Let f, g ∈ Lq (10 ) for some q > 2s and η, ψ ∈ Lr (B1+ , y 1−2s ) + 1−2s N+2−2s 1 . Let v ∈ H (B1 , y ) be a weak solution to for some r > 2



−div(y 1−2s ∇v) + y 1−2s ηv = y 1−2s ψ in B1+ , ∂v = f (x)v + g(x) on 10 . ∂ν 1−2s

+ Then, v ∈ C 0,α (B1/2 ) and, in addition,

vC 0,α (B +

1/2 )



≤ C vL2 (B + ) + gLq ( 0 ) + ηLr (B + ,y 1−2s ) , 1

1

1

with C, α > 0 depending only on N, s, f Lq ( 0 ) , ηLr (B + ,y 1−2s ) . 1

1

Proposition 4.2.9 ([181]) Let f, g ∈ C k (10 ) and ∇xi η, ∇xi ψ ∈ L∞ (B1+ ), for some k ≥ 1 and i = 0, . . . , k. Let v ∈ H 1 (B1+ , y 1−2s ) be a weak solution to 

−div(y 1−2s ∇v) + y 1−2s ηv = y 1−2s ψ in B1+ , ∂v = f (x)v + g(x) on 10 . ∂ν 1−2s

Then, for i = 1, . . . , k, we have that v ∈ C i,α (Br+ ) for some r ∈ (0, 1) depending only on k, and, in addition, k 

∇xi vC 0,α (B + ) ≤ C vL2 (B + ,y 1−2s ) + f C k ( 0 ) + gC k ( 0) 1

r

i=1

+

k 

r

r

 ∇xi η, ∇xi ψL∞ (B + ) 1

,

i=1

where C, α > 0 depend only on N, s, k, r, f L∞ ( 0 ) , ηL∞ ( 0 ) . 1/2

1/2

114

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Lemma 4.2.10 ([181]) Let g ∈ C 0,γ (10 ) for some γ ∈ [0, 2 − 2s) (when γ = 0 we mean g ∈ C 0,γ = L∞ ). Let v ∈ Hm1 (B1+ , y 1−2s ) be a weak solution to 

−div(y 1−2s ∇v) + m2 y 1−2s v = 0 in B1+ , ∂v =g on 10 . ∂ν 1−2s

Then, for every sufficiently small t0 > 0 there exist positive constants C and α ≥ 0 (with α > 0 if γ > 0), depending only on N, s, t0 , m, γ , such that    1−2s ∂v  y   ∂y 

0 ×[0,t )) C 0,α (1/8 0

≤ C vL2 (B + ,y 1−2s ) + gC γ ( 0 ) . 1

1/2

We also have the following weak Harnack inequality. N Proposition 4.2.11 ([181]) Let f, g ∈ Lq (10 ) for some q > 2s and η, ψ ∈ + 1−2s + 1−2s N+2−2s r 1 L (B1 , y ) for some r > . Let v ∈ H (B , y ) be a nonnegative weak 1 2 solution to  −div(y 1−2s ∇v) + y 1−2s ηv ≥ y 1−2s ψ in B1+ , ∂v ≥ f (x)v + g(x) on 10 . ∂ν 1−2s

Then, for some p0 > 0 and any 0 < r < r < 1, inf v + g− Lq ( 0 ) + ψ− Lr (B + ,y 1−2s ) ≥ CvLp0 (B + ,y 1−2s ) , B¯ r+

1

1

r

where C > 0 depends only on N, s, r, r , f− Lq ( 0 ) , η+ Lr (B + ,y 1−2s ) . 1

1

In the spirit of [124], we prove some maximum principles: Proposition 4.2.12 (Weak Maximum Principle) Let v ∈ Hm1 (BR+ , y 1−2s ) be a weak solution to ⎧ 1−2s ∇v) + m2 y 1−2s v ≥ 0 in B + , ⎪ ⎨ −div(y R ∂v 0 ≥ 0 on  1−2s R, ⎪ ⎩ ∂ν v≥0 on R+ . Then, v ≥ 0 in BR+ . Proof It is enough to use v− as a test function in the weak formulation of the above problem.  

4.2 Preliminaries

115

Remark 4.2.13 In addition, the following strong maximum principle holds true: either v ≡ 0, or v > 0 in BR+ ∪ R0 . In fact, v cannot vanish at an interior point, due to the classical strong maximum principle for strictly elliptic operators. Finally, the fact that v cannot vanish at a point in R0 follows from the Hopf principle that we establish below. Note that the same weak and strong maximum principles (and proofs) hold in other different than BR+ . bounded domains of RN+1 + Proposition 4.2.14 (Hopf Principle) Let CR,1 = R0 × (0, 1) and v ∈ Hm1 (CR,1 , y 1−2s ) ∩ C(CR,1 ) be a weak solution to ⎧ 1−2s ∇v) + m2 y 1−2s v ≥ 0 in C ⎪ R,1 , ⎨ −div(y v>0 in CR,1 , ⎪ ⎩ v(0, 0) = 0. Then lim sup −y 1−2s y→0

v(0, y) < 0. y

In addition, if y 1−2s ∂v ∂y ∈ C(CR,1 ), then ∂v (0, 0) < 0. ∂ν 1−2s Proof We modify in a convenient way the proof of Proposition 4.11 in [124]. Consider the function wA (x, y) = y 2s−1 (y + Ay 2 )ϕ(x), where A > 0 is a constant that will be chosen later and ϕ(x) is the first eigenfunction of 0 −x + m2 in R/2 with Dirichlet boundary conditions, that is 

0 , −x ϕ + m2 ϕ = λ1 ϕ in R/2

ϕ=0

0 on ∂R/2 .

0 so that ϕL∞ ( 0 ) = 1. Then Note that λ1 > 0 and that we can choose ϕ > 0 in R/2 R/2 wA satisfies the following problem:

⎧ 1−2s ∇w ) − m2 y 1−2s w = ϕ(x)[A(1 + 2s) − λ (y + Ay 2 )] in C ⎪ A A 1 R/2,1 , ⎨ div(y wA ≥ 0 in CR/2,1 , ⎪ ⎩ 0 on ∂R/2 × [0, 1). wA = 0

116

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Choosing A big enough, we get div(y 1−2s ∇wA ) − m2 y 1−2s wA ≥ 0

in CR/2,1.

Hence, for ε > 0, div(y 1−2s ∇(v − εwA )) − m2 y 1−2s (v − εwA ) ≤ 0 in CR/2,1, 0 and v − ε wA = v ≥ 0 on ∂R/2 × [0, 1). Moreover, taking ε > 0 small enough, we see that   1 0 × y= v ≥ ε wA on R/2 , 2

since v is continuous and positive on the closure of this set. Noting that wA = 0 on R0 × {y = 0}, we have 

div(y 1−2s ∇(v − εwA )) − m2 y 1−2s (v − εwA ) ≤ 0 in CR/2,1/2, on ∂CR/2,1/2. v − εwA ≥ 0

Now using the weak maximum principle we deduce that v − εwA ≥ 0

in CR/2,1/2,

which gives lim sup −y 1−2s y→0

v(0, y) wA (0, y) ≤ ε lim sup −y 1−2s = −εϕ(0) < 0. y y y→0

Now suppose that y 1−2s ∂v ∂y ∈ C(CR,1 ). Let y0 ≤ (v − ε wA )(0, 0) = 0, we have 

∂wA ∂v −ε ∂y ∂y

1 2.

Since v − ε wA ≥ 0 in [0, y0] and

 (0, y1 ) ≥ 0

for some y1 ∈ (0, y0 ). Repeating this argument for a sequence of y0 ’s tending to 0, we see 1−2s ∂wA at a sequence of points (0, y ) → (0, 0) as j → ∞. Using that −y 1−2s ∂v j ∂y ≤ −εy ∂y 1−2s ∂wA the continuity of y 1−2s ∂v ∂y up to {y = 0} and the fact that −εyj ∂y (0, yj ) → −εϕ(0),

we conclude that

∂v (0, 0) ∂ν 1−2s

< 0.

 

4.3 A Minimization Argument via the Extension Method

117

Corollary 4.2.15 Let d be a Hölder continuous function in R0 and v ∈ L∞ (BR+ ) ∩ Hm1 (BR+ , y 1−2s ) be a weak solution to ⎧ 1−2s ∇v) + m2 y 1−2s v = 0 in B + , ⎪ ⎨ −div(y R v≥0 on BR+ , ⎪ ⎩ ∂v + d(x)v = 0 on R0 . ∂ν 1−2s Then, v > 0 in BR+ ∪ R0 , unless v ≡ 0 in BR+ . 0,α up to the Proof In view of Proposition 4.2.8 and Lemma 4.2.10, v and y 1−2s ∂v ∂y are C 0 boundary. Hence, the equation ∂ν∂v 1−2s + d(x)v = 0 is satisfied pointwise on R . If v is not + + identically 0 in BR , then v > 0 in BR by the strong maximum principle for the strictly elliptic operator operator

−div(y 1−2s ∇v) + m2 y 1−2s v. If v(x0 , 0) = 0 at some point (x0 , 0) ∈ R0 , then a rescaled version of Proposition 4.2.14   yields ∂ν∂v 1−2s (x0 , 0) < 0. This gives a contradiction.

4.3

A Minimization Argument via the Extension Method

In this subsection we prove the existence of a ground state solution to (4.2.6). Let us consider the functional Im (v) =

1 v2 1 N+1 1−2s − Hm (R+ ,y ) 2 μ + v(·, 0)2L2 (RN ) − 2

m2s v(·, 0)2L2 (RN ) 2 1 p v(·, 0)Lp (RN ) , p

1−2s ). Firstly we note that defined for any v ∈ Hm1 (RN+1 + ,y



 RN+1 +

y 1−2s (|∇v|2 + m2 v 2 ) dxdy + (μ − m2s )

RN

v 2 (x, 0) dx

1−2s ), is equivalent to the standard norm in Hm1 (RN+1 + ,y

 v2

1−2s ) Hm1 (RN+1 + ,y

=

RN+1 +

y 1−2s (|∇v|2 + m2 v 2 ) dxdy.

(4.3.1)

118

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

In fact, if μ ≥ m2s , then 

 RN+1 +

y

1−2s

(|∇v| + m v ) dxdy + (μ − m ) 2

2 2

2s

RN

v 2 (x, 0) dx ≥ v2

1−2s ) Hm1 (RN+1 + ,y

and using (4.2.2) we get 

 RN+1 +

y 1−2s (|∇v|2 + m2 v 2 ) dxdy + (μ − m2s )

  μ − m2s 1+ v2 1 N+1 1−2s . Hm (R+ ,y ) m2s

RN

v 2 (x, 0) dx ≤

Now suppose that μ < m2s . Then 

 RN+1 +

y 1−2s (|∇v|2 + m2 v 2 ) dxdy + (μ − m2s )

RN

v 2 (x, 0) dx ≤ v2

1−2s ) Hm1 (RN+1 + ,y

and by (4.2.2) it follows that 

 RN+1 +

y

1−2s

2

2 2

2s

(|∇v| + m v ) dxdy + (μ − m )

  μ − m2s ≥ 1+ v2 1 N+1 1−2s . Hm (R+ ,y ) m2s

RN

v 2 (x, 0) dx

Thus C1 (m, s, μ)v2

1−2s ) Hm1 (RN+1 + ,y

≤ v2

1−2s ) Hm1 (RN+1 + ,y

+ (μ − m2s )v(·, 0)2L2 (RN )

≤ C2 (m, s, μ)v2

1−2s ) Hm1 (RN+1 + ,y

.

(4.3.2)

Set  v2e,m =

 RN+1 +

y 1−2s (|∇v|2 + m2 v 2 ) dxdy + (μ − m2s )

RN

v 2 (x, 0) dx.

In order to prove Theorem 4.1.1, we minimize Im on the following Nehari manifold 1−2s Nm = {v ∈ Hm1 (RN+1 ) \ {0} : Jm (v) = 0}, + ,y

4.3 A Minimization Argument via the Extension Method

119

(v), v, that is, where Jm (v) = Im p

Jm (v) = v2

1−2s ) Hm1 (RN+1 + ,y

− m2s v(·, 0)2L2 (RN ) + μv(·, 0)2L2 (RN ) − v(·, 0)Lp (RN ) p

= v2e,m − v(·, 0)Lp (RN ) . Finally, we define cm = min Im (v). v∈Nm

Proof of Theorem 4.1.1 We divide the argument into several steps. Step 1

The set Nm is not empty.

1−2s ) \ {0}. Then Fix v ∈ Hm1 (RN+1 + ,y

h(t) = Im (tv) =

t2 tp p v2e,m − v(·, 0)Lp (RN ) 2 p

achieves its maximum at some τ > 0. Differentiating h with respect to t we have

(τ v), τ v = 0, and so τ v ∈ N . τ −1 Im m Step 2

Selection of an adequate minimizing sequence.

Let (vj ) ⊂ Nm be a minimizing sequence for Im and set uj = vj (·, 0). Let u˜ j be the symmetric-decreasing rearrangement of |uj |. It is well known that (see [3, 249]) for all w ∈ Lq (RN ) ˜ Lq (RN ) wLq (RN ) = w

for any q ∈ [1, ∞].

Now, we prove that for all w ∈ Hms (RN ), |w| ˜ Hms (RN ) ≤ |w|Hms (RN ) , namely  RN

s

2 |(− + m2 ) 2 w(x)| ˜ dx ≤



s

RN

|(− + m2 ) 2 w(x)|2 dx.

(4.3.3)

The case m = 0 is proved in [3] (see also [200]). When m > 0 we argue as in [64, 248, 249, 308]. Since ||w(x)| − |w(y))| ≤ |w(x) − w(y)| and by (4.2.1), we may assume that

120

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

w is non-negative. Recalling that −y 1−2s θ (y) → κs as y → 0 (see Sect. 1.2.3), we note that   2 2s 2 |(− + m ) w(x)| dx = κs (|ξ |2 + m2 )s |F w(ξ )|2 dξ κs RN

 =−

RN

lim t

1−2s ∂θ (t



RN t →0

|ξ |2 + m2 ) |F w(ξ )|2 dξ ∂t

or, equivalently,  −2s

RN

  s [θ (t |ξ |2 + m2 ) − 1] 2 lim |F w(ξ )| dξ = κ |(− + m2 ) 2 w(x)|2 dx, s N t →0 t 2s R

where θ is defined as in Remark 1.2.9. Since r ∈ R+ → θ(r)−1 is decreasing (because r 2s Ks (r) is decreasing and positive [178, 336]), we can pass to the limit as t → 0 under the integral sign to obtain that  2s lim

t →0 RN

  s [1 − θ (t |ξ |2 + m2 )] 2 |F w(ξ )| dξ = κ |(− + m2 ) 2 w(x)|2 dx. s 2s t RN

Let t > 0 and define  Ist (w)

=

RN

 [1 − θ (t |ξ |2 + m2 )] |F w(ξ )|2 dξ. t 2s

By the Parseval identity, Ist (w) =

1 t 2s



 RN

w2 (x) dx −

R2N

 w(x)Pm (x − y, t)w(y) dxdy ,

(4.3.4)

m is defined as in (4.2.5). Since the L2 (RN ) norm of w does not change under where P rearrangements and the second term on the right-hand side in (4.3.4) increases by Riesz’s ˜ ≤ Ist (w) for all t > 0. Thus (4.3.3) rearrangement inequality [249], we obtain Ist (w) follows by letting t → 0. Now, let v˜j be the unique solution to 

−div(y 1−2s ∇ v˜j ) + m2 y 1−2s v˜j = 0 in RN+1 + , v˜j (·, 0) = u˜ j on ∂RN+1 + .

(4.3.5)

We recall that v˜j H 1 (RN+1 ,y 1−2s ) = |u˜ j |Hms (RN ) , m

+

(4.3.6)

4.3 A Minimization Argument via the Extension Method

121

thanks to Theorem 4.2.1-(ii) (we recall that we are assuming κs = 1). Taking into account Theorem 4.2.1-(ii), (4.3.3) and (4.3.6), we get v˜j H 1 (RN+1 ,y 1−2s ) = |u˜ j |Hms (RN ) ≤ |uj |Hms (RN ) ≤ vj H 1 (RN+1 ,y 1−2s ) , m

+

m

+

so we deduce that Jm (v˜j ) ≤ Jm (vj ) = 0

and

Im (v˜j ) ≤ Im (vj ).

Proceeding as in the proof of Step 1, we can find tj > 0 such that Jm (tj v˜j ) = 0. Since

(v), v, we have Jm (v˜j ) ≤ 0, we see that tj ≤ 1. Further, since Jm (v) = Im     1 1 0 = Jm (tj v˜j ) = Im (tj v˜j ) + |t v˜j (x, 0)|p dx − |tj v˜j (x, 0)|p dx , p RN 2 RN and because 0 < tj ≤ 1 we obtain  Im (tj v˜j ) =  ≤  =

1 1 − 2 p 1 1 − 2 p 1 1 − 2 p

 RN



RN

|tj v˜j (x, 0)|p dx |v˜j (x, 0)|p dx



RN

|vj (x, 0)|p dx = I(vj ).

m is a minimizing sequence. Therefore, we can assume Then (wj ) = (tj v˜j ) ⊂ Nm ∩ Xrad that for all j ∈ N, wj is non-negative and radially symmetric with respect to x.

Step 3

Passage to the limit.

By (4.3.2), we get 

   1 1 1 1 2 − − C1 (m, s, μ)wj  1 N+1 1−2s ≤ wj 2e,m = Im (wj ) ≤ C. Hm (R+ ,y ) 2 p 2 p

Using Theorem 4.2.6 we may assume that 1−2s wj  w in Hm1 (RN+1 ), + ,y

wj (·, 0) → w(·, 0) in Lq (RN )

(4.3.7) ∀q ∈ (2, 2∗s ).

(4.3.8)

122

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Thus (4.3.7) and (4.3.8) yield Im (w) ≤ cm and Jm (w) ≤ 0. Now we show that w is not identically zero. Using the fact that Jm (wj ) = 0 and (4.2.3), we can see that 

 |wj (x, 0)| dx =



p

RN

+1 RN +

y

1−2s

(|∇wj | + m 2

2

wj2 ) dxdy

+ (μ − m ) 2s

RN

|wj (x, 0)|2 dx

≥ C1 (m, s, μ)wj 2

+1 a Hm1 (RN + ,y )

 ≥ C(m, s, μ, N, p)

2 RN

|wj (x, 0)|p dx

p

,

1

that is, wj (·, 0)Lp (RN ) ≥ C(m, s, μ, N, p) p−2 > 0. By (4.3.8) and p ∈ (2, 2∗s ), we deduce that w(·, 0)Lp (RN ) > 0. Then, as above, we can find τ ∈ (0, 1] such that Im (τ w) ≤ cm and Jm (τ w) = 0. Since Im (τ w) ≥ cm , we have that Im (τ w) = cm . Step 4

Conclusion.

Let v be the minimizer obtained above. Using the fact that v ∈ Nm , we have Jm (v), v

  =2



 RN+1 +

y

1−2s

(|∇v| + m v ) dxdy + (μ − m ) 2

2 2



−p

RN

2s

|v(x, 0)| dx 2

RN

|v(x, 0)|p dx



= (2 − p)

RN

|v(x, 0)|p dx = 0.

Consequently, we can find a Lagrange multiplier λ ∈ R such that

Im (v), ϕ = λJm (v), ϕ

(4.3.9)

m . Taking ϕ = v in (4.3.9) we deduce that λ = 0 and v is a nontrivial for any ϕ ∈ Xrad solution to (4.2.6). Therefore, u = v(·, 0) is a ground state solution to (4.1.1). Now Lemma 4.4.1 and Theorem 4.5.4 (see also Theorem 1.2.8) imply that u ∈ C 0,α (RN ) for   some α > 0, and applying Proposition 4.2.11 we conclude that u > 0 in RN .

Remark 4.3.1 At the end of the proof of Theorem 4.1.1, we used the fact that, if U ∈ 1−2s ) is a ground state solution to (4.2.6), then U (·, 0) is a ground state Hm1 (RN+1 + ,y solution to (4.1.1). Conversely, if u ∈ Hms (RN ) is a ground state solution to (4.1.1), then the extension Extm (u) of u is a ground state solution of (4.2.6). The proofs of these facts 1−2s ) defined are essentially based on the fact that the map T : Hms (RN ) → Hm1 (RN+1 + ,y as u → Extm (u) is an isometry (if we insert the constant √1κs in the definition of the norm 1−2s )). in Hm1 (RN+1 + ,y

4.4 Regularity, Decay and Symmetry

123

Remark 4.3.2 If we consider (− + m2 )s u = h(u) in RN , where h is a continuous ∗ function such that h(t) = o(t) as t → 0 and h(t) = o(|t|2s −1 ) as |t| → ∞, it is not hard to t check that we can solve the constrained minimization problem (here H (t) = 0 h(τ ) dτ )  inf

RN

s 2





|(− + m ) u| dx : 2

2

RN

H (u) dx = 1

in the framework of radially-symmetric functions, but getting rid of the associated Lagrange multiplier is a complicated task. This difficulty is again caused by the lack of scaling invariance for the fractional operator (− + m2 )s . Recently, in [226], a mountain pass approach was used to study scalar field equations governed by (4.1.3) and involving general subcritical nonlinearities.

4.4

Regularity, Decay and Symmetry

In this section we analyze regularity, decay and symmetry properties of solutions to (4.1.1). 1−2s ) be a weak solution to Lemma 4.4.1 Let v ∈ Hm1 (RN+1 + ,y



−div(y 1−2s ∇v) + m2 y 1−2s v = 0 in RN+1 + , ∂v 2s = m v + f (v) on ∂RN+1 + , ∂ν 1−2s

(4.4.1)

where f (v(x, 0)) = −μv(x, 0) + |v(x, 0)|p−2 v. Then, v(·, 0) ∈ Lq (RN ) for all q ∈ [2, ∞] and v ∈ L∞ (RN+1 + ). Proof We combine a Brezis–Kato argument [112] with a Moser iteration [278]. Since v is a weak solution to (4.4.1), we know that 

 RN+1 +

y

1−2s

(∇v · ∇η + m vη) dxdy = 2

RN

[m2s v(x, 0) + f (v(x, 0))]η(x, 0) dx (4.4.2) 2β

N+1 1−2s 1−2s ). For K > 1 and β > 0, let w = vv 1 for all η ∈ Hm1 (RN+1 ), + ,y K ∈ Hm (R+ , y where vK = min{|v|, K}. Taking η = w in (4.4.2), we deduce that





2β

N+1 R+

y 1−2s vK (|∇v|2 + m2 v 2 ) dxdy +  = m2s

2β

RN

v 2 (x, 0)vK (x, 0) dx +



2β

DK

2βy 1−2s vK |∇v|2 dxdy 2β

RN

f (v(x, 0))v(x, 0)vK (x, 0) dx (4.4.3)

124

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

where DK = {(x, y) ∈ RN+1 : |v(x, y)| ≤ K}. It is easy to check that + 



β

RN+1 +

y 1−2s |∇(vvK )|2 dxdy =

2β

RN+1 +

y 1−2s vK |∇v|2 dxdy



2β

+ DK

(2β + β 2 )y 1−2s vK |∇v|2 dxdy.

(4.4.4)

Then, combining (4.4.3) and (4.4.4), we get 

β

vvK 2

1−2s ) Hm1 (RN+1 + ,y

 =

RN+1 +

= cβ

y 1−2s [|∇(vvK )|2 + m2 v 2 vK ] dxdy 

+ m v ] dxdy + 2 2

2β

RN+1 +



2β

β

RN+1 +

2β y 1−2s vK [|∇v|2

#  ≤ cβ

=

  β 2β y 1−2s vK |∇v|2 dxdy 2β 1 + 2 DK $ 

y 1−2s vK [|∇v|2 + m2 v 2 ] dxdy +

2β

DK

2β

RN

2βy 1−2s vK |∇v|2 dxdy

2β

m2s v 2 (x, 0)vK (x, 0) + f (v(x, 0))v(x, 0)vK (x, 0) dx,

(4.4.5)

where cβ = 1 +

β . 2

Now, we prove that there exist a constant c > 0 and a function h ∈ LN/2s (RN ), h ≥ 0 and independent of K and β, such that 2β

2β

2β

m2s v 2 (·, 0)vK (·, 0) + f (v(·, 0))v(·, 0)vK (·, 0) ≤ (c + h)v 2 (·, 0)vK (·, 0)

on RN . (4.4.6)

First, we note that 2β

2β

m2s v 2 (·, 0)vK (·, 0) + f (v(·, 0))v(·, 0)vK (·, 0) 2β

2β

≤ (m2s + C)v 2 (·, 0)vK (·, 0) + C|v(·, 0)|p−2 v 2 (·, 0)vK (·, 0) Moreover, |v(·, 0)|p−2 ≤ 1 + h

on RN ,

on RN .

4.4 Regularity, Decay and Symmetry

125

for some h ∈ LN/2s (RN ). In fact, we can observe that |v(·, 0)|p−2 = χ{|v(·,0)|≤1}|v(·, 0)|p−2 + χ{|v(·,0)|>1}|v(·, 0)|p−2 ≤ 1 + χ{|v(·,0)|>1}|v(·, 0)|p−2

on RN ,

N and that if (p − 2) 2s < 2, then





N

RN

χ{|v(·,0)|>1}|v(x, 0)| 2s (p−2) dx ≤

RN

χ{|v(·,0)|>1}|v(x, 0)|2 dx < ∞,

N N while if 2 ≤ (p − 2) 2s we deduce that (p − 2) 2s ∈ [2, 2∗s ]. Taking into account (4.4.5) and (4.4.6) we obtain that β vvK 2 1 N+1 1−2s H (R ,y ) +

m

 ≤ cβ

2β

RN

(c + h(x))v 2 (x, 0)vK (x, 0) dx,

and by the monotone convergence theorem (vK is nondecreasing with respect to K) we have, as K → ∞,  |v|β+1 2 1 N+1 1−2s Hm (R+ ,y )

≤ ccβ

 RN

|v(x, 0)|

2(β+1)

dx + cβ

RN

h(x)|v(x, 0)|2(β+1) dx. (4.4.7)

Fix M > 0 and let A1 = {h ≤ M} and A2 = {h > M}. Then,  RN

h(x)|v(x, 0)|2(β+1) dx ≤ M|v(·, 0)|β+1 2L2 (RN ) + ε(M)|v(·, 0)|β+1 2 2∗s

L (RN )

,

(4.4.8)  2s

 where ε(M) =

N/2s

h

N

dx

→ 0 as M → ∞. In view of (4.4.7) and (4.4.8),

A2

|v|β+1 2

+1 1−2s ) Hm1 (RN + ,y

≤ cβ (c + M)|v(·, 0)|β+1 2L2 (RN ) + cβ ε(M)|v(·, 0)|β+1 2 2∗s

L (RN )

.

(4.4.9) By (4.2.3), it follows that |v(·, 0)|β+1 2 2∗s

L (RN )

≤ C22∗s |v|β+1 2

1−2s ) Hm1 (RN+1 + ,y

.

(4.4.10)

126

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Then, choosing M large so that ε(M)cβ C22∗s
0  RN

h(x)|v(x, 0)|2(β+1) dx ≤ h

L

≤ h

N Ls

N s

(RN )

|v(·, 0)|β+1 L2 (RN ) |v(·, 0)|β+1 L2∗s (RN )

  1 β+1 2 β+1 2 |v(·, 0)|  +  λ|v(·, 0)| . ∗ L2 (RN ) L2s (RN ) (RN ) λ

Then, using (4.4.7) and (4.4.10), we deduce that |v(·, 0)|β+1 2 2∗s

L (RN )

≤ cβ C22∗s (c + h

N Ls

≤ C22∗s |v|β+1 2

+1 1−2s ) Hm1 (RN + ,y

(RN )

λ)|v(·, 0)|β+1 2L2 (RN ) + C22∗s

cβ h

L

λ

N s

(RN )

|v(·, 0)|β+1 2 2∗s

L (RN )

.

(4.4.12) Taking λ > 0 such that cβ h

L

N s

(RN )

λ

C22∗s =

1 , 2

we obtain that |v(·, 0)|β+1 2 2∗s

L (RN )

≤ 2cβ (c + h

L

N s

(RN )

λ)C22∗s |v(·, 0)|β+1 2L2 (RN )

= Mβ |v(·, 0)|β+1 2L2 (RN ) .

4.4 Regularity, Decay and Symmetry

127

Now we can control the dependence on β of Mβ as follows: Mβ ≤ Ccβ2 ≤ C(1 + β)2 ≤ M02 e2

√ β+1

,

which implies that 1

v(·, 0)L2∗s (β+1) (RN ) ≤ M0β+1 e

√1 β+1

v(·, 0)L2(β+1) (RN ) .

Iterating this last inequality and choosing β0 = 0 and 2(βn+1 + 1) = 2∗s (βn + 1), we deduce that n

v(·, 0)L2∗s (βn +1) (RN ) ≤ M0

1 i=0 βi +1

n

e

√1

i=0

βi +1

v(·, 0)L2(β0 +1) (RN ) .

N Since 1 + βn = ( N−2s )n , the series ∞  i=0

1 βi + 1

∞ 

and

1 √ βi + 1

i=0

converge and we get v(·, 0)L∞ (RN ) = lim v(·, 0)L2∗s (βn +1) (RN ) < ∞. n→∞

Now, using (4.4.12) with λ = 1 and the fact that v(·, 0)Lq (RN ) ≤ C˜ for all q ∈ [2, ∞], we have for all β > 0 |v|β+1 2

1−2s ) Hm1 (RN+1 + ,y

≤ cβ (c + h

L

N s

(RN )

)C˜ 2(β+1) + cβ h

L

N s

(RN )

C˜ 2(β+1) .

Since Lemma 1.3.9 yields 

  +1 RN +

y

1−2s

|v(x, y)|

2γ (β+1)

 

β+1 2γ (β+1)

=

dxdy

1

2γ

+1 RN +

y

1−2s

||v(x, y)|

|

β+1 2γ

dxdy

≤ C |v|β+1 H 1 (RN +1 ,y 1−2s ) m

where γ = 1 +

2 N−2s ,

+

for some constant C > 0 independent of β, it follows that

 

 2(β+1)

2γ (β+1)

RN+1 +

y

1−2s

|v(x, y)|

2γ (β+1)

dxdy

≤ C

cβ C˜ 2(β+1)

128

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

for some constant C

> 0 independent of β. Consequently, 

  RN+1 +

y

1−2s

|v(x, y)|

2γ (β+1)

1 2γ (β+1)

dxdy

1

˜ ≤ (C

cβ ) 2(β+1) C.

1

Note that (C

cβ ) 2(β+1) C˜ ≤ C¯ for all β > 0, since   β 1 1 log 1 + ≤ β +1 2 2

for all β > 0.

Fix k > C¯ and define Ak = {(x, y) ∈ RN+1 : |v(x, y)| > k}. Then we obtain that + C¯ ≥



  RN+1 +

≥k

γ (β+1)−1 γ (β+1)

y

1−2s

|v(x, y)|

2γ (β+1)

1 2γ (β+1)

dxdy

 

 y 1−2s |v(x, y)|2 dxdy

1 2γ (β+1)

,

Ak

whence 1  ¯ β+1− γ1   2γ C 1 1−2s 2 ≥ 1 y |v(x, y)| dxdy . k Ak C¯ γ Letting β → ∞ we have  y 1−2s |v(x, y)|2 dxdy = 0 $⇒ |Ak | = 0, Ak

i.e., v ∈ L∞ (RN+1 + ).

 

In order to study the Hölder regularity of solutions of (4.1.1), we introduce the Hölder/ N we set Zygmund (or Lipschitz) spaces α ; see [128, 241, 315, 326]. If α > 0 and α ∈ α = C [α],α−[α] (RN ). If α = k ∈ N we set α = ∗k , where 

∗1

|u(x + h) + u(x − h) − 2u(x)| 0 ∞

N

N



4.4 Regularity, Decay and Symmetry

129

if k = 1, and  ∗k = u ∈ C k−1 (RN ) : D γ u ∈ ∗1 for all |γ | ≤ k − 1 , if k ∈ N, k ≥ 2. Note that C 0,1 (RN ) = Λ∗1 and Λβ ⊂ Λα if 0 < α < β. Moreover, we have the following useful result: Theorem 4.4.2 ([128,315,323]) If α, β ≥ 0, then (1 − )−α is an isomorphism from Lβ p to Lβ+2α . If α ≥ 0 and β > 0, then (1 − )−α is an isomorphism from β to β+2α .

p

Next we prove a stronger regularity result for positive solutions of (4.1.1). Theorem 4.4.3 Let u ∈ H s (RN ) be a positive solution to (4.1.1). Then, u ∈ C 1,α (RN ) and u(x) → 0 as |x| → ∞. Proof Let g(u) = (m2s − μ)u + up−1 . By Lemma 4.4.1, g(u) ∈ Lr (RN ) for any r ∈ [2, ∞]. Then q

u = G2s,m ∗ g(u) ∈ L2s,m (RN ) = {G2s,m ∗ h|h ∈ Lq (RN )} for any q ∈ [2, ∞), where G2s,m is the Bessel kernel given by F G2s,m (ξ ) = (2π)− 2 (|ξ |2 + m2 )−s . N

Theorem 4.5.4 implies that u ∈ C 0,β (RN ) for all β ∈ (0, 2s) if 2s ≤ 1, and u ∈ C 1,β (RN ) for all β ∈ (0, 2s − 1) if 2s > 1. In particular, u ∈ C 0,β (RN ) ∩ L2 (RN ) implies that |u(x)| → 0 as |x| → ∞. Now, in view of Theorem 4.4.2 (see also Theorem 4.5.3), we can repeat the argument used in Theorem 3.2.11 for studying the regularity of solutions in the   case of the operator (−)s to obtain the desired result. Remark 4.4.4 We can also deduce the Hölder regularity of u arguing as follows. Let U (x, y) = (Pm (·, y) ∗ u)(x). Fix x ∈ RN . By Remark 4.2.5, the fact that θ (y) = 2 y s −s as y → 0, and Young’s inequality, we get (s) ( 2 ) Ks (y) ∼ 2 U L2 (B1 (x)×[0,1]) ≤ U L2 (RN ×[0,1]) = Pm (·, y) ∗ uL2 (RN ×[0,1]) 

1

≤ 0



1

≤ 0

Pm (·, y) ∗ u2L2 (RN ) dy

 12

Pm (·, y)2L1 (RN ) u2L2 (RN ) dy

 12

130

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

 = uL2 (RN )

1

2

 12

θ (my) dy 0

≤ Cm,s uL2 (RN ) . Then, using that g(u) ∈ Lr (RN ) for any r ∈ [2, ∞], Proposition 4.2.8 and the translation invariance of (4.1.1) with respect to x ∈ RN , we deduce that u ∈ C 0,α (RN ) for some α > 0. The Hölder regularity of ∇u follows from Proposition 4.2.9. Theorem 4.4.5 Let v be a positive solution to (4.2.6). Then, v ∈ C 0,α (RN+1 + ) and satisfies for all y ≥ 1 1

sup |v(x, y)| ≤ Cv(·, 0)L2 (RN ) y s− 2 e−my .

x∈RN

In particular, for any λ ∈ (0, m), |v(x, y)|eλy → 0 as y → ∞. Proof We argue as in [117, 152]. By Lemma 4.4.1 and Proposition 4.2.8, we deduce that v ∈ C 0,α (RN+1 + ); see also [180]. To prove the decay at infinity, note that v is a solution to (4.2.4), and then use the Fourier transform with respect to the variable x ∈ RN (see Remark 4.2.5) to obtain that  F v(ξ, y) = θ (y |ξ |2 + m2 )F u(ξ ), where u(x) = v(x, 0) ∈ L2 (RN ) and θ (y) ∈ H 1 (R+ , y 1−2s ) is defined as in Remark 1.2.9. Then,

m (·, y) ∗ u (x) = v(x, y) = P

 RN

Pm (x − z, y)u(z) dz

m (x, y) is given in (4.2.5). Therefore, by Parseval’s identity, where P  |v(x, y)| ≤ uL2 (RN )

RN

1  2 2 2 2 |θ (y |ξ | + m )| dξ .

Since θ (y) is continuous, for y ≥ 1 we have   √ 2s−1 2 2 |θ (y |ξ |2 + m2 )| ≤ C1 (y |ξ |2 + m2 ) 2 e−y |ξ | +m ,

4.4 Regularity, Decay and Symmetry

131

which gives  |v(x, y)| ≤ uL2 (RN ) C1

RN

1  √ 2 2s−1 −2y |ξ |2 +m2 2 2 (y |ξ | + m ) e dξ .

(4.4.13)

Now let us estimate  the integral in the right-hand side of (4.4.13). We first assume that s ∈ (0, 12 ]. Since y |ξ |2 + m2 ≥ my and 2s − 1 ≤ 0, we get  RN

  √ 2 2 (y |ξ |2 + m2 )2s−1 e−2y |ξ | +m dξ ≤ (my)2s−1

RN

e−2y

√

|ξ |2 +m2

dξ.

(4.4.14)

2 Next we study the integral on the right-hand side of (4.4.14). Take R > 0 such that |ξ | ≥ c c 2 3m for all ξ ∈ BR . Hence, y |ξ |2 + m2 ≥ 2my for all ξ ∈ BR . On the other hand,  y |ξ |2 + m2 ≥ |ξ |y. Combining these two estimates, we find

 −2y |ξ |2 + m2 ≤ −(2my + |ξ |y)

for all ξ ∈ BRc .

Consequently, for all y ≥ 1  RN

e−2y

√

|ξ |2 +m2



e−2y

dk = 

|ξ |2 +m2

BR

e−2my dξ +

≤



√

dξ +



BR

BRc

≤ e−2my |BR | + e−2my

e−2y

√



e−|ξ | dξ

BRc

(4.4.15)

Combining (4.4.13), (4.4.14) and (4.4.15), we see that

1 2 |v(x, y)| ≤ C1 uL2 (RN ) (my)2s−1C2 e−2my 1

≤ C3 uL2 (RN ) y s− 2 e−my for all y ≥ 1. Now, let us treat the case s ∈ ( 12 , 1). Note that

RN

  2s−1 √ 2 2 e−2y |ξ | +m dξ y |ξ |2 + m2

= y 2s−1



 RN

2s−1 |ξ |2 + m2

e−2y

dξ

e−(2my+|ξ |y) dξ

≤ C2 e−2my .



|ξ |2 +m2

BRc

√

|ξ |2 +m2

dξ

132

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

 ≤y

2s−1 RN

 = y 2s−1

RN

√

2 2 |ξ |2s−1 + m2s−1 e−2y |ξ | +m dξ |ξ |2s−1e−2y

√

|ξ |2 +m2

 dξ + (my)2s−1

e−2y

RN

√

|ξ |2 +m2

dξ.

As before, choosing R > 0 such that |ξ |2 ≥ 3m2 for all ξ ∈ BRc , we have  −2y |ξ |2 + m2 ≤ −(2my + |ξ |y)

for all ξ ∈ BRc .

Hence, for y ≥ 1, it holds that  RN

|ξ |2s−1 e−2y

√

|ξ |2 +m2



|ξ |2s−1 e−2y

dξ =

√

|ξ |2 +m2

 dξ +

BR

≤ e−2my R 2s−1 |BR | + e−2my

 RN

BRc

|ξ |2s−1 e−2y

√

|ξ |2 +m2

dξ

|ξ |2s−1 e−|ξ | dξ

≤ C3 e−2my .

This combined with (4.4.15) yields the required estimate in the case s ∈ ( 12 , 1).

 

Remark 4.4.6 If u is a positive solution to (4.1.1), then u has an exponential decay at infinity. Let m = 1 and fix δ ∈ (0, 1). Let h ∈ L2 (RN ) be such that h ≥ 0 in RN , h ≡ 0 and h has compact support. For instance, assume that supp(h) = Br for some r > 0. Let v ∈ H s (RN ) be the unique solution to (− + 1)s v − (1 − δ)v = h in RN . Then v = G2s ∗((1−δ)+h), v ∈ C 0,α (RN ) for some α > 0, and v > 0 in RN (remark that G2s > 0). By [63, 64], we know that the kernel B2s (x) = F −1 ([(|ξ |2 + 1)s − (1 − δ)]−1 ) ≤ Ce−c|x| for all |x| ≥ 2. Set R = max{r, 2}. Then, for all |x| ≥ 2R, we get 

 v(x) = Br

B2s (x − y)h(y) dy ≤ ChL∞ (RN )

where we used 2R ≥

R R−1 ,

|x|

e−c|x−y| dy ≤ C e−c R , Br

since R ≥ 2, and then

|x − y| ≥ |x| − |y| ≥ |x| − 1 ≥

|x| ≥2 R

for all |x| ≥ 2R.

From the continuity of v we deduce that 0 < v(x) ≤ C1 e−c2 |x| for all x ∈ RN . Next we use a comparison argument to deduce the desired estimate. Since u ∈ 0,α C (RN ) ∩ Lq (RN ) for all q ∈ [2, ∞], we infer that u(x) → 0 as |x| → ∞ and that the function h(x) = (g(u(x)) − (1 − δ0 )u(x))+ > 0 has compact support (note that limt →0 g(t)/t = 1 − μ < 1), for some δ0 ∈ (0, 1). Hence, (− + 1)s u − (1 − δ0 )u = g(u) − (1 − δ0 )u ≤ h = (− + 1)s v − (1 − δ0 )v

in RN ,

4.4 Regularity, Decay and Symmetry

133

and this implies that (− + 1)s (u − v) − (1 − δ0 )(u − v) ≤ 0 in RN . Taking (u − v)+ as a test function, and recalling the definition (4.1.3) and the elementary inequality (x − y)(x + − y + ) ≥ |x + − y + |2

for all x, y ∈ R,

we obtain u ≤ v in RN , that is, u(x) ≤ Ce−c|x| for all x ∈ RN . Finally we use the method of moving planes to deduce the symmetry of positive solutions to (4.1.1). Theorem 4.4.7 Every positive solution u ∈ H s (RN ) of (4.1.1) is radially symmetric and radially decreasing with respect to a point of RN . Proof Let v be the unique solution of (4.2.4) with boundary datum u. Let λ > 0 and consider the sets λ = {(x1 , . . . , xN , y) : x1 > λ, y ≥ 0} and Tλ = {(x1 , . . . , xN , y) : x1 = λ, y ≥ 0}. Let vλ (x, y) = v(2λ − x1 , . . . , xN , y) and wλ = vλ − v. Then wλ satisfies 

−div(y 1−2s ∇wλ ) + m2 y 1−2s wλ = 0 in RN+1 + , ∂wλ 2s = (Cλ (x) + m − μ)wλ on ∂RN+1 + , ∂ν 1−2s

(4.4.16)

where  Cλ (x) =

p−1

vλ

(x,0)−v p−1(x,0) vλ (x,0)−v(x,0) ,

0,

if vλ (x, 0) = v(x, 0), if vλ (x, 0) = v(x, 0).

Let wλ− = min{0, wλ }. Note that as λ → ∞, Cλ (x) → 0 uniformly in x ∈ {x ∈ : x1 > λ}, so that wλ− (x, 0) = 0 because lim|x|→∞ v(x, 0) = 0 and 0 ≤ vλ (x, 0) < v(x, 0) whenever wλ− (x, 0) = 0. Multiplying the weak formulation of (4.4.16) by wλ− and applying (4.2.2), we get RN

 y

1−2s

λ

 ≤

{x1 >λ}

[|∇wλ− |2

+m

2

(wλ− )2 ] dxdy

Cλ (x)(wλ− (x, 0))2 dx

 =

{x1 >λ}

[Cλ (x) + m2s − μ](wλ− (x, 0))2 dx



+ A(m, μ, s) λ

y 1−2s [|∇wλ− |2 + m2 (wλ− )2 ] dxdy

134

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

where  A(m, μ, s) =

1 − mμ2s , if m2s − μ > 0, 0, if m2s − μ ≤ 0.

Choosing λ > 0 sufficiently large, we deduce that wλ− ≡ 0 on λ and consequently wλ (x, y) ≥ 0

on λ

for such λ > 0. Now, we define ν = inf{τ > 0 : wλ ≥ 0 on λ for every λ ≥ τ }. We distinguish two cases. First, we assume that ν > 0 and prove that wν ≡ 0. We argue by contradiction. By continuity, wν ≥ 0 on ν , and by the strong maximum principle, wν > 0 on the set ν = {(x1 , . . . , xN , y) : x1 > ν, y > 0}. We also have wν (x, 0) ≥ 0 on the set {x ∈ RN : x1 ≥ ν} by continuity. Furthermore, by the Hopf principle, wν (x, 0) > 0 on the set {x ∈ RN : x1 > ν}. Indeed, if, by ¯ 0) = 0, then by the Hopf contradiction, there exists x¯ ∈ RN such that x¯1 > ν and wν (x, ν ( x, ¯ 0) < 0, which is in contrast with principle we have ∂ν∂w 1−2s ∂wν (x, ¯ 0) = (Cλ (x) + m2s − μ)wν (x, ¯ 0) = 0. ∂ν 1−2s Take λj < ν such that λj → ν as j → ∞. Let r0 > 0 be such that |Cν (x)| ≤ for every |x| > r0 . Note that, since vλj (·, 0)C 1 (RN ) is uniformly bounded, D = Cλj L∞ (RN ) < ∞ and |Cλj (x)| ≤ μ2 for every |x| > r0 and j ∈ N. Denote μ 4

Br0 (pj ) = {x ∈ RN : |x − pj | < r0 } ⊂ RN , where pj = (λj , 0, . . . , 0). As above, we obtain  λj

y 1−2s [|∇wλ−j |2 + m2 (wλ−j )2 ] dxdy ≤

≤ (D + m2s + μ)

 {x1 >λj }∩Br0 (pj )

 {x1 >λj }

(wλ−j (x, 0))2 dx+

(Cλj + m2s − μ)(wλ−j (x, 0))2 dx

4.4 Regularity, Decay and Symmetry

μ (m − ) 2



2s

{x1 >λj }\Br0 (pj )

135

(wλ−j (x, 0))2 dx



2s

≤ (D + m + μ)  B(m, μ, s)

{x1 >λj }∩Br0 (pj )

{x1 >λj }\Br0 (pj )

(wλ−j (x, 0))2 dx+

(wλ−j (x, 0))2 dx,

where  B(m, μ, s) =

m2s − μ2 , if m2s − 0, if m2s −

μ 2 μ 2

> 0, ≤ 0.

Since wν (x, 0) > 0 on the set {x ∈ RN : x1 > ν}, the measure of the support Ej of wλ−j (·, 0) on Br0 (pj ) goes to 0 as j → ∞. Then, using the Hölder and Sobolev inequalities, we see that  {x1 >λj }∩Br0 (pj )

(wλ−j (x, 0))2 dx

 =

{x1 >λj }

χEj (x)(wλ−j (x, 0))2 dx

≤ χEj LN/2s ({x1 >λj }) wλ−j (·, 0)2L2N/N−2s ({x >λ }) 1 j  ≤ o(1) y 1−2s |∇wλ−j |2 dxdy. λj

Therefore, if j is large, we have wλj ≥ 0 on λj . This gives a contradiction because of the minimality of ν. Hence, we conclude that wν ≡ 0 on ν and get the symmetry in the x1 -direction. Now assume ν = 0. We repeat the above argument for λ < 0 and wλ = vλ − v defined on

λ = {(x1 , . . . , xN , y) : x1 < λ, y ≥ 0}. Then wλ ≥ 0 for |λ| sufficiently large. Let ν = sup{τ < 0 : wλ ≥ 0 on λ for every λ ≤ τ }. If ν < 0, we get the symmetry as above. If ν = 0, then since ν = 0, we have v(−x1 , x2 , . . . , xN , y) ≥ v(x1 , x2 , . . . , xN , y)

on RN+1 + .

136

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Consequently, by replacing x1 with −x1 , we deduce that v(−x1 , x2 , . . . , xN , y) = v(x1 , x2 , . . . , xN , y)

on RN+1 + .  

Using the same approach in any arbitrary direction xi , we obtain the thesis.

We conclude this subsection by observing that every solution v of (4.4.1) satisfies the following Pohozaev-type identity: N − 2s 2

 RN+1 +

y 1−2s |∇v|2 dxdy + m2  = Nκs

RN

N + 2 − 2s 2

 RN+1 +

y 1−2s v 2 dxdy

G(v(x, 0)) dx,

(4.4.17)

where G(v(x, 0)) = m2s

v 2 (x, 0) + F (v(x, 0)). 2

Indeed, arguing as in the proof of Theorem 3.5.1, we have the additional term  A4 (R, δ) = −m2

+ DR,δ

  2  v dxdy y 1−2s (x, y) · ∇ + 2 DR,δ

 y 1−2s v ((x, y) · ∇v) dxdy = −m2

which in view of the divergence theorem can be written as   2   N + 2 − 2s v y 1−2s (x, y) · ∇ y 1−2s v 2 dxdy dxdy = m2 + + 2 2 DR,δ DR,δ   2 2 v v + m2 y 1−2s y dσ − m2 y 1−2s R dσ. 1 2 2 2 ∂DR,δ ∂DR,δ

 −m2

Then one can verify that lim lim A4 (R, δ) = m

n→∞ δ→0

2N

+ 2 − 2s 2

 RN+1 +

y 1−2s v 2 dxdy

and this proves our claim. On the other hand, as in [64,116,181,184,226,308], we can obtain a Pohozaev identity  for (− + m2 )s . Indeed, noting that F v(ξ, y) = F u(ξ )θ (y |ξ |2 + m2 ), where u =

4.4 Regularity, Decay and Symmetry

137

v(·, 0), we see that 

 RN+1 +

y 1−2s v 2 dxdy =

  RN

∞

|F u(ξ )|2 (|ξ |2 + m2 )s−1 dξ

 y 1−2s θ 2 (y) dy .

0

Recalling that θ is a solution to (1.2.10) and the asymptotic estimates (1.2.4)– (1.2.5), (1.2.11), an integration by parts yields (see [116]) 

y 0

∞



y 2−2s θ (y) dy = θ 2 (y) dy 2 − 2s 0  ∞ 1 θ (y)θ (y)y 2−2s dy =− 1−s 0    ∞ 1 1 − 2s

=− θ (y) + θ

(y) y 2−2s dy θ (y) 1−s 0 y  ∞  ∞ 1 1 − 2s y 1−2s (θ (y))2 dy − y 2−2s θ (y)θ

(y) dy =− 1−s 0 1−s 0  ∞  1 − 2s ∞ 1−2s

=− y (θ (y))2 dy + y 1−2s (θ (y))2 dy 1−s 0 0  ∞ s = y 1−2s (θ (y))2 dy 1−s 0  ∞ s s κs − = y 1−2s θ 2 (y) dy, 1−s 1−s 0 

∞

1−2s 2

that is, 

∞

y 1−2s θ 2 (y) dy = sκs .

0

Consequently, 

 RN+1 +

y 1−2s v 2 dxdy = sκs

RN

|F u(ξ )|2 (|ξ |2 + m2 )s−1 dξ.

Hence, by using Theorem 4.2.1 and (4.4.17), we discover that N − 2s 2



 |F u(ξ )| (|ξ | + m ) dξ + sm 2

RN

=N



RN

G(u) dx.

2

2 s

2 RN

|F u(ξ )|2 (|ξ |2 + m2 )s−1 dξ

138

4.5

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Passage to the Limit as m → 0

In this section we show that it is possible to pass to the limit in problem (4.2.6) and to find a nontrivial ground state to (4.1.4). To this end, we estimate cm from above and below uniformly in m > 0, provided that m is sufficiently small. Fix 0 < m < ( μ2 )1/2s . Using the characterization of the infimum cm on Nm we can verify that cm = inf Im (v) = v∈Nm

inf

max Im (tv).

v∈Xm,rad \{0} t >0

Let us prove that there exist λ > 0 and δ > 0 independent of m such that λ ≤ cm ≤ δ.

(4.5.1)

Let w(x, y) = v0 (x)

1 , y+1

where v0 is defined by setting ⎧ ⎪ if |x| ≤ 1, ⎨ 1, v0 (x) = 2 − |x|, if 1 ≤ |x| ≤ 2, ⎪ ⎩ 0, if |x| ≥ 2.

(4.5.2)

1−2s ) and Then w ∈ Hm1 (RN+1 + ,y

 w2 1 N+1 1−2s ) Hm (R+ ,y

∞

=

y 0

 +  ≤A

1−2s

RN

 

 RN

|∇x v0 |

2

+ m2 v02 dx

 1 y dy v02 dx (y + 1)4 RN   μ 1/s |∇x v0 |2 + v02 dx + B v02 dx = C. N 2 R

∞

0

1 dy (y + 1)2

1−2s

4.5 Passage to the Limit as m → 0

139

Therefore,  sup Im (tw) = t >0

1 1 − 2 p



p p−2 we,m 2

wLp−2 p (RN ) 

 ≤

 ≤

1 1 − 2 p



 w2 1 N+1 1−2s Hm (R+ ,y )

+ μv0 2L2 (RN )

p p−2

2

v0 Lp−2 p (RN )

& p % p−2  C + μv0 2 2 N 1 L (R )

1 − 2 p

2 p−2 p L (RN )

= δ.

v0 

Since Im (vm ) = cm and Jm (vm ) = 0,  cm = Im (vm ) =

1 1 − 2 p

 RN

|vm (x, 0)|p dx.

Then, to deduce a lower bound for cm , it is enough to estimate the Lp (RN ) norm of vm (·, 0). Since Jm (vm ) = 0, it follows from Remark 4.2.3 that 

p

vm (·, 0)Lp (RN ) =  ≥ ≥ κs

+1 RN +

+1 RN +

 2 y 1−2s (|∇vm |2 + m2 vm ) dxdy + (μ − m2s )

y 1−2s |∇vm |2 dxdy +

RN

|vm (x, 0)|2 dx

μ vm (·, 0)2L2 (RN ) 2

C(N, s) μ [vm (·, 0)]2s + vm (·, 0)2L2 (RN ) 2 2

≥ Cs,μ |vm (·, 0)|2H s (RN ) ≥ Cs,μ,p vm (·, 0)2Lp (RN ) ,

that is, vm (·, 0)Lp (RN ) ≥ (Cs,μ,p )

1 p−2

 and

cm ≥

 p 1 1 − (Cs,μ,p ) p−2 = λ. 2 p

(4.5.3)

Now, using (4.5.1), we are able to prove the following result: 1 (RN+1 , y 1−2s ) such that, as m → 0, v  v in Theorem 4.5.1 There exists v ∈ Hloc m + 2 N 1−2s 1−2s ) and v (·, 0) → L (R × [0, ε], y ) for any ε > 0, ∇vm  ∇v in L2 (RN+1 , y m + v(·, 0) in Lq (RN ) for any q ∈ (2, 2∗s ). In particular v(·, 0) is a nontrivial weak solution to (4.1.4).

140

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Proof Taking into account that J (vm ) = 0, cm ≤ δ, 0 < m < ( μ2 )1/2s and using Remark 4.2.3, we can see that  δ 1/p

1 1 − 2 p

−1/p

 1/p

≥ cm

 =  ≥ ≥ κs

1 1 − 2 p

+1 RN +

+1 RN +

−1/p

p

= vm (·, 0)Lp (RN ) 

2 y 1−2s (|∇vm |2 + m2 vm ) dxdy + (μ − m2s )

y 1−2s |∇vm |2 dxdy +

RN

|vm (x, 0)|2 dx

μ vm (·, 0)2L2 (RN ) 2

μ C(N, s) [vm (·, 0)]2s + vm (·, 0)2L2 (RN ) 2 2

≥ Cs,μ |vm (·, 0)|2H s (RN ) ,

that is,  RN+1 +

y 1−2s |∇vm |2 dxdy ≤ C1

and |vm (·, 0)|2H s (RN ) ≤ C2 . N Now, fix ε > 0 and let v ∈ Cc∞ (RN+1 + ). For any x ∈ R and y ∈ [0, ε], we have



y

v(x, y) = v(x, 0) + 0

∂v (x, t) dt. ∂y

Since (a + b)2 ≤ 2a 2 + 2b 2 for all a, b ≥ 0, we obtain 

y

|v(x, y)|2 ≤ 2|v(x, 0)|2 + 2 0

 2    ∂v  (x, t) dt ,   ∂y

and applying the Hölder inequality we deduce that #



y

|v(x, y)| ≤ 2 |v(x, 0)| + 2

2

t 0

1−2s

 2  2s $  ∂v   (x, t) dt y .  ∂y  2s

4.5 Passage to the Limit as m → 0

141

Multiplying both members by y 1−2s , we get #



y

y 1−2s |v(x, y)|2 ≤ 2 y 1−2s |v(x, 0)|2 + 0

 2  $   y ∂v . t 1−2s  (x, t) dt ∂y 2s

(4.5.4)

Integration of (4.5.4) over RN × [0, ε] yields v2L2 (RN ×[0,ε],y 1−2s )

 2  ε2−2s ε2  2  ∂v  v(·, 0)L2 (RN ) + ≤ . 1−s 2s  ∂y L2 (RN+1 1−2s ) + ,y

(4.5.5)

1−2s ). Then, replacing v by v , we can By density, (4.5.5) holds for all v ∈ Hm1 (RN+1 m + ,y infer that

vm 2L2 (RN ×[0,ε],y 1−2s ) ≤ C(ε, s)K(δ, p)2 1 (RN+1 , y 1−2s ) such that for any 0 < m < ( μ2 )1/2s . Hence, there exists v ∈ Hloc +

vm  v

in L2 (RN × [0, ε], y 1−2s ) for all ε > 0,

∇vm  ∇v

1−2s in L2 (RN+1 ), + ,y

vm (·, 0) → v(·, 0)

in Lq (RN )

∀q ∈ (2, 2∗s ),

(4.5.6) (4.5.7) (4.5.8)

as m → 0. Finally, we prove that v(·, 0) is a nontrivial weak solution to (4.1.4). We ∞ proceed as in [31]. Fix η ∈ Cc∞ (RN+1 + ) and let ψ ∈ C ([0, ∞)) be defined by ⎧ ⎪ if 0 ≤ y ≤ 1, ⎨ ψ = 1, 0 ≤ ψ ≤ 1, if 1 ≤ y ≤ 2, ⎪ ⎩ ψ = 0, if y ≥ 2.

(4.5.9)

1−2s ). Then taking ηψ in Let ψR (y) = ψ( Ry ) for R > 1. Clearly, ηψR ∈ Hm1 (RN+1 R + ,y the weak formulation of (4.2.6) we have

 RN+1 +

y 1−2s [∇vm · ∇(ηψR ) + m2 vm ηψR ] dxdy 



2s

+ (μ − m )

RN

vm (x, 0)η(x, 0) dx =

RN

|vm (x, 0)|p−2vm (x, 0)η(x, 0) dx.

142

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

Letting m → 0 and using (4.5.6)–(4.5.8) we find 



RN+1 +

y 1−2s ∇v · ∇(ηψR ) dxdy + μ



RN

v(x, 0)η(x, 0) dx =

RN

|v(x, 0)|p−2 v(x, 0)η(x, 0) dx.

Letting R → ∞ we deduce that 

 +1 RN +

y 1−2s ∇v · ∇η dxdy + μ

RN

 v(x, 0)η(x, 0) dx =

RN

|v(x, 0)|p−2 v(x, 0)η(x, 0) dx

for all η ∈ Cc∞ (RN+1 + ). Finally, v(·, 0) is not identically zero because of (4.5.3), (4.5.8)  and 2 < p < 2∗s . Applying Proposition 1.3.11-(ii) we deduce that v(·, 0) > 0 in RN . 

4.5.1

Final Comments on (− + m2 )s

Bearing in mind the asymptotic estimates (1.2.4) and (1.2.5) for Kν , we can now prove the analogue of Lemma 1.2.1 for the operator (− + m2 )s , with m > 0 and s ∈ (0, 1). Theorem 4.5.2 ([63]) Let s ∈ (0, 1) and m > 0. Then, for every u ∈ S(RN ), (−+m2 )s u(x) = m2s u(x)+

cN,s N +2s m 2 2



2u(x) − u(x + y) − u(x − y) |y|

RN

N +2s 2

K N +2s (m|y|) dy. 2

Proof Choosing the substitution z = y − x in (4.1.3), we obtain (− + m2 )s u(x) = m2s u(x) + cN,s m = m2s u(x) + cN,s m

N+2s 2

N+2s 2

 P.V.

u(x) − u(y)

 P.V.

|x − y|

RN

N+2s 2

K N+2s (m|x − y|) dy 2

u(x) − u(x + z)

RN

|z|

N+2s 2

K N+2s (m|z|) dz. 2

(4.5.10) Substituting z˜ = −z in the last term in (4.5.10), we get  P.V.

u(x + z) − u(x) RN

|z|

N+2s 2

 K N+2s (m|z|) dz = P.V. 2

RN

u(x − z˜ ) − u(x) |˜z|

N+2s 2

K N+2s (m|˜z|) d z˜ , 2

(4.5.11)

4.5 Passage to the Limit as m → 0

143

and so, after relabeling z˜ as z,  2P.V.

u(x + z) − u(x) |z|

RN



= P.V.

K N+2s (m|z|) dz 2

u(x + z) − u(x) |z|

RN



+ P.V.

N+2s 2

K N+2s (m|z|) dz 2

u(x − z) − u(x) RN

 = P.V.

N+2s 2

|z|

N+2s 2

K N+2s (m|z|) dz 2

u(x + z) + u(x − z) − 2u(x) |z|

RN

N+2s 2

(4.5.12)

K N+2s (m|z|) dz. 2

Hence, if we rename z as y in (4.5.10) and (4.5.12), we can write (− + m2 )s as (− + m2 )s u(x) =

cN,s N+2s m 2 P.V. 2



2u(x) − u(x + y) − u(x − y) |y|

RN

N+2s 2

K N+2s (m|y|) dy 2

+ m2s u(x).

(4.5.13)

Now, a second-order Taylor expansion shows that    2u(x) − u(x + y) − u(x − y)  D 2 u ∞ N   L (R ) K (m|y|) K N+2s (m|y|). N+2s  ≤ N+2s N+2s−4 2 2   2 |y| |y| 2 From (1.2.4) we deduce that |D 2 u|∞ |y|

N+2s−4 2

K N+2s (m|y|) ∼ 2

C

as |y| → 0

|y|N+2s−2

which is integrable near 0. On the other hand, using (1.2.5), we get    2u(x) − u(x + y) − u(x − y)  Cu ∞ N   L (R ) K (m|y|) K N+2s (m|y|) N+2s  ≤ N+2s N+2s 2 2   |y| 2 |y| 2 ∼

C |y|

N+2s+1 2

e−m|y|

as |y| → ∞

which is integrable near ∞. Therefore, we can remove the P.V. in (4.5.13).

 

The next two theorems can be seen as the analogues of Propositions 1.3.1 and 1.3.2, respectively, for the operator (− + m2 )s with m > 0 and s ∈ (0, 1) (more precisely,

144

4 Ground States for a Pseudo-Relativistic Schrödinger Equation

compare with Theorem 15 in [316] concerning the case m = 0). The first result is a direct consequence of Theorem 4.4.2 and provides Schauder-Zygmund estimates. Theorem 4.5.3 Let s ∈ (0, 1), m > 0 and α ∈ (0, 1). Assume that f ∈ C 0,α (RN ) and that u ∈ L∞ (RN ) is a solution to (− + m2 )s u = f in RN . • • • •

If α + 2s < 1, then u ∈ C 0,α+2s (RN ). If 1 < α + 2s < 2, then u ∈ C 1,α+2s−1 (RN ). If 2 < α + 2s < 3, then u ∈ C 2,α+2s−2 (RN ). If α + 2s = k ∈ {1, 2}, then u ∈ ∗k .

The next result gives Schauder-Hölder-Zygmund estimates for bounded solutions to the equation (− + m2 )s u = f in RN with f ∈ L∞ (RN ). To prove it one can use the characterization of Lipschitz spaces in terms of the Poisson kernel P 1 (x, y) for RN+1 + 2 defined in Section 1.2 (see [241, 315] for more details). We argue as in [64]. Assume for simplicity m = 1. Define U (x, y) = P 1 (x, y) ∗ u(x) = (P 1 (x, y) ∗ G2s (x)) ∗ f (x) = G2s (x, y) ∗ f (x). 2

2

By using the formula (59) with l = 2 and β = 2s at pag. 149 in [315], we can find C > 0 such that   2   ∂ G2s 2s−2   (x, y) ,  1 N ≤ Cy  ∂y 2 L (R )

y > 0.

Since ∂∂yU2 (x, y) = ∂∂yG22s (x, y) ∗ f (x) and f ∈ L∞ (RN ), we use Young’s inequality and Theorem 15.6 with α = 2s and k = 2 in [241], to deduce that u ∈ Λ2s . Therefore, we have proved the following result: 2

2

Theorem 4.5.4 Let s ∈ (0, 1) and m > 0. Assume that f ∈ L∞ (RN ) and that u ∈ L∞ (RN ) is a solution to (− + m2 )s u = f in RN . • If 2s < 1, then u ∈ C 0,2s (RN ). • If 2s = 1, then u ∈ ∗1 . • If 2s > 1, then u ∈ C 1,2s−1 (RN ).

5

Ground States for a Superlinear Fractional Schrödinger Equation with Potentials

5.1

Nonlinearities with Subcritical Growth

5.1.1

Introduction

In this section we focus our attention on the study of the following fractional Schrödinger equation: 

(−)s u + V (x)u = f (x, u) in RN , u ∈ H s (RN ),

(5.1.1)

where s ∈ (0, 1), N ≥ 2, the potential V : RN → R satisfies the assumption (V 1) V ∈ C(RN , R) and α ≤ V (x) ≤ β, and the nonlinearity f : RN × R → R fulfills the following conditions: (f 1) f ∈ C(RN × R, R) is 1-periodic in x and f (x, t) = 0, ∗ |t |→∞ |t|2s −1 lim

uniformly in x ∈ RN ;

(f 2) f (x, t) = o(t) as |t| → 0, uniformly in x ∈ RN .

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_5

145

146

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

When s = 1, the equation in (5.1.1) formally reduces to the classical nonlinear Schrödinger equation − u + V (x)u = f (x, u)

in RN ,

(5.1.2)

which has been extensively studied in the last 20 years. Since we cannot review the huge bibliography, we just mention [28,90,165,197,299,330,340] and references therein, where several existence and multiplicity results are obtained under different assumptions on the potential V and the nonlinearity f . To deal with (5.1.2), many authors assume that the nonlinear term satisfied the following condition due to Ambrosetti and Rabinowitz [29] ∃ μ > 2, R > 0 : 0 < μF (x, t) ≤ f (x, t)t

∀ |t| ≥ R,

(AR)

t where F (x, t) = 0 f (x, τ ) dτ . Roughly speaking, the role of (AR) is to guarantee the boundedness of Palais–Smale sequences for the functional associated with the problem under consideration. However, although (AR) is a quite natural condition when we deal with superlinear elliptic problems, it is somewhat restrictive. In fact, by a direct integration of (AR), we can deduce that (f 3) lim

|t |→∞

F (x, t) = ∞ uniformly in x ∈ RN . t2

Of course, also condition (f 3) characterizes the nonlinearity f to be superlinear at infinity. Anyway, if we consider the function f (x, t) = t log(1 + |t|), then it is easy to prove that f fulfills (f 3) but does not satisfy (AR). This means that there are functions that are superlinear at infinity and do not verify (AR). For this reason, in several works concerning superlinear problems, some authors tried to drop the condition (AR); see for instance [151, 231, 259, 271, 305] and references therein. For instance, Jeanjean in [231] introduced the following assumption: (f 4) There exists λ ≥ 1 such that G(x, θ t) ≤ λG(x, t)

for (x, t) ∈ RN × R and θ ∈ [0, 1],

where G(x, t) = f (x, t)t − 2F (x, t). In view of the above discussion and motivated by the fact that recently several authors [172, 183, 306] studied (5.1.1) assuming the well-known condition (AR), here we investigate solutions of (5.1.1) without requiring this condition. Our first main result can

5.1 Nonlinearities with Subcritical Growth

147

be stated as follows: Theorem 5.1.1 ([32]) Assume that f satisfies (f 1)–(f 4) and V fulfills (V 1) and (V 2) V (x) is 1-periodic. Then there exists a nontrivial ground state solution u ∈ H s (RN ) to (5.1.1). In order to study our problem, we will look for the critical points for the functional     1 2 2 J (u) = [u]s + V (x)u dx − F (x, u) dx. 2 RN RN Thanks to the assumptions on f , it is easy to see that J has a mountain pass geometry. Namely, setting  = {γ ∈ C([0, 1], H s (RN )) : γ (0) = 0 and J (γ (1)) < 0}, we have  = ∅ and c = inf max J (γ (t)) γ ∈ t ∈[0,1]

is the mountain pass level for J . The Ekeland variational principle [177] guarantees the existence of a Cerami sequence at the level c. Hence, by using some suitable variational arguments inspired by Jeanjean and Tanaka [233] and Liu [259] and the ZN -invariance of the problem (5.1.1), we will prove that every Cerami sequence for J is bounded and admits a subsequence which converges to a critical point for J . Finally, we will also consider the potential well case. We will assume that V (x) satisfies, in addition to (V 1), the following condition: (V 3) V (x) < V∞ = lim V (y) < ∞ ∀x ∈ RN . |y|→∞

We will also assume that f (x, u) = b(x)f (u), where b ∈ C(RN ) and 0 < b∞ = lim b(y) ≤ b(x) ≤ b¯ < ∞ |y|→∞

(5.1.3)

for any x ∈ RN and that f satisfies (f 1)–(f 4). Therefore our problem becomes 

(−)s u + V (x)u = b(x)f (u) in RN , u ∈ H s (RN ).

(5.1.4)

148

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

To study (5.1.4), we will use the energy comparison method presented in [233]. More precisely, by introducing the energy functional at infinity,     1 2 2 V∞ u dx − b∞ F (u) dx, J∞ (u) = [u]s + 2 RN RN we will show that, under the above assumptions on f and V , J has a nontrivial critical point provided that (5.1.5)

c < m∞ , where

m∞ = inf{J∞ (u) : u = 0 and J∞ (u) = 0}.

In order to prove (5.1.5), we use the fact that our problem at infinity is autonomous: (−)s u = −V∞ u + b∞ f (u)

in RN ,

so it admits a least one energy solution satisfying the Pohozaev identity. This information will be fundamental for deducing the existence of a path γ ∈  such that max J (γ (t)) < m∞ . Combining these facts, we will be able to prove our second result:

t ∈[0,1]

Theorem 5.1.2 ([32]) Assume that V satisfies (V 1) and (V 3), and that f fulfills assumptions (f 1)–(f 4). Then (5.1.4) has a ground state.

5.1.2

Preliminaries

We start with the concept of weak solution for the equation (−)s u + V (x)u = g

in RN .

(5.1.6)

Definition 5.1.3 Given g ∈ L2 (RN ), we say that u is a weak solution to (5.1.6) if u ∈ H s (RN ) and satisfies  R2N

(u(x) − u(y))(v(x) − v(y)) dxdy + |x − y|N+2s

for all v ∈ H s (RN ).



 RN

V (x)uv dx =

RN

gv dx

5.1 Nonlinearities with Subcritical Growth

149

In order to deal with (5.1.1), we consider the functional on H s (RN ) given by     1 2 2 V (x)u dx − F (x, u) dx. J (u) = [u]s + 2 RN RN By (V 1), it follows that   u = [u]2s +

1 RN

V (x)u2 dx

2

is a norm that is equivalent to the standard one in H s (RN ). For this reason, we will always write  1 F (x, u) dx. J (u) = u2 − 2 RN In particular, by the assumptions on f , we deduce that J ∈ C 1 (H s (RN ), R). Moreover, J possesses a mountain pass geometry. More precisely, we have the following result whose simple proof is omitted. Lemma 5.1.4 Under assumptions (f 1)–(f 4), there exist r > 0 and v0 ∈ H s (RN ) such that v0  > r and b = inf J (u) > J (0) = 0 ≥ J (v0 ). u=r

(5.1.7)

In particular, J (u), u = u2 + o(u2 ) as u → 0, J (u) = 12 u2 + o(u2 )

as u → 0,

and, consequently (i) there exists η > 0 such that if v is a critical point for J , then v ≥ η; (ii) for every c > 0 there exists ηc > 0 such that if J (vn ) → c, then vn  ≥ ηc . Therefore, by Lemma 5.1.4, it follows that  = {γ ∈ C([0, 1], H s (RN )) : γ (0) = 0 and J (γ (1)) < 0} = ∅ and we can define the mountain pass level c = inf max J (γ (t)). γ ∈ t ∈[0,1]

(5.1.8)

150

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Let us point out that, by (5.1.7), c is positive. Then, Theorem 2.2.15 ensures that there exists a Cerami sequence (vn ) at the level c for J , that is J (vn ) → c

(1 + vn )J (vn )H −s (RN ) → 0.

and

We conclude this subsection by proving that the primitive F (x, t) of f (x, t) is nonnegative. Lemma 5.1.5 Assume that f satisfies (f 1), (f 2), and (f 4). Then F ≥ 0 in RN × R. Proof First we observe that by (f 4) we have G(x, t) = f (x, t)t − 2F (x, t) ≥ 0

for all (x, t) ∈ RN × R.

Fix t > 0. For x ∈ RN , let us compute the derivative of ∂ ∂t



F (x, t) t2

 =

F (x,t ) t2

with respect to t:

f (x, t) t 2 − 2t F (x, t) ≥ 0. t4

(5.1.9)

Moreover, by (f 2), we get F (x, t) = 0. t →0 t2 lim

(5.1.10)

Combining (5.1.9) and (5.1.10) we deduce that F (x, t) ≥ 0 for all (x, t) ∈ RN × [0, ∞).   Analogously, we obtain that F (x, t) ≥ 0 for all (x, t) ∈ RN × (−∞, 0].

5.1.3

Periodic Potentials

In this subsection we give the proof of the Theorem 5.1.1. We begin with the following result which guarantees the boundedness of Cerami sequences for the functional J . Lemma 5.1.6 Assume that (V 1) and (f 1)–(f 4) hold. Let c ∈ R. Then every Cerami sequence for J is bounded. Proof Let (vn ) be a Cerami sequence for J . Assume, by contradiction, that (vn ) is unbounded. Going if necessary to a subsequence, we may assume that J (vn ) → c,

vn  → ∞,

J (vn )H −s (RN ) vn  → 0.

(5.1.11)

5.1 Nonlinearities with Subcritical Growth

151

vn . Clearly, (wn ) is bounded in H s (RN ) and its elements have vn  unit norm. We claim that (wn ) vanishes, i.e., that Now denote wn =

 lim sup

n→∞

B2 (z)

z∈RN

wn2 dx = 0.

(5.1.12)

Indeed, if (5.1.12) does not hold, then there exists δ > 0 such that  sup B2 (z)

z∈RN

wn2 dx ≥ δ > 0.

Consequently, we can choose a sequence of points (zn ) ⊂ RN such that  B2 (zn )

wn2 dx ≥

δ . 2

Since the number of points in ZN ∩ B2 (zn ) is less than 4N , there exists a point ξn ∈ ZN ∩ B2 (zn ) such that  B2 (ξn )

wn2 dx ≥ K > 0,

(5.1.13)

where K = δ2−(2N+1) . Set w˜ n = wn (· + ξn ). Using (V 1) and that wn has unit norm, we deduce that  V (x)w˜ n2 dx w˜ n 2 = [w˜ n ]2s + RN



≤

[w˜ n ]2s

+β

RN

V (x)w˜ n2 dx

 = [wn ]2s + β β ≤ α

RN

wn2 dx

  2 [wn ]s + α

 RN

wn2 dx

   β β ≤ V (x)wn2 dx = , [wn ]2s + N α α R

that is, (w˜ n ) is bounded. By Lemma 1.1.8, we may assume, passing if necessary to a subsequence, that w˜ n → w˜

in L2loc (RN ),

w˜ n (x) → w(x) ˜

for a.e. x ∈ RN .

(5.1.14)

152

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

By (5.1.13) and (5.1.14), 

 w˜ 2 dx = lim



n→∞ B 2

B2

w˜ n2 dx = lim

n→∞ B (ξ ) 2 n

wn2 dx ≥ K > 0,

(5.1.15)

which implies that w˜ = 0. Let v˜n = vn w˜ n . Since w˜ = 0, the set A = {x ∈ RN : w˜ = 0} has positive Lebesgue measure and |v˜n (x)| → ∞. In particular, by (f 3), we get F (x, v˜n (x)) (w˜ n (x))2 → ∞. (v˜n (x))2

(5.1.16)

Let us observe that f (x, t) is 1-periodic with respect to x, so 



RN

F (x, vn ) dx =

RN

F (x, v˜n ) dx.

(5.1.17)

By (5.1.11), (5.1.16), (5.1.17) and Lemma 5.1.5, it follows that 1 c + o(1) − = 2 vn 2

 RN

F (x, vn ) dx = vn 2

 RN

F (x, v˜n ) dx ≥ vn 2

 A

F (x, v˜n ) 2 w˜ n dx → ∞, v˜n2 (5.1.18)

which gives a contradiction. Therefore (5.1.12) holds true. In particular, by Lemma 1.4.4, we get ∀q ∈ (2, 2∗s ).

wn → 0 in Lq (RN )

Now, let ρ be a positive real number. By (f 1)–(f 3) and Lemma 5.1.5, we deduce that for every ε > 0 there exists Cε > 0 such that ∗

0 ≤ F (x, ρt) ≤ ε(|t|2 + |t|2s ) + Cε |t|q .

(5.1.19)

Since wn  = 1, by (1.1.1) there exists c˜ > 0 such that 2∗s

wn 2L2 (RN ) + wn 

∗

L2s (RN )

≤ c. ˜

(5.1.20)

Taking into account (5.1.19) and (5.1.20), we obtain  lim sup n→∞

RN

≤ εc, ˜

% 2∗ F (x, ρwn ) dx ≤ lim sup ε(wn 2L2 (RN ) + wn  s2∗s n→∞

L (RN )

q

) + Cε (wn Lq (RN )

&

5.1 Nonlinearities with Subcritical Growth

153

and in view of the arbitrariness of ε we conclude that  F (x, ρwn ) dx = 0. lim n→∞ RN

(5.1.21)

Now, let (tn ) ⊂ [0, 1] be a sequence such that J (tn vn ) = max J (tvn ). t ∈[0,1]

(5.1.22)

 Using (5.1.11), we can verify that 2 j vn −1 ∈ (0, 1) for n sufficiently large and j ∈ N. √ Taking ρ = 2 j in (5.1.21), we have   J (tn vn ) ≥ J (2 j wn ) = 2j −

RN

 F (x, 2 j wn ) dx ≥ j

for n large enough and for all j ∈ N. Then J (tn vn ) → ∞.

(5.1.23)

Since J (0) = 0 and J (vn ) → c we deduce that tn ∈ (0, 1). By (5.1.22), we obtain   d = 0. J (tn vn ), tn vn  = tn J (tvn ) dt t =tn

(5.1.24)

Indeed, (5.1.11), (5.1.24) and (f 4) show that

2 1 J (tn vn ) = 2J (tn vn ) − J (tn vn ), tn vn  λ λ  1 = (f (x, tn vn )tn vn − 2F (x, tn vn )) dx λ RN  1 = G(x, tn vn ) dx λ RN  G(x, tn vn ) dx ≤ RN

 =

RN

(f (x, vn )vn − 2F (x, vn )) dx

= 2J (vn ) − J (vn ), vn  → 2c, which is incompatible with (5.1.23). Thus (vn ) is bounded.

 

154

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Remark 5.1.7 We stress that the conclusion of Lemma 5.1.6 holds true if we consider f (x, t) = b(x)f (t) with b ∈ C(RN , R) and 0 < b0 ≤ b(x) ≤ b1 < ∞ for any x ∈ RN . In fact, in this case, the contradiction in (5.1.18) is obtained replacing (5.1.17) by  RN

b0 b(x)F (vn ) dx ≥ b1

 RN

b(x)F (v˜n ) dx.

Now let us prove that, up to a subsequence, our bounded Cerami sequence (un ) converges weakly to a non-trivial critical point for J . Proof of Theorem 5.1.1 Let c be the mountain pass level defined in (5.1.8). We know that c > 0 and that there exists a Cerami sequence (un ) for J , which is bounded in H s (RN ) by Lemma 5.1.6. Denote  δ = lim sup n→∞

z∈RN

B2 (z)

u2n dx.

If δ = 0, then Lemma 1.4.4 implies that un → 0 in Lq (RN ) for all q ∈ (2, 2∗s ). Analogously to (5.1.21), we can see that  lim

n→∞ RN

F (x, un ) dx = 0,



lim

n→∞ RN

f (x, un )un dx = 0.

Consequently,  0 = lim

n→∞ RN



   1

1 f (x, un )vn − F (x, un ) dx = lim J (un ) − J (un ), un  = c n→∞ 2 2

which is impossible because c > 0. Therefore, δ > 0. As for (5.1.15), we can find a sequence (ξn ) ⊂ ZN and a positive constant K such that 

 B2

wn2 dx

= B2 (ξn )

u2n dx > K,

(5.1.25)

where wn = un (· + ξn ). Note that wn  = un , so (wn ) is bounded in H s (RN ). By Lemma 1.1.8, we can assume, up to a subsequence, that wn  w

in H s (RN ),

wn → w

in L2loc (RN ),

5.1 Nonlinearities with Subcritical Growth

155

and by using (5.1.25) we have w = 0. Since (5.1.1) is ZN -invariant, (wn ) is a Cerami sequence for J . Then, J (w), φ = lim J (wn ), φ = 0 n→∞

for all φ ∈ Cc∞ (RN ), that is, J (w) = 0 and w is a nontrivial solution to (5.1.1). Now we want to prove that the problem (5.1.1) has a ground state. Let m = inf{J (v) : v = 0 and J (v) = 0} and suppose that v is an arbitrary critical point for J . By (f 4), G(x, t) ≥ 0 ∀(x, t) ∈ RN × R, which implies that 1 1 J (v) = J (v) − J (v), v = 2 2

 RN

G(x, v) dx ≥ 0.

Hence, m ≥ 0. Now, let (un ) be a sequence of nontrivial critical points for J such that J (un ) → m. By Lemma 5.1.4, there exists η > 0 such that un  ≥ η.

(5.1.26)

Since un is a critical point for J , we have that (1 + un )J (un )H −s (RN ) → 0. Therefore, (un ) is a Cerami sequence at the level m and, by Lemma 5.1.6, (un ) is bounded in H s (RN ). Let  δ = lim sup n→∞

z∈RN

B2 (z)

u2n dx.

As before, if δ = 0, then  lim

n→∞ RN

f (x, un )un dx = 0,

whence un 2 = J (un ), un  +

 RN

f (x, un )un dx → 0,

(5.1.27)

156

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

which is impossible because of (5.1.26). Thus δ > 0. Arguing as before, we can find the translated sequence wn (x) = un (x + ξn ) such that J (wn ) = 0,

J (wn ) = J (un ) → m,

(5.1.28)

and (wn ) weakly converges to a nonzero critical point w for J . Hence, by (5.1.28), the fact that G ≥ 0 and Fatou’s lemma, 1 J (w) = J (w) − J (w), w 2  1 = G(x, w) dx 2 RN  1 ≤ lim inf G(x, wn ) dx n→∞ 2 RN   1 = lim inf J (wn ) − J (wn ), wn  = m. n→∞ 2 Thus, w is a nontrivial critical point for J such that J (w) = m.

5.1.4

(5.1.29)

 

Bounded Potentials

In this section we give the proof of the Theorem 5.1.2. The main ingredient of our proof is the following result which takes advantage of the Pohozaev identity for (−)s . Proposition 5.1.8 Let u ∈ H s (RN ) be a nontrivial critical point for  1 2 I(u) = [u]s − G(u) dx. 2 RN Then there exists γ ∈ C([0, 1], H s (RN )) such that γ (0) = 0, I(γ (1)) < 0, u ∈ γ ([0, 1]) and max I(γ (t)) = I(u).

t ∈[0,1]

Proof Let u ∈ H s (RN ) be a nontrivial critical point for I. We set for t > 0 x

. ut (x) = u t By the Pohozaev identity, N − 2s 2 [u]s = N 2

 RN

G(u) dx,

5.1 Nonlinearities with Subcritical Growth

157

which implies that t N−2s 2 I(u ) = [u]s − t N 2





t

RN

G(u) dx =

 1 N−2s N − 2s N − [u]2s . t t 2 2N

Therefore, we deduce that max I(ut ) = I(u), I(ut ) → −∞ as t → ∞, and t >0

ut 2H s (RN ) = t N−2s [u]2s + t N u2L2 (RN ) → 0 as t → 0. Choosing α > 1 such that I(uα ) < 0 and setting  γ (t) =

uαt , for t ∈ (0, 1], 0, for t = 0,  

we get the conclusion. Now we consider the following functionals: 1 J (u) = u2 − 2

 RN

b(x)F (u) dx

and J∞ (u) =

    1 V∞ u2 dx − b∞ F (u) dx. [u]2s + 2 RN RN

By (V 3), it follows that J (u) < J∞ (u)

for any u ∈ H s (RN ) \ {0}.

(5.1.30)

Taking into account Proposition 5.1.8, we prove the following result: Lemma 5.1.9 Assume that V (x) satisfies (V 1) and (V 3) and f fulfills (f 1)–(f 4). Then J has a nontrivial critical point. Proof Let c be the mountain pass level for J . We know that J has a Cerami sequence (un ) at the level c, which is bounded by Lemma 5.1.6. Then un  u in H s (RN ) and J (u) = 0. We claim that u ≡ 0. Assume, by contradiction, that u = 0. Thanks to (V 3), the fact that un → u in L2loc (RN ), Lemma 5.1.5 and (5.1.3),  |J∞ (un ) − J (un )| ≤

 |V∞ − V (x)||un | dx + 2

RN

RN

|b(x) − b∞ |F (un ) dx → 0

158

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

and

J∞ (un ) − J (un )H −s (RN )        ≤ sup [V∞ − V (x)]un φ dx  +   φ∈H s (RN )

φH s (RN ) =1

RN

R

  [b∞ − b(x)]f (un )φ dx  → 0, N

that is, (un ) is a Palais–Smale sequence for J∞ at the level c. Now denote  u2n dx. δ = lim sup n→∞

ξ ∈RN

(5.1.31)

B2 (ξ )

If δ = 0, then proceeding similarly to (5.1.27), we deduce that un 2 → 0, which contradicts Lemma 5.1.4. Therefore, δ > 0 and there exists (ξn ) ⊂ ZN such that  B2 (ξn )

u2n dx ≥

δ > 0. 2

(5.1.32)

Let vn = un (x + ξn ). Then vn  = un , J∞ (vn ) = J∞ (un ),



J∞ (vn ) = J∞ (un ).

Therefore, (vn ) is a bounded Palais–Smale sequence for J∞ . As in the proof of Theorem 5.1.1, we use (5.1.32) to deduce that vn  v in H s (RN ) and v is a nontrivial critical point for J∞ . Moreover, proceeding as in (5.1.29), we have J∞ (v) ≤ c. Using Proposition 5.1.8 with g(t) = b∞ f (t) − V∞ t, we can find γ∞ ∈ C([0, 1], H s (RN )) such that γ∞ (0) = 0, J∞ (γ∞ (1)) < 0, v ∈ γ∞ ([0, 1]) and max J∞ (γ∞ (t)) = J∞ (v).

t ∈[0,1]

Since 0 ∈ / γ∞ ((0, 1]), by (5.1.30), it follows that, for all t ∈ (0, 1] J (γ∞ (t)) < J∞ (γ∞ (t)).

(5.1.33)

5.2 Nonlinearities with Critical Growth

159

In particular, J (γ∞ (1)) ≤ J∞ (γ∞ (1)) < 0, so γ∞ ∈ . Then, taking into account that J (0) = J∞ (0) = 0, (5.1.33) and that c > 0, we deduce that c ≤ max J (γ∞ (t)) < max J∞ (γ∞ (t)) = J∞ (v) ≤ c, t ∈[0,1]

t ∈[0,1]

 

which gives a contradiction.

Remark 5.1.10 Since (un ) is a Cerami sequence for J at the level c and un  u in H s (RN ), a similar argument as in (5.1.29) shows that J (u) ≤ c. Finally, we give the proof of Theorem 5.1.2. Proof of Theorem 5.1.2 Let m = inf{J (u) : u = 0 and J (u) = 0} and let u denote the nontrivial critical point for J obtained in the previous lemma. Then (see Remark 5.1.10) one can verify that 0 ≤ m ≤ J (u) ≤ c.

(5.1.34)

Now, let (un ) be a sequence of nontrivial critical points for J such that J (un ) → m. As in the proof of Theorem 5.1.1, we have that (un ) is a Cerami bounded sequence at the level m and δ > 0, where δ is defined via (5.1.31). Extracting a subsequence, un  u˜ in H s (RN ), and u˜ is a critical point for J satisfying J (u) ˜ ≤ m as in (5.1.29). Now, if u˜ = 0, (un ) is a bounded Palais–Smale sequence for J∞ at the level m. Since δ > 0, we deduce that (vn ), which is a suitable translation of (un ), converges weakly to some critical point v = 0 for J∞ and J∞ (v) ≤ m. Arguing as in the proof of Lemma 5.1.9, it follows from Proposition 5.1.8 that there exists γ∞ ∈ ∞ ∩  such that c ≤ max J (γ∞ (t)) < max J∞ (γ∞ (t)) = J∞ (v) ≤ m, t ∈[0,1]

t ∈[0,1]

which is a contradiction because of (5.1.34). Thus, u˜ is a nontrivial critical point for J such that J (u) ˜ = m.  

5.2

Nonlinearities with Critical Growth

5.2.1

Introduction

This section is devoted to the existence of ground state solutions for the following nonlinear fractional elliptic problem:  (−)s u + V (x)u = f (u) in RN , (5.2.1) u ∈ H s (RN ), u > 0 in RN ,

160

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

with s ∈ (0, 1) and N ≥ 2. We assume that the potential V : RN → R satisfies the following conditions: (V 1) (V 2) (V 3) (V 4)

V ∈ C 1 (RN , R); there exists V0 > 0 such that infx∈RN V (x) ≥ V0 ; V (x) ≤ V∞ = lim|x|→∞ V (x) < ∞ for all x ∈ RN ;  max{x · ∇V (x), 0} N N < 2sS∗ . L 2s (R )

Concerning the nonlinear term f , we assume that f (t) = 0 for t ≤ 0 and (f 1) f ∈ C 1 (R+ , R); f (t) (f 2) lim = 0; t →0 t f (t) (f 3) lim 2∗ −1 = K > 0; t →∞ t s

 (f 4) there exist D > 0 and max 2,

4s N − 2s



< q < 2∗s such that

∗

f (t) ≥ Kt 2s −1 + Dt q−1 for all t ≥ 0. We observe that assumptions (f 3) and (f 4) on the nonlinearity f enable us to consider the critical growth case. In the case s = 1, assumption (f 4) was introduced in [345]. We point out that (f 4) plays an important role in ensuring the existence of solutions for ∗ the problem (5.2.1). In fact, if we take f (t) = (t + )2s −1 , then f satisfies (f 1)–(f 3), and by using the Pohozaev identity for the fractional Laplacian, we can see that there are no nontrivial solutions to (5.2.1). Our first main result concerns the existence of ground state solutions to (5.2.1) in the case of constant potentials. Theorem 5.2.1 ([67]) Let s ∈ (0, 1) and N ≥ 2. Assume that f satisfies (f 1)–(f 4) and V (x) ≡ V > 0 is constant. Then (5.2.1) possesses a positive ground state solution u ∈ H s (RN ). Let us sketch the proof of Theorem 5.2.1. In order to obtain the existence of a nontrivial solution to (5.2.1), we look for critical points of the Euler-Lagrange functional associated with (5.2.1), that is, I(u) =

1 2



RN



s |(−) 2 u|2 + V (x)u2 dx −

RN

F (u) dx

t for any u ∈ H s (RN ), where F (t) = 0 f (τ ) dτ . In view of the assumptions on f , it is clear that I has a mountain pass geometry, but it is hard to verify the boundedness of

5.2 Nonlinearities with Critical Growth

161

Palais–Smale sequences of I. To overcome this difficulty, we use the idea in [231]. For λ ∈ [ 12 , 1], let us introduce the family of functionals Iλ (u) =

1 2

 RN



s |(−) 2 u|2 + V (x)u2 dx − λ

RN

F (u) dx.

As a first step, we prove that for every λ ∈ [ 12 , 1], Iλ has a mountain pass geometry and that Iλ admits a bounded Palais–Smale sequence (un ) at the mountain pass level cλ . More precisely, we use the following abstract version of Struwe’s monotonicity trick [319] developed by Jeanjean [231]: Theorem 5.2.2 ([231]) Let (X,  · ) be a Banach space and  ⊂ R+ an interval. Let (Iλ )λ∈ be a family of C 1 -functionals on X of the form Iλ (u) = A(u) − λB(u),

for λ ∈ ,

where B(u) ≥ 0 for all u ∈ X, and either A(u) → ∞ or B(u) → ∞ as u → ∞. We assume that there exist v1 , v2 ∈ X such that cλ = inf max Iλ (γ (t)) > max{Iλ (v1 ), Iλ (v2 )}, γ ∈ t ∈[0,1]

for all λ ∈ ,

where  = {γ ∈ C([0, 1], X) : γ (0) = v1 , γ (1) = v2 }. Then, for almost every λ ∈ , there is a sequence (un ) ⊂ X such that (i) (un ) is bounded; (ii) Iλ (un ) → cλ ; (iii) Iλ (un ) → 0 in X∗ . Moreover, the map λ → cλ is non-increasing and continuous from the left. Since we are dealing with the critical case, we are able to prove that for any λ ∈ [ 12 , 1] N

s S∗2s 0 < cλ < . N λ N−2s 2s Secondly, in the spirit of [236] (see also [260, 346]), we establish a global compactness result in the critical case, which gives a description of the bounded Palais–Smale sequences of Iλ . Then, by using the fact that every solution of (5.2.1) satisfies the Pohozaev identity

162

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

and the compactness lemma, we prove the existence of a bounded Palais–Smale sequence of I which converges to a positive solution to (5.2.1). Now, we state our second main result which deals with the existence of ground state of (5.2.1) in the case in which V is not a constant. Theorem 5.2.3 ([67]) Let s ∈ (0, 1) and N ≥ 2. Assume that f fulfills (f 1)–(f 4) and V satisfies (V 1)–(V 4), and V (x) ≡ V∞ . Then (5.2.1) admits a positive ground state solution u ∈ H s (RN ). To deal with the non-autonomous case, we enlist some ideas developed in [236]. We consider the previous family of functionals Iλ , and, since Iλ satisfies the assumptions j of Theorem 5.2.2, we can deduce the existence of a Palais–Smale sequence (un ) at the j mountain pass level cλj , where λj → 1. Therefore, un  uj in H s (RN ), where uj is a critical point of Iλj . This time, the boundedness of the sequence (uj ) follows by the assumption (V 4). Moreover, we prove that (uj ) is a bounded Palais–Smale sequence of I. To show that the bounded sequence (uj ) converges to a nontrivial weak solution of (5.2.1), we show that c1 is strictly less than the least energy level m∞ of the functional I ∞ associated to the “problem at infinity” (−)s u + V∞ u = f (u)

in RN .

Together with an accurate description of the sequence as a sum of translated critical points, this allows us to infer that uj  u in H s (RN ), for some nontrivial critical point u of I. Let us recall that when f is an odd function satisfying (f 1)–(f 4) and V is constant, the existence of a radial positive ground state to (5.2.1) was proved in [36] via a minimization s (RN ) which is compactly argument and by working in the space of radial functions Hrad p N ∗ embedded into L (R ) for all p ∈ (2, 2s ). Here we present a different proof of this result (see Theorem 5.2.1) which is based on the global compactness lemma, which will be also useful for proving Theorem 5.2.3. In fact, we believe that the global compactness lemma is not only interesting for the aim of this chapter, but it can be also used to deal with other problems similar to (5.2.1). Let us also point out that by using the methods developed here, we are able to study (5.2.1) dealing with radial and non-radial potentials by a unified approach. We conclude this introduction by mentioning that further interesting results for fractional elliptic problems with critical growth in RN can be found in [170,174,216,311,344], and [41, 87, 310, 324] for problems in bounded domains.

5.2.2

Splitting Lemmas

In this subsection we collect some useful splitting results; see [1, 113, 154, 169, 340, 347] for some classical results.

5.2 Nonlinearities with Critical Growth

163

Lemma 5.2.4 Assume that un  u in H s (RN ). Then we have    

' RN

|un |

2∗s −2

un − |u|

2∗s −2

u − |un − u|

2∗s −2

  ( (un − u) w dx  = on (1)wH s (RN )

where on (1) → 0 as n → ∞, uniformly for any w ∈ Cc∞ (RN ). Proof We follow [262]. Using the mean value theorem, the boundedness of (un ) in ∗ H s (RN ), the generalized Hölder inequality, the fact that u ∈ L2s (RN ), and Theorem 1.1.8, we deduce that for every ε > 0 there exists R = R(ε) > 0 such that    

  |un | un − |u| u − |un − u| (un − u) w dx  RN \BR     % &     2∗s −2 2∗s −2 2∗s −2    ≤ |un | un − |un − u| (un − u) w dx  +  |u| uw dx  RN \BR RN \BR  

∗ ∗ ∗ ≤C |un |2s −2 + |un − u|2s −2 |uw| dx + |u|2s −1 |w| dx %

2∗s −2

2∗s −2

&

2∗s −2

RN \BR

RN \BR

2∗s −2 ∗ L2s (RN )

≤ Cun 

2∗s −1

uL2∗s (RN \B ) wL2∗s (RN ) + Cu

∗

L2s (RN \BR )

R

wL2∗s (RN )

≤ C ε wH s (RN ) .

(5.2.2)

On the other hand, for every r > 0, we have    

% BR

 &  ∗ ∗ ∗ |un |2s −2 un − |u|2s −2 u − |un − u|2s −2 (un − u) w dx 



≤

BR ∩{|un −u|≤r}

  ∗ ∗ ∗   |un |2s −2 un − |u|2s −2 u − |un − u|2s −2 (un − u) |w| dx



+

BR ∩{|un −u|≥r}

  ∗ ∗ ∗   |un |2s −2 un − |u|2s −2 u − |un − u|2s −2 (un − u) |w| dx

= I1 + I2 .

(5.2.3) 1 ∗

Now, there exists r = r(R) such that r|BR | 2s ≤ ε. This fact together with the mean value theorem, the boundedness of (un ) in H s (RN ) and the generalized Hölder inequality implies that  I1 ≤ C

BR ∩{|un −u|≤r} 2∗s −2

≤ Cun 

∗

L2s (RN )



∗ ∗ ∗ |un |2s −2 + |u|2s −2 + |un − u|2s −2 |(un − u)w| dx

un − uL2∗s (B

R ∩{|un −u|≤r})

wL2∗s (RN )

164

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . . 2∗s −2

+ Cu

∗

L2s (RN )

un − uL2∗s (B

2∗s −2

+ Cun − u

∗

L2s (RN )

R ∩{|un −u|≤r})

un − uL2∗s (B

wL2∗s (RN )

R ∩{|un −u|≤r})

wL2∗s (RN )

1 ∗

≤ Cr|BR | 2s wH s (RN ) ≤ C ε wH s (RN ) .

(5.2.4)

For such r, R fixed above, un strongly converges to u in L2 (BR ) and then |BR ∩{|un −u| ≥ r}| → 0 as n → ∞. Therefore, by the dominated convergence theorem, 

∗

BR ∩{|un −u|≥r}

|u|2s dx → 0

as n → ∞,

which implies that, for n large enough,  I2 ≤ C

BR ∩{|un −u|≥r} 2∗s −2

≤ Cun 

∗

L2s (RN )



2∗s −2 2∗s −2 + |un − u| |un | |uw| dx +

uL2∗s (B

2∗s −2

+ Cun − u

∗

L2s (RN )

R ∩{|un −u|≥r})

uL2∗s (B

2∗s −1

+ Cu

∗

L2s (BR ∩{|un −u|≥r})

≤ CuL2∗s (B

R ∩{|un −u|≥r})

∗

BR ∩{|un −u|≥r}

|u|2s −1 |w| dx

wH s (RN )

R ∩{|un −u|≥r})

wH s (RN )

wH s (RN ) 2∗s −1

wH s (RN ) + Cu

∗

L2s (BR ∩{|un −u|≥r})

wH s (RN )

≤ C ε wH s (RN ) .

(5.2.5)

Combining (5.2.2), (5.2.3), (5.2.4), and (5.2.5) we get the thesis. Lemma 5.2.5 Let f1 : R → R be a continuous function such that f1 (t) = 0, ∗ |t |→∞ |t|2s −1 lim

and ∗

|f1 (t)| ≤ C(|t| + |t|2s −1 )

for all t ∈ R.

If un  u in H s (RN ) and un → u a.e. in RN , then  lim

n→∞

 RN

F1 (un ) dx −

RN

 F1 (u) dx −

RN

 F1 (un − u) dx = 0,

 

5.2 Nonlinearities with Critical Growth

where F1 (t) =

t

165

f (τ ) dτ .

0

Proof The proof is standard. However, for the reader’s convenience, we show it here. For R > 0, the mean value theorem shows that 

 RN

F1 (un ) dx =



BR

F1 (un ) dx + 

 =

BR

F1 (un ) dx +

BRc

F1 (un ) dx F1 ((un − u) + u) dx



 =

BRc

BR

F1 (un ) dx +

#

 BRc

F1 (un − u) dx +

We now write     F (u ) dx − 1 n  N

BRc

1

0

$ f1 (un − u + θu)u dθ dx.

  F1 (u) dx − F1 (un − u) dx  R RN RN          ≤  (F1 (un ) − F1 (u)) dx  +  F1 (u) dx  c   BR BR      1       +  F1 (un − u) dx  +  f1 (un − u + θ u)u dθ dx  . c   0 BR B 

R

From the growth assumption on f and applying the Hölder inequality we obtain that      1   f (u − u + θu)u dθ dx    Bc 0 1 n  R #  ≤C ⎡ ≤C⎣

BRc

(|un | + |u|)|u| dx +

 (|un | + |u|) dx 2

RN

⎡  ≤C⎣

BRc

1 |u| dx

+

(|un | + |u|)

 1  2 BRc



2

2

BRc

BRc

$ |u| dx

1



2

|u| dx 2

 2∗s

2∗s −1

|u| dx

1 2∗ s

+

2∗s

RN

(|un | + |u|) dx

  2∗s −1 ∗

 2∗s

2s

BRc

|u| dx

1 2∗ s

⎤ ⎦

⎤ ⎦,

∗

where in the last inequality we used the boundedness of (un ) in L2 (RN ) ∩ L2s (RN ). Moreover,   $ #    ∗   F1 (u) dx  ≤ C |u|2 dx + |u|2s dx .   Bc  Bc Bc R

R

R

166

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Hence, for every ε > 0 there exists R = Rε > 0 such that          1     F1 (u) dx  +  f1 (un − u + θ u)u dθ dx  ≤ ε .   Bc   Bc 0  R

(5.2.6)

R

∗

On the other hand, it follows from Lemma 1.4.2 (with P (t) = F (t) and Q(t) = t 2 + |t|2s ) that  lim (F1 (un ) − F1 (u)) dx = 0 (5.2.7) n→∞ B R

and  lim

n→∞ B R

F1 (un − u) dx = 0.

Now the result follows from (5.2.6), (5.2.7), and (5.2.8).

(5.2.8)  

Lemma 5.2.6 Let f1 : R → R be a continuous function such that lim

|t |→0

f1 (t) f1 (t) = 0 = lim . ∗ |t |→∞ |t|2s −1 |t|

Let (un ) ⊂ H s (RN ) be a sequence such that un  u in H s (RN ). Then we have    

RN

  (f1 (un ) − f1 (u) − f1 (un − u)) w dx  = on (1)wH s (RN ) ,

where on (1) → 0 as n → ∞, uniformly for any w ∈ Cc∞ (RN ). Proof We argue as in [7, 347]. For any fixed η > 0, since f (t) = o(|t|) as |t| → 0, we can choose r0 = r0 (η) ∈ (0, 1) such that |f1 (t)| ≤ η|t|

for |t| ≤ 2r0 .

(5.2.9)

∗

On the other hand, since f (t) = o(|t|2s −1 ) as |t| → ∞, we can pick r1 = r1 (η) > 2 such that ∗

|f1 (t)| ≤ η|t|2s −1

for |t| ≥ r1 − 1.

(5.2.10)

By the continuity of f , there exists δ = δ(η) ∈ (0, r0 ) such that |f1 (t1 ) − f1 (t2 )| ≤ r0 η

for |t1 − t2 | ≤ δ, |t1 |, |t2 | ≤ r1 + 1.

(5.2.11)

5.2 Nonlinearities with Critical Growth

167

∗

Moreover, since f (t) = o(|t|2s −1 ) as |t| → ∞, there exists a positive constant c = c(η) such that ∗

|f1 (t)| ≤ c(η)|t| + η|t|2s −1

for t ∈ R.

(5.2.12)

In what follows we estimate the integral:  RN \BR

|f1 (un − u) − f1 (un ) − f1 (u)| |w| dx. ∗

Using (5.2.12) and u ∈ L2 (RN ) ∩ L2s (RN ) we can find R = R(η) > 0 such that 

 RN \B

|f1 (u)w| dx ≤ c

2∗s

RN \B

R

|u| dx

2s

wL2∗s (RN )

R

 +c

 2∗s −1 ∗

1 |u| dx 2

RN \B

2

wL2 (RN )

R

≤ cηwH s (RN ) . Set An = {x ∈ RN \ BR : |un (x)| ≤ r0 }. In view of (5.2.9), the Hölder inequality shows that    f1 (un ) − f1 (un − u)|w| dx ≤ η(un  2 N + un − u 2 N )w 2 N L (R )

An ∩{|u|≤δ}

L (R )

≤ cηwH s (RN ) .

L (R )

(5.2.13)

Let Bn = {x ∈ RN \ BR : |un (x)| ≥ r1 }. Then (5.2.10) and Hölder’s inequality yield  Bn ∩{|u|≤δ}

∗   f1 (un ) − f1 (un − u)|w| dx ≤ η(un 2s −1 2∗ s

L

(RN )

+ un − u

2∗s −1 ∗

L2s (RN )

)wL2∗s (RN )

≤ cηwH s (RN ) .

(5.2.14)

Finally, define Cn = {x ∈ RN \ BR : r0 ≤ |un (x)| ≤ r1 }. Since un ∈ H s (RN ), we know that |Cn | < ∞. Then (5.2.11) gives  Cn ∩{|u|≤δ}

  f1 (un ) − f1 (un − u)|w| dx ≤ r0 ηw 2 N |Cn | 12 L (R ) ≤ ηun L2 (RN ) wL2 (RN ) ≤ cηwH s (RN ) .

(5.2.15)

168

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Combining (5.2.13), (5.2.14), and (5.2.15), we have that  (RN \BR )∩{|u|≤δ}

  f1 (un ) − f1 (un − u)|w| dx ≤ cηw s N H (R )

for all n ∈ N. (5.2.16)

Now, we note that (5.2.12) implies ∗

∗

|f1 (un ) − f1 (un − u)| ≤ η(|un |2s −1 + |un − u|2s −1 ) + c(η)(|un | + |un − u|), and consequently  (RN \BR )∩{|u|≥δ}

|f1 (un ) − f1 (un − u)||w| dx



≤

%

(RN \BR )∩{|u|≥δ}

≤ cηwH s (RN ) +

& ∗ ∗ η(|un |2s −1 + |un − u|2s −1 )|w| + c(η)(|un | + |un − u|)|w| dx

 (RN \BR )∩{|u|≥δ}

c(η)(|un | + |un − u|)|w| dx.

Since u ∈ H s (RN ), we get |(RN \ BR ) ∩ {u ≥ δ}| → 0 as R → ∞. Then choosing R = R(η) large enough we infer that  (RN \BR )∩{|u|≥δ}

c(η)(|un | + |un − u|)|w| dx

2∗ s −2 ∗ ≤ c(η) un L2∗s (RN ) + un − uL2∗s (RN ) wL2∗s (RN ) |(RN \ BR ) ∩ {|u| ≥ δ}| 2s ≤ ηwH s (RN ) , where we used the generalized Hölder inequality. Therefore  (RN \BR )∩{|u|≥δ}

  f1 (un ) − f1 (un − u)|w| dx ≤ cηw s N H (R )

for all n ∈ N,

which combined with (5.2.16) yields  RN \BR

  f1 (un ) − f1 (u) − f1 (un − u)|w| dx ≤ cηw s N H (R )

for all n ∈ N. (5.2.17)

Now, recalling that un  u in H s (RN ) we may assume that, up to a subsequence, un → u strongly in L2 (BR ) and there exists h ∈ L2 (BR ) such that |un (x)|, |u(x)| ≤ |h(x)| a.e.

5.2 Nonlinearities with Critical Growth

169

x ∈ BR . It is clear that  BR

|f1 (un − u)||w| dx ≤ cηwH s (RN )

(5.2.18)

as long as n is large enough. Set Dn = {x ∈ BR : |un (x) − u(x)| ≥ 1}. Thus  Dn

  f1 (un ) − f1 (u)|w| dx ≤

 Dn



∗ ∗ c(η)(|u| + |un |) + η(|un |2s −1 + |u|2s −1 ) |w| dx

≤ cηwH s (RN ) + 2c(η)

 Dn

|h||w| dx

 ≤ cηwH s (RN ) + 2c(η)

1 2

Dn

|h| dx

2

wL2 (RN ) .

Observing that |Dn | → 0 as n → ∞, we deduce that  Dn

  f1 (un ) − f1 (u)|w| dx ≤ cηw s N . H (R )

(5.2.19)

Since u ∈ H s (RN ), we know that |{|u| ≥ L}| → 0 as L → ∞, so there exists L = L(η) > 0 such that for all n  (BR \Dn )∩{|u|≥L}

|f1 (un ) − f1 (u)||w| dx



≤

(BR \Dn )∩{|u|≥L}

% & ∗ ∗ η(|un |2s −1 + |u|2s −1 )|w| + c(η)(|un | + |u|)|w| dx

≤ cηwH s (RN ) + c(η)(un L2∗s (RN ) + uL2∗s (RN ) ) wL2∗s (RN ) |(BR \ Dn ) ∩ {|u| ≥ L}| ≤ cηwH s (RN ) .

2∗ s −2 2∗ s

(5.2.20)

On the other hand, by the dominated convergence theorem,  (BR \Dn )∩{|u|≤L}

|f1 (un ) − f1 (u)|2 dx → 0

as n → ∞.

Consequently,  (BR \Dn )∩{|u|≤L}

  f1 (un ) − f1 (u)|w| dx ≤ cηw s N H (R )

(5.2.21)

170

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

for n large enough. Combining (5.2.19), (5.2.20), and (5.2.21) we have  BR

  f1 (un ) − f1 (u)|w| dx ≤ cηw s N . H (R )

This and (5.2.18) yield  BR

  f1 (un ) − f1 (u) − f1 (un − u)|w| dx ≤ cηw s N . H (R )

(5.2.22)

Finally, (5.2.17) and (5.2.22) imply that for n large enough  RN

  f1 (un ) − f1 (u) − f1 (un − u)|w| dx ≤ cηw s N . H (R )  

5.2.3

A Compactness Lemma when the Potential is Constant

In this subsection we provide the proof of Theorem 5.2.1. For simplicity, we may assume that K = 1 in (f 3). Consider the space 





H = u ∈ H (R ) : s

V (x)u dx < ∞ , 2

N

RN

endowed with the norm   u = [u]2s +

1

RN

V (x)u2 dx

2

.

Clearly, H is a Hilbert space with respect to the inner product  u, vH =

R2N

(u(x) − u(y))(v(x) − v(y)) dxdy + |x − y|N+2s

 RN

V (x)uv dx

∀u, v ∈ H.

Using the assumptions (V 2) and (V 3), it is easy to prove that  ·  is equivalent to the standard norm in H s (RN ). In what follows, we use the symbol H ∗ to denote the topological dual of H . The functional I : H → R associated with (5.2.1) is defined as I(u) =

1 u2 − 2

 RN

F (u) dx.

5.2 Nonlinearities with Critical Growth

171

By Theorem 1.1.8 and the assumptions on f , it is clear that I is well defined, I ∈ C 1 (H, R), and the differential of I is given by Iλ (u), ϕ

 = u, ϕH −

RN

f (u)ϕ dx

for all u, ϕ ∈ H . Moreover, the critical points of I are weak solutions to (5.2.1). For λ ∈ [ 12 , 1], we introduce the family of functionals Iλ : H → R defined by Iλ (u) =

1 u2 − λ 2

 RN

F (u) dx.

Clearly, I1 = I. Let us prove that Iλ satisfies the assumptions of Theorem 5.2.2. Lemma 5.2.7 Assume that (V 1)–(V 2) and (f 1)–(f 4) hold. Then, for almost every λ ∈ [ 12 , 1], there is a sequence (un ) ⊂ H such that (i) (un ) is bounded; (ii) Iλ (un ) → cλ = infγ ∈ maxt ∈[0,1] Iλ (γ (t)), where  = {γ ∈ C([0, 1], H ) : γ (0) = 0, γ (1) = v2 } for some v2 ∈ H \ {0} such that Iλ (v2 ) < 0 for all λ ∈ [ 12 , 1]; (iii) Iλ (un ) → 0 on H ∗ . Moreover, if V ∈ L∞ (RN ), then N

s S∗2s . cλ < N λ N−2s 2s

(5.2.23)

1 1 2 Proof We  aim to apply Theorem 5.2.2 with X = H ,  = [ 2 , 1], A(u) = 2 u , and B(u) = RN F (u) dx. Clearly, A(u) → ∞ as u → ∞, and by the assumption (f 4), it follows that B(u) ≥ 0 for any u ∈ H . Now, assumptions (f 1)–(f 3) show that for every ε > 0 there exists Cε > 0 such that ∗

|F (t)| ≤ εt 2 + Cε |t|22

for all t ∈ R.

Then, since λ ∈ [ 12 , 1], Theorem 1.1.8 and (V 2) imply that Iλ (u) ≥

& % 1 2∗ u2 − λ εu2L2 (RN ) + Cε u 22∗s N L (R ) 2 ∗

≥

2 1 ε ∗ − s u2 − u2 − Cε S∗ 2 u22 , 2 V0

172

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

so there exist α > 0 and r > 0, independent of λ, such that Iλ (u) ≥ α > 0

for any u

with u = r.

Condition (f 4) and the fact that λ ∈ [ 12 , 1] imply that Iλ (u) ≤

1 N − 2s + 2∗s D + q u2 − u  2∗s N − u Lq (RN ) L (R ) 2 4N 2q

(5.2.24)

so, taking ϕ ∈ H such that ϕ ≥ 0 and ϕ = 0, we can see that Iλ (tu) → −∞ as t → ∞. Hence, there exists t0 > 0 such that t0 ϕ > r and Iλ (t0 ϕ) < 0 for all λ ∈ [ 12 , 1]. Since Iλ (0) = 0, we set v1 = 0 and v2 = t0 ϕ. Therefore, Iλ satisfies the assumptions of Theorem 5.2.2, and we can find a bounded Palais–Smale sequence for Iλ at the level cλ . Finally, we prove the estimate in (5.2.23). Let η ∈ Cc∞ (RN ) be a cut-off function such that 0 ≤ η ≤ 1, η = 1 on Br and η = 0 on RN \ B2r , where Br denotes the ball in RN of radius r centered at the origin. As in [310], for ε > 0, we define uε (x) = η(x)Uε (x), where N−2s

κε− 2 Uε (x) = ⎛ 2 ⎞ N−2s  2    ⎝μ2 +  x 1  ⎠  ε S 2s  ∗

and κ ∈ R and μ > 0 are fixed constants. We recall (see Remark 1.1.9) that the value S∗ is achieved by Uε . Now we set vε =

uε uε L2∗s (RN )

.

As proved in [174, 310], vε satisfies the following useful estimates: [vε ]2s ≤ S∗ + O(εN−2s ),

vε 2L2 (RN )

(5.2.25)

⎧ 2s ⎪ if N > 4s, ⎨ O(ε ), = O(ε2s | log(ε)|), if N = 4s, ⎪ ⎩ O(εN−2s ), if N < 4s,

(5.2.26)

and  q vε Lq (RN )

=

O(ε

2N−(N−2s)q 2

O(ε

(N−2s)q 2

),

), if q > if q
0 and p ≥ 1, and the assumption that V ∈ L∞ (RN ), we have N

s S∗2s sup Iλ (tvε ) ≤ + O(εN−2s ) + C1 N λ N−2s t ≥τ 2s



q

RN

V (x)vε2 dx − C0 vε Lq (RN )

N

s S∗2s q ≤ + O(εN−2s ) + C2 vε 2L2 (RN ) − C0 vε Lq (RN ) . N λ N−2s 2s

174

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

We distinguish the following cases: If N > 4s, then q ∈ (2, 2∗s ) and, in particular, q > and (5.2.27), we can see that

N N−2s .

Hence, by using (5.2.26)

N 2N−(N−2s)q s S∗2s 2 + O(εN−2s ) + O(ε2s ) − O(ε ). sup Iλ (tvε ) ≤ N−2s N t ≥τ λ 2s

Since 2N − (N − 2s)q < 2s < N − 2s, 2 there exists ε0 > 0 such that for any ε ∈ (0, ε0 ), N

s S∗2s sup Iλ (tvε ) < . N λ N−2s t ≥τ 2s If now N = 4s, then q ∈ (2, 4) and, in particular, q > and (5.2.27) we deduce that

(5.2.31) N N−2s

= 2, so from (5.2.26)

N

s S∗2s sup Iλ (tvε ) ≤ + O(ε2s ) + O(ε2s | log(ε)|) − O(ε4s−sq ). N λ N−2s t ≥τ 2s Since ε4s−sq = ∞, ε→0 ε 2s (1 + | log(ε)|) lim

for any ε sufficiently small we have N

s S∗2s sup Iλ (tvε ) < . N λ N−2s t ≥τ 2s 4s Finally, if 2s < N < 4s, then q ∈ ( N−2s , 2∗s ) and, in particular, q >

observing that

2N−(N−2s)q 2 N

(5.2.32) N N−2s .

Hence,

< N − 2s, we get N

2N−(N−2s)q s S∗2s s S∗2s 2 + O(εN−2s ) + O(εN−2s ) − O(ε )< sup Iλ (tvε ) ≤ N−2s N λ 2s N λ N−2s t ≥τ 2s (5.2.33)

5.2 Nonlinearities with Critical Growth

175

for any ε > 0 small enough. Combining (5.2.28), (5.2.30) and (5.2.31)–(5.2.33), we can conclude that (5.2.23) holds.   Remark 5.2.8 Let us note that for λ ∈ [ 12 , 1], if (un ) ⊂ H is such that un  ≤ C,

Iλ (un ) → cλ ,

Iλ (un ) → 0 in H ∗ ,

− then we may assume that un ≥ 0. In fact, using the equality Iλ (un ), u− n  = μn , un , − ∗ where u = min{u, 0} and μn → 0 in H , and the fact that f (t) = 0 if t ≤ 0, we can infer that  % & s s − 2 (−) 2 un (−) 2 u− dx = μn , u− n + V (x)(un ) n . RN

On the other hand, using the fact that |x − y|(x − − y − ) ≥ |x − − y − |2

∀x, y ∈ R,

we see that 

(−) 2 un (−) 2 u− n dx = s

RN

s

 

≥

R2N

− (un (x) − un (y))(u− n (x) − un (y)) dxdy |x − y|N+2s

R2N

− 2 |u− n (x) − un (y)| 2 dxdy = [u− n ]s . |x − y|N+2s

+ + Then, u− n  = on (1), which also yields that un  ≤ C. Now, we prove that Iλ (un ) → cλ

+ ∗ 2 + 2 and Iλ (un ) → 0 in H as n → ∞. Clearly, un  = un  +on (1). Using the conditions − (f 2)–(f 3), the mean value theorem, the decomposition un = u+ n + un , and Hölder’s inequality, we deduce that

   

 

 RN

F (un ) dx −

RN

 F (u+ n ) dx 

 ≤C

∗

RN

(|un | + |un |2s −1 )|u− n | dx

− ≤ C(u− n L2 (RN ) + un L2∗s (RN ) ) = on (1)

+ ∗ and this shows that Iλ (u+ n ) → cλ . We claim that Iλ (un ) → 0 in H . Fix ϕ ∈ H such that + ϕ ≤ 1. Then, since u− n = un − un , we have

Iλ (un ), ϕ − Iλ (u+ n ), ϕ   s s (−) 2 un (−) 2 ϕ dx − = RN

RN

2 (−) 2 u+ n (−) ϕ dx s

s

176

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

 +

 RN

 =

RN

V (x)un ϕ dx −

RN

V (x)u+ n ϕ dx

2 (−) 2 u− n (−) ϕ dx + s

s

 RN

 −λ

RN

(f (un ) − f (u+ n ))ϕ dx

V (x)u− n ϕ dx − λ

 RN

f (u− n )ϕ dx + μn , ϕ,

for some μn → 0 in H ∗ . Therefore, by (f 2)–(f 3), (V 2), ϕ ≤ 1, and Hölder’s inequality, we get 

 I (un ), ϕ − Iλ (u+ ), ϕ λ n √ − √ − ≤ [u− n ]s [ϕ]s +  V un L2 (RN )  V ϕL2 (RN ) + λ[C1 un L2 (RN ) ϕL2 (RN ) 2∗s −1

+ C2 u− n

∗

L2s (RN )

ϕL2∗s (RN ) ] + μn ∗ ϕ ∗

− 2s −1 ≤ C3 u− + μn ∗ = on (1). n  + C4 un  ∗ Since Iλ (un ) → 0 in H ∗ , we can deduce that Iλ (u+ n ) → 0 in H .

Arguing as in Theorem 3.5.1 (see also [307]), we obtain the following fractional Pohozaev identity: Lemma 5.2.9 For λ ∈ [ 12 , 1], if uλ is a critical point of Iλ , then uλ satisfies the following Pohozaev identity: N − 2s 1 [uλ ]2s + 2 2



 RN

∇V (x) · x u2λ dx = N

RN

  1 λF (u) − V (x)u2λ dx. 2

(5.2.34)

Remark 5.2.10 It is easy to check that if (V 1)–(V 2) and (f 1)–(f 3) hold, then there exists β > 0 independent of λ ∈ [ 12 , 1] such that any nontrivial critical point uλ of Iλ satisfies uλ  ≥ β > 0. In fact, by using (f 1)–(f 3), we can see that for every ε > 0 there exists Cε > 0 such that ∗

|F (t)| ≤ εt 2 + Cε |t|2s

for all t ∈ R.

Taking into account that Iλ (uλ ), uλ  = 0, λ ≤ 1, F ≥ 0, Theorem 1.1.8, (V 1)–(V 2), we have  ∗ 2 F (uλ ) dx ≤ εC1 uλ 2 + Cε C2 uλ 2s uλ  = λ RN

where C1 , C2 > 0 depend only on V0 and the best constant S∗ . Choosing ε > 0 sufficiently small and by using that uλ = 0, we deduce that there exists β > 0 such that uλ  ≥ β > 0.

5.2 Nonlinearities with Critical Growth

177

Now, we establish the following compactness lemma which will be useful in proving Theorem 5.2.1. Lemma 5.2.11 Assume that V (x) ≡ V and f satisfies (f 1)–(f 4). For λ ∈ [ 12 , 1], let (un ) ⊂ H be a bounded sequence in H such that un ≥ 0, Iλ (un ) → cλ , Iλ (un ) → 0 in H ∗ . Moreover, N

s S∗2s cλ < . N λ N−2s 2s Then there exist a subsequence of (un ), which we denote again by (un ), and an integer j k ∈ N ∪ {0} and wλ ∈ H for 1 ≤ j ≤ k, such that (i) un  uλ in H and Iλ (uλ ) = 0, j j (ii) wλ =  0 and Iλ (wλ ) = 0 for 1 ≤ j ≤ k, k j (iii) cλ = Iλ (uλ ) + j =1 Iλ (wλ ), j

where we agree that in the case k = 0, the above holds without wλ . Proof We divide the proof into five steps. Step 1 Passing to a subsequence if necessary, we can assume that un  uλ in H , where uλ is a critical point of Iλ . Since (un ) is bounded in H , up to a subsequence, we can suppose that un  uλ in H , and, in view of Theorem 1.1.8, un → uλ in Lrloc (RN ) for all r ∈ [1, 2∗s ). Then, for any ϕ ∈ Cc∞ (RN ), Iλ (un ), ϕ − Iλ (uλ ), ϕ =un − uλ , ϕH − λ  −λ

 RN

[g(un ) − g(u)]ϕ dx

∗

RN

∗

(|un |2s −2 un − |uλ |2s −2 uλ )ϕ dx, (5.2.35)

∗

where g(t) = f (t) − (t + )2s −1 . Since un  uλ in H , we get un − uλ , ϕH → 0. ∗

(5.2.36) 2∗ s ∗

∗

Moreover, the sequence (|un |2s −2 un − |uλ |2s −2 uλ ) is bounded in L 2s −1 (RN ) and ∗

∗

|un |2s −2 un → |uλ |2s −2 uλ

a.e. in RN ,

178

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

so we conclude that  ∗ ∗ (|un |2s −2 un − |uλ |2s −2 uλ )ϕ dx → 0.

(5.2.37)

RN

Using Lemma 1.4.2 and conditions (f 2) and (f 3), we infer that  RN

[g(un ) − g(u)]ϕ dx → 0.

(5.2.38)

Combining (5.2.35), (5.2.36), (5.2.37), (5.2.38) and the fact that Iλ (un ) → 0, we see that Iλ (uλ ), ϕ = 0 for any ϕ ∈ Cc∞ (RN ). Since Cc∞ (RN ) is dense in H s (RN ), we deduce that Iλ (uλ ) = 0, that is, (i) is satisfied. Now, we set vn1 = un − uλ .  Step 2 If limn→∞ supz∈RN B1 (z) |vn1 |2 dx = 0, then un → uλ in H and Lemma 5.2.11 holds with k = 0. In view of Lemma 1.4.4, ∀t ∈ (2, 2∗s ).

vn1 → 0 in Lt (RN ),

(5.2.39)

Now, we observe that Iλ (un ), vn1  = un , vn1 H − λ



 RN

f (un )vn1 dx = vn1 2 + uλ , vn1 H − λ

RN

f (un )vn1 dx,

that is, vn1 2 = Iλ (un ), vn1  − uλ , vn1 H + λ

 RN

f (un )vn1 dx.

By Iλ (uλ ), vn1  = 0, Iλ (un ), vn1  = o(1) and the definition of g, we have vn1 2 = Iλ (un ), vn1  + λ  =λ

RN

 RN

(f (un ) − f (uλ ))vn1 dx 

(g(un ) − g(uλ ))vn1 dx + λ

RN

∗

∗

(|un |2s −2 un − |uλ |2s −2 uλ )vn1 dx + o(1).

Now, by (f 1)–(f 3), we know that for every ε > 0 there exists Cε > 0 such that ∗

|g(t)| ≤ ε(|t| + |t|2s −1 ) + Cε |t|q−1

for all t ∈ R.

(5.2.40)

5.2 Nonlinearities with Critical Growth

179

Therefore, taking into account (5.2.39) and (5.2.40), we see that  vn1 2 = λ

∗

RN

∗

(|un |2s −2 un − |uλ |2s −2 uλ )vn1 dx + o(1).

By Lemma 5.2.4, it follows that    

RN

  ∗ ∗ ∗ [|un |2s −2 un − |uλ |2s −2 uλ − |un − uλ |2s −2 (un − uλ )]ϕ dx  = o(1)ϕ,

∀ϕ ∈ H.

(5.2.41) Taking ϕ = vn1 = un − uλ in (5.2.41), we deduce that  vn1 2

∗

|vn1 |2s dx + o(1).

(5.2.42)

vn1 2 = un 2 − uλ 2 + o(1)

(5.2.43)

=λ

RN

Now Brezis-Lieb lemma [113] yields

and  RN

∗



|vn1 |2s dx =

∗

RN

|un |2s dx −



∗

RN

|uλ |2s dx + o(1),

(5.2.44)

G(uλ ) dx + o(1).

(5.2.45)

and applying Lemma 5.2.5 we obtain  RN

 G(vn1 ) dx =

 RN

G(un ) dx −

RN

Next, combining (5.2.43)–(5.2.45), we have cλ − Iλ (uλ ) = Iλ (un ) − Iλ (uλ ) + o(1)    1 λ 2∗s 2 un  − λ = G(un ) dx − ∗ un  2∗s N L (R ) 2 2s RN    λ 1 2∗s 2 uλ  − λ G(uλ ) dx − ∗ uλ  2∗s N + o(1) − L (R ) 2 2s RN  1 λ 2∗ = vn1 2 − λ G(vn1 ) dx − ∗ vn1  s2∗s N + o(1). L (R ) 2 2s RN

(5.2.46)

(5.2.47)

180

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Using (5.2.39), (5.2.40), and (5.2.46) we infer that cλ − Iλ (uλ ) =

1 1 2 λ v  − ∗ 2 n 2s



∗

RN

|vn1 |2s dx + o(1).

Since Iλ (uλ ) = 0, it follows from Lemma 5.2.9 that Iλ (uλ ) = in view of (5.2.48), we get

(5.2.48) s 2 N [uλ ]s

≥ 0. Then,

N

s S∗2s cλ − Iλ (uλ ) < . N λ N−2s 2s Now, we may assume that vn1 2 → L ≥ 0. By (5.2.42), it follows that 2∗

λvn1  s2∗s N → L. L (R ) Suppose that L > 0. Then, by Theorem 1.1.8, vn1 2 2∗s

L (RN )

S∗ ≤ vn1 2 ,

N

so we can deduce that L ≥

S∗2s λ

N−2s 2s

. This fact and (5.2.48) yield N

s S∗2s s cλ − Iλ (uλ ) = L ≥ N N λ N−2s 2s which gives a contradiction. Hence, vn1  → 0 as n → ∞. Step 3 If there exists (zn ) ⊂ RN such that B1 (zn ) |vn1 |2 dx → d > 0, then, up to a subsequence, the following conditions hold: (1) |zn | → ∞, (2) un (· + zn )  wλ = 0 in H , (3) Iλ (wλ ) = 0. We may assume that there exists (zn ) ⊂ RN such that  B1 (zn )

|vn1 |2 dx ≥

d > 0. 2

Set v˜n1 (x) = vn1 (x+zn ). Then v˜n1 is bounded in H and we may assume that v˜n1  v˜ 1 in H . Since  d |v˜n1 |2 dx ≥ , 2 B1

5.2 Nonlinearities with Critical Growth

181

we get  |v˜ 1 |2 dx ≥ B1

d , 2

that is, v˜ 1 = 0. Using the fact that vn1  0 in H , we deduce that (zn ) is unbounded, so we may assume that |zn | → ∞. Now, we set u˜ n (x) = un (x + zn )  wλ = 0. As in Step 1, we can see that Iλ (u˜ n ), ϕ−Iλ (wλ ), ϕ → 0, for all ϕ ∈ Cc∞ (RN ). On the other hand, since |zn | → ∞, we have Iλ (u˜ n ), ϕ = Iλ (un ), ϕ(· − zn ) → 0 for all ϕ ∈ Cc∞ (RN ), so we conclude that Iλ (wλ ), ϕ = 0, for all ϕ ∈ Cc∞ (RN ). Step 4 If there exist m ≥ 1, (ynk ) ⊂ RN , wλk ∈ H for 1 ≤ k ≤ m such that (i) |ynk | → ∞, |ynk − ynh | → ∞ if k = h, (ii) un (· + ynk )  wλk = 0 in H , for any 1 ≤ k ≤ m, (iii) wλk ≥ 0 and Iλ (wλk ) = 0 for any 1 ≤ k ≤ m, then one of the following conclusions must hold:     k k 2 (1) If supz∈RN B1 (z) un − u0 − m k=1 wλ (· − yn ) dx → 0, then   m     k k  wλ (· − yn ) → 0. un − u0 −   k=1

(2) If there exists (zn ) ⊂ RN such that 

2  m     k k  wλ (· − yn ) dx → d > 0, u n − u 0 −  B1 (zn )  k=1

then up to a subsequence, the following conditions hold: (i) |zn | → ∞, |zn − ynk | → ∞ for any 1 ≤ k ≤ m, (ii) un (· + zn )  wλm+1 = 0 in H , (iii) wλm+1 ≥ 0 and Iλ (wλm+1 ) = 0.  k k Assume that (1) holds. Set ξn = un − u0 − m k=1 wλ (· − yn ). Then, using Lemma 1.4.4, we see that in Lt (RN ) for all t ∈ (2, 2∗s ).

ξn → 0

(5.2.49)

By the definition of ξn and the fact that Iλ (uλ ), ξn  = 0 = Iλ (wλk ), ξn , ξn 2 = Iλ (un ), ξn  + λ

 RN

m 

f (un ) − f (uλ ) ξn dx − λ



N k=1 R

f (wλk )ξn (· + ynk ) dx.

182

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

In view of (5.2.40) and (5.2.49), we deduce that  ξn 2 = λ

RN

m 

 ∗ ∗ |un |2s −2 un − |uλ |2s −2 uλ ξn dx − λ

∗

N k=1 R

|wλk |2s −2 wλk ξn (· + ynk ) dx + o(1).

Recalling (5.2.41), we observe that  ξn 2 = λ

∗

|un − uλ |2s −2 (un − uλ )ξn dx − λ

RN

−λ  =λ

m  

|wλ1 |2s −2 wλ1 ξn (· + yn1 ) dx

|wλk |2s −2 wλk ξn (· + ynk ) dx + o(1)

 2∗ −2

 s un (· + yn1 ) − uλ (· + yn1 ) ξn (· + yn1 ) dx un (· + yn1 ) − uλ (· + yn1 )



−λ

∗

RN

∗

N k=2 R

RN



RN

∗

|wλ1 |2s −2 wλ1 ξn (· + yn1 ) dx − λ

m   N k=2 R

∗

|wλk |2s −2 wλk ξn (· + ynk ) dx + o(1).

Since |yn1 | → ∞, and un (· + yn1 )  wλ1 in H , we deduce that un (· + yn1 ) − uλ (· + yn1 )  wλ1 in H . Consequently,  ξn 2 = λ

RN

−λ

 2∗ −2

 s un (· + yn1 ) − uλ (· + yn1 ) − wλ1 ξn (· + yn1 ) dx un (· + yn1 ) − uλ (· + yn1 ) − wλ1 

m   N k=2 R

∗

|wλk |2s −2 wλk ξn (· + ynk ) dx + o(1).

Iterating this procedure, we conclude that  ξn  = λ

∗

2

RN

|ξn |2s dx + o(1).

(5.2.50)

Now, since un (· + yn1 ) − uλ (· + yn1 )  wλ1 in H , we can argue as in Step 2 to see that cλ − Iλ (uλ ) = =

1 un − uλ 2 − λ 2

 RN

λ 2∗s 

G(un − uλ ) dx −

 RN

∗

|un − uλ |2s dx + o(1)

1 un (· + yn1 ) − uλ (· + yn1 ) − wλ1 2 − λ G(un (· + yn1 ) − uλ (· + yn1 ) − wλ1 ) dx 2 RN   2∗s λ   − ∗ un (· + yn1 ) − uλ (· + yn1 ) − wλ1  dx + Iλ (wλ1 ) + o(1). 2s RN

5.2 Nonlinearities with Critical Growth

183

Continuing this process, we obtain that cλ − Iλ (uλ ) −

m 

Iλ (wλk ) =

k=1

1 ξn 2 − λ 2

 RN

G(ξn ) dx −

λ 2∗s



∗

RN

|ξn |2s dx + o(1), (5.2.51)

which together with (5.2.49) yields cλ − Iλ (uλ ) −

m 

Iλ (wλk ) =

k=1

1 λ ξn 2 − ∗ 2 2s



∗

RN

|ξn |2s dx + o(1).

(5.2.52)

Then, taking into account (5.2.50) and (5.2.52), we can argue as in Step 2 to infer that   m      wλk (· − ynk ) = ξn  → 0 as n → ∞.  un − u0 −   k=1

Now, we assume that (2) holds. The proof of this is standard (see [236]), so we skip the details here. Step 5 Conclusion. Using Step 1, we can verify that Lemma 5.2.11 (i) holds. If the assumption of Step 2 holds, then Lemma 5.2.11 holds with k = 0. Otherwise, the assumption of Step 3 holds. We set (yn1 ) = (zn ) and wλ1 = wλ . Now, if (1) of Step 4 holds with m = 1, from (5.2.52), we obtain the conclusion of Lemma 5.2.11. Otherwise, item (2) of Step 4 must hold, and by setting (yn2 ) = (zn ), wλ2 = wλ2 , we iterate Step 4. Then, to conclude the proof, we have to show that item (1) of Step 4 must occur after a finite number of iterations. Note that, for all m ≥ 1, we have  lim

2

n→∞

2

un  − uλ  −

m 

 wλk 2

k=1

2  m     k k  wλ (· − yn ) ≥ 0. = lim un − uλ − n→∞   k=1

In fact, by using items (i) and (ii) of Step 4 and the fact that un  uλ in H , we see that  2 m      wλk (· − ynk ) un − uλ −   k=1

= un 2 + uλ 2 +

m  k=1

wλk 2 − 2un , uλ H − 2

m  k=1

un , wλk (· − ynk )H

184

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

+2

m 

uλ , wλk (· − ynk )H + 2

k=1

 wλh (· − ynh ), wλk (· − ynk )H h,k

= un 2 − uλ 2 −

m 

wλk 2 + o(1).

(5.2.53)

k=1

On the other hand, by Remark 5.2.10, wλk  ≥ β for some β > 0 that does not depend on λ. Thus, using (5.2.53) and the fact that (un ) is bounded in H , we deduce that (1) in Step 4 must occur after a finite number of iterations. This together with (5.2.52) shows that Lemma 5.2.11 holds.   Before giving the proof of the main result of this subsection, we prove the following lemma. Lemma 5.2.12 Under the assumptions of Theorem 5.2.1, for almost every λ ∈ [ 12 , 1], Iλ has a positive critical point. Proof By Lemma 5.2.7, for almost every λ ∈ [ 12 , 1] there exists a bounded sequence (un ) ⊂ H such that Iλ (un ) → 0

Iλ (un ) → cλ ,

in H ∗ .

(5.2.54)

In the light of Remark 5.2.8, we may assume that un ≥ 0 in H . In addition, ⎞ N 2s S s ∗ ⎠. cλ ∈ ⎝0, N λ N−2s 2s ⎛

Then, up to a subsequence, we may assume that un  uλ in H . If uλ = 0, then we are finished. Otherwise, we may assume that un  0 in H . Now, we aim to show that there exists δ > 0 such that  |un |2 dx ≥ δ > 0. (5.2.55) lim sup n→∞

y∈RN

B1 (y)

Suppose (5.2.55) fails. Then, by Lemma 1.4.4, un → 0 in Lt (RN ), ∀t ∈ (2, 2∗s ).

(5.2.56)

5.2 Nonlinearities with Critical Growth

185

Using (5.2.40) and (5.2.56), we see that o(1). This and (5.2.54) yield 1 λ un 2 − ∗ 2 2s

 RN

G(un ) dx = o(1) and



∗

RN

|un |2s dx = cλ + o(1)

 RN

g(un )un dx =

(5.2.57)

and  un 2 − λ

∗

RN

|un |2s dx = o(1).

(5.2.58)

Since cλ > 0, we may assume that un 2 → L for some L > 0. By the Sobolev embedding, we can infer that N

L≥

S∗2s λ

N−2s 2s

,

which in conjunction with (5.2.57) and (5.2.58) implies that N

s S∗2s cλ ≥ ; N λ N−2s 2s we reached a contradiction. Thus, (5.2.55) holds true, and we can find a sequence (yn ) ⊂  RN such that |yn | → ∞ and B1 (yn ) |un |2 dx ≥ 2δ > 0. Set vn = un (· + yn ). Using (5.2.54), we derive that Iλ (vn ) → cλ and Iλ (vn ) → 0. In view of (5.2.55), we can deduce that vn  vλ = 0 in H and Iλ (vλ ) = 0. Since Iλ (vλ ), vλ−  = 0, it is easy to check that vλ ≥ 0 in RN . By Lemma 3.2.14 and combining Proposition 1.3.1 with Proposition 1.3.2, we deduce that vλ ∈ C 1,α (RN ). Finally, by   Theorem 1.3.5, we obtain that vλ > 0 in RN . Now, we are ready to prove the existence of a positive ground state to (5.2.1) when V is constant. Proof of Theorem 5.2.1 Using Lemma 5.2.12, for almost every λ ∈ [ 12 , 1], there exists   N (un ) ⊂ H such that un ≥ 0 in H , Iλ (un ) → cλ ∈ 0, Ns

S∗2s

λ

N−2s 2s

un  uλ > 0 in H . In view of Lemma 5.2.11, we can see that cλ = Iλ (uλ ) +

k  j =1

j

Iλ (wλ ),

, Iλ (un ) → 0 in H ∗ , and

186

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Iλ (uλ ) = 0 and Iλ (wλ ) = 0 for 1 ≤ j ≤ k. Then Lemma 5.2.9 implies that Iλ (uλ ) > 0 j and Iλ (wλ ) ≥ 0 for 1 ≤ j ≤ k, so we have cλ ≥ Iλ (uλ ) > 0. Thus, there exists (λn ) ⊂ [ 12 , 1] such that λn → 1, uλn ∈ H , uλn > 0, Iλ n (uλn ) = 0, cλn ≥ Iλn (uλn ) > 0   N j

and cλn ∈ 0, Ns

S∗2s

. Using the fact that Iλ n (uλn ) = 0 and Lemma 5.2.9, we infer that

N−2s λn 2s

cλn ≥ Iλn (uλn ) =

s [uλn ]2s > 0. N

Moreover, in view of (1.1.1) we have uλn L2∗s (RN ) ≤ C for all n ∈ N. Putting together (f 1)–(f 3) and Lemma 5.2.9, we see that for every ε > 0 there exists Cε > 0 such that 1 N − 2s [uλn ]2s + 2N 2



 RN

V u2λn dx = λn

RN

F (uλn ) dx ≤ εuλn 2L2 (RN ) + Cε uλn 

2∗s

∗

L2s (RN )

,

which implies that V 2

 RN

u2λn dx ≤ εuλn 2L2 (RN ) + Cε C.

Therefore, choosing ε ∈ (0, V2 ), we deduce that (uλn ) is bounded in H . Now, we may assume that limn→∞ Iλn (uλn ) exists. Since the map λ → cλ is left continuous (see Theorem 5.2.2), we have 0 ≤ lim Iλn (uλn ) ≤ c1 < n→∞

N s 2s S∗ . N

Then, using the fact that  I(uλn ) = Iλn (uλn ) + (λn − 1)

RN

F (uλn ) dx

and uλn  ≤ C, we infer that 0 ≤ lim I(uλn ) ≤ c1 < n→∞

s 2sN S∗ N

(5.2.59)

and lim I (uλn ) = 0.

n→∞

(5.2.60)

In view of Remark 5.2.10, there exists β > 0 independent of λn such that uλn  ≥ β. Moreover, we know that (uλn ) is bounded in H , so we can argue as in the proof of Lemma 5.2.12 to obtain the existence of a positive solution u0 to (5.2.1).

5.2 Nonlinearities with Critical Growth

187

By Lemma 5.2.11, we can also see that I(u0 ) ≤ lim I(uλn ) ≤ c1 < n→∞

N s 2s S∗ . N

Let us define m = inf{I(u) : u ∈ H, u = 0, I (u) = 0}. Since I (u0 ) = 0, we have that m ≤ I(u0 ) < N 2s

N

s 2s N S∗ ,

and using Lemma 5.2.9, we get that

0 ≤ m < Ns S∗ . By the definition of m, we can find (un ) ⊂ H such that I(un ) → m and I (un ) = 0. Taking into account Remark 5.2.10, we deduce that un  ≥ β > 0 for some β independent of n. Moreover, it is easy to see that (un ) is bounded in H . In virtue of Remark 5.2.8, we may assume that un ≥ 0 in H . Then, bearing in mind that un  ≥ β > 0, we can proceed as in the proof of Lemma 5.2.12 to show that there exists a sequence (vn ) ⊂ H such that vn ≥ 0 in H , vn  v0 = 0 in H , I(vn ) → m and I (vn ) = 0. Using Lemma 5.2.11, we infer that I(v0 ) ≤ m and I (v0 ) = 0. Since I (v0 ) = 0, we also have I(v0 ) ≥ m. Consequently, v0 ≥ 0 in RN , v0 ≡ 0, and satisfies I(v0 ) = m and I (v0 ) = 0. By Lemma 3.2.14 and combining Proposition 1.3.1 with Proposition 1.3.2, we deduce that v0 ∈ C 1,α (RN ). Finally, by using Theorem 1.3.5, we   obtain that v0 > 0 in RN .

5.2.4

Ground States when the Potential is Not Constant

In this subsection we establish the existence of a ground state to (5.2.1) under the assumption that V is a non-constant potential. Thus, we will assume that V (x) ≡ V∞ . For λ ∈ [ 12 , 1], we introduce the following family of functionals defined for u ∈ H : Iλ∞ (u) =

1 2 [u]s + V∞ u2L2 (RN ) − λ 2

 RN

F (u) dx.

Arguing as in the proof of Theorem 3.2.13 we can derive the following result. Lemma 5.2.13 For λ ∈ [ 12 , 1], if wλ ∈ H is a nontrivial critical point of Iλ∞ , then there exists γλ ∈ C([0, 1], H ) such that γλ (0) = 0, I ∞ 1 (γλ (1)) < 0, wλ ∈ γλ ([0, 1]), 0∈ / γλ ((0, 1]) and maxt ∈[0,1] Iλ∞ (γλ (t)) = Iλ∞ (wλ ).

2

Proof We only give a sketch of the proof. Let us define  γλ (t)(x) =

wλ ( xt ), for t > 0, 0, for t = 0.

188

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Then, γλ (t)2 = t N−2s [wλ ]2s + t N V∞ wλ 2L2 (RN ) , Iλ∞ (γλ (t)) =

t N−2s tN [wλ ]2s + V∞ wλ 2L2 (RN ) − t N λ 2 2

 RN

F (wλ ) dx.

(5.2.61)

Hence, γλ ∈ C([0, ∞), H ). By Lemma 5.2.9, we know that  λ

RN

F (wλ ) dx =

N − 2s 1 [wλ ]2s + V∞ wλ 2L2 (RN ) , 2N 2

(5.2.62)

which together with (5.2.61) and (5.2.62) yields Iλ∞ (γλ (t))

 =

N − 2s t N−2s − tN 2 2N

 [wλ ]2s .

Then the proof follows along the lines of Theorem 3.2.13.

 

Remark 5.2.14 By Theorem 5.2.1, we know that if (f 1)–(f 4) hold, then for λ ∈ [ 12 , 1], Iλ∞ has a ground state. Lemma 5.2.15 Under the same assumptions of Theorem 5.2.3, for almost every λ ∈ [ 12 , 1], the functional Iλ has a positive critical point. Proof In view of Lemma 5.2.7 and Remark 5.2.8, we may assume that for almost every λ ∈ [ 12 , 1], there exists a sequence (un ) ⊂ H such that un ≥ 0 in H , un  uλ in H , N

Iλ (un ) → cλ ∈ (0, Ns

S∗2s

N−2s λ 2s

) and Iλ (un ) → 0 in H ∗ .

Now, let us prove that uλ = 0. We argue by contradiction and suppose that uλ = 0. As in the proof of Lemma 5.2.12, we can find a sequence (yn ) ⊂ RN such that |yn | → ∞ and vn = un (· + yn )  vλ = 0 in H . Furthermore, using the fact that un  0 in H , we can see that Iλ∞ (un ) → cλ and (Iλ∞ ) (un ) → 0. Thus, Iλ∞ (vn ) → cλ and (Iλ∞ ) (vn ) → 0. Since vn  vλ = 0 in H , we have that (Iλ∞ ) (vλ ) = 0. In view of Lemma 5.2.11, we get cλ ≥ Iλ∞ (vλ ). It follows from Remark 5.2.14 that Iλ∞ has a ground state wλ . Thus, cλ ≥ Iλ∞ (wλ ). By Lemma 5.2.13, we can find a path γλ ∈ C([0, 1], H ) such that γλ (0) = 0, / γλ ((0, 1]) and maxt ∈[0,1] Iλ∞ (γλ (t)) = Iλ∞ (wλ ). I∞ 1 (γλ (1)) < 0, wλ ∈ γλ ([0, 1]), 0 ∈ 2

Consequently, cλ ≥ Iλ∞ (wλ ) = max Iλ∞ (γλ (t)). t ∈[0,1]

(5.2.63)

5.2 Nonlinearities with Critical Growth

189

Taking into account (V 3), V ≡ V∞ and 0 ∈ / γλ ((0, 1]), we see that Iλ (γλ (t)) < Iλ∞ (γλ (t)) for all t ∈ (0, 1]. Now, we take v1 = 0 and v2 = γλ (1) in Theorem 5.2.2. Then, using the definition of cλ and (5.2.63), we see that cλ ≤ max Iλ (γλ (t)) < max Iλ∞ (γλ (t)) ≤ cλ , t ∈[0,1]

t ∈[0,1]

(5.2.64)

which gives a contradiction. Consequently, uλ = 0 and, in particular, uλ ≥ 0 in RN . Now, by (V 2), we see that vλ is a weak subsolution of (−)s uλ + V0 uλ = λf (uλ ) in RN , and using Lemma 3.2.14 we deduce that uλ ∈ L∞ (RN ). Propositions 1.3.1 and 1.3.2 imply that uλ ∈ C 1,α (RN ), and by applying Theorem 1.3.5 we get uλ > 0 in RN .   At this point we establish the following lemma which will play a fundamental role in the proof of Theorem 5.2.3. Lemma 5.2.16 Assume that (V 1)–(V 3) and (f 1)–(f 4) are satisfied. For λ ∈ [ 12 , 1], let (un ) ⊂ H   be a bounded sequence in H such that un ≥ 0 in H , Iλ (un ) → cλ ∈ N

0, Ns

S∗2s λ

N−2s 2s

and Iλ (un ) → 0 in H ∗ .

Then there exists a subsequence of (un ), which we denote again by (un ), such that (i) un  u in H and Iλ (uλ ) = 0, (ii) Iλ (uλ ) ≤ cλ . Proof Since (un ) is bounded in H , up to a subsequence, we may suppose that un  uλ in H . Then, proceeding as in the proof of Step 1 in Lemma 5.2.11, and using the assumption (V 3), we can see that Iλ (uλ ) = 0, that is, (i) is satisfied. Set wn1 = un − uλ . Similarly to the proof of Lemma 5.2.11, we deduce that cλ − Iλ (uλ ) =

1 1 2 w  − λ 2 n

 RN

G(wn1 ) dx −

λ 2∗ wn1  s2∗s N + o(1). ∗ L (R ) 2s

(5.2.65)

Using Lemma 5.2.6, we see that for any ϕ ∈ H    

RN



 g(un ) − g(uλ ) − g(wn1 ) ϕ dx  = o(1)ϕ.

Next, for λ ∈ [ 12 , 1], we introduce the following functionals on H : Hλ (u) =

1 u2 − λ 2

 RN

G(u) dx −

λ 2∗s



∗

RN

|u|2s dx,

(5.2.66)

190

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Hλ∞ (u)

1 = 2

Jλ∞ (u) =

1 2

 

RN

RN



s 2 2 2 |(−) u| + V∞ u dx − λ

RN

λ G(u) dx − ∗ 2s



s λ ∗ |(−) 2 u|2 + V∞ u2 dx − ∗ |u|2s dx. 2s RN



∗

RN

|u|2s dx,

Thus (5.2.65) becomes cλ − Iλ (uλ ) = Hλ (wn1 ) + o(1).

(5.2.67)

Using (5.2.41) and (5.2.66) we have, for any ϕ ∈ H , |Iλ (un ) − Iλ (uλ ), ϕ − Hλ (wn1 ), ϕ| = o(1)ϕ,

(5.2.68)

which gives Hλ (wn1 ) = o(1).

(5.2.69)

Taking into account (5.2.67), (5.2.69), (V 3) and the fact that wn1  0 in H , we see that cλ − Iλ (uλ ) = Hλ∞ (wn1 ) + o(1)

(5.2.70)

(Hλ∞ ) (wn1 ) = o(1).

(5.2.71)

and

Now we distinguish two cases.  Case 1: limn→∞ supy∈RN B1 (y) |wn1 |2 dx = 0. By Lemma 1.4.4, wn1 → 0

in Lt (RN ), ∀t ∈ (2, 2∗s ).

Combining (5.2.40) and (5.2.70)–(5.2.72), we deduce that cλ − Iλ (uλ ) = Jλ∞ (wn1 ) + o(1)

and

Jλ (wn1 ) = o(1)

which gives cλ − Iλ (uλ ) = and then cλ ≥ Iλ (uλ ).

λs 1 2∗s wn  2∗s N + o(1), L (R ) N

(5.2.72)

5.2 Nonlinearities with Critical Growth

191



1 2 B1 (y) |wn | dx ≥ δ1 for some δ1 > 0.  Thus, there exists yn1 ∈ RN , |yn1 | → ∞, such that B1 (y 1) |wn1 |2 dx n wn1 (· + yn1 )  wλ1 = 0 in H ,

Case 2: limn→∞ supy∈RN

δ1 2.

≥

cλ − Iλ (uλ ) = Hλ∞ (wn1 (· + yn1 )) + o(1)

Then

(5.2.73)

and (Hλ∞ ) (wn1 (· + yn1 )) = o(1).

(5.2.74) N

By (5.2.74) we have (Hλ∞ ) (wn1 ) = 0. Now, if cλ − Iλ (uλ )
0.

(5.2.78)

lim sup

n→∞

y∈RN

B1 (y)

or there exists δ2 > 0 such that  lim sup

n→∞

y∈RN

B1 (y)

Suppose (5.2.77) holds. Then, by Case 1, we deduce that cλ − Iλ (uλ ) − Hλ∞ (wλ1 ) ≥ 0, and by Lemma 5.2.9 we get Hλ∞ (wλ1 ) ≥ 0. This two facts give cλ − Iλ (uλ ) ≥ 0.

192

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Now, suppose (5.2.78) holds. Repeating this procedure, we can find wni ∈ H , yni ∈ RN , |yni | → ∞, i ∈ N, such that wni (· + yni )  wλi = 0 in H , (Hλ∞ ) (wλi ) = 0, cλ − Iλ (uλ ) −

j 

j +1

Hλ∞ (wλi ) + o(1) = Hλ∞ (wn

(5.2.79)

)

i=1

and j +1

) = o(1),

j

j

(Hλ∞ ) (wn

(5.2.80)

where j +1

wn

j

= wn (· + yn ) − wλ ,

j ∈ N.

Since (Hλ∞ ) (wλi ) = 0, Lemma 5.2.9 implies that Hλ∞ (wλi ) =

s i 2 [w ] . N λs

(5.2.81)

Now we show that there exists α > 0 independent of i such that [wλi ]s ≥ α.

(5.2.82)

In fact, using that (Hλ∞ ) (wλi ) = 0, λ ∈ [ 12 , 1], and conditions (f 1)–(f 3), we can see that for any ε > 0 there exists Cε > 0 such that  RN



s |(−) 2 wλi |2 + V∞ |wλi |2 dx ≤ ε ≤

 |wλi |2 dx + Cε

RN

ε V∞

 RN

∗

RN

|wλi |2s dx

V∞ |wλi |2 dx + Cε



∗

RN

|wλi |2s dx.

Choosing ε ∈ (0, V∞ ), we can infer that 2∗s

[wλi ]2s ≤ Cwλi 

∗

L2s (RN )

,

which together with (1.1.1) gives (5.2.82). Then, combining (5.2.81) and (5.2.82), for some j = k, we obtain that cλ − Iλ (uλ ) −

j 

N

Hλ∞ (wλi )

i=1

The conclusion follows by applying Lemma 5.2.11.

s S∗2s < . N λ N−2s 2s  

5.2 Nonlinearities with Critical Growth

193

We end this subsection by giving the Proof of Theorem 5.2.3 In view of Lemma 5.2.15, for almost every λ ∈ [ 12 , 1], there   N exists a sequence (un ) ⊂ H such that Iλ (un ) → cλ ∈ 0, Ns

S∗2s

λ

N−2s 2s

, Iλ (un ) → 0 in H ∗ ,

un  uλ = 0 in H . By Lemma 5.2.16, we deduce that Iλ (uλ ) ≤ cλ and Iλ (uλ ) = 0. Hence, we can find N

a sequence (λn ) ⊂ [ 12 , 1] such that λn → 1, cλn ∈ (0, Ns

S∗2s λ

N−2s 2s

), uλn ∈ H such that

Iλ n (uλn ) = 0, Iλn (uλn ) ≤ cλn . Let us show that there exists a positive constant C such that uλn  ≤ C

for all n ∈ N.

(5.2.83)

By (V 4), there exists θ ∈ (0, 2s) such that  max{x · ∇V , 0}

N

L 2s (RN )

≤ θ S∗ .

(5.2.84)

Since Iλn (uλn ) ≤ c 1 and Iλ n (uλn ) = 0, Lemma 5.2.9, the Hölder inequality, Theo2 rem 1.1.8 and (5.2.84) imply that s[uλn ]2s =

N N [uλn ]2s + 2 2



1 = N Iλn (uλn ) + 2

RN

V (x)u2λn dx +



RN

1 2



 RN

x · ∇V (x)u2λn dx − λn N

x · ∇V (x)u2λn dx ≤ Nc 1 + 2

θ [uλ ]2 , 2 n s

RN

F (uλn ) dx

(5.2.85)

and consequently [uλn ]s ≤ C for any n ∈ N. Next, combining Iλ n (uλn ) = 0, λn ∈ [ 12 , 1], (f 1)–(f 3) and (1.1.1), we see that for any ε ∈ (0, V0 )  V0 uλn 2L2 (RN ) ≤  ≤

RN

RN



s |(−) 2 uλn |2 + V (x)u2λn dx f (uλn )uλn dx 2∗s

≤ εuλn 2L2 (RN ) + Cε uλn 

∗

L2s (RN )

2∗

≤ εuλn 2L2 (RN ) + Cε [uλn ]s s .

(5.2.86)

194

5 Ground States for a Superlinear Fractional Schrödinger Equation with. . .

Using [uλn ]s ≤ C, the estimate (5.2.86) yields uλn L2 (RN ) ≤ C for all n ∈ N. In view of conditions (V 2) and (V 3), we deduce that  0≤

RN

V (x)u2λn dx ≤ V∞ uλn 2L2 (RN ) ≤ V∞ C 2 ,

which completes the proof of (5.2.83). Now, we note that  I(uλn ) = Iλn (uλn ) + (λn − 1)

RN

F (uλn ) dx,

so we can infer that lim I(uλn ) ≤ c1
0 independent of λn such that uλn  ≥ β. Since uλn  ≤ C for all n ∈ N, we can proceed as in the proof of Lemma 5.2.15 to show that uλn  u0 = 0 in H . Then, by using Lemma 5.2.16, we see that I(u0 ) ≤ lim I(uλn ) ≤ c1 < n→∞

N s 2s S∗ N

and

I (u0 ) = 0.

Now set m = inf{I(u) : u ∈ H, u = 0, I (u) = 0}. N

Since I (u0 ) = 0, it is clear that m ≤ I(u0 ) < Ns S∗2s . Now, by using the definition of m, we can find a sequence (vn ) ⊂ H such that vn = 0, I(vn ) → m and I (vn ) = 0. Arguing as in (5.2.85) and (5.2.86), we can show that (vn ) is bounded in H , and that there exists β > 0 independent of n such that vn  ≥ β. This means that m > −∞. Proceeding in much the same way as in the proof of Lemma 5.2.15, we can see that vn  v0 = 0 in H . Then, by using Lemma 5.2.16, we deduce that I (v0 ) = 0 and I(v0 ) ≤ m. Since I (v0 ) = 0, we also have that I(v0 ) ≥ m. Therefore, we have proved that v0 = 0 is such that I(v0 ) = m and I (v0 ) = 0, that is, v0 is a ground state of (5.2.1). Standard arguments   show that v0 > 0 in RN .

Fractional Schrödinger Equations with Rabinowitz Condition

6.1

Introduction

In the first part of this chapter we focus our attention on the existence, multiplicity and concentration of positive solutions for the fractional elliptic problem 

ε2s (−)s u + V (x)u = f (u) in RN , u ∈ H s (RN ), u > 0 in RN ,

(6.1.1)

where ε > 0 is a small parameter, s ∈ (0, 1), N > 2s. We require that V : RN → R is a continuous function satisfying the following condition introduced by Rabinowitz [299]: V∞ = lim inf V (x) > V0 = inf V (x) > 0, |x|→∞

x∈RN

(V )

and we consider both cases V∞ < ∞ and V∞ = ∞. Concerning the nonlinearity f : R → R we assume that (f1 ) f ∈ C(R, R) and f (t) = 0 for all t < 0; f (t) (f2 ) lim = 0; t →0 t f (t) (f3 ) there exists q ∈ (2, 2∗s ) such that lim q−1 = 0; |t |→∞ t (f4 ) there exists ϑ ∈ (2, q) such that 

t

0 < ϑF (t) = ϑ

f (τ ) dτ ≤ tf (t)

for all t > 0;

0

(f5 ) the map t →

f (t) is increasing in (0, ∞). t

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_6

195

6

196

6 Fractional Schrödinger Equations with Rabinowitz Condition

When s = 1, equation (6.1.1) boils down to the classical nonlinear Schrödinger equation −ε2 u + V (x)u = f (u) in RN ,

(6.1.2)

for which the existence and the multiplicity of solutions have been extensively studied in the last 30 years by many authors; see [13, 28, 90, 100, 146, 197, 284, 299, 318, 330]. For instance, Rabinowitz in [299] investigated the existence of positive solutions to (6.1.2) for ε > 0 small enough, under the assumption that f satisfies (f4 ) and the potential V (x) fulfills (V ). Wang [330] showed that these solutions concentrate at global minimum points of V (x). We recall that a solution uε of (6.1.2) is said to concentrate at x0 ∈ RN as ε → 0 if ∀δ > 0,

∃ ε0 , R > 0 : uε (x) ≤ δ,

∀|x − x0 | ≥ ε R, ε < ε0 .

Using a local mountain pass approach, del Pino and Felmer in [165] proved the existence of a single-spike solution to (6.1.2) which concentrates around a local minimum of V , by assuming that there exists a bounded open set  in RN such that inf V (x) < min V (x),

x∈

x∈∂

and considering nonlinearities f satisfying (f 4) and the monotonicity assumption on t → f (t ) t . Subsequently, many authors introduced some new variational methods to extend the results obtained in [165] to a wider class of nonlinearities. For more details we refer to [121, 123, 235, 341]. In the nonlocal setting, there are only few results concerning the existence and the concentration phenomena of solutions for the fractional equation (6.1.1), maybe because many important techniques developed in the local framework cannot be adapted so easily to the fractional case. Next, we recall some recent results related to the concentration phenomenon of solutions for the nonlinear fractional Schrödinger equation (6.1.1). Chen and Zheng [141] studied, via the Lyapunov–Schmidt reduction method, the concentration phenomenon for solutions of (6.1.1) with f (t) = |t|α t, and under suitable limitations on the dimension N of the space and the fractional powers s, Davila et al. [159] showed via Lyapunov–Schmidt reduction that if the potential V satisfies V ∈ C 1,α (RN ) ∩ L∞ (RN ) and inf V (x) > 0, x∈RN

then (6.1.1) has multi-peak solutions. Fall et al. [182] established necessary and sufficient conditions for the smooth potential V to produce concentration of solutions of (6.1.1) when the parameter ε converges to zero. In particular, when V is coercive and has a unique global minimum, then ground-states concentrate at this point. The multiplicity of positive

6.1 Introduction

197

solutions to (6.1.1) under condition (V ) and involving subcritical and critical nonlinearities was considered in [6, 193] and [311], respectively. Alves and Miyagaki [19] investigated the existence and the concentration of positive solutions to (6.1.1), via a penalization approach and extension method [127]. He and Zou [216] used variational methods and the ∗ Lusternik–Schnirelman theory to study (6.1.1) when f (t) = g(t) + t 2s −1 and g satisfies the conditions (f1 ), (f2 ), (f3 ) and (f4 ). In [35] the author extended the results in [19] and [216], obtaining the existence and the multiplicity of solutions to (6.1.1) when f has subcritical, critical or supercritical growth. Finally, we would like also to mention the paper [158] in which the concentration phenomenon for a nonlocal Schrödinger equation with Dirichlet datum is considered. Let us denote M = {x ∈ RN : V (x) = V0 }

and

Mδ = {x ∈ RN : dist(x, M) ≤ δ}, for δ > 0.

Our first main result can be stated as follows: Theorem 6.1.1 ([74]) Let N > 2s, and suppose that V satisfies (V ) and f fulfills (f1 )– (f5 ). Then, for every δ > 0, there exists εδ > 0 such that, for any ε ∈ (0, εδ ), problem (6.1.1) has at least catMδ (M) positive solutions. Moreover, if uε denotes one of these solutions and xε ∈ RN is a global maximum point of uε , then lim V (xε ) = V0 .

ε→0

Exploiting the variational nature of problem (6.1.1), we look for critical points of the functional Jε (u) =

1 2

 R2N

|u(x) − u(y)|2 1 dxdy + N+2s |x − y| 2



 RN

V (ε x)u2 dx −

RN

F (u) dx,

defined on a suitable subspace of H s (RN ). Since f is only continuous, we cannot apply standard Nehari manifold arguments for C 1 functionals; see [27, 280, 340]. Indeed, we cannot proceed as in [14], where the authors considered the corresponding local problem to (6.1.1) (with p = 2) under the assumptions f ∈ C 1 and (f6 ) there exist C > 0 and σ ∈ (2, 2∗s ) such that f (t)t 2 − f (t)t ≥ Ct σ

for all t ≥ 0.

198

6 Fractional Schrödinger Equations with Rabinowitz Condition

To overcome the indicated difficulty, we use some variants of critical point theorems due to Szulkin and Weth [321, 322]; see Sect. 2.4. As usual, the presence of the fractional Laplacian operator makes our analysis more delicate and intriguing. In order to obtain multiple critical points, we employ a technique introduced by Benci and Cerami [95], which consists in making precise comparisons between the category of some sublevel sets of Jε and the category of the set M. Then, after proving that the levels of compactness are strongly related to the behavior of the potential V (x) at infinity (see Proposition 6.3.14), we can apply Lusternik–Schnirelman theory to deduce the existence of multiple positive solutions uε ’s of (6.1.1). Finally, we show that each uε concentrates around a global minimum point of V as ε → 0. To do this, we first adapt the Moser iteration technique [278] in the fractional setting to obtain L∞ -estimates (independent of ε) for the translated sequence vε = uε (· + y˜ε ) of uε , with ε y˜ε → y ∈ M. Then, taking into account some elliptic regularity estimates presented in Chap. 1, we infer that vε (x) → 0 as |x| → ∞ uniformly in ε. This step will be fundamental in deducing the desired concentration result. Secondly, we consider the following fractional problem involving the critical Sobolev exponent: 

∗

ε2s (−)s u + V (x)u = f (u) + |u|2s −2 u in RN , u ∈ H s (RN ), u > 0 in RN .

(6.1.3)

In order to deal with the critical growth of the nonlinearity, we assume that f satisfies (f1 )–(f5 ), and the following technical condition: (f6 ) there exist q1 ∈ (2, 2∗s ) and λ > 0 such that f (t) ≥ λt q1 −1 for any t > 0, where λ is such that • λ > 0 if either N ≥ 4s, or 2s < N < 4s and 2∗s − 2 < q1 < 2∗s , • λ is sufficiently large if 2s < N < 4s and 2 < q1 ≤ 2∗s − 2. Then we are able to obtain our second result: Theorem 6.1.2 ([74]) Let N > 2s, and suppose that (V ) and (f1 )–(f5 ) and (f6 ) hold. Then, for every δ > 0, there exists εδ > 0 such that, for any ε ∈ (0, εδ ), problem (6.1.3) has at least catMδ (M) positive solutions. Moreover, if uε denotes one of these solutions and xε ∈ RN is a global maximum point of uε , then lim V (xε ) = V0 .

ε→0

We note that Theorem 6.1.2 improves and extends, in the fractional setting, Theorem 1.1 in [186], where the author assumed that f ∈ C 1 . The approach developed in this case follows the arguments used to analyze the subcritical case. Anyway, this new problem

6.2 Preliminaries

199

presents an extra difficulty due to the fact that the level of non-compactness is affected by the critical growth of the nonlinearity. To overcome this hitch we adapt some calculations performed in [310] and we prove that the functional associated with (6.1.3) satisfies the Palais–Smale condition at every level 0 0 and introduce the fractional Sobolev space   Hε = u ∈ H s (RN ) :

 RN

V (ε x)u2 dx < ∞

endowed with the norm   uε = [u]2s +

1 RN

V (ε x)u2 dx

2

.

Clearly, Hε is a Hilbert space with the inner product  u, vε = u, vDs,2 (RN ) +

RN

V (ε x)uv dx

∀u, v ∈ Hε .

In view of assumption (V ), Theorems 1.1.8 and 1.1.7, it is easy to check that the following results hold.

200

6 Fractional Schrödinger Equations with Rabinowitz Condition

Lemma 6.2.1 The space Hε is continuously embedded in H s (RN ). Therefore, Hε is continuously embedded in Lr (RN ) for any r ∈ [2, 2∗s ] and compactly embedded in Lrloc (RN ) for any r ∈ [1, 2∗s ). Lemma 6.2.2 Cc∞ (RN ) is dense in Hε . Moreover, when V is coercive, we get the following compactness lemma. Lemma 6.2.3 Let V∞ = ∞. Then Hε is compactly embedded in Lr (RN ) for any r ∈ [2, 2∗s ). Proof For simplicity, we assume that ε = 1. Let r = 2. From Lemma 6.2.1 we know that H1 ⊂ L2 (RN ). Let (un ) be a sequence such that un  0 in H1 . Then, un  0 in H s (RN ). Let us define M = sup un 1 < ∞.

(6.2.1)

n∈N

Since V is coercive, for every η > 0 there exists R = Rη > 0 such that 1 ≤ η, V (x)

for any |x| ≥ R.

(6.2.2)

Since un → 0 in L2 (BR ), it follows that there exists n0 > 0 such that  BR

u2n dx ≤ η

for any n ≥ n0 .

Hence, for any n ≥ n0 , by (6.2.1)–(6.2.3), we have 

 RN

u2n dx



= BR

u2n dx

≤η+η

+

 RN \BR

RN \BR

u2n dx

V (x)u2n dx ≤ η(1 + M 2 ).

Therefore, un → 0 in L2 (RN ). For r ∈ (2, 2∗s ), the interpolation inequality and Theorem 6.2.1 yield un Lr (RN ) ≤ C[un ]αs un 1−α , L2 (RN )

(6.2.3)

6.2 Preliminaries

201

where α 1−α 1 = + , r 2 2∗s and using the conclusion with r = 2 we get the required result.

 

Finally, we prove a splitting lemma for the functions f and F . This result is consequence of Lemmas 5.2.5 and 5.2.6 proved in Sect. 5.2.2, but here we prefer to give an alternative proof which can also be used to obtain splitting results in other situations. Lemma 6.2.4 Let (un ) ⊂ Hε be a sequence such that un  u in Hε , and vn = un − u. Then we have 

(6.2.4) F (vn ) − F (un ) + F (u) dx = on (1) RN

and  sup

wε ≤1 RN

  (f (vn ) − f (un ) + f (u))w dx = on (1).

(6.2.5)

Proof We follow [74] (see also [113]). We begin by proving (6.2.4). Let us note that 

1

F (vn ) − F (un ) = 0

d F (un − tu) dt = − dt



1

uf (un − tu) dt.

0

In view of (f2 ) and (f3 ), for any δ > 0 there exists cδ > 0 such that ∗

|f (t)| ≤ 2δ|t| + cδ |t|2s −1 ∗

|F (t)| ≤ δ|t|2 + cδ |t|2s

for all t ∈ R,

(6.2.6)

for all t ∈ R.

(6.2.7)

Using (6.2.6) with δ = 1 and (|a| + |b|)r ≤ C(r)(|a|r + |b|r ) for any a, b ∈ R and r ≥ 1, we can see that ∗

∗

|F (vn ) − F (un )| ≤ C|un ||u| + C|u|2 + C|un |2s −1 |u| + C|u|2s .

(6.2.8)

Fix η > 0. Applying the Young inequality ab ≤ ηa r + C(η)br for all a, b > 0, with r, r ∈ (1, ∞) such that 1r + r1 = 1, to the first and third term on the right-hand side of (6.2.8), we deduce that ∗

∗

|F (vn ) − F (un )| ≤ η(|un |2 + |un |2s ) + Cη (|u|2 + |u|2s )

202

6 Fractional Schrödinger Equations with Rabinowitz Condition

which together with (6.2.7) with δ = η implies that ∗

∗

|F (vn ) − F (un ) + F (u)| ≤ η(|un |2 + |un |2s ) + Cη (|u|2 + |u|2s ). Let  ∗ Gη,n (x) = max |F (vn ) − F (un ) + F (u)| − η(|un |2 + |un |2s ), 0 . ∗

Then Gη,n → 0 a.e. in RN as n → ∞, and 0 ≤ Gη,n ≤ Cη (|u|2 + |u|2s ) ∈ L1 (RN ). By the dominated convergence theorem,  RN

Gη,n (x) dx → 0

as n → ∞.

On the other hand, the definition of Gη,n implies that ∗

|F (vn ) − F (un ) + F (u)| ≤ η(|un |2 + |un |2s ) + Gη,n ∗

which together with the boundedness of (un ) in L2 (RN ) ∩ L2s (RN ) yields  lim sup n→∞

RN

|F (vn ) − F (un ) + F (u)| dx ≤ Cη.

By the arbitrariness of η > 0 we deduce that (6.2.4) holds. Now we prove (6.2.5). Here we follow [1,74,169]. We claim that there is a subsequence (unj ) of (un ) such that, for all η > 0 there exists rη > 0 satisfying  lim sup j →∞

Bj \Br

|unj |τ dx ≤ η

(6.2.9)

τ ∈ [2, 2∗s ) is fixed. To verify (6.2.9) note that, for each j ∈ N, for  all rτ ≥ rη , where  τ ˆ j ∈ N such that Bj |un | dx → Bj |u| dx as n → ∞, so there exists n  (|un |τ − |u|τ ) dx < Bj

1 j

for all n = nˆ j + i, i = 1, 2, 3, . . . .

Without loss of generality we can assume that nˆ j +1 ≥ nˆ j . In particular, for nj = nˆ j + j we have  1 (|unj |τ − |u|τ ) dx < . j Bj

6.2 Preliminaries

203

Observe that there is rη > 0 such that  RN \Br

|u|τ dx < η

(6.2.10)

for all r ≥ rη . Since 

 Bj \Br

|unj |τ dx = ≤

Bj

 (|unj |τ − |u|τ ) dx +

1 + j



 RN \Br

1 ≤ +η+ j

Bj \Br

|u|τ dx + Br



Br

 |u|τ dx + Br

(|u|τ − |unj |τ ) dx

(|u|τ − |unj |τ ) dx

(|u|τ − |unj |τ ) dx,

(6.2.9) now follows. Let first (unj ) be a subsequence of (un ) such that (6.2.9) holds for τ = 2. Repeating the argument, we can then find a subsequence (unjh ) of (unj ) such that (6.2.9) holds for τ = q. Therefore, for notational convenience, we assume in what follows that (6.2.9) holds for both τ = 2 and τ = q with the same subsequence. Let φ : [0, ∞) → [0, 1] be a smooth function such that φ(t) = 1 if t ∈ [0, 1] and u(x). Applying Lemma 1.4.8 and the φ(t) = 0 if t ∈ [2, ∞), and define u˜ j (x) = φ 2|x| j dominated convergence theorem, we can see that u˜ j → u

in H s (RN ).

(6.2.11)

Set hj = u − u˜ j . Observe that for all w ∈ Hε one has that  RN

[f (unj ) − f (vnj ) − f (u)]w dx =  =  +

RN

RN

[f (unj ) − f (unj − u˜ j ) − f (u˜ j )]w dx  [f (vnj + hj ) − f (vnj )]w dx +

RN

= Ij + I Ij + I I Ij .

[f (u˜ j ) − f (u)]w dx (6.2.12)

Clearly, (6.2.11) ensures that lim

sup |I I Ij | = 0.

j →∞ wε ≤1

(6.2.13)

204

6 Fractional Schrödinger Equations with Rabinowitz Condition

Let us show that lim

sup |Ij | = 0.

(6.2.14)

j →∞ wε ≤1

Invoking Lemma 6.2.1 and conditions (f2 )–(f3 ) we have, for all r > 0, lim

    sup  [f (unj ) − f (unj − u˜ j ) − f (u˜ j )]w dx  = 0.

j →∞ wε ≤1

(6.2.15)

Br

For any η > 0 let rη > 0 be so large that (6.2.9) and (6.2.10) hold. Then, it follows from (6.2.10) and (6.2.11) that 

 lim sup j →∞

|u˜ j | dx ≤ τ

Bj \Br

RN \Br

|u|τ dx ≤ η

(6.2.16)

for all r ≥ rη . Hence, since (f2 )–(f3 ) and u˜ j = 0 on RN \ Bj for all j ∈ N, the Hölder inequality, Lemma 6.2.1, (6.2.9), and (6.2.16) imply that   lim sup  j →∞

RN \Br

  [f (unj ) − f (unj − u˜ j ) − f (u˜ j )]w dx 

      = lim sup  [f (unj ) − f (unj − u˜ j ) − f (u˜ j )]w dx    B \B j →∞ r j % & ≤ C lim sup unj L2 (Bj \Br ) + u˜ j L2 (Bj \Br ) wε j →∞

% & q−1 q−1 + C lim sup unj Lq (Bj \Br ) + u˜ j Lq (Bj \Br ) wε j →∞

1

≤ Cη 2 + Cη

q−1 q

(6.2.17)

.

Combining (6.2.15) and (6.2.17), we obtain (6.2.14) holds true. Finally we verify that lim

sup |I Ij | = 0.

j →∞ wε ≤1

Set  g(t) =

f (t ) |t | ,

0,

if t = 0, if t = 0.

(6.2.18)

6.2 Preliminaries

205

In the light of (f1 )–(f3 ), we can see that g ∈ C(R) and |g(t)| ≤ C(1 + |t|q−2 ) for all t ∈ R. For any a > 0 and any j ∈ N, we set Cja = {x ∈ RN : |vnj (x)| ≤ a} and Dja = RN \ Cja . Since (vnj ) is bounded in L2 (RN ), we can see that |Dja | ≤

1 a2

 Dja

|vnj |2 dx ≤

C → 0 as a → ∞. a2

Then, in view of (f2 )–(f3 ), the Hölder inequality and the boundedness of (hj ) we have that       [f (vnj + hj ) − f (vnj )]w dx    Da  j  ≤C [|vnj | + |vnj |q−1 + |hj | + |hj |q−1 ]|w| dx Dja

≤ C(|Dja |

2∗ s −2 2∗ s

+ C(|Dja | ≤ C(|Dja |

vnj L2∗s (RN ) wL2∗s (RN ) + |Dja |

2∗ s −2 2∗ s

2∗ s −2 2∗ s

2∗ s −q 2∗ s

hj L2∗s (RN ) wL2∗s (RN ) + |Dja |

+ |Dja |

2∗ s −q 2∗ s

q−1 wL2∗s (RN ) ) ∗ L2s (RN )

vnj 

2∗ s −q 2∗ s

q−1 wL2∗s (RN ) ) ∗ L2s (RN )

hj 

)wε ,

which implies that there exists a˜ = a˜ η > 0 such that       [f (vnj + hj ) − f (vnj )]w dx  ≤ η   D a˜ 

(6.2.19)

j

uniformly in wε ≤ 1. Since g is uniformly continuous in the interval [−a, ˜ a], ˜ there exists δ > 0 such that |g(t + h) − g(t)| ≤ η

for all t ∈ [−a, ˜ a] ˜ and |h| ≤ δ.

(6.2.20)

Let Vjδ = {x ∈ RN : |hj (x)| ≤ δ} and Wjδ = RN \ Vjδ . Noting that |Cja˜

∩ Wjδ |

≤

|Wjδ |

1 ≤ 2 δ

 Wjδ

|hj |2 dx ≤

1 hj 2L2 (RN ) → 0 δ2

as j → ∞,

206

6 Fractional Schrödinger Equations with Rabinowitz Condition

we can argue as before to infer that there exists j0 ∈ N such that       [f (vnj + hj ) − f (vnj )]w dx  ≤ η   C a˜ ∩W δ  j

for all j ≥ j0 ,

(6.2.21)

j

uniformly in wε ≤ 1. Now observe that [f (vnj + hj ) − f (vnj )]w = g(vnj + hj )[|vnj + hj | − |vnj |]w + [g(vnj + hj ) − g(vnj )]|vnj |w.

In view of (6.2.11), we can find j1 ∈ N such that j1 ≥ j0 and hj Lτ (RN ) < η

for all j ≥ j1 .

(6.2.22)

Taking into account (6.2.20), (6.2.22) and the boundedness of (vnj ), we can see that for all j ≥ j1       [g(vnj + hj ) − g(vnj )]|vnj |w dx     C a˜ ∩V δ j j  ≤η |vnj ||w| dx Cja˜ ∩Vjδ

≤ ηvnj L2 (RN ) wL2 (RN ) ≤ Cη, uniformly in wε ≤ 1. On the other hand, using the fact that |g(t)| ≤ C(1 + |t|q−2 ), the Hölder inequality and (6.2.22), we have       g(vnj + hj )[|vnj + hj | − |vnj |]w dx     C a˜ ∩V δ j j  (1 + |vnj + hj |q−2 )|hj ||w| dx ≤C RN

q−2

q−1

≤ C[hj L2 (RN ) wL2 (RN ) + vnj Lq (RN ) hj Lq (RN ) wLq (RN ) + hj Lq (RN ) wLq (RN ) ] ≤ C[η + ηq−1 ].

This estimates in conjunction with (6.2.21) and the equality Cja˜ = (Cja˜ ∩ Vjδ ) ∪ (Cja˜ ∩ Wjδ ) shows that       [f (vnj + hj ) − f (vnj )]w dx  ≤ C(η + ηq−1 ) for all j ≥ j1 ,    C a˜ j

uniformly in wε ≤ 1, which together with (6.2.19) yields (6.2.18).

 

6.3 The Subcritical Case

207

6.3

The Subcritical Case

6.3.1

The Nehari Method for (6.1.1)

Making the change of variable x → ε x, we are led to consider the problem 

(−)s u + V (ε x)u = f (u) in RN , u ∈ H s (RN ), u > 0 in RN .

(Pε )

To find weak solutions of (Pε ), we look for critical points of the functional Jε (u) =

1 u2ε − 2

 RN

F (u) dx,

which is well defined on Hε . It is standard to check that (f2 )–(f3 ) ensure that given ξ > 0 there exists Cξ > 0 such that |f (t)| ≤ ξ |t| + Cξ |t|q−1 |F (t)| ≤

ξ 2 Cξ q t + |t| 2 q

∀t ∈ R,

(6.3.1)

∀t ∈ R.

(6.3.2)

is increasing for any t > 0.

(6.3.3)

On the other hand, hypothesis (f5 ) implies that t →

1 f (t)t − F (t) 2

By Lemma 6.2.1, it is readily seen that Jε ∈ C 1 (Hε , R) and its differential Jε is given by Jε (u), ϕ

 = u, vε −

RN

f (u)ϕ dx,

for any u, ϕ ∈ Hε . Now, we introduce the Nehari manifold associated with Jε , that is, ! " Nε = u ∈ Hε \ {0} : Jε (u), u = 0 . Then  u2ε =

RN

f (u)u dx

208

6 Fractional Schrödinger Equations with Rabinowitz Condition

for all u ∈ Nε , which together with (f4 ) implies that 1 Jε (u) = Jε (u) − Jε (u), u = 2

 RN



 1 f (u)u − F (u) dx ≥ 0 2

(6.3.4)

for all u ∈ Nε Since f is merely continuous, the next results are very important for overcoming the non-differentiability of Nε . We begin by proving some properties for the functional Jε . Lemma 6.3.1 Under assumptions (V ) and (f1 )–(f5 ), for ε > 0 we have: (i) Jε maps bounded sets in Hε into bounded sets in Hε . (ii) Jε is weakly sequentially continuous in Hε . (iii) Jε (tn un ) → −∞ as tn → ∞, where un ∈ K and K ⊂ Hε \ {0} is a compact subset. Proof (i) Let (un ) be a bounded sequence in Hε . Then, for all v ∈ Hε , we deduce from (6.3.1) and Lemma 6.2.1 that Jε (un ), v ≤ Cun ε vε + Cun q−1 ε vε ≤ C. (ii) Let un  u in Hε . By Lemma 6.2.1, un → u in Lrloc (RN ) for all r ∈ [1, 2∗s ) and un → u a.e. in RN . Then, for all v ∈ Cc∞ (RN ), it follows from (6.3.1) and the dominated convergence theorem that Jε (un ), v → Jε (u), v.

(6.3.5)

In view of Lemma 6.2.2, for any v ∈ Hε we can take (vj ) ⊂ Cc∞ (RN ) such that vj − vε → 0 as j → ∞. Note that (6.3.1) and Lemma 6.2.1 yield |Jε (un ), v − Jε (u), v| ≤ |Jε (un ) − Jε (u), vj | + |Jε (un ) − Jε (u), v − vj |  (|un | + |u| + |un |q−1 + |u|q−1 )|v − vj | dx ≤ |Jε (un ) − Jε (u), vj | + C RN

≤ |Jε (un ) − Jε (u), vj | + Cvj − vε .

6.3 The Subcritical Case

209

For any ξ > 0, fix j ∈ N such that vj − vε < that |Jε (un ) − Jε (u), vj |
0 we have: (i) for every u ∈ Sε , there exists a unique tu > 0 such that tu u ∈ Nε . Moreover, mε (u) = tu u is the unique maximum of Jε on Hε , where Sε = {u ∈ Hε : uε = 1}. (ii) The set Nε is bounded away from 0. Furthermore, Nε is closed in Hε . (iii) There exists α > 0 such that tu ≥ α for each u ∈ Sε and, for each compact subset W ⊂ Sε , there exists a constant CW > 0 such that tu ≤ CW for all u ∈ W . (iv) Nε is a regular manifold diffeomorphic to the unit sphere Sε . (v) cε = infNε Jε ≥ ρ > 0 and Jε is bounded below on Nε , where ρ is independent of ε.

Proof (i) For each u ∈ Sε and t > 0, set h(t) = Jε (tu). It is clear that h(0) = 0 and that h(t) < 0 for t large. Let us prove that h(t) > 0 for t > 0 small. Indeed, by (6.3.2) and Lemma 6.2.1, it follows that t2 h(t) = u2ε − 2

 RN

F (tu) dx

210

6 Fractional Schrödinger Equations with Rabinowitz Condition

≥

t2 ξt2 tq q u2ε − u2L2 (RN ) − Cξ uLq (RN ) 2 2 q

≥

tq t2 ξt2 u2ε − u2ε − Cξ uqε . 2 2V0 q

Choosing ξ ∈ (0, V0 ) we deduce that h(t) > 0 for t > 0 sufficiently small. Therefore, maxt ≥0 h(t) is achieved at some tu > 0 satisfying h (tu ) = 0 and tu u ∈ Nε . Next, we prove the uniqueness of such a tu . Suppose, by contradiction, that there exist tu > tu > 0 such that tu u, tu u ∈ Nε . Then  tu2 u2ε

=

RN

and

f (tu u)tu u dx

(tu )2 u2ε

 =

RN

f (tu u)tu u dx,

whence 

1 1 − tu

tu





 u2ε =

RN

 f (tu u) f (tu u) − u2 dx. tu u tu u

Using (f5 ) and tu > tu > 0, we deduce that this identity makes no sense. (ii) By virtue of (6.3.1) and Lemma 6.2.1, we can see that for any u ∈ Nε  upε =

RN

f (u)u dx ≤ Cξ u2ε + Cξ uqε ,

which implies that uε ≥ κ for some κ > 0. Now we prove that the set Nε is closed in Hε . Let (un ) ⊂ Nε be such that un → u in Hε . In view of Lemma 6.3.1, Jε (un ) is bounded, and we deduce that Jε (un ), un  − Jε (u), u = Jε (un ), un − u + Jε (un ) − Jε (u), u → 0, that is, Jε (u), u = 0. This fact combined with uε ≥ κ implies that uε = lim un ε ≥ κ > 0, n→∞

and then u ∈ Nε . (iii) For (un ) ⊂ Sε there exists tun > 0 such that tun un ∈ Nε . The proof of (ii) shows that tun = tun un ε ≥ κ, which implies that tun  0 as n → ∞. Now let us prove that tu ≤ CW for all u ∈ W ⊂ Sε . Assume, by contradiction, that there exists a sequence (un ) ⊂ W ⊂ Sε such that tun → ∞. Since W is compact, we can find u ∈ W such that un → u in Hε

6.3 The Subcritical Case

211

and un → u a.e. in RN . Using Lemma 6.3.1-(iii), we deduce that Jε (tun un ) → −∞ as n → ∞, which gives a contradiction because of (6.3.4). (iv) Define the maps m ˆ ε : Hε \ {0} → Nε and mε : Sε → Nε by setting m ˆ ε (u) = tu u

and

mε = m ˆ ε |Sε .

(6.3.6)

In view of (i)–(iii) and Proposition 2.4.2, we deduce that mε is a homeomorphism u between Sε and Nε and the inverse of mε is given by m−1 ε (u) = uε . Therefore, Nε is a regular manifold diffeomorphic to Sε . (v) For ε > 0, t > 0 and u ∈ Hε \ {0}, we see that (6.3.2) yields t2 Jε (tu) ≥ 2

  ξ 1− u2ε − t q Cξ uqε . V0

Choosing ξ > 0 small enough, we can find ρ > 0 such that Jε (tu) ≥ ρ > 0 for t > 0 small enough. On the other hand, we deduce from (i)–(iii) that (see (2.4.1)) cε = inf Jε (u) = u∈Nε

inf

max Jε (tu) = inf max Jε (tu)

u∈Hε \{0} t >0

u∈Sε t >0

(6.3.7)

which implies that cε ≥ ρ and Jε |Nε ≥ ρ.

 

ˆ ε : Hε \ {0} → R and ε : Sε → R defined by Now we introduce the functionals  Ψˆ ε (u) = Jε (m ˆ ε (u))

and

ˆ ε |Sε , ε = 

where m ˆ ε (u) = tu u is given in (6.3.6). As in [322] (see Proposition 2.4.3 and Corollary 2.4.4) we have the following result: Lemma 6.3.3 Under the assumptions of Lemma 6.3.1, for ε > 0 we have: (i) ε ∈ C 1 (Sε , R), and ε (w), v = mε (w)ε Jε (mε (w)), v

for all v ∈ Tw Sε .

(ii) (wn ) is a Palais–Smale sequence for ε if and only if (mε (wn )) is a Palais–Smale sequence for Jε . If (un ) ⊂ Nε is a bounded Palais–Smale sequence for Jε , then (m−1 ε (un )) is a Palais–Smale sequence for ε . (iii) u ∈ Sε is a critical point of ε if and only if mε (u) is a critical point of Jε . Moreover, the corresponding critical values coincide and inf ε = inf Jε = cε . Sε

Nε

212

6 Fractional Schrödinger Equations with Rabinowitz Condition

Finally, let us show that Jε possesses a mountain pass geometry [29]. Lemma 6.3.4 The functional Jε satisfies the following conditions: (i) There exist α, ρ > 0 such that Jε (u) ≥ α, with uε = ρ. (ii) There exists e ∈ Hε such that eε > ρ and Jε (e) < 0. Proof (i) Using (6.3.2), the inequality V0 ≤ V (ε x), and Lemma 6.2.1, we get   1 ξ Cξ 2 2 Jε (u) ≥ uε − V (ε x)u dx − |u|q dx 2 2V0 RN q RN   Cξ Cq 1 ξ ≥ − uqε . u2ε − 2 2V0 q Choosing ξ ∈ (0, V0 ), there exist α, ρ > 0 such that Jε (u) ≥ α > 0,

with uε = ρ.

(ii) By (f4 ), F (t) ≥ C1 t ϑ − C2

for any t ≥ 0,

for some C1 , C2 > 0. Taking ϕ ∈ Cc∞ (RN ) such that ϕ ≥ 0 and ϕ ≡ 0, we have

Jε (tϕ) ≤

t2 ϕ2ε − t ϑ C1 2

 ϕ ϑ dx + C2 |supp(ϕ)| → −∞

as t → ∞.

supp(ϕ)

  By Lemma 6.3.4 and using a variant of the mountain pass theorem without the Palais– Smale condition (see Remark 2.2.10), we see that there exists a Palais–Smale sequence (un ) ⊂ Hε for Jε at the level cε , that is, Jε (un ) → cε

and

Jε (un ) → 0 in Hε∗ ,

where cε = inf max Jε (γ (t)) γ ∈ε t ∈[0,1]

6.3 The Subcritical Case

213

and ε = {γ ∈ C([0, 1], Hε ) : Jε (0) = 0, Jε (γ (1)) < 0}. Motivated by Rabinowitz [299], we use an equivalent characterization of cε that is more adequate for our purpose, namely cε =

inf

max Jε (tu) = cε ,

u∈Hε \{0} t >0

where in the last equality we used (6.3.7). We note that, by using (f4 ), any Palais–Smale sequence (un ) of Jε is bounded in Hε . Indeed, for any n ∈ N, we have 1 C(1 + un ε ) ≥ Jε (un ) − Jε (un ), un  ϑ    1 1 1 − = un 2ε + (f (un )un − ϑF (un )) dx 2 ϑ ϑ RN   1 1 − un 2ε , ≥ 2 ϑ which gives the desired result. Remark 6.3.5 Arguing as in Remark 5.2.8, we may always suppose that un ≥ 0 in RN for all n ∈ N.

6.3.2

The Autonomous Subcritical Problem

Let us consider the family of autonomous problems related to (Pε ), that is, for μ > 0 

(−)s u + μu = f (u) in RN , u ∈ H s (RN ), u > 0 in RN .

The functional energy corresponding to (Pμ ) is given by Iμ (u) =

1 u2μ − 2

 RN

F (u) dx

(Pμ )

214

6 Fractional Schrödinger Equations with Rabinowitz Condition

which is well defined on the space Xμ = H s (RN ) endowed with the norm

1 2 uμ = [u]2s + μu2L2 (RN ) . Clearly, Iμ ∈ C 1 (Xμ , R) and its differential Iμ is given by Iμ (u), ϕ

 = u, vDs,2 (RN ) + μ

 RN

uϕ dx −

RN

f (u)ϕ dx

for any u, ϕ ∈ Xμ . Let us introduce the Nehari manifold associated with Iμ , that is, ! " Mμ = u ∈ Xμ \ {0} : Iμ (u), u = 0 . Note that assumption (f4 ) implies that 

 Iμ (u) =

RN

   1 1 1 f (u)u − F (u) dx ≥ − u2μ 2 2 ϑ

for all u ∈ Mμ .

(6.3.8)

Arguing as in the previous section and using (6.3.8), it is easy to prove the following lemma. Lemma 6.3.6 Under the assumptions of Lemma 6.3.1, for μ > 0 we have: (i) for all u ∈ Sμ , there exists a unique tu > 0 such that tu u ∈ Mμ . Moreover, mμ (u) = tu u is the unique maximum of Iμ on Xμ , where Sμ = {u ∈ Xμ : uμ = 1}. (ii) The set Mμ is bounded away from 0. Furthermore, Mμ is closed in Xμ . (iii) There exists α > 0 such that tu ≥ α for each u ∈ Sμ and, for each compact subset W ⊂ Sμ , there exists CW > 0 such that tu ≤ CW for all u ∈ W . (iv) Mμ is a regular manifold diffeomorphic to the unit sphere Sμ . (v) dμ = infMμ Iμ > 0 and Iμ is bounded below on Mμ by some positive constant. ˆ μ : Xμ \ {0} → R and μ : Sμ → R by Now we define the following functionals  setting ˆ μ (u)) Ψˆ μ (u) = Iμ (m

and

ˆ μ |Sμ . μ = 

The inverse of the mapping mμ to Sμ is given by m−1 μ : Mμ → Sμ ,

m−1 μ (u) =

u . uμ

6.3 The Subcritical Case

215

We have the next result: Lemma 6.3.7 Under the assumptions of Lemma 6.3.1, for μ > 0 it holds that: (i) μ ∈ C 1 (Sμ , R), and μ (w), v = mμ (w)μ Iμ (mμ (w)), v

for all v ∈ Tw Sμ .

(ii) (wn ) is a Palais–Smale sequence for μ if and only if (mμ (wn )) is a Palais–Smale sequence for Iμ . If (un ) ⊂ Mμ is a bounded Palais–Smale sequence for Iμ , then (m−1 μ (un )) is a Palais–Smale sequence for μ . (iii) u ∈ Sμ is a critical point of μ if and only if mμ (u) is a critical point of Iμ . Moreover, the corresponding critical values coincide and inf μ = inf Iμ = dμ . Mμ

Sμ

Remark 6.3.8 As in (6.3.7), items (i)–(iii) of Lemma 6.3.2 imply that cμ admits the following minimax characterization: dμ = inf Iμ (u) = u∈Mμ

max Iμ (tu) = inf max Iμ (tu).

inf

u∈Xμ \{0} t >0

u∈Sμ t >0

(6.3.9)

Arguing as in the proof of Lemma 6.3.4, it is easy to verify that Iμ has a mountain pass geometry. Thus we can use a variant of the mountain pass theorem without the Palais– Smale condition (see Remark 2.2.10) to find a sequence (un ) in Xμ such that Iμ (un ) → dμ = inf max Iμ (γ (t)) γ ∈μ t ∈[0,1]

and

Iμ (un ) → 0 in X∗μ ,

where μ = {γ ∈ C([0, 1], Xμ ) : Iμ (0) = 0, Iμ (γ (1)) < 0}. By using (f4 ), we can see that (un ) is bounded in Xμ . In what follows, we use the equivalent characterization of dμ

given by dμ =

inf

max Iμ (tu) = dμ ,

u∈Xμ \{0} t >0

where in the last equality we used (6.3.9). We now study the minimizing sequences for Iμ . Lemma 6.3.9 Let (un ) ⊂ Mμ be such that Iμ (un ) → dμ . Then, (un ) is bounded in Xμ . Moreover, there exist a sequence (yn ) ⊂ RN and constants R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

216

6 Fractional Schrödinger Equations with Rabinowitz Condition

Proof By using (f4 ), we have 1 dμ = Iμ (un ) − Iμ (un ), un  ϑ    1 1 1 2 − un μ + = (f (un )un − ϑF (un )) dx 2 ϑ ϑ RN   1 1 − ≥ un 2μ , 2 ϑ that is, (un ) is bounded in Xμ . Now, to prove the second conclusion of the lemma, we argue by contradiction. Assume that for any R > 0 it holds that  lim sup

n→∞

y∈RN

BR (y)

u2n dx = 0.

Since (un ) is bounded in Xμ , it follows from Lemma 1.4.4 that un → 0 in Lt (RN )

for any t ∈ (2, 2∗s ).

(6.3.10)

Fix ξ ∈ (0, μ). Since Iμ (un ), un  = 0, (6.3.1) and the boundedness of (un ) in Xμ imply that   ξ q 0 ≤ un 2μ ≤ ξ u2n dx + Cξ |un |q dx ≤ un 2μ + Cξ un Lq (RN ) , μ RN RN whence   ξ q 1− un 2μ ≤ Cξ un Lq (RN ) . μ In the light of (6.3.10), we obtain that un → 0 in Xμ and then Iμ (un ) → 0, which is a   contradiction because dμ > 0. Next we prove the main result for the autonomous problem (Pμ ). Lemma 6.3.10 The problem (Pμ ) has at least one positive ground state solution. Proof By item (v) of Lemma 6.3.6, dμ > 0 for each μ > 0. Moreover, if u ∈ Mμ satisfies Iμ (u) = dμ , then m−1 μ (u) is a minimizer of μ and therefore a critical point of μ . In view of Lemma 6.3.7, we see that u is a critical point of Iμ . It remains to show that there exists a minimizer of Iμ |Mμ . Theorem 2.2.1 yields a sequence (νn ) ⊂ Sμ such that μ (νn ) → dμ and μ (νn ) → 0 as n → ∞. Let un = mμ (νn ) ∈ Mμ . Then,

6.3 The Subcritical Case

217

Iμ (un ) → dμ and Iμ (un ) → 0 in X∗μ as n → ∞. It is easy to see that (un ) is bounded in Xμ and un  u in H s (RN ). Clearly, Iμ (u) = 0. Assume that u = 0. Then u ∈ Mμ . If we can show that un μ → uμ ,

(6.3.11)

then we can use the fact that Xμ is a Hilbert space to deduce that un → u in Xμ . Consequently, Iμ (u) = dμ . Let us verify that (6.3.11) holds. By Fatou’s lemma, u2μ ≤ lim inf un 2μ .

(6.3.12)

u2μ < lim sup un 2μ .

(6.3.13)

n→∞

Suppose, by contradiction, that

n→∞

Let us note that 1 dμ + on (1) = Iμ (un ) − Iμ (un ), un  ϑ      1 1 1 2 − f (un )un − F (un ) dx. = un μ + 2 ϑ RN ϑ Then, recalling that lim sup (an + bn ) ≥ lim sup an + lim inf bn n→∞

n→∞

n→∞

and ϑ > 2, Fatou’s lemma, (6.3.13), (6.3.14), (f4 ) and u ∈ Mμ imply that  dμ ≥  >

1 1 − 2 ϑ 1 1 − 2 ϑ

= Iμ (u) −



 1 f (un )un − F (un ) dx n→∞ RN ϑ   1 f (u)u − F (u) dx ϑ 



lim sup un 2μ + lim inf 

n→∞

u2μ



+

RN

1

I (u), u = Iμ (u) ≥ dμ , ϑ μ

which gives a contradiction. Consequently, by (6.3.12), u2μ ≤ lim inf un 2μ ≤ lim sup un 2μ ≤ u2μ n→∞

and this implies that (6.3.11) holds true.

n→∞

(6.3.14)

218

6 Fractional Schrödinger Equations with Rabinowitz Condition

Finally, we consider the case u = 0. Arguing as in the proof of Lemma 6.3.9, we can find a sequence (yn ) ⊂ RN and constants R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Set vn = un (· + yn ). Then, using that Mμ and Iμ are invariant under translations, we infer that (vn ) is a Palais–Smale sequence for Iμ at the level dμ , (vn ) ⊂ Mμ and vn  v = 0 in H s (RN ). Thus we can proceed as above to deduce that vn → v strongly in Xμ and thus v ∈ Mμ and Iμ (v) = dμ . Let u be the ground state solution of (6.1.1) obtained above. Then u is positive. Indeed, since Iμ (u), u−  = 0, f (t) = 0 for t ≤ 0 and (x − y)(x − − y − ) ≥ |x − − y − |2 for x, y ∈ R, where x − = min{x, 0}, we see that u− 2μ ≤ u, u− Ds,2 (RN ) +

 RN

μuu− dx =

 RN

f (u)u− dx = 0

which implies that u− = 0, that is, u ≥ 0. By Proposition 3.2.14, u ∈ L∞ (RN ), and using Proposition 1.3.2 we can see that u ∈ C 0,α (RN ). Applying Proposition 1.3.11-(ii)   (or Theorem 1.3.5), we conclude that u > 0 in RN . Following the above arguments, we obtain the next compactness result: Theorem 6.3.11 Let (un ) ⊂ MV0 be a sequence such that IV0 (un ) → dV0 . Then we have either (i) (un ) has a subsequence that converges strongly in H s (RN ), or (ii) there exists a sequence (y˜n ) ⊂ RN such that, up to a subsequence, vn = un (· + y˜n ) converges strongly in H s (RN ). In particular, there exists a minimizer for dV0 . Proof By Lemma 6.3.9, we know that (un ) is bounded in XV0 . From Lemma 6.3.7, νn = m−1 ε (un ) is a minimizing sequence of V0 . By Theorem 2.2.1, we may assume that V0 (νn ) → dV0 and V 0 (νn ) → 0. Then IV0 (un ) → dV0 , IV 0 (un ) → 0 and IV 0 (un ), un  = 0 where un = mV0 (νn ). Hence, up to a subsequence, we may assume that there exists u ∈ H s (RN ) such that un  u in H s (RN ). At this point we can argue as in the proof of Lemma 6.3.10 by considering the cases u = 0 and u = 0.  

6.3 The Subcritical Case

6.3.3

219

An Existence Result for (6.1.1)

In this section we focus on the existence of solutions to (6.1.1) under the assumption that ε is sufficiently small. We begin with the next useful lemma. Lemma 6.3.12 Let (un ) ⊂ Nε be a sequence such that Jε (un ) → c and un  0 in Hε . Then either (a) un → 0 in Hε , or (b) there are a sequence (yn ) ⊂ RN and constants R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Proof Assume that (b) does not hold true. Then, for any R > 0,  lim sup

n→∞

y∈RN

BR (y)

u2n dx = 0.

Since (un ) is bounded in Hε , Lemma 1.4.4 shows that un → 0 in Lt (RN )

for any t ∈ (2, 2∗s ).

(6.3.15)

Now, we can argue as in the proof of Lemma 6.3.9 to deduce that un ε → 0 as n → ∞.   To obtain a compactness result for Jε , we need the following auxiliary lemma. Lemma 6.3.13 Assume that V∞ < ∞. Let (vn ) ⊂ Nε be a sequence such that Jε (vn ) → d with vn  0 in Hε . If vn → 0 in Hε , then d ≥ dV∞ , where dV∞ is the infimum of IV∞ over MV∞ . Proof Let (tn ) ⊂ (0, ∞) be such that (tn vn ) ⊂ MV∞ . Claim 1 We have lim sup tn ≤ 1. n→∞

Suppose, by contradiction, that there exist δ > 0 and a subsequence, still denoted by (tn ), such that tn ≥ 1 + δ

for all n ∈ N.

(6.3.16)

220

6 Fractional Schrödinger Equations with Rabinowitz Condition

Since Jε (vn ), vn  = 0, we deduce that  [vn ]2s

+

 RN

=

V (ε x)vn2 dx

f (vn )vn dx.

RN

(6.3.17)

Further, since tn vn ∈ MV∞ , we also have 

 tn2 [vn ]2s

+ tn2 V∞

RN

vn2

dx =

RN

f (tn vn )tn vn dx.

(6.3.18)

Combining (6.3.17) and (6.3.18) we obtain 

 RN

  f (tn vn ) f (vn ) 2 vn dx = − (V∞ − V (ε x)) vn2 dx. tn vn vn RN

Hypothesis (V ) ensures that, given ζ > 0 there exists R = R(ζ ) > 0 such that V (ε x) ≥ V∞ − ζ

for any |x| ≥ R.

(6.3.19)

Now, since vn → 0 in L2 (BR ) and since the sequence (vn ) is bounded in Hε , we infer that  RN

(V∞ − V (ε x)) vn2 dx 

 = BR

≤ V∞

(V∞ − V (ε x)) vn2 dx + 

 BR

≤ on (1) +

vn2 dx + ζ

BRc

BRc

(V∞ − V (ε x)) vn2 dx

vn2 dx

ζ vn 2ε ≤ on (1) + ζ C. V0

Thus 

 RN

 f (tn vn ) f (vn ) 2 vn dx ≤ ζ C + on (1). − tn vn vn

(6.3.20)

Since vn → 0 in Hε , we apply Lemma 6.3.12 to deduce the existence of a sequence (yn ) ⊂ RN and two positive numbers R¯ and β such that  BR¯ (yn )

vn2 dx ≥ β > 0.

(6.3.21)

6.3 The Subcritical Case

221

Define v¯n = vn (x + yn ). By condition (V ) and the boundedness of (vn ) in Hε ,  v¯n 2V0 = vn 2V0 ≤ [vn ]2s +

RN

V (ε x)vn2 dx = vn 2ε ≤ C,

therefore (v¯n ) is bounded in H s (RN ). Taking into account that H s (RN ) is a reflexive Banach space, we may assume that v¯n  v¯ in H s (RN ). By (6.3.21), there exists ⊂ RN with positive measure and such that v¯ > 0 in . Using (6.3.16), assumption (f5 ) and (6.3.20), we can infer that   0<

 f ((1 + δ)v¯n ) f (v¯n ) 2 − v¯n dx ≤ ζ C + on (1). ((1 + δ)v¯n ) v¯n

Letting n → ∞ and applying Fatou’s lemma we obtain   0<

 f (v) ¯ f ((1 + δ)v) ¯ − v¯ 2 dx ≤ ζ C ((1 + δ)v) ¯ v¯

for any ζ > 0, which is a contradiction. Now, we distinguish the following cases: Case 1 Assume that lim supn→∞ tn = 1. Thus, there exists (tn ) such that tn → 1. Recalling that Jε (vn ) → d, we have d + on (1) = Jε (vn ) = Jε (vn ) − IV∞ (tn vn ) + IV∞ (tn vn ) ≥ Jε (vn ) − IV∞ (tn vn ) + dV∞ .

(6.3.22)

Let us compute the difference Jε (vn ) − IV∞ (tn vn ): Jε (vn ) − IV∞ (tn vn ) (1 − tn2 ) 1 [vn ]2s + = 2 2



 RN

(V (ε x) − tn2 V∞ )vn2 dx

+

(F (tn vn ) − F (vn )) dx.

RN

(6.3.23) Using condition (V ), vn → 0 in L2 (BR ), tn → 1, (6.3.19), and V (ε x) − tn2 V∞ = (V (ε x) − V∞ ) + (1 − tn2 )V∞ ≥ −ζ + (1 − tn2 )V∞

for |x| ≥ R,

222

6 Fractional Schrödinger Equations with Rabinowitz Condition

we get  RN



V (ε x) − tn2 V∞ vn2 dx



= BR

≥



2 2 V (ε x) − tn V∞ vn dx + 

(V0 − tn2 V∞ )

BRc



BR

vn2 dx

−ζ



BRc

vn2

V (ε x) − tn2 V∞ vn2 dx 

dx +

V∞ (1 − tn2 )

BRc

vn2 dx

≥ on (1) − ζ C.

(6.3.24)

On the other hand, since (vn ) is bounded in Hε , (1 − tn2 ) [vn ]2s = on (1). 2

(6.3.25)

Putting together (6.3.23), (6.3.24), and (6.3.25), we obtain  Jε (vn ) − IV∞ (tn vn ) =

RN

(F (tn vn ) − F (vn )) dx + on (1) − ζ C.

(6.3.26)

Now, we claim that  RN

(F (tn vn ) − F (vn )) dx = on (1).

(6.3.27)

Indeed, applying the mean value theorem and (6.3.1) we have  RN

q |F (tn vn ) − F (vn )| dx ≤ C|tn − 1| vn 2L2 (RN ) + vn Lq (RN ) ,

so thanks to the boundedness of (vn ) in Hε the claim is proved. Now (6.3.22), (6.3.26), and (6.3.27) imply that d + on (1) ≥ on (1) − ζ C + dV∞ , and passing to the limit as ζ → 0 we get d ≥ dV∞ . Case 2 Assume that lim supn→∞ tn = t0 < 1. Then there is a subsequence, still denoted by (tn ), such that tn → t0 and tn < 1 for any n ∈ N. Let us observe that 1 d + on (1) = Jε (vn ) − Jε (vn ), vn  = 2



 RN

1 f (vn )vn − F (vn ) 2

 dx. (6.3.28)

6.3 The Subcritical Case

223

Using the facts that tn vn ∈ MV∞ , (6.3.3) and (6.3.28), we obtain 1 dV∞ ≤ IV∞ (tn vn ) = IV∞ (tn vn ) − IV ∞ (tn vn ), tn vn  2    1 = f (tn vn )tn vn − F (tn vn ) dx RN 2    1 f (vn )vn − F (vn ) dx = d + on (1). ≤ RN 2 Taking the limit as n → ∞ we get d ≥ dV∞ .

 

At this point we are able to prove the following compactness result. Proposition 6.3.14 Let (un ) ⊂ Nε be such that Jε (un ) → c, where c < dV∞ if V∞ < ∞, and c ∈ R if V∞ = ∞. Then (un ) has a subsequence that converges strongly in Hε . Proof It is readily seen that (un ) is bounded in Hε . Then, up to a subsequence, we may assume that un  u, in Hε , q

un → u, in Lloc (RN )

∀q ∈ [1, 2∗s ),

(6.3.29)

un → u, a.e. in RN . Using assumptions (f2 )–(f3 ), (6.3.29) and Lemma 6.2.2, it is routine to check that Jε (u) = 0. Now, let vn = un − u. By the Brezis-Lieb lemma [113] and Lemma 6.2.4, Jε (vn ) =

u2ε un 2ε − − 2 2



 RN

F (un ) dx +

RN

F (u) dx + on (1)

= Jε (un ) − Jε (u) + on (1) = c − Jε (u) + on (1) = d + on (1),

(6.3.30)

and Jε (vn ) = on (1). Assume that V∞ < ∞. It follows from (6.3.30) and (6.3.4) that d ≤ c < dV∞ which together with Lemma 6.3.13 gives vn → 0 in Hε , that is, un → u in Hε .

224

6 Fractional Schrödinger Equations with Rabinowitz Condition

Let us consider the case V∞ = ∞. Then we use Lemma 6.2.3 to deduce that vn → 0 in Lr (RN ) for all r ∈ [2, 2∗s ). This fact combined with assumptions (f2 ) and (f3 ) implies that  f (vn )vn dx = on (1). (6.3.31) RN

Since Jε (vn ), vn  = 0 and applying (6.3.31) we infer that vn 2ε = on (1), which yields un → u in Hε .

 

We end this section by establishing the existence of a positive solution to (Pε ) when ε > 0 is small enough. Theorem 6.3.15 Assume that (V ) and (f1 )–(f5 ) hold. Then there exists ε0 > 0 such that, for any ε ∈ (0, ε0 ), problem (Pε ) admits a positive ground state solution. Proof From item (v) of Lemma 6.3.2, we know that cε ≥ ρ > 0 for each ε > 0. Moreover, if u ∈ Nε satisfies Jε (u) = cε , then m−1 ε (u) is a minimizer of ε and it is a critical point of ε . In view of Lemma 6.3.3, we see that u is a critical point of Jε . Let us show that there exists a minimizer of Jε |Nε . Theorem 2.2.1 yields a sequence (vn ) ⊂ Sε such that ε (vn ) → cε and ε (vn ) → 0 as n → ∞. Let un = mε (vn ) ∈ Nε . Then, by Lemma 6.3.3, Jε (un ) → cε , Jε (un ), un  = 0 and Jε (un ) → 0 as n → ∞. Therefore, (un ) is a Palais–Smale sequence for Jε at level cε . Consequently, (un ) is bounded in Hε , and we denote by u its weak limit. It is easy to verify that Jε (u) = 0. When V∞ = ∞, we use Lemma 6.2.3 to deduce that Jε (u) = cε and Jε (u) = 0. Now, we deal with the case V∞ < ∞. By virtue of Proposition 6.3.14, it is enough to show that cε < dV∞ for small ε > 0. Without loss of generality, we may suppose that V (0) = V0 = inf V (x). x∈RN

Let μ ∈ R be such that μ ∈ (V0 , V∞ ). Clearly, dV0 < dμ < dV∞ . By Lemma 6.3.10, it follows that the autonomous problem (PV0 ) admits a positive ground state w ∈ H s (RN ). c . Let us Let ηr ∈ Cc∞ (RN ) be a cut-off function such that ηr = 1 in Br and ηr = 0 in B2r define wr (x) = ηr (x)w(x), and take tr > 0 such that Iμ (tr wr ) = max Iμ (twr ). t ≥0

We claim that there exists r sufficiently large for which Iμ (tr wr ) < dV∞ .

6.3 The Subcritical Case

225

Assume, by contradiction, that Iμ (tr wr ) ≥ dV∞ for any r > 0. Since wr → w in H s (RN ) as r → ∞ thanks to Lemma 1.4.8, tr wr and w belong to Mμ and using (f5 ), we have that tr → 1. Therefore, dV∞ ≤ lim inf Iμ (tr wr ) = Iμ (w) = dμ , r→∞

which leads to a contradiction because dV∞ > dμ . Hence, there exists r > 0 such that Iμ (tr wr ) = max Iμ (τ (tr wr )) τ ≥0

and

Iμ (tr wr ) < dV∞ .

(6.3.32)

Condition (V ) implies that there exists ε0 > 0 such that V (ε x) ≤ μ

for all x ∈ supp(wr ),

ε ∈ (0, ε0 ).

(6.3.33)

Therefore, using (6.3.32) and (6.3.33), we deduce that for all ε ∈ (0, ε0 ) cε ≤ max Jε (τ (tr wr )) ≤ max Iμ (τ (tr wr )) = Iμ (tr wr ) < dV∞ τ ≥0

τ ≥0

which implies that cε < dV∞ for any ε > 0 sufficiently small. Now, if u is a nonnegative ground state of (Pε ), then we can use a Moser iteration 0,α (RN ) and that argument (see Lemma 6.3.23 below) to deduce that u ∈ L∞ (RN ) ∩ Cloc u(x) → 0 as |x| → ∞. By Theorem 1.3.5, u > 0 in RN , as needed.  

6.3.4

A Multiplicity Result for (6.1.1)

This section deals with the multiplicity of solutions to (6.1.1). We begin by proving the following result which will be needed to implement the barycenter machinery. Proposition 6.3.16 Let εn → 0 and (un ) = (uεn ) ⊂ Nεn be such that Jεn (un ) → dV0 . Then there exists (y˜n ) = (y˜εn ) ⊂ RN such that vn (x) = un (x + y˜n ) has a subsequence that converges in H s (RN ). Moreover, up to a subsequence, yn → y ∈ M, where yn = εn y˜n . Proof Since Jε n (un ), un  = 0 and Jεn (un ) → dV0 , we deduce that (un ) is bounded in Hε . From dV0 > 0, we can infer that un εn → 0. Therefore, as in the proof of Lemma 6.3.12, we can find a sequence (y˜n ) ⊂ RN and constants R, β > 0 such that  lim inf n→∞

BR (y˜n )

u2n dx ≥ β.

(6.3.34)

226

6 Fractional Schrödinger Equations with Rabinowitz Condition

Let us define vn (x) = un (x + y˜n ). Thanks to the boundedness of (un ) and (6.3.34), we may assume that vn  v in H s (RN ) for some v = 0. Let (tn ) ⊂ (0, ∞) be such that wn = tn vn ∈ MV0 , and we set yn = εn y˜n . Thus, using the change of variables z → x + y˜n , the fact that V (x) ≥ V0 and the translation invariance, we can see that dV0 ≤ IV0 (wn ) ≤ Jεn (tn vn ) ≤ Jεn (un ) = dV0 + on (1). Hence, IV0 (wn ) → dV0 . This fact and (wn ) ⊂ MV0 imply that there exists K > 0 such that wn V0 ≤ K for all n ∈ N. Moreover, we can prove that the sequence (tn ) is bounded in R. In fact, vn → 0 in H s (RN ), so there exists α > 0 such that vn V0 ≥ α. Consequently, |tn |α ≤ tn vn V0 = wn V0 ≤ K, for all n ∈ N, and so |tn | ≤ K α for all n ∈ N. Therefore, up to a subsequence, we may suppose that tn → t0 ≥ 0. Let us show that t0 > 0. Otherwise, if t0 = 0, from the boundedness of (vn ), we get wn = tn vn → 0 in H s (RN ), that is IV0 (wn ) → 0, in contrast with dV0 > 0. Thus t0 > 0. Let w be the weak limit of wn in H s (RN ). Since tn → t0 > 0 and vn  v = 0 in H s (RN ), by the uniqueness of the weak limit we have that wn  w = t0 v = 0 in H s (RN ). Hence, we have proved that IV0 (wn ) → dV0

wn  w = 0 in H s (RN ).

and

Now Theorem 6.3.11 implies that wn → w in H s (RN ), and then w ∈ MV0 and vn → v = 0 in H s (RN ). Let us show that (yn ) has a subsequence such that yn → y ∈ M. First, we prove that (yn ) is bounded in RN . Assume, by contradiction, that (yn ) is not bounded, that is, there exists a subsequence, still denoted by (yn ), such that |yn | → ∞. First, we deal with the case V∞ = ∞. Since (un ) ⊂ Nεn , a change of variable shows that   V (εn x + yn )vn2 dx ≤ [vn ]2s + V (εn x + yn )vn2 dx RN

RN



= un 2εn =

RN

 f (un )un dx =

RN

f (vn )vn dx.

6.3 The Subcritical Case

227

Then Fatou’s lemma and the fact that vn → v in H s (RN ) imply that  ∞ = lim inf n→∞



 RN

V (εn x +

yn )vn2 dx

≤ lim inf n→∞

RN

f (vn )vn dx =

RN

f (v)v dx < ∞,

which gives a contradiction. Let us consider the case V∞ < ∞. Taking into account that w ∈ MV0 , that wn → w strongly in H s (RN ), condition (V ), Fatou’s lemma and using the translation invariance, we have dV0 = IV0 (w) < IV∞ (w)     1 1 2 2 ≤ lim inf [wn ]s + V (εn x + yn )wn dx − F (wn ) dx n→∞ 2 2 RN RN  2    tn tn2 2 2 [un ]s + V (εn x)un dx − F (tn un ) dx = lim inf n→∞ 2 2 RN RN = lim inf Jεn (tn un ) ≤ lim inf Jεn (un ) = dV0 , n→∞

(6.3.35)

n→∞

which is impossible. Thus (yn ) is bounded in RN and, up to a subsequence, we may assume / M, then V0 < V (y) and we can argue as above to get a contradiction. that yn → y. If y ∈ Therefore, we conclude that y ∈ M.   Let δ > 0 be fixed. Let ψ be a smooth nonincreasing cut-off function defined in [0, ∞) such that ψ = 1 in [0, 2δ ], ψ = 0 in [δ, ∞), 0 ≤ ψ ≤ 1 and |ψ | ≤ c for some c > 0. For any y ∈ M, we define 

εx −y ϒε,y (x) = ψ(| ε x − y|)ω ε

 ,

where ω ∈ H s (RN ) is a positive ground state solution to (PV0 ), the existence of which is guaranteed by Lemma 6.3.10. Let tε > 0 be the unique positive number such that Jε (tε ϒε,y ) = max Jε (tϒε,y ) t ≥0

and define the map ε : M → Nε by setting ε (y) = tε ϒε,y . By construction, ε has compact support for any y ∈ M. Lemma 6.3.17 The functional ε has the property that lim Jε (ε (y)) = dV0 ,

ε→0

uniformly in y ∈ M.

(6.3.36)

228

6 Fractional Schrödinger Equations with Rabinowitz Condition

Proof Suppose, by contradiction, that there exist δ0 > 0, (yn ) ⊂ M and εn → 0 such that |Jεn (εn (yn )) − dV0 | ≥ δ0 .

(6.3.37)

Let us note that Lemma 1.4.8 and the dominated convergence theorem imply that lim ϒε n ,yn 2εn = ω2V0 ∈ (0, ∞).

(6.3.38)

n→∞

Since Jε n (tεn ϒεn ,yn ), tεn ϒεn ,yn  = 0, we can use the change of variable z = see that  f (tεn ϒεn )tεn ϒεn dx tεn ϒεn ,yn 2εn =  =

εn x−yn εn

to

RN

RN

f (tεn ψ(| ε n z|)ω(z))tεn ψ(| ε n z|)ω(z) dz.

(6.3.39)

Now, let us prove that tεn → 1. First we show that tεn → t0 < ∞. Again by contradiction, suppose that |tεn | → ∞. Since ψ(|x|) = 1 for x ∈ B δ and B δ ⊂ B δ for n sufficiently 2 2 2 εn large, (6.3.39) and (f5 ) give  ϒεn ,yn 2εn ≥

Bδ

f (tεn ω(z)) 2 f (tεn ω(¯z)) ω (z) dz ≥ tεn ω(z) tεn ω(¯z)

2

 ω2 (z) dz,

(6.3.40)

Bδ

2

where z¯ ∈ RN is such that ω(¯z) = min{ω(z) : |z| ≤ 2δ } > 0. From (f4 ) and tεn → ∞, we see that (6.3.40) implies that ϒε n ,yn 2εn → ∞, which contradicts (6.3.38). Therefore, up to a subsequence, we may assume that tεn → t0 ≥ 0. If t0 = 0, we can use the growth assumptions on f , (6.3.38) and (6.3.39) to get a contradiction. Hence, t0 > 0. Let us show that t0 = 1. Indeed, taking the limit as n → ∞ in (6.3.39), we obtain that  ω2V0 =

RN

f (t0 ω) ω dx, t0

so recalling that ω ∈ MV0 and using (f5 ), we get t0 = 1. This fact and the dominated convergence theorem yield 

 lim

n→∞ RN

F (tεn ϒεn ,yn ) dx =

RN

F (ω) dx.

Passing to the limit as n → ∞ in Jε (εn (yn )) =

tε2n 2

 ϒεn ,yn 2εn −

RN

F (tεn ϒεn ,yn ) dx,

(6.3.41)

6.3 The Subcritical Case

229

and using (6.3.38) and (6.3.41), we infer that lim Jεn (εn (yn )) = IV0 (ω) = dV0

n→∞

 

which is impossible in view of (6.3.37).

Now, we are in the position to introduce the barycenter map. For any δ > 0, let ρ = ρ(δ) > 0 be such that Mδ ⊂ Bρ , and we consider χ : RN → RN given by  χ(x) =

x, ρx |x| ,

if |x| < ρ, if |x| ≥ ρ.

We define the barycenter map βε : Nε → RN by the formula  βε (u) =

2 RNχ(ε x)u (x) dx . 2 RN u (x) dx

Lemma 6.3.18 The functional ε has the property that lim βε (ε (y)) = y,

ε→0

uniformly in y ∈ M.

(6.3.42)

Proof Suppose, by contradiction, that there exist δ0 > 0, (yn ) ⊂ M and εn → 0 such that |βεn (εn (yn )) − yn | ≥ δ0 .

(6.3.43)

n Using the definitions of εn (yn ), βεn , ψ and the change of variable z = εn x−y εn , we see that  2 N [χ(ε n z + yn ) − yn ](ψ(| ε n z|)ω(z)) dz  βεn (εn (yn )) = yn + R . 2 RN (ψ(| ε n z|)ω(z)) dz

Since (yn ) ⊂ M ⊂ Bρ , the dominated convergence theorem shows that |βεn (εn (yn )) − yn | = on (1), which contradicts (6.3.43). ε of Nε : At this point, we introduce the following subset N ε = {u ∈ Nε : Jε (u) ≤ dV0 + h(ε)}, N

 

230

6 Fractional Schrödinger Equations with Rabinowitz Condition

where h(ε) = supy∈M |Jε (ε (y)) − dV0 |. From Lemma 6.3.17, we know that h(ε) → 0 ε as ε → 0. By the definition of h(ε), we deduce that, for all y ∈ M and ε > 0, ε (y) ∈ N ε = ∅. and N Lemma 6.3.19 For any δ > 0, lim sup dist(βε (u), Mδ ) = 0.

ε→0

ε u∈N

εn such that Proof Let εn → 0 as n → ∞. By definition, there exists (un ) ⊂ N sup

ε u∈N n

inf |βεn (u) − y| = inf |βεn (un ) − y| + on (1).

y∈Mδ

y∈Mδ

Therefore, it suffices to prove that there exists (yn ) ⊂ Mδ such that lim |βεn (un ) − yn | = 0.

(6.3.44)

n→∞

εn ⊂ Nεn , we deduce that Recalling that (un ) ⊂ N dV0 ≤ cεn ≤ Jεn (un ) ≤ dV0 + h(εn ), which implies that Jεn (un ) → dV0 . Using Proposition 6.3.16, there exists (y˜n ) ⊂ RN such that yn = εn y˜n ∈ Mδ for n sufficiently large. Thus  βεn (un ) = yn +

RN [χ(ε n z + yn ) − yn ](un (z + y˜n )) 2 RN (un (z + y˜n )) dz

2 dz

.

Since un (· + y˜n ) converges strongly in H s (RN ) and εn z + yn → y ∈ Mδ , we infer that   βεn (un ) = yn + on (1), that is, (6.3.44) holds. Now we show that (Pε ) admits at least catMδ (M) positive solutions. To this end, we recall the following result for critical points involving Lusternik–Schnirelman category. For more details one can consult [147, 267]. Theorem 6.3.20 Let U be a C 1,1 complete Riemannian manifold (modeled on a Hilbert space). Assume that h ∈ C 1 (U, R) is bounded from below and satisfies −∞ < infU h < d < k < ∞. Moreover, assume that h satisfies the Palais–Smale condition on the sublevel {u ∈ U : h(u) ≤ k} and that d is not a critical level for h. Then card{u ∈ hd : ∇h(u) = 0} ≥ cathd (hd ), where hd = {u ∈ U : h(u) ≤ d}.

6.3 The Subcritical Case

231

With a view to apply Theorem 6.3.20, the following abstract lemma provides a very useful tool in that it relates the topology of some sublevel of a functional to the topology of some subset of the space RN . For the proof, an easy application of the definitions of category and of homotopy equivalence between maps, we refer to [95, 147]. Lemma 6.3.21 Let I , I1 and I2 be closed sets with I1 ⊂ I2 , and let π : I → I2 and ψ : I1 → I be two continuous maps such that π ◦ ψ is homotopy equivalent to the embedding j : I1 → I2 . Then catI (I ) ≥ catI2 (I1 ). Since Nε is not a C 1 submanifold of Hε , we cannot apply directly Theorem 6.3.20. Fortunately, by Lemma 6.3.2, we know that the mapping mε is a homeomorphism between Nε and Sε , and Sε is a C 1 submanifold of Hε . So we can apply Theorem 6.3.20 to ε (u) = ˆ ε (u))|Sε = Jε (mε (u)), where ε is given in Lemma 6.3.3. Jε (m Theorem 6.3.22 Assume that (V ) and (f1 )–(f5 ) hold. Then, for every δ > 0 there exists ε¯ δ > 0 such that, for any ε ∈ (0, ε¯ δ ), problem (Pε ) has at least catMδ (M) positive solutions. Proof For any ε > 0, we define αε : M → Sε by setting αε (y) = m−1 ε (ε (y)). Using Lemma 6.3.17 and the definition of ε , we see that lim ε (αε (y)) = lim Jε (ε (y)) = dV0 ,

ε→0

ε→0

uniformly in y ∈ M.

Set  Sε = {w ∈ Sε : ε (w) ≤ dV0 + h(ε)}, where h(ε) = supy∈M |ε (αε (y)) − dV0 | → 0 Sε for all y ∈ M, and this yields  Sε = ∅ for all ε > 0. as ε → 0. Thus, αε (y) ∈  By Lemmas 6.3.2, 6.3.3, 6.3.17, and 6.3.19, we can find ε¯ = ε¯ δ > 0 such that the following diagram ε

m−1 ε

mε

βε

ε −→  ε −→ Mδ Sε −→ N M −→ N is well defined for any ε ∈ (0, ε¯ ). By Lemma 6.3.18, for ε > 0 small enough, we can write βε (ε (y)) = y + θ (ε, y) for y ∈ M, where |θ (ε, y)| < 2δ uniformly for y ∈ M. Let H (t, y) = y + (1 − t)θ (ε, y). Then we see that H : [0, 1] × M → Mδ is continuous, H (0, y) = βε (ε (y)) and H (1, y) = y for all y ∈ M. Hence, H (t, y) is a homotopy between βε ◦ ε = (βε ◦ mε ) ◦ αε and the inclusion map id : M → Mδ . This fact and Lemma 6.3.21 imply that catSε ( Sε ) ≥ catMδ (M). On the other hand, let us choose a function h(ε) > 0 such that h(ε) → 0 as ε → 0 and dV0 + h(ε) is not a critical level for Jε . For ε > 0 small enough, we deduce from Theorem 6.3.15 that Jε satisfies ε . Then, by item (ii) of Lemma 6.3.3, we infer that ε the Palais–Smale condition in N satisfies the Palais–Smale condition in  Sε . Applying Theorem 6.3.20 we see that ε has

232

6 Fractional Schrödinger Equations with Rabinowitz Condition

  at least cat Sε (Sε ) critical points on Sε . In view of item (iii) of Lemma 6.3.3, we conclude that Jε admits at least catMδ (M) critical points.  

6.3.5

Concentration Phenomenon for (6.1.1)

Let us prove the following result which plays a fundamental role in the study of the behavior of maximum points of solutions to (6.1.1). Lemma 6.3.23 Let vn be a solution of the following problem: 

(−)s vn + Vn (x)vn = f (vn ) in RN , vn ∈ H s (RN ), vn > 0 in RN ,

(6.3.45)

where Vn (x) ≥ V0 for all x ∈ RN . Assume that vn → v in H s (RN ) for some v ≡ 0. Then, vn ∈ L∞ (RN ) and there exists C > 0 such that vn L∞ (RN ) ≤ C for all n ∈ N. In addition, vn (x) → 0 as |x| → ∞, uniformly in n ∈ N. Proof We use a Moser iteration argument [278]. For any L > 0 and β > 1, let t 1 2(β−1) γ (t) = ttL for t ≥ 0, where tL = min{t, L}, and (t) = 0 (γ (τ )) 2 dτ . Thus 2(β−1) ∈ Hε , where vL,n = min{vn , L}. Moreover, arguing as in the proof of γ (vn ) = vn vL,n Proposition 3.2.14, we have the following estimates: 2(β−1)

|(vn (x)) − (vn (y))|2 ≤ (vn (x) − vn (y))((vn vL,n

2(β−1)

)(x) − (vn vL,n

)(y)),

and [(vn )]2s

≥

S∗ (vn )2 2∗s N L (R )

2(β−1)

Choosing γ (vn ) = vn vL,n

 2 1 β−1 ≥ S∗ vn vL,n 2 2∗s N . L (R ) β

as test function in (6.3.45) we deduce that

 2  1 β−1 2(β−1) S∗ vn vL,n 2 2∗s N + Vn (x)vn2 vL,n dx L (R ) β RN   (vn (x) − vn (y)) 2(β−1) 2(β−1) 2(β−1) ≤ ((v v )(x) − (v v )(y)) dxdy + Vn (x)vn2 vL,n dx n n L,n L,n |x − y|N +2s R2N RN  2(β−1) f (vn )vn vL,n dx. (6.3.46) = RN

6.3 The Subcritical Case

233

On the other hand, by (f2 )–(f3 ), we know that for any ξ > 0 there exists Cξ > 0 such that ∗

|f (t)| ≤ ξ |t| + Cξ |t|2s −1

for all t ∈ R.

(6.3.47)

Taking ξ ∈ (0, V0 ), and combining (V ), (6.3.46) and (6.3.47), we see that β−1 vn vL,n 2 2∗s N L (R )

Now, we take β =  RN

2∗s 2

=  ≤

=

{vn R}

vns

 dx +

{vn >R}

2∗ s −2

(vn vL,n2 )2 dx

2∗ vns dx

 2∗s −2  ∗

2∗ s −2 2

2s

RN

2∗s

(vn vL,n ) dx

 2∗ 2s

.

(6.3.49) Since vn → v in H s (RN ), for R > 0 sufficiently large, independent of n, we have that  {vn >R}

2∗ vns dx

 2∗s −2 ∗ 2s

1 . 2Cβ 2

≤

(6.3.50)

Putting together (6.3.48), (6.3.49), and (6.3.50) we get 

2∗ s −2 2

RN

2∗s

 2∗

(vn vL,n ) dx

2s

 ≤ Cβ

∗

2 RN

2∗

R 2s −2 vns dx < ∞ 2 (2∗ s)

and taking the limit as L → ∞, we conclude that vn ∈ L 2 (RN ). Next, since 0 ≤ vL,n ≤ vn , passing to the limit as L → ∞ in (6.3.48), we have  RN

2∗ β vns

 2∗ dx

2s

 ≤ Cβ 2

2∗ +2(β−1)

RN

vns

dx,

234

6 Fractional Schrödinger Equations with Rabinowitz Condition

which implies that  RN

2∗ β vns

 dx

1 2∗ s (β−1)

≤ (Cβ)

1 β−1

 RN

2∗ +2(β−1) vns dx



1 2(β−1)

(6.3.51)

.

For m ≥ 1 we define βm+1 inductively so that 2∗s + 2(βm+1 − 1) = 2∗s βm and β1 =

2∗s . 2

Therefore,  βm+1 = 1 +

2∗s 2

m (β1 − 1),

which together with (6.3.51) yields 

2∗ βm+1

RN

vns

 dx

1 2∗ s (βm+1 −1)

1



≤ (Cβm+1 ) βm+1 −1

2∗ βm

RN

vns

 dx

1 2∗ s (βm −1)

.

Let us define  Dm =

RN

2∗ βm

vns

 dx

1 2∗ s (βm −1)

.

Using an iteration argument, we can find C0 > 0 independent of m such that Dm+1 ≤

m 1

1

(Cβk+1 ) βk+1 −1 D1 ≤ C0 D1 .

k=1

Taking the limit as m → ∞ we get vn L∞ (RN ) ≤ K for all n ∈ N. In particular, by using interpolation in Lp spaces, vn Lr (RN ) ≤ C for all n ∈ N and r ∈ [2, ∞].

(6.3.52)

From vn → v in H s (RN ) and (6.3.52), we see that vn → v in Lr (RN ) for all r ∈ [2, ∞). Let wn (x, y) = Ext(vn ) = Ps (x, y) ∗ vn (x) be the s-harmonic extension of vn . Note that wn solves (see Sect. 1.2.3 in Chap. 1) ⎧ 1−2s ∇w ) = 0 ⎪ in RN+1 n ⎨ − div(y + , wn (·, 0) = vn on ∂RN+1 + , ⎪ ⎩ ∂w = −V (x)v + f (v ) on ∂RN+1 . n n n + ∂ν 1−2s

6.3 The Subcritical Case

Since

 RN

235

Ps (z, y) dz = 1 for all y > 0, inequality (6.3.52) implies that 

|wn (x, y)| ≤ vn L∞ (RN )

RN

Ps (z, y) dz ≤ K for all (x, y) ∈ RN+1 + , n ∈ N.

Using the above estimate and (6.3.52), it follows from Proposition 1.3.11-(iii) that vn ∈ 0,α Cloc (RN ) (alternatively, one can combine (6.3.52) with Corollary 5.5 in [222] to deduce the Hölder continuity of vn ). Now, recalling that 

κs C(N, s) [ω]s = Ext(ω)Xs (RN+1 ) + 0 2

for all ω ∈ H s (RN ),

the convergence vn → v in H s (RN ) and Lemma 1.3.9-(i) show that wn → Ext(v) in 2 1−2s ), where γ = 1 + N L2γ (RN+1 + ,y N−2s . Fix x0 ∈ R and δ ∈ (0, V0 ). Since Vn ≥ 2∗ −1

V0 > 0 in RN and |f (vn )| ≤ δvn + Cδ vns , we deduce that wn is a weak subsolution 2∗ −1 to (1.3.3) in Q1 (x0 , 0) = B1 (x0 ) × (0, 1), with a(x) = 0 and b(x) = Cδ vns (x). By Proposition 1.3.11-(i) (with ν = 2γ ), 2∗ −1

vn (x0 ) ≤ C(wn L2γ (Q1 (x0 ,0),y 1−2s ) + |vns

|Lq (B1 (x0 )) )

for all n ∈ N,

N is fixed and C > 0 is a constant depending only on N, s, q, γ and not on where q > 2s N . Using the strong n ∈ N and x0 . Note that q(2∗s − 1) ∈ (2, ∞) because N > 2s and q > 2s ∗ −1) N+1 2γ q(2 N convergence of (wn ) in L (R+ ) and (vn ) in L s (R ), respectively, we conclude   that vn (x0 ) → 0 as |x0 | → ∞ uniformly in n ∈ N.

Lemma 6.3.24 Under the assumptions of Lemma 6.3.23, there exists δ > 0 such that vn L∞ (RN ) ≥ δ for all n ∈ N. Proof Suppose, by contradiction, that vn L∞ (RN ) → 0 as n → ∞. Using (f2 ), there exists n0 ∈ N such that

f (vn L∞ (RN ) ) vn L∞ (RN )


0 there exists ε¯ δ > 0 such that, for any ε ∈ (0, ε¯ δ ), problem (Pε ) has at least catMδ (M) positive solutions. Let uεn be a solution to (Pεn ). Then vn (x) = uεn (x + y˜ n ) is a solution to (6.3.45) with Vn (x) = V (εn x + εn y˜n ), where (y˜n ) is given in Proposition 6.3.16. Moreover, up to a subsequence, it follows from Proposition 6.3.16 that vn → v = 0 in H s (RN ) and yn = εn y˜n → y ∈ M. Let pn be a global maximum point of vn . Using Lemmas 6.3.23 and 6.3.24, we see that pn ∈ BR for some R > 0. Consequently, zεn = pn + y˜n is a global maximum point of uεn and then εn zεn = εn pn + εn y˜n → y ∈ M because (pn ) is bounded. This fact and the continuity of V imply that lim V (εn zεn ) = V (y) = V0 .

n→∞

Now, if uε is a positive solution to (Pε ), then wε (x) = uε ( xε ) is a positive solution to (6.1.1). Thus, the maximum points ηε and zε of wε and uε , respectively, satisfy the equality ηε = ε zε , and lim V (ηε ) = V0 .

ε→0

 

This ends the proof of theorem.

6.4

The Critical Case

6.4.1

The Nehari Method for (6.1.3)

In this section we deal with the critical problem (6.1.3). Since many calculations are adaptations to those presented earlier, we will emphasize only the differences between the subcritical and the critical cases. Using a change of variable, we can consider the following problem 

∗

(−)s u + V (ε x)u = f (u) + |u|2s −2 u in RN , u ∈ H s (RN ), u > 0 in RN .

We consider the energy functional Jε : Hε → R given by Jε (u) =

1 u2ε − 2

 RN

F (u) dx −

1 2∗ u s2∗s N ∗ L (R ) 2s

and introduce the Nehari manifold associated with Jε , that is ! " Nε = u ∈ Hε \ {0} : Jε (u), u = 0 .

(Pε∗ )

6.4 The Critical Case

237

Arguing as in the previous section we can prove that the next lemmas hold true. Lemma 6.4.1 The functional Jε satisfies the following conditions: (i) There exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ. (ii) There exists e ∈ Hε such that eε > ρ and Jε (e) < 0. Lemma 6.4.2 Under assumptions (V ) and (f1 )–(f5 ) and (f6 ), for ε > 0 we have: (i) Jε maps bounded sets in Hε into bounded sets in Hε . (ii) Jε is weakly sequentially continuous in Hε . (iii) Jε (tn un ) → −∞ as tn → ∞, where un ∈ K and K ⊂ Hε \ {0} is a compact subset. Lemma 6.4.3 Under the assumptions of Lemma 6.4.2, for ε > 0 we have: (i) For every u ∈ Sε , there exists a unique tu > 0 such that tu u ∈ Nε . Moreover, mε (u) = tu u is the unique maximum of Jε on Hε , where Sε = {u ∈ Hε : uε = 1}. (ii) The set Nε is bounded away from 0. Furthermore, Nε is closed in Hε . (iii) There exists α > 0 such that tu ≥ α for each u ∈ Sε and, for each compact subset W ⊂ Sε , there exists CW > 0 such that tu ≤ CW for all u ∈ W . (iv) Nε is a regular manifold diffeomorphic to the unit sphere Sε . (v) cε = infNε Jε ≥ ρ > 0 and Jε is bounded below on Nε , where ρ is independent of ε. ˆ ε : Hε \ {0} → R and ε : Sε → R defined by Now, we introduce the functionals  ˆ ε (u) = Jε (m  ˆ ε (u))

and

ˆ ε |Sε . ε = 

Since the inverse of the mapping mε to Sε is given by m−1 ε : Nε → Sε ,

m−1 ε (u) =

u , uε

the following result holds true: Lemma 6.4.4 Under the assumptions of Lemma 6.4.2, for ε > 0, we have: (i) ε ∈ C 1 (Sε , R), and ε (w), v = mε (w)ε Jε (mε (w)), v

for v ∈ Tw Sε .

238

6 Fractional Schrödinger Equations with Rabinowitz Condition

(ii) (wn ) is a Palais–Smale sequence for ε if and only if (mε (wn )) is a Palais–Smale sequence for Jε . If (un ) ⊂ Nε is a bounded Palais–Smale sequence for Jε , then (m−1 ε (un )) is a Palais–Smale sequence for ε . (iii) u ∈ Sε is a critical point of ε if and only if mε (u) is a critical point of Jε . Moreover, the corresponding critical values coincide and inf ε = inf Jε = cε . Nε

Sε

6.4.2

The Autonomous Critical Problem

Let us consider the following family of autonomous critical problems, with μ > 0, 

∗

(−)s u + μu = f (u) + |u|2s −2 u in RN , u ∈ H s (RN ), u > 0 in RN .

(Pμ∗ )

Let us introduce the energy functional Iμ : Xμ → R defined as 1 Iμ (u) = u2μ − 2

 RN

F (u) dx −

1 2∗s u , ∗ L2s (RN ) 2∗s

and the Nehari manifold associated with Iμ , given by ! " Mμ = u ∈ Xμ \ {0} : Iμ (u), u = 0 . Similarly to the autonomous subcritical case, we have the following result: Lemma 6.4.5 Under the assumptions of Lemma 6.4.2, for μ > 0 we have: (i) for every u ∈ Sμ , there exists a unique tu > 0 such that tu u ∈ Mμ . Moreover, mμ (u) = tu u is the unique maximum of Iμ on Wε , where Sμ = {u ∈ Xμ : uμ = 1}. (ii) The set Mμ is bounded away from 0. Furthermore, Mμ is closed in Xμ . (iii) There exists α > 0 such that tu ≥ α for each u ∈ Sμ and, for each compact subset W ⊂ Sμ , there exists CW > 0 such that tu ≤ CW for all u ∈ W . (iv) Mμ is a regular manifold diffeomorphic to the unit sphere Sμ . (v) dμ = infMμ Iμ > 0 and Iμ is bounded below on Mμ by some positive constant. ˆ μ : Xμ \ {0} → R and μ : Sμ → R by setting Let us define the functionals  Ψˆ μ (u) = Iμ (m ˆ μ (u))

and

ˆ μ |Sμ . μ = 

6.4 The Critical Case

239

Then we obtain the following result: Lemma 6.4.6 Under the assumptions of Lemma 6.4.2, for μ > 0 we have: (i) μ ∈ C 1 (Sμ , R), and μ (w), v = mμ (w)μ Iμ (mμ (w)), v

for all v ∈ Tw Sμ .

(ii) (wn ) is a Palais–Smale sequence for μ if and only if (mμ (wn )) is a Palais–Smale sequence for Iμ . If (un ) ⊂ Mμ is a bounded Palais–Smale sequence for Iμ , then (m−1 μ (un )) is a Palais–Smale sequence for μ . (iii) u ∈ Sμ is a critical point of μ if and only if mμ (u) is a critical point of Iμ . Moreover, the corresponding critical values coincide and inf μ = inf Iμ = dμ . Mμ

Sμ

Remark 6.4.7 As in the previous section, we have the following variational characterization of the infimum of Iμ over Mμ : dμ = inf Iμ (u) = u∈Mμ

inf

max Iμ (tu) = inf max Iμ (tu).

u∈Xμ \{0} t >0

u∈Sμ t >0

It is easy to check that Iμ has a mountain pass geometry. Hence, we can define the mountain pass level dμ = inf max Iμ (γ (t)), γ ∈μ t ∈[0,1]

where μ = {γ ∈ C([0, 1], Xμ ) : Iμ (0) = 0, Iμ (γ (1)) < 0}. In what follows we use the following equivalent characterization of dμ which is more appropriate for our purpose: dμ =

inf

max Iμ (tu) = dμ ,

u∈Xμ \{0} t >0

where in the last equality we used (6.3.9). Standard arguments show that any Palais–Smale sequence of Iμ is bounded in Xμ . In order to obtain the existence of a nontrivial solution to the autonomous critical problem, we need to prove the next fundamental result. Lemma 6.4.8 For any μ > 0, there exists v ∈ Xμ \ {0} such that max Iμ (tv) < t ≥0

s 2sN S∗ . N

240

6 Fractional Schrödinger Equations with Rabinowitz Condition N

In particular, 0 < dμ
0, S∗ is achieved by Uε (x) = 

κ ε−   μ2 + 

N−2s 2

x 1 ε S∗2s

2  

 N−2s , 2

where κ ∈ R and μ > 0 are fixed constants. Let η ∈ Cc∞ (RN ) be a cut-off function such that η = 1 in B1 , supp(η) ⊂ B2 and 0 ≤ η ≤ 1. Set vε (x) = η(x)Uε (x). It follows from [310] that 2∗s

N

[vε ]2s = S∗2s + O(εN−2s )

vε 

and

∗ L2s (RN )

N

= S∗2s + O(εN ).

Define uε =

vε . vε L2∗s (RN )

(6.4.1)

Then we have the following estimates (see [174, 310]): [uε ]2s ≤ S∗ + O(εN−2s ),

uε 2L2 (RN )

(6.4.2)

⎧ 2s ⎪ ⎨ O(ε ), if N > 4s, = O(ε2s log 1ε ), if N = 4s, ⎪ ⎩ O(εN−2s ), if N < 4s,

(6.4.3)

and

q uε Lq (RN )

⎧ N− (N−2s)q ⎪ 2 ⎪ ), if q > ⎨ O(ε N ≥ O(ε 2 | log(ε)|), if q = ⎪ ⎪ ⎩ O(ε (N−2s)q 2 ), if q
0 such that limt →∞ Iμ (tuε ) = −∞ uniformly for ε ∈ (0, ε0 ). Then, there exists t2 > 0 such that for ε ∈ (0, ε0 ), max Iμ (tuε )
0, r ≥ 1,

and gathering the estimates (6.4.2), (6.4.3), (6.4.4), we get ⎧ N s 2s N−2s ) + O(ε 2s ) − λCu q1 , if N > 4s, ⎪ S ⎪ ε q1 ∗ ⎨ N N + O(ε

max Iμ (tuε ) ≤ s S∗2s + ε2s (1 + | log ε |) − λCuε qq1 , if N = 4s, 1 N ⎪ t ∈[t1 ,t2 ] ⎪ N ⎩ s 2s N−2s ) − λCu q1 , S + O(ε if N < 4s. ε q1 N ∗ In what follows, we show that, for ε > 0 small enough, max Iμ (tuε )
4s. Then q1 > 2 > max Iμ (tuε ) ≤

t ∈[t1 ,t2 ]

N N−2s

N s 2s S∗ . N

and using (6.4.4), we have

N (N−2s) s 2s S∗ + O(εN−2s ) + O(ε2s ) − O(εN− 2 q1 ). N

(6.4.8)

242

6 Fractional Schrödinger Equations with Rabinowitz Condition

Since (N − 2s) q1 < 2s < N − 2s, 2

N− we infer that

max Iμ (tuε )
0 is sufficiently small. N If N = 4s, then q1 > 2 = N−2s , and in view of (6.4.4), we obtain that max Iμ (tuε ) ≤

t ∈[t1 ,t2 ]

s 2sN S∗ + O(ε2s (1 + | log ε |)) − O(ε4s−sq1 ). N

Observing that q1 > 2 yields ε4s−sq1 = ∞, ε→0 ε 2s (1 + | log ε |) lim

we get the conclusion for ε > 0 small enough. Now let us consider the case N < 4s. First, we assume that 2∗s − 2 < q1 < 2∗s . Then, q1 > 2∗s − 2 >

N N − 2s

which combined with (6.4.4) gives max Iμ (tuε ) ≤

t ∈[t1 ,t2 ]

(N−2s) s 2sN S∗ + O(εN−2s ) − O(εN− 2 q1 ). N

Since (N − 2s) q1 < N − 2s < 2s, 2

N− we have for ε > 0 small enough

max Iμ (tuε )
2s − (N−2s) q1 , we have the thesis for ε > 0 sufficiently 2 small. Putting together (6.4.6), (6.4.7), and (6.4.8), we conclude that max Iμ (tuε ) < t ≥0

N s 2s S∗ . N

Since dμ ≤ maxt ≥0 Iμ (tuε ), we obtain the desired estimate.

 

Now, we prove the following lemma. Lemma 6.4.9 Let the sequence (un ) ⊂ Mμ be such that Iμ (un ) → dμ . Then, (un ) is bounded in Xμ , and there are a sequence (yn ) ⊂ RN and constants R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Proof It is easy to check that (un ) is bounded in Xμ . Now, we assume that for any R > 0,  lim sup

n→∞

y∈RN

BR (y)

u2n dx = 0.

244

6 Fractional Schrödinger Equations with Rabinowitz Condition

The boundedness of (un ) and Lemma 1.4.4, imply that for any r ∈ (2, 2∗s ).

un → 0 in Lr (RN )

(6.4.9)

Using (6.3.1), (6.3.2), and (6.4.9), we deduce that    

RN

  f (un )un dx  ≤ ξ un 2L2 (RN ) + on (1)

(6.4.10)

  F (un ) dx  ≤ Cξ un 2L2 (RN ) + on (1).

(6.4.11)

and    

RN

Letting ξ → 0 in (6.4.10) and (6.4.11), we see that 

 RN

f (un )un dx = on (1)

and

RN

F (un ) dx = on (1).

(6.4.12)

Taking into account that Jε (un ), un  = 0 and using (6.4.12), we obtain 2∗s

un 2μ − un 

∗

L2s (RN )

= on (1).

Since (un ) is bounded in Xμ , we may assume that un 2μ → % ≥ 0

and

2∗s

un 

∗

L2s (RN )

→ % ≥ 0.

(6.4.13)

Suppose, by contradiction, that % > 0. By virtue of (6.4.12) and (6.4.13), dμ = Iμ (un ) + on (1) = = that is, % =

1 un 2μ − 2

 RN

F (un ) dx −

1 2∗ un  s2∗s N + on (1) ∗ L (R ) 2s

% % s − ∗ + on (1) = % + on (1), 2 2s N N s dμ .

On the other hand, by Theorem 1.1.8, S∗ un 2 2∗s

L (RN )

≤ [un ]2s + μun 2L2 (RN ) = un 2μ

and taking the limit as n → ∞ we obtain that 2 ∗

S∗ % 2s ≤ %.

6.4 The Critical Case

Since % =

N s dμ ,

245

we get dμ ≥

N

s 2s N S∗ ,

which is impossible in view of Lemma 6.4.8.

 

Now we prove the main result for the autonomous critical case. Lemma 6.4.10 The problem (Pμ∗ ) has at least one positive ground state solution. Proof The proof follows the arguments used in Lemma 6.3.10. We only need to replace (6.3.14) by 1

I (un ), un  ϑ μ        1 1 1 1 1 2∗ = − un 2μ + f (un )un − F (un ) dx + − ∗ un  s2∗s N , L (R ) 2 ϑ ϑ 2s RN ϑ

dμ + on (1) = Iμ (un ) −

and then recalling that lim sup (an + bn + cn ) ≥ lim sup an + lim inf(bn + cn ) n→∞

n→∞

n→∞

≥ lim sup an + lim inf bn + lim inf cn , n→∞

n→∞

n→∞

we deduce that      1 1 1 dμ ≥ − f (un )un − F (un ) dx lim sup un 2μ + lim inf n→∞ RN ϑ 2 ϑ n→∞   1 1 2∗ − ∗ lim inf un  s2∗s N . + L (R ) ϑ 2s n→∞ Moreover, we use Lemma 6.4.9 instead of Lemma 6.3.9. Thus, we obtain the existence of a ground state solution to (Pμ∗ ). Let us note that all the calculations above can be repeated word by word, replacing Iμ by Iμ+ (u)

1 = u2μ − 2

 RN

F (u+ ) dx −

1 + 2∗s u  2∗s N . L (R ) 2∗s

In this way we find a ground state solution u ∈ Xμ to the equation ∗

(−)s u + μu = f (u+ ) + (u+ )2s −1 and standard arguments show that u > 0 in RN .

in RN ,  

246

6 Fractional Schrödinger Equations with Rabinowitz Condition

Finally, similarly to Theorem 6.3.11, we have the following compactness result for the critical autonomous case: Theorem 6.4.11 Let (un ) ⊂ MV0 be a sequence such that IV0 (un ) → dV0 . Then either (i) (un ) has a subsequence that converges strongly in H s (RN ), or (ii) there exists a sequence (y˜n ) ⊂ RN such that, up to a subsequence, vn = un (· + y˜n ) converges strongly in H s (RN ). In particular, there exists a minimizer for dV0 .

6.4.3

An Existence Result for the Critical Case

Arguing as in the proof of Lemma 6.4.9, we can prove the “critical” version of Lemma 6.3.12. N

Lemma 6.4.12 Let 0 < d < Ns S∗2s and let (un ) ⊂ Nε be a sequence such that Jε (un ) → d and un  0 in Hε . Then either (a) un → 0 in Hε , or (b) there are a sequence (yn ) ⊂ RN and constants R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

The next result is obtained following the lines of the proof of Lemma 6.3.13. Lemma 6.4.13 Assume that V∞ < ∞ and let (vn ) ⊂ Nε be a sequence such that N

Jε (vn ) → d with 0 < d < Ns S∗2s and vn  0 in Hε . If vn → 0 in Hε , then d ≥ dV∞ , where dV∞ is the infimum of IV∞ over MV∞ . Now, we establish a compactness result in the critical case. Proposition 6.4.14 Let (un ) ⊂ Nε be a sequence such that Jε (un ) → c, where c < dV∞ if V∞ < ∞ and c < Hε .

N

s 2s N S∗

if V∞ = ∞. Then (un ) admits a convergent subsequence in

Proof Since Jε (un ) → c and Jε (un ) = 0, we can see that (un ) is bounded in Hε and, up to a subsequence, we may assume that un  u in Hε . Clearly, Jε (u) = 0.

6.4 The Critical Case

247

Let vn = un − u. Using the Brezis-Lieb lemma [113] and Lemma 3.3 in [269], we know that 2∗s

vn 

2∗s

∗

L2s (RN )

= un 

2∗s

∗

L2s (RN )

− u

∗

L2s (RN )

+ on (1)

and  RN

  2∗s  2∗s −2 2∗s −2 2∗s −2  2∗s −1 vn − |un | un + |u| u dx = on (1). |vn |

Then, arguing as in Proposition 6.3.14, we see that Jε (vn ) = Jε (un ) − Jε (u) + on (1) = c − Jε (u) + on (1) = d + on (1) and Jε (vn ) = on (1). Note that, by (f4 ), it holds that 1 Jε (u) = Jε (u) − Jε (u), u = 2



 RN

   1 1 1 2∗ f (u)u − F (u) dx + − ∗ u s2∗s N ≥ 0. L (R ) 2 2 2s (6.4.14)

If V∞ < ∞, then from (6.4.14) we deduce that d ≤ c < dV∞ . In view of Lemma 6.4.13 we have that vn → 0 in Hε , that is un → u in Hε . Let us consider the case V∞ = ∞. Then, we can use Lemma 6.2.3 to deduce that vn → 0 in Lr (RN ) for all r ∈ [2, 2∗s ), which in conjunction with (f2 ) and (f3 ) implies that   f (vn )vn dx = on (1) and F (vn ) dx = on (1). (6.4.15) RN

RN

Putting together (6.4.15) and the fact that Jε (vn ), vn  = on (1), we deduce that 2∗s

vn 2ε = vn 

∗

L2s (RN )

+ on (1). 2∗

Since (vn ) is bounded in Hε , we may assume that vn 2ε → % and vn  s2∗s N → %, for L (R ) some % ≥ 0. Let us show that % = 0. If, by contradiction, % > 0, then since Jε (vn ) = d + on (1), we get 1 1 2∗ vn 2ε − ∗ vn  s2∗s N = d + on (1), L (R ) 2 2s

248

6 Fractional Schrödinger Equations with Rabinowitz Condition

whence s vn 2ε = d + on (1). N Taking the limit as n → ∞ we see that Theorem 1.1.8, vn 2ε ≥ S∗ vn 2 2∗s

L (RN )

s N%

= d, that is, % = d Ns . Therefore, by

2∗ = S∗ vn  s2∗s

2∗ 2s

L (RN )

N

and letting n → ∞ we get % ≥ S∗2s . Thus we have d ≥ get a contradiction. Hence, % = 0 and un → u in Hε .

N

s 2s N S∗ .

,

Since d ≤ c
0 small enough. Theorem 6.4.15 Assume that (V ) and (f1 )–(f5 ) and (f6 ) hold. Then there exists ε0 > 0 such that, for any ε ∈ (0, ε0 ), problem (Pε∗ ) admits a ground state solution. Proof It is enough to proceed as in the proof of Theorem 6.3.15 once Lemmas 6.3.2, 6.3.3, Proposition 6.3.14 and Lemma 6.3.10 are replaced by Lemmas 6.4.3, 6.4.4, Proposition 6.4.14 and Lemma 6.4.10, respectively.  

6.4.4

A Multiplicity Result for (6.1.3)

In this section we study the multiplicity of solutions to (6.1.3). Arguing as in the proof of Proposition 6.3.16 and using Lemma 6.4.11 instead of Lemma 6.3.11, we can deduce the following result. Proposition 6.4.16 Let ε n → 0 and (un ) = (uεn ) ⊂ Nεn be such that Jεn (un ) → dV0 . Then there exists (y˜n ) = (y˜εn ) ⊂ RN such that vn (x) = un (x + y˜n ) has a convergent subsequence in H s (RN ). Moreover, up to a subsequence, yn → y ∈ M, where yn = εn y˜n . Now, fix δ > 0 and let ω ∈ H s (RN ) be a positive ground state solution to problem (PV∗0 ), the existence of which is guaranteed by Lemma 6.4.10. Let ψ be a smooth nonincreasing cut-off function defined in [0, ∞), such that ψ = 1 in [0, 2δ ], ψ = 0 in [δ, ∞), 0 ≤ ψ ≤ 1 and |ψ | ≤ c for some c > 0. For any y ∈ M, we define  ϒε,y (x) = ψ(| ε x − y|)ω

εx −y ε

 ,

6.4 The Critical Case

249

and then take tε > 0 such that Jε (tε ϒε,y ) = max Jε (tε ϒε,y ). t ≥0

Also, define ε : M → Nε by ε (y) = tε ϒε,y . Lemma 6.4.17 The functional ε has the property that lim Jε (ε (y)) = dV0 ,

ε→0

uniformly in y ∈ M.

(6.4.16)

Proof Suppose, by contradiction, that there exist δ0 > 0, (yn ) ⊂ M and εn → 0 such that |Jεn (εn (yn )) − dV0 | ≥ δ0 .

(6.4.17)

Using Lemma 1.4.8 we know that lim ϒε n ,yn 2εn = ω2V0 .

(6.4.18)

n→∞

On the other hand, by the definition of tε , we see that Jε n (tεn ϒεn ,yn ), tεn ϒεn ,yn  = 0 which implies  tεn ϒεn ,yn 2εn =  =

 RN

RN

f (tεn ϒεn )tεn ϒεn dx +

∗

RN

(tεn ϒεn )2s dx

f (tεn ψ(| εn z|)ω(z))tεn ψ(| ε n z|)ω(z) dz

2∗



+ tεns

∗

RN

(ψ(| ε n z|)ω(z))2s dz.

(6.4.19)

Let us prove that tεn → 1. First, we show that tεn → t0 < ∞. If, by contradiction, |tεn | → ∞, then since ψ(|x|) = 1 for x ∈ B δ and B δ ⊂ B δ for n sufficiently large, we 2 2 2 εn see that (6.4.19) yields 2∗ −2



ϒεn ,yn 2εn ≥ tεns

∗

(ω(z))2s dz Bδ

2

≥





2∗ −2   tεns B δ  2

2∗s

 min ω(z)

|z|≤ 2δ

→ ∞,

(6.4.20)

which is impossible in view of (6.4.18). Hence, we can suppose that tεn → t0 ≥ 0. From the growth assumptions on f and (6.4.18), we deduce that t0 > 0.

250

6 Fractional Schrödinger Equations with Rabinowitz Condition

Let us prove that t0 = 1. Taking the limit as n → ∞ in (6.4.19), we can see that  ω2V0 =

RN

f (t0 ω) 2∗ −2 ω dx + t0 s t0



∗

RN

ω2s dx,

and using that ω ∈ MV0 and condition (f5 ), we deduce that t0 = 1. This and the dominated convergence theorem show that 

 lim

n→∞ RN

F (tεn ϒεn ,yn ) dx =

RN

F (ω) dx

and 



2∗s

lim

n→∞ RN

(tεn ϒεn ,yn ) dx =

∗

RN

ω2s dx.

Consequently, lim Jεn (εn (yn )) = IV0 (ω) = dV0

n→∞

 

which leads to a contradiction because of (6.4.17). For any δ > 0, let ρ = ρ(δ) > 0 be such that Mδ ⊂ Bρ . Define χ : RN → RN by  χ(x) =

x, ρx |x| ,

if |x| < ρ, if |x| ≥ ρ.

Consider the barycenter map βε : Nε → RN given by  βε (u) =

2 RNχ(ε x)u (x) dx . 2 RN u (x) dx

Arguing as in the proof of Lemma 6.3.18 we obtain the next result. Lemma 6.4.18 The functional ε has the property that lim βε (ε (y)) = y,

ε→0

uniformly in y ∈ M.

Define ε = {u ∈ Nε : Jε (u) ≤ dV0 + h(ε)}, N

(6.4.21)

6.5 A Remark on the Ambrosetti-Rabinowitz Condition

251

where h(ε) = supy∈M |Jε (ε (y)) − dV0 |. From Lemma 6.4.17 we deduce that h(ε) → 0 ε as ε → 0. By the definition of h(ε), we know that, for every y ∈ M and ε > 0, ε (y) ∈ N ε = ∅. In addition, proceeding as in the proof of Lemma 6.3.19, we get the following and N lemma: Lemma 6.4.19 For any δ > 0, there holds that lim sup dist(βε (u), Mδ ) = 0.

ε→0

ε u∈N

Finally, we can prove the main result related to (6.1.3). Proof of Theorem 6.1.2 Given δ > 0 and taking into account Lemmas 6.4.3, 6.4.4, 6.4.17, 6.4.18, 6.4.19, Proposition 6.4.14, Theorem 6.4.15 and recalling that 0 < dV0 < N

s 2s N S∗

(see Lemma 6.4.8), we can argue as in the proof of Theorem 6.3.22 to deduce that (6.1.3) admits at least catMδ (M) positive solutions for all ε > 0 sufficiently small. The concentration of solutions is obtained by means of the arguments used in the proof of Theorem 6.1.1. Indeed, the proof of Lemma 6.3.23 works also in the critical case and the proof of Lemma 6.3.24 can be easily adapted to this case.  

6.5

A Remark on the Ambrosetti-Rabinowitz Condition

As discussed in Chap. 5, the Ambrosetti-Rabinowitz condition plays a fundamental role in the verification of the boundedness of Palais–Smale sequences of the energy functional associated with the problem under consideration. Anyway, as proved in [75] (see also [6]), we can see that Theorems 6.1.1 and 6.1.2 are still true if we replace (f4 ) by the condition (f4 )

F (t ) t2

→ ∞ as t → ∞.

In what follows, we only prove some technical results which are used to implement the arguments used in the previous subsections. First, we need the following variant of Lions lemma inspired by Ramos et al. [300]. Lemma 6.5.1 Let (un ) be a bounded sequence in H s (RN ) such that 

∗

|un |2s dx = 0,

lim sup

n→∞

y∈RN

BR (y)

for some R > 0. Then, un → 0 in Lt (RN ) for all t ∈ (2, 2∗s ].

252

6 Fractional Schrödinger Equations with Rabinowitz Condition

Proof Applying the Hölder inequality we have 

2∗s



2∗s

|un | dx ≤ BR (y)

|un | dx

 2∗s −2  ∗

2∗s

2s

BR (y)

|un | dx

 2∗ 2s

.

BR (y)

Now, covering RN by balls of radius R in such a way that each point of RN is contained in at most N + 1 balls and using the fact that (un ) is bounded in H s (RN ), we obtain that 2∗s

un 

 ∗

L2s (RN )

2∗s

≤ C sup

|un | dx

 2∗s −2 ∗ 2s

→0

as n → ∞.

BR (y)

y∈RN

An interpolation argument concludes the proof of lemma.

 

In order to apply the Lusternik–Schnirelman category theory, we have to prove that Jε satisfies the Palais–Smale condition on Nε . Since we do not assume (AR), we prove the following auxiliary result to deduce the boundedness of Palais–Smale sequences. Lemma 6.5.2 Let (un ) ⊂ Nε be a sequence such that Jε (un ) → c. Then, (un ) is bounded in Hε . Proof Since we study both subcritical and critical cases, we consider the nonlinearity ∗ f (t) + γ |t|2s −2 t, where γ ∈ {0, 1}. Assume, by contradiction, that, up to subsequences, un ε → ∞ as n → ∞. Set vn = un un −1 ε . Note that (vn ) is bounded in Hε (since vn ε = 1 for all n ∈ N). Let us prove that 

∗

|vn |2s dx = 0

lim sup

n→∞

y∈RN

BR (y)

for all R > 0. Suppose this is not the case. Then there are R, δ > 0, (yn ) ⊂ RN such that 

∗

|vn |2s dx ≥ δ > 0.

lim inf n→∞

BR (yn )

Set v˜n = vn (· + y˜n ). Then, (v˜n ) is bounded in Hε and v˜n  v˜ in Hε and strongly in Ltloc (RN ) for all t ∈ [1, 2∗s ), for some v˜ ≡ 0. Define u˜ n = un ε v˜n . By (f4 ) we deduce that F (u˜ n ) 2 v˜ → ∞ u˜ 2n n

a.e.

in A = {x ∈ RN : v˜ = 0}.

6.6 Further Generalizations: The Fractional p-Laplacian Operator

253

Then using Fatou’s lemma we have that 1 c + on (1) − = 2 un 2ε



∗

F (un ) + γ |un |2s dx ≥ un 2ε

RN

 A

F (u˜ n ) 2 v˜ dx → ∞, u˜ 2n n

which gives a contradiction. Therefore, by Lemma 6.5.1, we infer that vn → 0 in Lt (RN ) for all t ∈ (2, 2∗s ]. Now, using conditions (f2 )–(f3 ), we see that for every r > 1 and for every τ > 0 there exists a constant Cτ > 0 such that ∗

|F (rt)| ≤ τ |rt|2 + Cτ |rt|2s

for all t ∈ R.

Therefore,   ∗   |rvn |2s   lim sup  F (rvn ) + γ dx  ≤ τ r2 ∗   2s n→∞ RN

for all τ > 0,

which implies that  lim sup n→∞

RN

∗

F (rvn ) + γ

|rvn |2s dx = 0. 2∗s

(6.5.1)

On the other hand, un −1 ε r ∈ (0, 1) for all n sufficiently large, and we have Jε (un ) = max Jε (tun ) ≥ Jε (run −1 ε un ) = Jε (rvn ) = t ≥0

$ ∗  # r2 |rvn |2s − F (rvn ) + γ dx, 2 2∗s RN

which together with (6.5.1) yields lim inf Jε (un ) ≥ r 2 n→∞

for all r > 1.

Letting r → ∞ we deduce that ∞ > c = lim inf Jε (un ) = ∞ n→∞

a contradiction.

6.6

 

Further Generalizations: The Fractional p-Laplacian Operator

One can show (see [74, 75]) that the existence and multiplicity results obtained in the previous subsections can be extended to the following fractional p-Laplacian type

254

6 Fractional Schrödinger Equations with Rabinowitz Condition

problem: 

∗

εsp (−)sp u + V (x)|u|p−2u = f (u) + γ |u|ps −2 u in RN , u ∈ W s,p (RN ), u > 0 in RN ,

(6.6.1)

where ε > 0 is a parameter, γ ∈ {0, 1}, s ∈ (0, 1), 1 < p < ∞, N > sp, and f is a continuous function with subcritical growth at infinity. Here, (−)sp is the fractional pLaplacian operator which may be defined, up to a normalization constant depending on N, p and s, by setting  (−)sp u(x) = 2 lim

r→0 RN \Br (x)

|u(x) − u(y)|p−2 (u(x) − u(y)) dy |x − y|N+sp

(x ∈ RN )

for any u ∈ Cc∞ (RN ). The above operator can be regarded as the fractional analogue of the p-Laplacian operator p u = div(|∇u|p−2 ∇u). We recall that the recent years have seen a surge of interest in nonlocal and fractional problems involving the fractional p-Laplacian operator because of the presence of two features: the nonlinearity of the operator and its nonlocal character. For this reason, several existence and regularity results have been established by many authors. Franzina and Palatucci [201] discussed some basic properties of the eigenfunctions of a class of nonlocal operators whose model is (−)sp (see also [252]). Mosconi et al. [277] used an abstract linking theorem based on the cohomological index to obtain nontrivial solutions to the Brezis–Nirenberg problem [114] for the fractional p-Laplacian operator. Di Castro et al. [167] established interior Hölder regularity results for fractional p-minimizers (see also [222]). In [33] the author obtained the existence of infinitely many solutions for a superlinear fractional p-Laplacian equation with sign-changing potential. We also mention [56, 110, 163, 164, 329, 335] for other interesting contributions. We stress that the operator (−)sp is not linear when p = 2, so more technical difficulties arise in the study of (6.6.1). For instance, we cannot make use of the s-harmonic extension by Caffarelli and Silvestre [127] to apply well-known variational techniques in the study of local degenerate elliptic problems. Clearly, the arguments developed in the case p = 2 are not trivially adaptable and some technical lemmas are needed to overcome the non-Hilbertian structure of the fractional Sobolev spaces W s,p (RN ) when p = 2. In particular, when γ = 1, more refined estimates coming from [110, 277] are used to obtain the corresponding version of Lemma 6.4.8. For more details we refer the interested reader to [74].

7

Fractional Schrödinger Equations with del Pino-Felmer Assumptions

7.1

Introduction

This chapter is concerned with the existence and concentration of positive solutions for the following fractional Laplacian problem 

∗

ε2s (−)s u + V (x)u = f (u) + γ |u|2s −2 u u ∈ H s (RN ), u > 0 in RN ,

in RN ,

(7.1.1)

2N where ε > 0 is a small parameter, s ∈ (0, 1), N > 2s, 2∗s = N−2s is the fractional critical Sobolev exponent, γ ∈ {0, 1}. Throughout the chapter we will assume that V : RN → R is a continuous potential satisfying the following assumptions due to del Pino and Felmer [165]:

(V1 ) there exists V1 > 0 such that V1 = infx∈RN V (x); (V2 ) there exists a bounded open set  ⊂ RN such that 0 < V0 = inf V (x) < min V (x). x∈

x∈∂

The nonlinearity f : R → R is a continuous function such that f (t) = 0 for t ≤ 0 and fulfills the following conditions if γ = 0: (f1 ) f (t) = o(t) as t → 0; (f2 ) there exists q ∈ (2, 2∗s ) such that lim

f (t)

t →∞ t q−1

= 0;

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_7

255

256

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions



t

(f3 ) there exists ϑ ∈ (2, q) such that 0 < ϑF (t) = ϑ (f4 ) the function t →

f (τ ) dτ ≤ tf (t) for all t > 0;

0

f (t) is increasing in (0, ∞). t

When γ = 1 we require that f satisfies (f1 ), (f3 ), (f4 ) and the following technical condition: (f2 ) there exist q, σ ∈ (2, 2∗s ) and λ > 0 such that f (t) ≥ λt q−1

∀t > 0,

lim

f (t)

t →∞ t σ −1

= 0,

where λ is such that • λ > 0 if either N > 4s, or 2s < N < 4s and 2∗s − 2 < q < 2∗s , • λ is sufficiently large if 2s < N < 4s and 2 < q ≤ 2∗s − 2. Differently from Chap. 6, here we focus on the existence and concentration of positive solutions to (7.1.1) under the local conditions (V1 )-(V2 ) on the potential V . We observe that no restriction on the global behavior of V is imposed other than (V1 ). In particular, V is not required to be bounded or to belong to a Kato class. Moreover, we consider the cases when f has subcritical, critical or supercritical growth. Our first main result is the following: Theorem 7.1.1 ([19, 35, 68]) Assume that (V1 )-(V2 ) and (f1 )–(f4 ) hold. Then, there exists ε0 > 0 such that, for all ε ∈ (0, ε0 ), problem (7.1.1) admits a positive solution. Moreover, if xε ∈ RN denotes a global maximum point of uε , then lim V (xε ) = V0 ,

ε→0

and there exists C > 0 such that 0 < uε (x) ≤

C εN+2s ε N+2s +|x − xε |N+2s

for all x ∈ RN .

The proof of Theorem 7.1.1 is obtained by using some variational techniques inspired by [19, 35, 68, 165]. Since we don’t have any information about the behavior of the potential V at infinity, we adapt the penalization method introduced by del Pino and Felmer in [165]. It consists in making a suitable modification on f , solving a modified problem and then checking that, for ε > 0 small enough, the solutions of the new problem are indeed solutions of the original one. This last step is rather difficult and it will be obtained by combining a Moser iteration argument [278] and some useful regularity elliptic estimates.

7.1 Introduction

257

Our second result concerns the critical case. Theorem 7.1.2 ([68, 216]) Assume that (V1 )-(V2 ) and (f1 ), (f2 ), (f3 ) with ϑ ∈ (2, σ ), (f4 ) hold. Then, there exists ε0 > 0 such that, for all ε ∈ (0, ε0 ), problem (7.1.1) admits a positive solution. Moreover, if xε ∈ RN denotes a global maximum point of uε , then lim V (xε ) = V0 .

ε→0

The proof of Theorem 7.1.2 follows some arguments in [12, 35, 68, 190, 216]. In this case an additional difficulty appears in the study of (7.1.1), due to the presence of the critical exponent. Anyway, we will see that the approach used in the subcritical case works, after a more careful analysis, in the critical case. Moreover, in order to prove some compactness properties for the energy modified functional, we make use of the concentration-compactness lemma established in Section 1.5. Finally, we consider a supercritical version of the problem under consideration, namely ∗

ε2s (−)s u + V (x)u = |u|q−2 u + λ|u|2s −2 u

in RN ,

(7.1.2)

where 2 < q < 2∗s ≤ r and λ > 0. In this case, our main result can be stated as follows: Theorem 7.1.3 ([35]) Assume that (V1 )-(V2 ) hold. Then there exists λ0 > 0 with the following property: for any λ ∈ (0, λ0 ) there exists ελ > 0 such that, for all ε ∈ (0, ελ ), problem (7.1.2) admits a positive solution. Moreover, if xε ∈ RN denotes a global maximum point of uε , then lim V (xε ) = V0 .

ε→0

The proof of this result is based on some arguments developed in [136, 188, 297]. Since r > 2∗s , we cannot use directly variational techniques because the corresponding functional is not well defined on the Sobolev space H s (RN ). To overcome this difficulty, we first truncate suitably the nonlinearity involving the supercritical exponent, in order to deal with a new subcritical problem. Taking into account Theorem 7.1.1, we know that an existence result for this truncated problem holds true. Then, we deduce a priori bounds for these solutions, and by using a Moser iteration technique [278], we are able to show that for λ > 0 sufficiently small the solutions of the truncated problem also satisfy the original problem (7.1.1).

258

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

7.2

The Subcritical Case

7.2.1

The Modified Problem

Using the change of variable x → ε x, we reduce the study of (7.1.1) to the investigation of the problem 

(−)s u + V (ε x)u = f (u) in RN , u ∈ H s (RN ), u > 0 in RN .

(7.2.1)

Now, we introduce a penalized function in the spirit of [165]. First of all, without loss of generality, we may assume that 0∈ Take K >

ϑ ϑ−2

and

V (0) = V0 = inf V . 

and a > 0 such that f (a) =  f˜(t) =

V1 K a,

and set

f (t), if t ≤ a, V1 K t, if t > a,

and  g(x, t) =

χ (x)f (t) + (1 − χ (x))f˜(t), if t ≥ 0, 0, if t < 0.

It is easy to check that g : RN ×R → R is a Carathéodory function satisfying the following properties: g(x, t) = 0 uniformly with respect to x ∈ RN ; t →0 t (g2 ) g(x, t) ≤ f (t) for all x ∈ RN , t > 0;

(g1 ) lim

t

(g3 ) (i) 0 < ϑG(x, t) = ϑ

g(x, τ ) dτ ≤ g(x, t)t for all x ∈  and t > 0,

0

V1 2 t for all x ∈ RN \  and t > 0; K g(x, t) (g4 ) for each x ∈  the function t → is increasing in (0, ∞), and for each x ∈ t g(x, t) is increasing in (0, a). RN \  the function t → t (ii) 0≤ 2G(x, t) ≤ g(x, t)t ≤

7.2 The Subcritical Case

259

Then, we consider the following modified problem: 

(−)s u + V (ε x)u = g(ε x, u) u ∈ H s (RN ), u > 0 in RN ,

in RN ,

(7.2.2)

In view of the definition of g, we will look for weak solutions to (7.2.2) having the property u(x) ≤ a

for any x ∈ RN \ ε ,

where ε = / ε. In order to study the problem (7.2.2), we seek the critical points of the functional  1 Jε (u) = u2ε − G(ε x, u) dx, 2 RN which is well defined for all u : RN → R belonging to the fractional space   s N Hε = u ∈ H (R ) :

 2

RN

V (ε x)u dx < ∞ ,

endowed with the norm   uε = [u]2s +

1 RN

V (ε x)u2 dx

2

.

Clearly, Hε is a Hilbert space with the inner product  u, vε = u, vDs,2 (RN ) +

RN

V (ε x)uv dx.

Standard arguments show that the functional Jε belongs to C 1 (Hε , R) and that its differential is given by Jε (u), v

 = u, vε −

RN

g(ε x, u)v dx

for any u, v ∈ Hε . Now let us show that Jε possesses a mountain pass geometry [29]: Lemma 7.2.1 The functional Jε has a mountain pass geometry: (a) there exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ; (b) there exists e ∈ Hε such that eε > ρ and Jε (e) < 0.

260

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Proof (a) By (g1 ), (g2 ) and (f2 ), we see that for every ξ > 0 there exists Cξ > 0 such that ∗

|g(x, t)| ≤ ξ |t| + Cξ |t|2s −1

for any (x, t) ∈ RN × R.

Therefore, Jε (u) ≥

1 u2ε − 2

 RN

G(ε x, u) dx ≥

1 2∗ u2ε − ξ Cu2ε − Cξ Cuε s , 2

and we can find α, ρ > 0 such that Jε (u) ≥ α with uε = ρ. (b) Using (g3 )-(i), we deduce that for any u ∈ Cc∞ (RN ) such that u ≥ 0, u ≡ 0 and supp(u) ⊂ ε Jε (τ u) ≤

τ2 u2ε − 2

 G(ε x, τ u) dx ε

τ2 ≤ u2ε − C1 τ ϑ 2

 uϑ dx + C2

for any τ > 0,

ε

for some positive constants C1 and C2 . Since ϑ ∈ (2, 2∗s ), we see that Jε (τ u) → −∞ as τ → ∞.   Invoking a variant of the mountain pass theorem without the Palais-Smale condition (see Remark 2.2.10), we can find a sequence (un ) ⊂ Hε such that Jε (un ) → cε

and

Jε (un ) → 0 in Hε∗ ,

where cε = inf max Jε (γ (t)) γ ∈ε t ∈[0,1]

and

ε = {v ∈ Hε : Jε (0) = 0, Jε (γ (1)) < 0}.

As in [299], we can use the following equivalent characterization of cε that is more appropriate for our purposes: cε =

inf

max Jε (tu).

u∈Hε \{0} t ≥0

Moreover, in view of the monotonicity of g, it is easy to check that for any non-negative u ∈ Hε \ {0} there exists a unique t0 = t0 (u) > 0 such that Jε (t0 u) = max Jε (tu). t ≥0

7.2 The Subcritical Case

261

In the next lemma we prove that every Palais-Smale sequence of Jε is bounded. Lemma 7.2.2 Let (un ) ⊂ Hε be a Palais-Smale sequence at the level c for Jε . Then (un ) is bounded in Hε . Proof Since (un ) is a Palais-Smale sequence at the level c, we have Jε (un ) → c

and

Jε (un ) → 0 in Hε∗ .

By (g3 ) we deduce that for all n ∈ N 1 C(1 + un ε ) ≥ Jε (un ) − Jε (un ), un  ϑ    ϑ −2 1 2 = [g(ε x, un )un − ϑG(ε x, un )] dx un ε + 2ϑ ϑ RN \ε  1 + [g(ε x, un )un − ϑG(ε x, un )] dx ϑ ε    ϑ −2 1 ≥ [g(ε x, un )un − ϑG(ε x, un )] dx un 2ε + 2ϑ ϑ RN \ε      ϑ −2 1 ϑ −2 V (ε x)u2n dx ≥ un 2ε − 2ϑ 2ϑ K RN \ε    ϑ −2 1 ≥ 1− un 2ε . 2ϑ K Since ϑ > 2 and K > 1, we conclude that (un ) is bounded in Hε .

 

Remark 7.2.3 Arguing as in Remark 5.2.8, we may always assume that the Palais-Smale sequence (un ) is non-negative in RN . Lemma 7.2.4 Jε satisfies the (PS)c condition at any level c ∈ R. Proof Let (un ) ⊂ Hε be a Palais-Smale sequence for Jε at the level c. Our aim is to prove that for any ξ > 0 there exists R = Rξ > 0 such that  lim sup n→∞

RN \BR

 RN

|un (x) − un (y)|2 dxdy + |x − y|N+2s

 RN \BR

V (ε x)u2n dx < ξ.

(7.2.3)

Assume that the above claim is true and we show how it can be used to finish the proof of lemma. By Lemma 7.2.2, we may assume that un  u in Hε . Since Hε is compactly embedded in Lp (K) for all p ∈ [1, 2∗s ) and compact sets K ⊂ RN , g has subcritical

262

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

growth, and Cc∞ (RN ) is dense in Hε , it is easy to check that Jε (u) = 0. In particular,  u2ε =

RN

g(ε x, u)u dx.

Recalling that Jε (un ), un  = on (1), we have that  un 2ε =

RN

g(ε x, un )un dx + on (1).

Using (7.2.3) and (V1 ), we can see that for all ξ > 0 there exists R = Rξ > 0 such that  lim sup n→∞

RN \BR

u2n dx ≤

1 lim sup V1 n→∞

 RN \BR

V (ε x)u2n dx
0 larger, that  RN \BR

u2 dx < ξ.

These last two inequalities combined with the fact that Hε is compactly embedded in L2 (BR ) imply that % lim sup un − u2L2 (RN ) = lim sup un − u2L2 (B n→∞

n→∞

= lim un − u2L2 (B n→∞

R)

R

+ un − u2L2 (RN \B )

& R)

+ lim sup un − u2L2 (RN \B n→∞

R)

≤ Cξ, and letting ξ → 0 we see that un → u in L2 (RN ). By interpolation on the Lp -spaces ∗ and the boundedness of (un ) in L2s (RN ), we conclude that un → u in Lp (RN ) for all p ∈ [2, 2∗s ). Consequently, by (g1 ), (g2 ), and the dominated convergence theorem, 

 lim

n→∞ RN

g(ε x, un )un dx =

RN

g(ε x, u)u dx.

Therefore, lim un 2ε = u2ε ,

n→∞

and using the fact that Hε is a Hilbert space we deduce that un → u in Hε as n → ∞.

7.2 The Subcritical Case

263

Now we prove that (7.2.3) holds. Let ηR ∈ C ∞ (RN ) be such that 0 ≤ ηR ≤ 1, ηR = 0 in B R , ηR = 1 in RN \ BR and ∇ηR L∞ (RN ) ≤ C R for some C > 0 independent of R. 2 Since Jε (un ), ηR un  = on (1), we have  un , un ηR Ds,2 (RN ) +

 RN

V (ε x)ηR u2n dx

=

RN

g(ε x, un )un ηR dx + on (1).

Fix R > 0 such that ε ⊂ BR/2 . Using (g3 )-(ii), we have  R2N

ηR (x)

 

≤− +

1 K

|un (x) − un (y)|2 dxdy + |x − y|N+2s un (y)

R2N



RN

 RN

V (ε x)u2n ηR dx

(un (x) − un (y))(ηR (x) − ηR (y)) dxdy |x − y|N+2s



V (ε x)u2n ηR dx + on (1).

(7.2.4)

By Hölder’s inequality and the boundedness of (un ) in Hε it follows that     

R2N

  (un (x) − un (y))(ηR (x) − ηR (y)) un (y) dxdy  |x − y|N+2s

  ≤

R2N

|un (x) − un (y)|2 dxdy |x − y|N+2s

  ≤C

R2N

 12   R2N

|ηR (x) − ηR (y)| |un (y)|2 |x − y|N+2s

|ηR (x) − ηR (y)| |un (y)|2 |x − y|N+2s

 12

2

dxdy

 12

2

dxdy

.

(7.2.5)

Further, by Lemma 1.4.5,  lim lim sup

R→∞ n→∞

R2N

|un (y)|2

|ηR (x) − ηR (y)|2 dxdy = 0. |x − y|N+2s

(7.2.6)

Filially, (7.2.4), (7.2.5) and (7.2.6), imply that  lim lim sup

R→∞ n→∞

RN \BR

 RN

   |un (x) − un (y)|2 1 2 dxdy + 1 − V (ε x)un dx = 0 K |x − y|N+2s RN \BR

which establishes (7.2.3). Now, we are ready to provide an existence result for (7.2.2).

 

264

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Theorem 7.2.5 Assume that (V1 )-(V2 ) and (f1 )–(f4 ) hold. Then, for all ε > 0, problem (7.2.2) admits a positive mountain pass solution. Proof Taking into account Lemma 7.2.1, Lemma 7.2.4 and applying Theorem 2.2.9, we can see that there exists u ∈ Hε such that Jε (u) = cε and Jε (u) = 0. Since Jε (u), u−  = 0, where u− = min{u, 0}, it is easy to check that u ≥ 0 in RN . Indeed, since g(x, t) = 0 for t ≤ 0 and (x − y)(x − − y − ) ≥ |x − − y − |2 , where x − = min{x, 0}, we get u− 2ε



−

≤ u, u Ds,2 (RN ) +



−

RN

V (ε x)uu dx =

RN

g(ε x, u)u− dx = 0,

which implies that u− = 0, that is, u ≥ 0. Moreover, proceeding as in the proof of 0,α Lemma 7.2.9 below, we see that u ∈ L∞ (RN ) ∩ Cloc (RN ), and applying Theorem 1.3.5, N   we can conclude that u > 0 in R . Now, we deal with the following family of autonomous problems, with μ > 0: 

(−)s u + μu = f (u) in RN , u ∈ H s (RN ), u > 0 in RN .

(7.2.7)

It is clear that the Euler-Lagrange functional associated with (7.2.7) is given by     1 2 2 [u]s + μ Iμ (u) = u dx − F (u) dx. 2 RN RN Let us denote by Xμ the fractional Sobolev space H s (RN ), endowed with the norm u2μ = [u]2s + μu2L2 (RN ) . The Nehari manifold associated with Iμ is given by Mμ = {u ∈ Xμ \ {0} : Iμ (u), u = 0}. As in Chap. 6, it is easy to check that Iμ possesses a mountain pass geometry, and denoting by mμ the corresponding mountain pass level, we have that dμ = inf Iμ = Mμ

inf

max Iμ (tu).

u∈Xμ \{0} t ≥0

In view of Lemma 6.3.10, we can state Theorem 7.2.6 For all μ > 0, problem (7.2.7) admits a positive ground state solution.

7.2 The Subcritical Case

265

In what follows, we establish a very useful relation between cε and dV0 : Lemma 7.2.7 The following relation holds: lim sup cε ≤ dV0 . ε→0

Proof Let ωε (x) = ψε (x)ω(x), where ω is the positive ground state of (7.2.7) whose existence is established by Theorem 7.2.6 with μ = V0 , and ψε (x) = ψ(ε x), with ψ ∈ Cc∞ (RN ) such that, 0 ≤ ψ ≤ 1, ψ(x) = 1 if |x| ≤ 12 and ψ(x) = 0 if |x| ≥ 1. For simplicity, we assume that supp(ψ) ⊂ B1 ⊂ . By Lemma 1.4.8 and the dominated convergence theorem, ωε → ω

in H s (RN )

and

IV0 (ωε ) → IV0 (ω) = dV0

(7.2.8)

as ε → 0. Now, for each ε > 0 there exists tε > 0 such that Jε (tε ωε ) = max Jε (tωε ). t ≥0

Then, Jε (tε ωε ), ωε  = 0 and this implies that  [ωε ]2s +

 RN

V (ε x)ωε2 dx =

RN

f (tε ωε ) 2 ωε dx. tε ωε

(7.2.9)

From (7.2.8), (7.2.9) and the growth assumptions on f , we obtain that tε → t0 ∈ (0, ∞). Taking the limit as ε → 0 in (7.2.9) we get  [ω]2s

+

 V0 ω dx = 2

RN

RN

f (t0 ω) 2 ω dx t0 ω

(7.2.10)

which together with (f4 ), ω ∈ MV0 and (7.2.10) implies that t0 = 1. On the other hand, cε ≤ max Jε (tωε ) = Jε (tε ωε ) = IV0 (tε ωε ) + t ≥0

tε2 2

 RN

(Vε (x) − V0 )ωε2 dx.

Since V (ε ·) is bounded on the support of ωε , we use the dominated convergence theorem, (7.2.8) and the above inequality to obtain the desired result.   We conclude this section by proving the following compactness result which will be fundamental for showing that, for ε > 0 small enough, the solutions of the modified problem are also solutions of the original one.

266

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Lemma 7.2.8 Let εn → 0 and (un ) = (uεn ) ⊂ Hεn be such that Jεn (un ) = cεn and Jε n (un ) = 0. Then there exists (y˜n ) = (y˜εn ) ⊂ RN such that u˜ n (x) = un (x + y˜n ) has a convergent subsequence in H s (RN ). Moreover, up to a subsequence, yn = εn y˜n → y0 for some y0 ∈  such that V (y0 ) = V0 . Proof Using Jε n (un ), un  = 0 and (g1 ), (g2 ), it is easy to see that there is κ > 0 such that un εn ≥ κ > 0

for all n ∈ N.

Taking into account that Jεn (un ) = cεn , Jε n (un ), un  = 0 and Lemma 7.2.7, we can argue as in the proof of Lemma 7.2.2 to deduce that (un ) is bounded in Hεn . Moreover, we can find a sequence (y˜n ) ⊂ RN and constants R, α > 0 such that  lim inf n→∞

BR (y˜n )

u2n dx ≥ α.

(7.2.11)

Indeed, if this limit inequality does not hold, we can use Lemma 1.4.4 to deduce that un → 0 in Lp (RN ) for all p ∈ (2, 2∗s ). In view of Jε n (un ), un  = 0 and the growth assumptions on g, it follows that un εn → 0 as n → ∞, which gives a contradiction. Set u˜ n (x) = un (x + y˜n ). Then, (u˜ n ) is bounded in H s (RN ), and we may assume that u˜ n  u˜

weakly in H s (RN ).

(7.2.12)

Moreover, u˜ = 0 because  u˜ 2 dx ≥ α.

(7.2.13)

BR

Now, we define yn = εn y˜n . Let us begin by proving that (yn ) is bounded in RN . To this end, it is enough to show the following claim: Claim 1 limn→∞ dist(yn , ) = 0. Indeed, assuming that this is not the case, there exist a δ > 0 and a subsequence of (yn ), still denoted (yn ), such that dist(yn , ) ≥ δ

for all n ∈ N.

Then we can find r > 0 such that Br (yn ) ⊂ c for all n ∈ N. Since u˜ ≥ 0 and Cc∞ (RN ) is dense in H s (RN ), we can find a sequence (ψj ) ⊂ Cc∞ (RN ) such that ψj ≥ 0 and ψj → u˜

7.2 The Subcritical Case

267

in H s (RN ). Fixing j ∈ N and using ψ = ψj as test function in Jε n (un ), ψ = 0 we get  u˜ n , ψj Ds,2 (RN ) +



RN

V (εn x + εn y˜n )u˜ n ψj dx =

RN

g(ε n x + εn y˜n , u˜ n )ψj dx. (7.2.14)

Recalling that un , ψj ≥ 0 and the definition of g, we have  RN

g(ε n x + εn y˜n , u˜ n )ψj dx 

 g(εn x + εn y˜n , u˜ n )ψj dx +

= Br/ ε n

≤

V1 K

RN \Br/ ε n





u˜ n ψj dx + Br/ ε n

RN \Br/ ε n

g(ε n x + εn y˜n , u˜ n )ψj dx

f (u˜ n )ψj dx,

which together with (7.2.14) implies that  u˜ n , ψj Ds,2 (RN ) + A

 RN

u˜ n ψj dx ≤

RN \Br/ εn

f (u˜ n )ψj dx,

(7.2.15)

where we set A = V1 (1 − K1 ). By (7.2.12), the fact that ψj has compact support in RN and since εn → 0, we deduce that as n → ∞ u˜ n , ψj Ds,2 (RN ) → u, ˜ ψj Ds,2 (RN ) and  RN \Br/ ε n

f (u˜ n )ψj dx → 0.

The above limits and (7.2.15) show that  u, ˜ ψj Ds,2 (RN ) + A

RN

uψ ˜ j dx ≤ 0,

and taking the limit as j → ∞ we obtain u ˜ 2A = [u] ˜ 2s + Au ˜ 2L2 (RN ) ≤ 0 which contradicts (7.2.13). Hence, there exists a subsequence of (yn ) such that yn → y0 ∈ . Claim 2 y0 ∈ .

268

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

From (g2 ) and (7.2.14) we can see that  u˜ n , ψj Ds,2 (RN ) +

 RN

V (εn x + εn y˜n )u˜ n ψj dx ≤

RN

f (u˜ n )ψj dx.

Letting n → ∞ yields  u, ˜ ψj Ds,2 (RN ) +



RN

V (y0 )uψ ˜ j dx ≤

RN

f (u)ψ ˜ j dx,

and passing to the limit as j → ∞ we have  [u] ˜ 2s +

 RN

V (y0 )u˜ 2 dx ≤

RN

f (u) ˜ u˜ dx.

Then there exists τ ∈ (0, 1) such that τ u˜ ∈ MV (y0 ) . Therefore, denoting by dV (y0 ) the mountain pass level associated with IV (y0 ) , we have dV (y0 ) ≤ IV (y0 ) (τ u) ˜ ≤ lim inf Jεn (un ) = lim inf cεn ≤ dV0 , n→∞

n→∞

from which we deduce that V (y0 ) ≤ V (0) = V0 . Since V0 = inf¯ V , we can infer that / ∂, that is y0 ∈ . V (y0 ) = V0 . Using (V2 ), we obtain that y0 ∈ s N Claim 3 u˜ n → u˜ in H (R ) as n → ∞. Let us define ˜ n =  − εn y˜n  εn and consider  χ˜ n1 (x)

=

˜ n, 1, if x ∈  ˜ n, 0, if x ∈ RN \ 

χ˜ n2 (x) = 1 − χ˜ n1 (x). Now, we introduce the following functions for all x ∈ RN  h1n (x)

= 

1 1 − 2 ϑ

 V (εn x + εn y˜n )u˜ 2n (x)χ˜ n1 (x),

 1 1 − V (y0 )u˜ 2 (x), 2 ϑ   1 1 1 2 − V (εn x + εn y˜n )u˜ 2n (x) + g(ε n x + εn y˜n , u˜ n (x))u˜ n (x) hn (x) = 2 ϑ ϑ h1 (x) =

7.2 The Subcritical Case

269

 − G(εn x + εn y˜n , u˜ n (x)) χ˜ n2 (x)  ≥ 

1 1 − 2 ϑ



1 − 2K

 V (εn x + εn y˜n )u˜ 2n (x)χ˜ n2 (x),

 1 g(ε n x + εn y˜n , u˜ n (x))u˜ n (x) − G(εn x + εn y˜n , u˜ n (x)) χ˜n1 (x) ϑ   1 f (u˜ n (x))u˜ n (x) − F (u˜ n (x)) χ˜ n1 (x), = ϑ

h3n (x) =

h3 (x) =

1 f (u(x)) ˜ u(x) ˜ − F (u(x)). ˜ ϑ

In view of (f3 ) and (g3 ), the above functions are non-negative. Moreover, by (7.2.12) and Claim 2, u˜ n (x) → u(x) ˜

a.e. x ∈ RN ,

yn = εn y˜n → y0 ∈ , which implies that χ˜n1 (x) → 1, h1n (x) → h1 (x), h2n (x) → 0 and h3n (x) → h3 (x) a.e. x ∈ RN . Then, using Lemma 7.2.7, Fatou’s lemma and a change of variable, we obtain that   1 dV0 ≥ lim sup cεn = lim sup Jεn (un ) − Jε n (un ), un  ϑ n→∞ n→∞     1 1 2 1 2 3 ≥ lim sup − (hn + hn + hn ) dx [u˜ n ]s + 2 ϑ n→∞ RN     1 1 2 1 2 3 ≥ lim inf − [u˜ n ]s + (hn + hn + hn ) dx n→∞ 2 ϑ RN    1 1 − ≥ (h1 + h3 ) dx ≥ dV0 . [u] ˜ 2s + 2 ϑ RN Accordingly, lim [u˜ n ]2s = [u] ˜ 2s

(7.2.16)

n→∞

and h1n → h1 , h2n → 0

and

h3n → h3

in L1 (RN ).

270

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Hence, 

 lim

n→∞ RN

V (εn x + εn y˜n )u˜ 2n dx =

RN

V (y0 )u˜ 2 dx,

from which we deduce that lim u˜ n 2L2 (RN ) = u ˜ 2L2 (RN ) .

n→∞

(7.2.17)

Finally, (7.2.16) and (7.2.17) and the fact that H s (RN ) is a Hilbert space imply that u˜ n − u ˜ 2V0 = u˜ n 2V0 − u ˜ 2V0 + on (1) = on (1),  

which ends the proof of lemma.

7.2.2

Proof of Theorem 7.1.1

This last section is devoted to the proof of Theorem 7.1.1. First, we use a Moser iteration argument [278] to prove the following useful L∞ -estimate for the solutions of the modified problem (7.2.2). Lemma 7.2.9 Let (u˜ n ) be the sequence given in Lemma 7.2.8. Then, u˜ n ∈ L∞ (RN ) and there exists C > 0 such that u˜ n L∞ (RN ) ≤ C

for all n ∈ N.

Moreover, |u˜ n (x)| → 0 as |x| → ∞, uniformly in n ∈ N. Proof Arguing as in the first part of the proof of Lemma 6.3.23 we see that  2  1 β−1 2(β−1) S∗ u˜ n u˜ L,n 2 2∗s N + Vn (x)u˜ 2n u˜ L,n dx L (R ) β RN  (u˜ n (x) − u˜ n (y)) 2(β−1) 2(β−1) ((u˜ n u˜ L,n )(x) − (u˜ n u˜ L,n )(y)) dxdy ≤ N+2s 2N |x − y| R  2(β−1) + Vn (x)u˜ 2n u˜ L,n dx  =

RN

2(β−1)

RN

gn (x, u˜ n )u˜ n u˜ L,n

dx,

(7.2.18)

7.2 The Subcritical Case

271

where we used the notations Vn (x) = V (εn x +εn y˜n ) and gn (x, u˜ n ) = g(ε n x +εn y˜n , u˜ n ). On the other hand, from (g1 ) and (g2 ), we know that for any ξ > 0 there exists Cξ > 0 such that ∗

|gn (x, u˜ n )| ≤ ξ |u˜ n | + Cξ |u˜ n |2s −1 .

(7.2.19)

Taking ξ ∈ (0, V1 ), and using (V1 ) and (7.2.19), we see that (7.2.18) implies that β−1 u˜ n u˜ L,n 2 2∗s N L (R )

 ≤ Cβ

∗

2 RN

2(β−1)

|u˜ n |2s u˜ L,n

dx.

Now we can exploit the arguments in Lemma 6.3.23 to finish the proof.

 

Remark 7.2.10 Next we prove in a different way that lim|x|→∞ u˜ n (x) = 0 uniformly in n ∈ N. From (V1 ), we know that u˜ n is a subsolution to the equation (−)s u˜ n + V1 u˜ n = hn in RN , where hn (x) = gn (x, u˜ n ). Since u˜ n → u˜ = 0 in H s (RN ) (by Lemma 7.2.8) and u˜ n L∞ (RN ) ≤ C for all n ∈ N (by Lemma 7.2.9), interpolation on the Lp spaces shows that hn → h = f (u) ˜ in Lq (RN ) for any q ∈ [2, ∞), and there exists a constant c1 > 0 such that hn L∞ (RN ) ≤ c1 for all n ∈ N. Now, let zn be the unique solution of (−)s zn + V1 zn = hn in RN . Then we can write  zn (x) = (K ∗ hn )(x) =

RN

K(x − y)hn (y) dy,

(7.2.20)

where the kernel K(x) = F −1 ((|ξ |2s + V1 )−1 ) satisfies the following properties (see [183, 199] and Remark 3.2.18): (b1 ) K is positive, radially symmetric and smooth in RN \ {0}; 1 (b2 ) there exists a positive constant C > 0 such that K(x) ≤ |x|KN+2s for all x ∈ RN \ {0}; N ). (b3 ) K ∈ Lq (RN ) for any q ∈ [1, N−2s

272

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

At this point we borrow some arguments used in [19] to show that zn (x) → 0 as |x| → ∞ uniformly in n ∈ N. Note that, for any δ > 0, it holds  0 ≤ zn (x) = (K ∗ hn )(x) =

 Aδ

K(x − y)hn (y) dy +

Bδ

K(x − y)hn (y) dy (7.2.21)

where   1 N Aδ = y ∈ R : |y − x| ≥ δ

and

  1 N Bδ = y ∈ R : |y − x| < . δ

From (b2 ) we deduce that 



Aδ

K(x − y)hn (y) dy ≤ K1 hn L∞ (RN )  ≤ c1 δ K1 2s

|ξ |≥1

dy N+2s Aδ |x − y| dξ = C1 δ 2s . |ξ |N+2s

(7.2.22)

On the other hand, 

 Bδ

K(x − y)|hn (y)| dy ≤

 Bδ

K(x − y)|hn (y) − h(y)| dy +

Bδ

K(x − y)|h(y)| dy.

Fix q > 1 with q ≈ 1 and q > 2 such that q1 + q1 = 1. From (b3 ) and Hölder’s inequality we have that  K(x − y)|hn (y)| dy ≤ KLq (RN ) hn − hLq (RN ) + KLq (RN ) hLq (Bδ ) . Bδ

Since hn − hLq (RN ) → 0 as n → ∞ and hLq (Bδ ) → 0 as |x| → ∞, we deduce that there exist R > 0 and n0 ∈ N such that  Bδ

K(x − y)|hn (y)| dy ≤ δ

(7.2.23)

for all n ≥ n0 and |x| ≥ R. Putting together (7.2.22) and (7.2.23) we obtain that  RN

for all n ≥ n0 and |x| ≥ R.

K(x − y)|hn (y)| dy ≤ C1 δ 2s + δ.

(7.2.24)

7.2 The Subcritical Case

273

The same approach can be used to prove that for each n ∈ {1, . . . , n0 − 1}, there is Rn > 0 such that  RN

K(x − y)|hn (y)| dy ≤ C1 δ 2s + δ

for all |x| ≥ Rn . Hence, increasing R if necessary, we must have  RN

K(x − y)|hn (y)| dy ≤ C1 δ 2s + δ

for |x| ≥ R, uniformly in n ∈ N. Letting δ → 0 we get the desired result for zn . Since by comparison we see that 0 ≤ u˜ n ≤ zn in RN , we obtain the assertion. Now we are ready to give the proof of the main result of this section. Proof of Theorem 7.1.1 We begin by showing that there exists ε0 > 0 such that for any ε ∈ (0, ε0 ) and any mountain pass solution uε ∈ Hε of (7.2.2), it holds uε L∞ (RN \ε ) < a.

(7.2.25)

Assume, by contradiction, that for some subsequence (εn ) such that εn → 0, we can find un = uεn ∈ Hεn such that Jεn (un ) = cεn , Jε n (un ) = 0 and un L∞ (RN \εn ) ≥ a.

(7.2.26)

In view of Lemma 7.2.8, we can find (y˜n ) ⊂ RN such that u˜ n = un (· + y˜n ) → u˜ in H s (RN ) and εn y˜n → y0 for some y0 ∈  such that V (y0 ) = V0 . Now, if we choose r > 0 so that Br (y0 ) ⊂ B2r (y0 ) ⊂ , we can see that B εr ( εyn0 ) ⊂ n εn . Then, for any y ∈ B εr (y˜n ), n

        y − y0  ≤ |y − y˜n | + y˜n − y0  < 1 (r + on (1)) < 2r    εn εn  εn εn

for n sufficiently large.

Therefore, RN \ εn ⊂ RN \ B εr (y˜n ) n

(7.2.27)

for any n big enough. Using Lemma 7.2.9 we see that u˜ n (x) → 0

as |x| → ∞

(7.2.28)

274

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

uniformly in n ∈ N. Therefore, there exists R > 0 such that u˜ n (x) < a

for any |x| ≥ R, n ∈ N.

Hence, un (x) < a for any x ∈ RN \ BR (y˜n ) and n ∈ N. On the other hand, by (7.2.27), there exists ν ∈ N such that for any n ≥ ν and r/ ε n > R, RN \ εn ⊂ RN \ B εr (y˜n ) ⊂ RN \ BR (y˜n ), n

which implies that un (x) < a for any x ∈ RN \ εn and n ≥ ν. This is impossible in view of (7.2.26). Since uε ∈ Hε satisfies (7.2.25), by the definition of g it follows that uε is a solution of (7.2.1) for all ε ∈ (0, ε0 ). Consequently, uˆ ε (x) = uε (x/ ε) is a solution to 0,α (7.1.1) for ε ∈ (0, ε0 ). We also notice that uˆ ε ∈ L∞ (RN ) ∩ Cloc (RN ). In what follows, we study the behavior of the maximum points of solutions to problem (7.1.1). Take εn → 0 and consider a sequence (un ) ⊂ Hεn of solutions to (7.2.1) as above. Let us observe that (g1 ) implies that there exists ω ∈ (0, a) such that V1 2 t K

g(ε x, t)t = f (t)t ≤

for any x ∈ RN , 0 ≤ t ≤ ω.

(7.2.29)

Arguing as before, we can find R > 0 such that un L∞ (RN \BR (y˜n )) < ω.

(7.2.30)

Moreover, up to a subsequence, we may assume that un L∞ (BR (y˜n )) ≥ ω.

(7.2.31)

Indeed, if (7.2.31) does not hold, then in view of (7.2.30), un L∞ (RN ) < ω. Then, using that Jε n (un ), un  = 0 and (7.2.29) we infer  un 2εn

=

RN

V1 g(εn x, un )un dx ≤ K

 RN

u2n dx

which yields un εn = 0, which is impossible. Hence, (7.2.31) holds true. Let pn ∈ RN be a global maximum point of un . Taking into account (7.2.30) and (7.2.31), we deduce that pn ∈ BR (y˜n ). Therefore, pn = y˜n + qn for some qn ∈ BR . Consequently, ηn = ε n y˜n + ε n qn is a global maximum point of uˆ n (x) = un (x/ εn ). Since |qn | < R for all n ∈ N and εn y˜n → y0 , the continuity of V implies that lim V (ηn ) = V (y0 ) = V0 .

n→∞

7.2 The Subcritical Case

275

Finally, we prove a decay estimate for uˆ n . By Lemma 3.2.17 and scaling, there exists a positive continuous function w such that 0 < w(x) ≤

C 1 + |x|N+2s

for all x ∈ RN ,

(7.2.32)

and w satisfies in the classical sense (−)s w +

V1 w=0 2

in RN \ B R1 ,

(7.2.33)

for a suitable R1 > 0. Using (g1 ) and (7.2.28), we can find R2 > 0 sufficiently large such that   V1 V1 u˜ n = gn (x, u˜ n ) − Vn − u˜ n (−)s u˜ n + 2 2 ≤ gn (x, u˜ n ) −

V1 u˜ n ≤ 0 in RN \ B R2 . 2

(7.2.34)

Let R3 = max{R1 , R2 } > 0 and set c = min w > 0 and w˜ n = (d + 1)w − cu˜ n ,

(7.2.35)

B R3

where d = supn∈N u˜ n L∞ (RN ) < ∞. Let us show that w˜ n ≥ 0 in RN .

(7.2.36)

First, we observe that (7.2.33), (7.2.34) and (7.2.35) yield w˜ n ≥ cd + w − cd > 0 in B R3 , (−)s w˜ n +

V1 w˜ n ≥ 0 in RN \ B R3 . 2

(7.2.37) (7.2.38)

Then we can use Lemma 1.3.8 with = RN \ B R3 to verify that (7.2.36) holds. In view of (7.2.32) and (7.2.36),  0 < u˜ n (x) ≤

 C˜ d +1 w(x) ≤ c 1 + |x|N+2s

for all x ∈ RN , n ∈ N,

(7.2.39)

276

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

for some constant C˜ > 0. Since uˆ n (x) = un ( εxn ) = u˜ n ( εxn − y˜n ) and ηn = εn y˜n + εn qn , we can use (7.2.39) to deduce that  0 < uˆ n (x) = un ≤ = ≤

x εn

1 + | εxn



 = u˜ n

x − y˜n εn



C˜ − y˜n |N+2s C˜ εN+2s n

εN+2s n

+|x − εn y˜n |N+2s C˜ εN+2s n

for all x ∈ RN .

εN+2s +|x − ηn |N+2s n

 

This ends the proof of Theorem 7.1.2.

7.3

The Critical Case

7.3.1

The Modified Critical Problem and the Local (PS) Condition

To establish the existence of a nontrivial solution to (7.1.1) in the case γ = 1, we first modify the nonlinearity in a suitable way. Using the change of variable x → ε x, we consider the following problem: 

∗

(−)s u + V (ε x)u = f (u) + |u|2s −2 u u ∈ H s (RN ), u > 0 in RN .

in RN ,

Without loss of generality we may assume that 0∈ Take K >

ϑ ϑ−2

V (0) = V0 = inf V .

and



∗

and a > 0 such that f (a) + a 2s −1 =  f˜(t) =

V1 K a,

and define

∗

f (t) + (t + )2s −1 , if t ≤ a, V1 if t > a, K t,

and  g(x, t) =

∗

χ (x)(f (t) + t 2s −1 ) + (1 − χ (x))f˜(t), if t ≥ 0, 0, if t < 0.

7.3 The Critical Case

277

It is easy to check that g satisfies the following properties: g(x, t) = 0 uniformly with respect to x ∈ RN ; t ∗ (k2 ) g(x, t) ≤ f (t) + t 2s −1 for all x ∈ RN , t > 0;

(k1 ) lim

t →0

t

(k3 ) (i) 0 < ϑG(x, t) = ϑ

g(x, τ ) dτ ≤ g(x, t)t for all x ∈  and t > 0,

0

V1 2 t for all x ∈ RN \  and t > 0; K g(x, t) is increasing in (0, ∞), and for each x ∈ (k4 ) for each x ∈  the function t → t g(x, t) is increasing in (0, a). RN \  the function t → t (ii) 0≤ 2G(x, t) ≤ g(x, t)t ≤

Then, we consider the following modified problem: 

(−)s u + V (ε x)u = g(ε x, u) u ∈ H s (RN ), u > 0 in RN .

in RN ,

(7.3.1)

Next we look for weak solutions of (7.3.1) having the property u(x) ≤ a

for any x ∈ RN \ ε .

Let us introduce the functional Jε : Hε → R defined as Jε (u) =

1 u2ε − 2

 RN

G(ε x, u) dx,

As in the proof of Lemma 7.2.1, we can see that Jε has a mountain pass geometry; we denote by cε the corresponding mountain pass level. In order to study the compactness properties of Jε , we need to prove some technical results. N

Lemma 7.3.1 It holds, 0 < cε
1 and ϑ > 2 we get the boundedness. Then we may assume that un  u in Hε . Since Jε (un ), un  = on (1), we can see that  un 2ε =

RN

g(ε x, un )un dx + on (1).

(7.3.2)

On the other hand, standard calculations show that u is a critical point of Jε and thus  u2ε

+

 V (ε x)u dx = 2

RN

RN

(7.3.3)

g(ε x, u)u dx.

Let us show that (un ) strongly converges to u in Hε . To this end, it is enough to show that un ε → uε , which in view of (7.3.2) and (7.3.3) means to verify that 

 lim

n→∞ RN

g(ε x, un )un dx =

RN

(7.3.4)

g(ε x, u)u dx.

We begin by showing that for each ξ > 0 there exists R = Rξ > 0 such that  lim sup n→∞

RN \B

 R

RN

|un (x) − un (y)|2 dxdy + |x − y|N+2s

 RN \B

V (ε x)u2n dx < ξ.

(7.3.5)

R

We may assume that R is chosen so that ε ⊂ B R . Let ηR be a cut-off function such that 2

ηR = 0 on B R , ηR = 1 on RN \ BR , 0 ≤ η ≤ 1 and ∇ηR L∞ (RN ) ≤ 2 bounded Palais-Smale sequence, we have Jε (un ), ηR un  = on (1).

C R.

Since (un ) is a

7.3 The Critical Case

279

From (k3 )-(ii), we get    (ηR (x) − ηR (y))(un (x) − un (y)) |un (x) − un (y)|2 ηR (x) dxdy + un (y) dxdy |x − y|N+2s |x − y|N+2s R2N R2N    1 V (ε x)u2n ηR dx = g(ε x, un )ηR un dx + on (1) ≤ V (ε x)u2n ηR dx + on (1). + K RN RN RN



Then, using the definition of ηR , the fact that K > 1, the Hölder inequality, the boundedness of (un ) in Hε , and Remark 1.4.6, we have that   |un (x) − un (y)|2 1 dxdy + 1 − V (ε x)u2n dx N+2s N N N |x − y| K R \BR R R \BR    2 1 |un (x) − un (y)| ηR (x) dxdy + 1 − V (ε x)u2n ηR dx ≤ N+2s 2N |x − y| K R RN    (ηR (x) − ηR (y))(un (x) − un (y)) u (y) dxdy + on (1) ≤− n |x − y|N+2s R2N 



  ≤C ≤

R2N

|ηR (x) − ηR (y)|2 |un (y)|2 dxdy |x − y|N+2s

 12

(7.3.6)

+ on (1)

C + on (1), Rs

so we deduce that  lim lim sup

R→∞ n→∞

R2N

|ηR (x) − ηR (y)|2 |un (y)|2 dxdy = 0. |x − y|N+2s

(7.3.7)

Putting together (7.3.6) and (7.3.7), we infer that (7.3.5) holds. Now, note that the fractional Sobolev inequality (1.1.1) in Theorem 1.1.8, shows that 

2∗s

RN \BR

|un | dx

 2∗ 2s

 ≤

2∗s

RN

|un ηR | dx

 2∗ 2s

≤ C[un ηR ]2s .

From 0 ≤ ηR ≤ 1, (7.3.6) and Remark 1.4.6, we see that 

|(un (x) − un (y))ηR (x) + (ηR (x) − ηR (y))un (y)|2 dxdy |x − y|N+2s R2N $ #   |un (x) − un (y)|2 2 |ηR (x) − ηR (y)|2 2 ηR (x) dxdy + |un (y)| dxdy ≤C |x − y|N+2s |x − y|N+2s R2N R2N

[un ηR ]2s =

280

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions #  ≤C ≤

R2N

|un (x) − un (y)|2 C ηR (x) dxdy + 2s |x − y|N+2s R

$

C C + on (1) + 2s , Rs R

and thus  lim lim sup

R→∞ n→∞

∗

RN \B

|un |2s dx = 0.

(7.3.8)

|un |2 dx = 0,

(7.3.9)

R

Now, we note that (7.3.5) and (V1 ) yield  lim lim sup

R→∞ n→∞

RN \BR

∗

and using interpolation on Lp -spaces and the boundedness of (un ) in L2s (RN ), we also deduce that for all p ∈ (2, 2∗s )  lim lim sup

R→∞ n→∞

RN \BR

|un |p dx = 0.

(7.3.10)

Consequently, using (k2 ), (f1 ), (f2 ), (7.3.8), (7.3.9) and (7.3.10), we see that for every ξ > 0 there exists R = Rξ > 0 such that  lim sup n→∞

 RN \B

g(ε x, un )un dx ≤ C lim sup n→∞

R

∗

RN \B

(|un |2 + |un |σ + |un |2s ) dx R

≤ Cξ.

(7.3.11)

On the other hand, choosing R > 0 large enough, we may assume that  RN \BR

g(ε x, u)u dx < ξ.

(7.3.12)

Then, (7.3.11) and (7.3.12) yield   lim sup  n→∞

 RN \BR

g(ε x, un )un dx −

RN \BR

  g(ε x, u)u dx  < Cξ

for all ξ > 0,

which implies that 

 lim

n→∞ RN \B R

g(ε x, un )un dx =

RN \BR

g(ε x, u)u dx.

(7.3.13)

7.3 The Critical Case

281

Using the definition of g, it follows that ∗

g(ε x, un )un ≤ f (un )un + a 2s +

V1 2 u for any x ∈ RN \ ε . K n

Since BR ∩(RN \ε ) is bounded, we can use the above estimate, (f1 ), (f2 ), Theorem 1.1.8, and the dominated convergence theorem to infer that, as n → ∞, 

 lim

n→∞ B ∩(RN \ ) ε R

g(ε x, un )un dx =

BR ∩(RN \ε )

g(ε x, u)u dx.

(7.3.14)

At this point, we aim to show that  lim

n→∞  ε

2∗s (u+ n)



∗

(u+ )2s dx.

dx =

(7.3.15)

ε

Indeed, if we assume that (7.3.15) is true, then from (k2 ), (f1 ), (f2 ), Theorem 1.1.8, and the dominated convergence theorem we deduce that 

 lim

n→∞  ∩B ε R

g(ε x, un )un dx =

ε ∩BR

g(ε x, u)u dx.

(7.3.16)

Now combining (7.3.13), (7.3.14) and (7.3.16), we conclude that (7.3.4) holds. So let us prove that (7.3.15) is satisfied. Since (u+ n ) is bounded in Hε , we may assume s 2  μ and (u+ )2∗s  ν, where μ and ν are two bounded non-negative | that |(−) 2 u+ n n measures on RN . Applying Lemma 1.5.1 we can find an at most countable index set I and sequences (xi )i∈I ⊂ RN , (μi )i∈I , (νi )i∈I ⊂ (0, ∞) such that s

μ ≥ |(−) 2 u+ |2 +



μi δ x i ,

i∈I ∗

ν = |u+ |2s +



νi δxi

and

2 2∗

S∗ νi s ≤ μi

∀i ∈ I,

(7.3.17)

i∈I

where δxi is the Dirac mass at the point xi . Let us show that (xi )i∈I ∩ ε = ∅. Assume, by i contradiction, that xi ∈ ε for some i ∈ I . For any ρ > 0, we define ψρ (x) = ψ( x−x ρ ) ∞ N N where ψ ∈ Cc (R ) is such that ψ = 1 in B1 , ψ = 0 in R \ B2 , 0 ≤ ψ ≤ 1 and ∇ψL∞ (RN ) ≤ 2. We suppose that ρ > 0 is such that supp(ψρ ) ⊂ ε . Since the

282

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

+ sequence (ψρ u+ n ) is bounded in Hε , we see that Jε (un ), ψρ un  = on (1), and so

 R2N

ψρ (y)  ≤

+ 2 |u+ n (x) − un (y)| dxdy N+2s |x − y|

R2N

ψρ (y)

  ≤−  +

RN

+ (un (x) − un (y))(u+ n (x) − un (y)) dxdy |x − y|N+2s

(ψρ (x) − ψρ (y))(un (x) − un (y)) + un (x) dxdy 2N |x − y|N+2s R  2∗s ψρ f (un )u+ dx + ψρ (u+ n n ) dx + on (1),



RN

(7.3.18)

where we used that (x − y)(x + − y + ) ≥ |x + − y + |2 for all x, y ∈ R, and (V1 ). Since f has subcritical growth and ψρ has compact support, we have 

ψρ f (un )u+ n dx = lim

lim lim

ρ→0 n→∞ RN

ρ→0 RN

Now we show that   lim lim sup ρ→0 n→∞



R2N

ψρ f (u)u+ dx = 0.

 (ψρ (x) − ψρ (y))(un (x) − un (y)) + un (x) dxdy = 0. |x − y|N+2s

(7.3.19)

(7.3.20)

Using the Hölder inequality and the fact that (un ) is bounded in Hε , we obtain that     

R2N

  (ψρ (x) − ψρ (y))(un (x) − un (y)) + un (x) dxdy  N+2s |x − y|

  ≤C

R2N

2 |ψρ (x) − ψρ (y)| |u+ n (x)| N+2s

|x − y|

 12

2

dxdy

.

By Lemma 1.4.7, we deduce that  lim lim sup

ρ→0 n→∞

R2N

2 |u+ n (x)|

|ψρ (x) − ψρ (y)|2 dxdy = 0 |x − y|N+2s

(7.3.21)

and this implies that (7.3.20) holds. Therefore, using (7.3.17) and taking the limit as n → ∞ and ρ → 0 in (7.3.18), we deduce that (7.3.19) and (7.3.20) yield νi ≥ μi . From the last statement in (7.3.17) it

7.3 The Critical Case

283

2 2∗

follows that νi ≥ S∗ s , and using (f4 ) and (k3 ) we get 1 c = Jε (un ) − Jε (un ), un  + on (1) 2       1 1 g(ε x, un )un − G(ε x, un ) dx + f (un )un − F (un ) dx = RN \ε 2 ε 2  s ∗ + (u+ )2s dx + on (1) N ε n  s ∗ ≥ (u+ )2s dx + on (1) N ε n  s 2∗s ≥ ψρ (u+ n ) dx + on (1). N ε Then, by (7.3.17) and letting n → ∞, we see that c≥

s N



ψρ (xi )νi =

{i∈I : xi ∈ε }

s N

 {i∈I : xi ∈ε }

νi ≥

s 2sN S∗ , N  

which gives a contradiction. This ends the proof of (7.3.15).

Remark 7.3.3 Arguing as in Remark 5.2.8, we may always suppose that the Palais-Smale sequence (un ) is non-negative in RN . Now Lemma 7.3.1, Lemma 7.3.2 and Theorem 2.2.9, yield the following result: Theorem 7.3.4 Assume that (V1 )-(V2 ) and (f1 ), (f2 ), (f3 ), (f4 ) hold. Then, for all ε > 0, problem (7.3.1) admits a positive mountain pass solution.

7.3.2

Proof of Theorem 7.1.2

Consider the following family of autonomous problems, with μ > 0: 

∗

(−)s u + μu = f (u) + |u|2s −2 u u ∈ H s (RN ), u > 0 in RN .

in RN ,

By Lemma 6.4.10 we can state the following result. Theorem 7.3.5 Problem (7.3.22) admits a positive ground state solution.

(7.3.22)

284

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Arguing as in the proof of Lemma 7.2.7 (and recalling Lemma 6.4.8), we have the following relation between cε and dV0 : Lemma 7.3.6 The following relation holds: lim sup cε ≤ dV0 < ε→0

s 2sN S∗ . N

As in the previous section, we need to prove the following compactness result which will be fundamental for showing that, for ε > 0 small enough, the solutions of the modified problem are also solutions of the original problem. Lemma 7.3.7 Let εn → 0 and (un ) = (uεn ) ⊂ Hεn be such that Jεn (un ) = cεn and Jε n (un ) = 0. Then there exists a sequence (y˜n ) = (y˜εn ) ⊂ RN such that u˜ n (x) = un (x + y˜n ) has a convergent subsequence in H s (RN ). Moreover, up to a subsequence, yn = εn y˜n → y0 for some y0 ∈  such that V (y0 ) = V0 . Proof The proof of this result can be done essentially as in Lemma 7.2.8. However, due to the presence of the fractional critical exponent, some modifications are needed. More precisely, to find a sequence (y˜n ) ⊂ RN and constants R, α > 0 such that  lim inf n→∞

BR (y˜n )

u2n dx ≥ α,

we proceed as follows. Assume, by contradiction, that the above limit inequality does not hold. By Lemma 1.4.4, we see that un → 0 in Lr (RN ) for all r ∈ (2, 2∗s ). By (f1 ) and (f2 ), it follows that 

 RN

F (un ) dx =

RN

f (un )un dx = on (1).

This implies that  RN

1 G(εn x, un ) dx ≤ ∗ 2s

 εn ∪{un ≤a}

2∗s (u+ n)



V1 dx + 2K

cεn ∩{un >a}

u2n dx + on (1) (7.3.23)

and  RN

 g(εn x, un )un dx =

∗

εn ∪{un ≤a}

2s (u+ n ) dx +

V1 K

 cε n ∩{un >a}

u2n dx + on (1), (7.3.24)

7.3 The Critical Case

285

where cεn = RN \ εn . Taking into account that Jε n (un ), un  = 0 and (7.3.24), we deduce that   V1 2 2 2∗s u dx = (u+ (7.3.25) un εn − n ) dx + on (1). K cεn ∩{un >a} n ε n ∪{un ≤a} Let % ≥ 0 be such that un 2εn −

V1 K

 cεn ∩{un >a}

u2n dx → %.

It is easy to see that % > 0: otherwise, un εn → 0 and this is impossible because un εn ≥ κ > 0. It follows from (7.3.25) that 

∗

εn ∪{un ≤a}

Using Jεn (un ) −

1

2∗s Jε (un ), un 

2s (u+ n ) dx → %.

= cεn , (7.3.23) and (7.3.24) we see that s % ≤ lim inf cεn . n→∞ N

(7.3.26)

On the other hand, by the definition of S∗ , we have that un 2εn

V1 − K



 cεn ∩{un >a}

u2n dx

≥ S∗

εn ∪{un ≤a}

2∗s (u+ n)

 2∗ 2s

dx

,

and taking the limit as n → ∞ we infer that 2 ∗

% ≥ S∗ % 2s . Then, by % > 0, (7.3.26) and (7.3.27), we deduce that lim inf cεn ≥ n→∞

which contradicts Lemma 7.3.6.

N s 2s S∗ , N

(7.3.27)

286

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Finally, instead of the functions h3n and h3 introduced in Lemma 7.2.8, we need to consider the following functions:  1 g(ε n x + εn y˜n , u˜ n (x))u˜ n (x) − G(εn x + εn y˜n , u˜ n (x)) χ˜n1 (x) ϑ  

 1 1 2∗s 2∗s − F (u˜ n (x)) + ∗ (u˜ n (x)) f (u˜ n (x))u˜ n (x) + (u˜ n (x)) = χ˜ n1 (x) ϑ 2s 

 1 2∗s 2∗s ˜h3 (x) = 1 f (u(x)) ˜ u(x) ˜ + (u(x)) ˜ . − F (u(x)) ˜ + ∗ (u(x)) ˜ ϑ 2s

h˜ 3n (x) =



  Proof of Theorem 7.1.2 In the light of Lemma 7.3.7 and Lemma 7.2.9, we can argue as in the proof of Theorem 7.1.1 to obtain the desired result.  

7.4

A Supercritical Fractional Schrödinger Equation

7.4.1

The Truncated Problem

In this last section we study the existence of positive solutions for the following supercritical fractional problem: (−)s u + V (ε x)u = |u|q−2 u + λ|u|r−2 u

in RN .

(7.4.1)

Motivated by Chabrowski and Yang [136] and Rabinowitz [297], we truncate the nonlinearity f (u) = |u|q−2 u + λ|u|r−2 u as follows. Let K > 0 be a real number, whose value will be fixed later, and set ⎧ ⎪ if t ≤ 0, ⎨ 0, fλ (t) = t q−1 + λt r−1 , if 0 < t < K, ⎪ ⎩ r−q q−1 )t , if t ≥ K. (1 + λK Then it is readily verified that fλ satisfies the assumptions (f1 )–(f4 ) ((f3 ) holds with ϑ = q > 2). In particular, fλ (t) ≤ (1 + λK r−q )t q−1 for all t ≥ 0.

(7.4.2)

Now consider the following truncated problem: (−)s u + V (ε x)u = fλ (u)

in RN ,

(7.4.3)

7.4 A Supercritical Fractional Schrödinger Equation

287

and the corresponding functional Jε,λ : Hε → R, defined as Jε,λ (u) =

1 u2ε − 2

 RN

Fλ (u) dx.

We also introduce the autonomous functional IV0 ,λ : H s (RN ) → R, given by IV0 ,λ (u) =

1 u2V0 − 2

 RN

Fλ (u) dx.

Using Theorem 7.1.1, we know that for every λ ≥ 0 there exists ε¯ (λ) > 0 such that, for every ε ∈ (0, ε¯ (λ)), problem (7.4.3) admits a positive solution uε,λ . Next we prove an auxiliary result which shows that the Hε -norm of uε,λ can be estimated from above by a constant independent of λ. ¯ Lemma 7.4.1 There exists C¯ > 0 such that, for any ε > 0 sufficiently small, uε,λ ε ≤ C. Proof From the proof of Theorem 7.1.1, we know that uε,λ satisfies the inequality Jε,λ (uε,λ ) ≤ dV0 ,λ + hλ (ε), where dV0 ,λ is the mountain pass level related to the functional IV0 ,λ and hλ (ε) → 0 as ε → 0. Then, decreasing ε¯ (λ) if necessary, we may assume that Jε,λ (uε,λ ) ≤ dV0 ,λ + 1

(7.4.4)

for any ε ∈ (0, ε¯ (λ)). Since dV0 ,λ ≤ dV0 ,0 for any λ ≥ 0, we infer that Jε,λ (uε,λ ) ≤ dV0 ,0 + 1

(7.4.5)

for any ε ∈ (0, ε¯ (λ)). Moreover, using (f3 ), we see that 1

J (uε,λ ), uε,λ  q ε,λ    1 1 1 = − fλ (uε,λ )uε,λ − qFλ (uε,λ ) dx uε,λ 2ε + 2 q q RN   1 1 − uε,λ 2ε . ≥ 2 q

Jε,λ (uε,λ ) = Jε,λ (uε,λ ) −

(7.4.6)

288

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Finally, (7.4.5) and (7.4.6) show that  uε,λ ε ≤

 1 2 2q (dV0 ,0 + 1) = C¯ q −2

∀ ε ∈ (0, ε¯ (λ)).  

7.4.2

A Moser Type Iteration Argument

Let ε ∈ (0, ε¯ (λ)) and assume that ε¯ (λ) is small in such a way that the estimate in Lemma 7.4.1 holds true. We aim to prove that uε,λ is a positive solution of the original problem (7.1.2) for small λ. To this end, we will show that we can find K0 > 0 such that for any K ≥ K0 , there exists λ0 = λ0 (K) > 0 such that uε,λ L∞ (RN ) ≤ K for all λ ∈ [0, λ0 ]. In what follows we use a Moser iteration argument [278] (see also [136, 188, 297]). For simplicity we will write u instead of uε,λ . β−1

Proof of Theorem 7.1.3 For any L > 0, let uL = min{u, L} ≥ 0 and wL = uuL , where 2(β−1) u in (7.4.3) we see that β > 1 will be chosen later. Taking uL  

(u(x) − u(y)) 2(β−1) 2(β−1) (u(x)uL (x) − u(y)uL (y)) dxdy N+2s R2N |x − y|   2(β−1) 2(β−1) fλ (u)uuL dx − V (ε x)u2 uL dx. = RN



(7.4.7)

RN

Putting together (7.4.7), (7.4.2) and (V1 ) we get  

(u(x) − u(y)) 2(β−1) 2(β−1) (u(x)uL (x) − u(y)uL (y)) dxdy N+2s 2N |x − y| R  2(β−1) ≤ Cλ,K uq uL dx, RN



(7.4.8)

where Cλ,K = 1 + λK r−q . Arguing as in the first part of Lemma 3.2.14, we can see that  wL 2 2∗s

L (RN )

≤ C0 β 2

R2N

(u(x) − u(y)) 2(β−1) 2(β−1) (u(x)uL (x) − u(y)uL (y)) dxdy. |x − y|N+2s (7.4.9)

7.4 A Supercritical Fractional Schrödinger Equation 2(β−1)

Since uq uL deduce that

289

2 , we can use (7.4.8), (7.4.9), and the Hölder inequality to = uq−2 wL

 wL 2 2∗s

L (RN )

≤ C1 β 2 Cλ,K

∗

RN

u2s dx

  q−2 ∗



α∗

2s

RN

wLs dx

2 αs∗

(7.4.10)

,

where αs∗ =

22∗s ∈ (2, 2∗s ) 2∗s − (q − 2) ∗

and C1 > 0 is independent of ε and λ. Then, by Ds,2 (RN ) ⊂ L2s (RN ) and Lemma 7.4.1, we obtain wL 2 2∗s

L (RN )

≤ C2 β 2 Cλ,K C¯ q−2 wL 2 αs∗ L

(7.4.11)

(RN ) ∗

for some C2 > 0 that is independent of ε and λ. Note that, if |u|β ∈ Lαs (RN ), the definition of wL , the fact that uL ≤ u and (7.4.11) imply that wL 2 2∗s

L (RN )

≤ C2 β 2 Cλ,K C¯ q−2



∗

RN



uβαs dx

2 αs∗

< ∞.

(7.4.12)

Taking the limit as L → ∞ in (7.4.12) and using Fatou’s lemma, we have 1

1

uLβ2∗s (RN ) ≤ (C3 Cλ,K ) 2β β β uLβαs∗ (RN )

(7.4.13)

∗ whenever |u|βαs ∈ L1 (RN ), where C3 = C2 C¯ q−2 . ∗ ∗ 2 Set β = αs∗ > 1 and we note that, since u ∈ L2s (RN ), the above inequality holds for s

this choice of β. Then, observing that β 2 αs∗ = β2∗s , it follows that (7.4.13) holds with β replaced by β 2 , so we have uLβ 2 2∗s (RN ) ≤ (C3 Cλ,K )

1 2β 2

β

2 β2

uLβ 2 αs∗ (RN ) ≤ (C3 Cλ,K )

1 2



1 1 β + β2



1

ββ

+

2 β2

uLβαs∗ (RN ) .

Iterating this process and using the fact that βαs∗ = 2∗s we deduce that for every m ∈ N u

m ∗ Lβ 2s (RN )

≤ (C3 Cλ,K )

m

1 j=1 2β j

β

m j=1

jβ −j

uL2∗s (RN ) .

(7.4.14) ∗

Letting m → ∞ in (7.4.14) and using the embedding Ds,2 (RN ) ⊂ L2s (RN ) and

290

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Lemma 7.4.1, we obtain uL∞ (RN ) ≤ (C3 Cλ,K )γ1 β γ2 C4 ,

(7.4.15)

1

− where C4 = S∗ 2 C¯ and ∞

γ1 =

1 1 0 such that 1

(KC4−1 ) γ1 γ2

−1>0

C3 β γ1 and fix λ0 > 0 such that ⎡ λ ≤ λ0 ≤ ⎣

1

(KC4−1 ) γ1 C3 β

γ2 γ1

⎤ − 1⎦

1 K r−q

.

Then, using (7.4.15), we conclude that uL∞ (RN ) ≤ K for all λ ∈ [0, λ0 ].  

7.5

Some Extensions to Fractional Schrödinger Systems

It is also possible consider the nonlocal counterpart of the following elliptic system of Schrödinger equations (see for instance [4, 16, 17, 20, 80, 118, 189]): ⎧ 2 N ⎪ ⎨ − ε u + V (x)u = Gu (u, v) in R , 2 − ε v + W (x)v = Gv (u, v) in RN , ⎪ ⎩ u, v > 0 in RN .

7.5 Some Extensions to Fractional Schrödinger Systems

291

More precisely, we can study the existence, multiplicity and concentration phenomenon of positive solutions for the following nonlinear fractional Schrödinger system ⎧ 2s γ s N ⎪ ⎨ ε (−) u + V (x)u = Qu (u, v) + 2∗s Ku (u, v) in R , γ ε2s (−)s v + W (x)v = Qv (u, v) + 2∗ Kv (u, v) in RN , s ⎪ ⎩ u, v > 0 in RN ,

(7.5.1)

where ε > 0 is a small parameter, s ∈ (0, 1), N > 2s, γ ∈ {0, 1}, V : RN → R and W : RN → R are Hölder continuous potentials, Q and K are homogeneous functions, with K having critical growth. We assume that there exist a bounded open set  ⊂ RN , x0 ∈ RN and ρ0 > 0 such that: (H 1) V (x), W (x) ≥ ρ0 for any x ∈ ∂; (H 2) V (x0 ), W (x0 ) < ρ0 ; (H 3) V (x) ≥ V (x0 ) > 0, W (x) ≥ W (x0 ) > 0 for any x ∈ RN . Concerning the function Q ∈ C 2 (R2+ , R), where R2+ = [0, ∞) × [0, ∞), we suppose that it satisfies the following conditions: (Q1) there exists p ∈ (2, 2∗s ) such that Q(tu, tv) = t p Q(u, v) for any t > 0, (u, v) ∈ R2+ ; (Q2) there exists C > 0 such that |Qu (u, v)| + |Qv (u, v)| ≤ C(up−1 + v p−1 ) for any (u, v) ∈ R2+ ; (Q3) Qu (0, 1) = 0 = Qv (1, 0); (Q4) Qu (1, 0) = 0 = Qv (0, 1); (Q5) Q(u, v) > 0 for any u, v > 0; (Q6) Qu (u, v), Qv (u, v) ≥ 0 for any (u, v) ∈ R2+ . Further, we assume that K ∈ C 2 (R2+ , R) fulfills the following hypotheses: ∗

(K1) K(tu, tv) = t 2s K(u, v) for all t > 0, (u, v) ∈ R2+ ; ∗ ∗ (K2) the 1-homogeneous function G : R2+ → R given by G(u2s , v 2s ) = K(u, v) is concave; ∗ ∗ (K3) there exists c > 0 such that |Ku (u, v)| + |Kv (u, v)| ≤ c(u2s −1 + v 2s −1 ) for all (u, v) ∈ R2+ ; (K4) Ku (0, 1) = 0 = Kv (1, 0); (K5) Ku (1, 0) = 0 = Kv (0, 1); (K6) K(u, v) > 0 for any u, v > 0; (K7) Ku (u, v), Kv (u, v) ≥ 0 for all (u, v) ∈ R2+ .

292

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

Since we are interested in positive solutions of (7.5.1), we extend the functions Q and K to the whole of R2 by setting Q(u, v) = K(u, v) = 0 if u ≤ 0 or v ≤ 0. We note that the p-homogeneity of Q implies that the following identity holds: pQ(u, v) = uQu (u, v) + vQv (u, v)

for any (u, v) ∈ R2 ,

and p(p − 1)Q(u, v) = u2 Quu (u, v) + 2uvQuv (u, v) + v 2 Qvv (u, v)

for any (u, v) ∈ R2 .

As a model for Q, we present the following example given in [161]. Let q ≥ 1 and 

Pq (u, v) =

ai uαi v βi ,

αi +βi =q

where i ∈ {1, . . . , k}, αi , βi ≥ 1 and ai ∈ R. The following functions and their possible combinations, with an appropriate choice of the coefficients ai , satisfy assumptions (Q1)(Q5) on Q Q1 (u, v) = Pp (u, v),

Q2 (u, v) =

 r P% (u, v)

Q3 (u, v) =

and

P%1 (u, v) , P%2 (u, v)

with r = %p and %1 − %2 = p. In order to state precisely the main results, we need to introduce some notations. Fix ξ ∈ RN , and consider the following autonomous system: ⎧ γ s N ⎪ ⎨ (−) u + V (ξ )u = Qu (u, v) + 2∗s Ku (u, v) in R , (−)s v + W (ξ )v = Qv (u, v) + 2γ∗ Kv (u, v) in RN , s ⎪ ⎩ u, v > 0 in RN . Let Jξ : H s (RN ) × H s (RN ) → R be the Euler-Lagrange functional associated with the above problem, namely 1 Jξ (u, v) = (u, v)2ξ − 2

 RN

γ Q(u, v) dx − ∗ 2s

 RN

K(u, v) dx,

where  (u, v)2ξ =

RN



s s |(−) 2 u|2 + |(−) 2 v|2 dx +

RN

(V (ξ )u2 + W (ξ )v 2 ) dx.

7.5 Some Extensions to Fractional Schrödinger Systems

293

By the assumptions on Q and K, it follows that Jξ possesses a mountain pass geometry (see [52]), so we can consider the mountain pass value C(ξ ) = inf max Jξ (γ (t)), γ ∈ t ∈[0,1]

where   = γ ∈ C([0, 1], H s (RN ) × H s (RN )) : γ (0) = 0, Jξ (γ (1)) < 0 . Moreover, the function ξ → C(ξ ) is continuous; in addition, C(ξ ) can be also characterized as C(ξ ) =

inf

(u,v)∈Nξ

Jξ (u, v),

where Nξ is the Nehari manifold associated with Jξ . Then, for any fixed ξ ∈ RN , C(ξ ) is achieved, and in view of condition (H 3) we deduce that C(x0 ) ≤ C(ξ ) for any ξ ∈ RN , which implies that   N M = x ∈ R : C(x) = inf C(ξ ) = ∅. ξ ∈RN

Furthermore, we can prove that C ∗ = C(x0 ) = inf C(ξ ) < min C(ξ ). ξ ∈

ξ ∈∂

Then, we are able to state the main results for (7.5.1) whose proofs can be found in [54] when γ = 0, and [40] when γ = 1: Theorem 7.5.1 ([40, 54]) Assume that, if γ = 0, (H 1)–(H 3) and (Q1)–(Q6) hold. When γ = 1, we suppose that (H 1)–(H 3), (Q1)–(Q6) and (K1)–(K7) are satisfied. In addition, we make the following technical assumption on Q: ˜

(Q7) Q(u, v) ≥ λuα˜ v β for any (u, v) ∈ R2+ with 1 < α, ˜ β˜ < 2∗s , α˜ + β˜ = q1 ∈ (2, 2∗s ), and λ verifying • λ > 0 if either N ≥ 4s, or 2s < N < 4s and 2∗s − 2 < q1 < 2∗s ; • λ is sufficiently large if 2s < N < 4s and 2 < q1 ≤ 2∗s − 2. Then, for every δ > 0 satisfying Mδ = {x ∈ RN : dist(x, M) ≤ δ} ⊂ ,

294

7 Fractional Schrödinger Equations with del Pino-Felmer Assumptions

there exists εδ > 0 such that, for any ε ∈ (0, εδ ), system (7.5.1) admits at least catMδ (M) solutions. Moreover, if (uε , vε ) is a solution to (7.5.1) and Pε and Qε are global maximum points of uε and vε , respectively, then C(Pε ), C(Qε ) → C(x0 ) as ε → 0, and the following estimates hold uε (x) ≤

C εN+2s C εN+2s and v (x) ≤ ε εN+2s +|x − Pε |N+2s εN+2s +|x − Qε |N+2s

∀x ∈ RN .

The above theorem is obtained by combining a variant of the penalization approach [165] considered in [4] (see also [16, 17]), Lusternik-Schnirelman theory and some ideas used in [19, 35, 216] to deal with fractional Schrödinger equations. Further results for fractional elliptic systems can be found in [52, 210, 261, 332].

Fractional Schrödinger Equations with Superlinear or Asymptotically Linear Nonlinearities

8.1

Introduction

In this paper we investigate the existence and the concentration phenomenon of positive solutions for the following fractional problem: 

ε2s (−)s u + V (x)u = f (u) in RN , u ∈ H s (RN ), u > 0 in RN ,

(8.1.1)

where ε > 0 is a small parameter, s ∈ (0, 1) and N ≥ 2. The external potential V : RN → R is a Hölder continuous function and bounded below away from zero, that is, there exists V0 > 0 such that V (x) ≥ V0 > 0

for all x ∈ RN .

(8.1.2)

Concerning the nonlinearity f : R → R, we assume that it satisfies the following basic assumptions: (f 1) f ∈ C 1 (R, R); (f 2) limt →0 f (tt ) = 0; (f 3) there exists p ∈ (1, 2∗s − 1) such that lim

t →∞

f (t) = 0. tp

In this chapter we aim to study the existence of positive solutions to (8.1.1) concentrating around local minima of the potential V (x), under the assumptions that the nonlinearity f is asymptotically linear or superlinear at infinity, and without assuming the monotonicity of f (t)/t. We also consider superlinear nonlinearities which do not satisfy the Ambrosetti–

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_8

295

8

296

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Rabinowitz condition [29]; see also [6,37,42,52,75] for some related works. We recall that the condition (AR) and the assumption f (t)/t is increasing for t > 0 play a fundamental role when we have to verify the boundedness of Palais–Smale sequences and apply Nehari manifold arguments, respectively. Now, we state our main result: Theorem 8.1.1 ([47]) Let us assume that f (t) satisfies (f 1)–(f 3) and either (f 4) there exists μ > 2 such that 0 < μF (t) ≤ f (t)t for any t > 0,  t f (τ ) dτ , or the following condition (f 5): where F (t) = 0

(i) There exists a ∈ (0, ∞] such that limt →∞ (ii) There exists a constant D ≥ 1 such that

f (t ) t

(t) ≤ D F (t¯) F

= a.

0 ≤ t ≤ t¯,

(8.1.3)

(t) = 1 f (t)t − F (t). where F 2 Let  ⊂ RN be a bounded open set such that inf V < min V

(8.1.4)

inf V < a.

(8.1.5)



∂

and, when a < ∞ in (f 5), 

Then, there exists ε0 > 0 such that, for any ε ∈ (0, ε0 ], problem (8.1.1) admits a positive solution uε (x). Moreover, if xε denotes a global maximum point of uε , then we have (1) V (xε ) → infx∈ V (x); (2) there exists C > 0 such that uε (x) ≤

εN+2s

CεN+2s + |x − xε |N+2s

for all x ∈ RN .

We would like to note that Theorem 8.1.1 extends and improves Theorem 7.1.1, because we do not require any monotonicity assumption on f (t)/t, and we are able to deal with a more general class of nonlinearities, including the asymptotically linear case (see condition (f 5)). Moreover, our result is in clear agreement with that for the classical local counterpart, that is Theorem 1.1 in [235]. Clearly, a more involved and accurate analysis with respect to the case s = 1 will be needed due to the nonlocal character of (−)s .

8.2 Modification of the Nonlinearity

297

Now, we give the main ideas for the proof of Theorem 8.1.1. After rescaling equation (8.1.1) with the change of variable v(x) = u(ε x), we introduce a modified functional Jε and we prove that it satisfies a mountain pass geometry [29]. Then, we investigate the boundedness of Cerami sequences for Jε , and we give two types of boundedness results: one when ε is fixed, the other one to deduce uniform boundedness when ε → 0. Through a careful study of the behavior of bounded Cerami sequences (vε ), as ε → 0, we prove that there exists a subsequence (vεj ) which converges, in a suitable sense, to a sum of translated critical points of certain autonomous functionals. This concentration-compactness type result will be useful for showing that an appropriate translated sequence vεj (· + yεj ) converges to a least energy solution ω1 . Finally, we prove L∞ -estimates (uniformly in j ∈ N) and deduce some information about the behavior at infinity of the translated sequence, which allows one to obtain a positive solution of the rescaled problem.

8.2

Modification of the Nonlinearity

Since we are looking for positive solutions of (8.1.1), we can assume that f (t) = 0 for any t ≤ 0. Arguing as in [235], we can prove the following useful properties of the function f: Lemma 8.2.1 Assume that (f 1)–(f 3) hold. Then we have: (i) For every δ > 0 there exists Cδ > 0 such that |f (t)| ≤ δ|t| + Cδ |t|p (ii) If (f 4) holds, then f (t) ≥ 0 for all t ≥ 0. (t) ≥ 0, and (iii) If (f 5) holds, then f (t) ≥ 0, F (iv) If t →

f (t ) t

d dt

for all t ∈ R.

F (t ) t2



≥ 0 for all t ≥ 0.

is increasing for t ∈ (0, ∞), then (f 5) is satisfied with D = 1.

Now, assume that f (t) satisfies (f 1)–(f 3) and that f (t) ∈ (0, ∞]. ξ →∞ t

V0 < a = lim Take ν ∈ (0, V20 ) and we define f (t) =

⎧ ⎨min{f (t), νt},

if t ≥ 0,

⎩0,

if t < 0.

(8.2.1)

298

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Using (f 2) we can find rν > 0 such that f (t) = f (t)

for all |t| ≤ rν .

⎧ ⎨νt,

for large t ≥ 0,

⎩0,

for t ≤ 0.

Moreover, it holds that f (t) =

For technical reasons, it is convenient to choose ν as follows: If (f 4) holds, then we take ν > 0 such that ν 1 1 < − . 2V0 2 μ

(8.2.2)

When (f 5) is satisfied, we choose ν ∈ (0, V20 ) such that ν is a regular value of t ∈ (0, ∞) → f (tt ) . Since limt →0 f (tt ) = 0 and limt →∞ f (tt ) = a > V0 > ν, if ν is a regular value of f (tt ) we deduce that kν = card{t ∈ (0, ∞) : f (t) = νt} < ∞.

(8.2.3)

Now, let  ⊂ RN be a bounded open set such that ∂ ∈ C ∞ , and we assume that  satisfies (8.1.4). Take an open set  ⊂  with smooth boundary ∂ and define a function χ ∈ C ∞ (RN , R) such that inf V > inf V ,

\



min V > inf V = inf V , ∂

χ(x) = 1





for x ∈  ,

χ(x) ∈ (0, 1) χ(x) = 0

for x ∈  \  ,

for x ∈ RN \ .

Without loss of generality, we suppose that 0 ∈  and V (0) = inf V . Finally, we introduce the penalty function g(x, t) = χ(x)f (t) + (1 − χ(x))f (t)

for (x, t) ∈ RN × R,

8.2 Modification of the Nonlinearity

299

and set  F (t) =

t

f (τ ) dτ, 0

 G(x, t) =

t

g(x, τ ) dτ = χ(x)F (t) + (1 − χ(x))F (t).

0

As in [235], it is easy to check that the following properties concerning f (t) and g(x, t) hold. Lemma 8.2.2 (i) (ii) (iii) (iv) (v)

f (t) = 0, F (t) = 0 for all t ≤ 0. f (t) ≤ νt, F (t) ≤ F (t) for all t ≥ 0. f (t) ≤ f (t) for all t ≥ 0. If f (t) satisfies either (f 4) or (f 5), then f (t) ≥ 0 for all t ∈ R. (t) ≥ 0 for all t ≥ 0. If f (t) satisfies (f 5), then f (t) also satisfies (f 5). Moreover, F

Corollary 8.2.3 (i) g(x, t) ≤ f (t), G(x, t) ≤ F (t) for all (x, t) ∈ RN × R. (ii) g(x, t) = f (t) if |t| < rν . (iii) For every δ > 0 there exists a Cδ > 0 such that |g(x, t)| ≤ δ|t| + Cδ |t|p

for all (x, t) ∈ RN × R.

(iv) if f (t) satisfies (f 5)–(ii), then g(x, t) satisfies  t¯)  t) ≤ D kν G(x, G(x,

for all 0 ≤ t ≤ t¯,

 t) = 1 g(x, t)t − G(x, t), D ≥ 1 is given in (f 5)–(ii) and kν is given in where G(x, 2 (8.2.3). In what follows, we investigate the existence of positive solutions uε of the modified problem  ε2s (−)s u + V (x)u = g(x, u) in RN , (8.2.4) u ∈ H s (RN ), u > 0 in RN , with the property uε (x) ≤ rν

for x ∈ RN \  .

In view of the definition of g, these functions uε are also solutions of (8.1.1).

300

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

8.3

Mountain Pass Argument

Using the change of variable v(x) = u(εx), one can show that (8.2.4) is equivalent to the following problem: 

(−)s v + V (εx)v = g(εx, v) u ∈ H s (RN ), u > 0 in RN .

in RN ,

(8.3.1)

The energy functional associated with (8.3.1) is given by Jε (v) =

1 2



RN



s |(−) 2 v|2 + V (εx)v 2 dx −

RN

G(εx, v) dx

∀v ∈ Hε

where the fractional space   Hε = v ∈ H s (RN ) :

 RN

V (εx)v 2 dx < ∞

is endowed with the norm  vε =

RN



 12 s 2 2 2 |(−) v| + V (εx)v dx .

Since V0 > 0, we can equip H s (RN ) with the equivalent norm  v0 =

RN

s 2

|(−) v| + V0 v 2

2

1



2

dx

.

Clearly, v0 ≤ vε ,

(8.3.2)

so Hε ⊂ H s (RN ) and Hε is continuously embedded in Lr (RN ) for any 2 ≤ r ≤ 2∗s , and there exists Cr > 0 such that vLr (RN ) ≤ Cr v0 .

(8.3.3)

In what follows, we denote by Hε∗ the dual of Hε . Let us prove that Jε possesses a mountain pass geometry [29] that is uniform with respect to ε.

8.3 Mountain Pass Argument

301

Lemma 8.3.1 Jε ∈ C 1 (Hε , R) and satisfies the following properties: (i) Jε (0) = 0; (ii) there exist ρ0 > 0 and δ0 > 0, independent of ε ∈ (0, 1], such that Jε (v) ≥ δ0 Jε (v) > 0

for all v0 = ρ0 , for all 0 < v0 ≤ ρ0 ;

(iii) there exist v0 ∈ Cc∞ (RN ) and ε0 > 0 such that Jε (v0 ) < 0 for all ε ∈ (0, ε0 ]. Proof Obviously, Jε ∈ C 1 (Hε , R) and Jε (0) = 0. Since F ≤ F , taking δ = we get 1 v2ε − 2

Jε (v) =

 

RN

in (8.2.1)



χ(εx)F (v) + (1 − χ(εx))F (v) dx

≥

1 v2ε − 2

≥

1 V0 p+1 v20 − v2L2 (RN ) − C V0 vLp+1 (RN ) 2 4 2

≥

v20 p+1 − C˜ p+1 C V0 v0 , 4 2

RN

V0 2

F (v) dx

where we used (8.3.2) and (8.3.3) with r = p + 1. Thus (ii) is satisfied. To verify that (iii) holds, we take v0 ∈ Cc∞ (RN ) such that 1 2

 RN



s |(−) 2 v0 |2 + V (0)v02 dx −

RN

F (v0 ) dx < 0.

This choice is lawful due to the fact that V (0) < limz→∞ f (z) z , so the existence of a such v0 is guaranteed by Theorem 3.1.1 (see Lemma 8.4.2), where is proved that v →

1 2

 RN



s |(−) 2 v|2 + V (0)v 2 dx −

RN

F (v) dx

has a mountain pass geometry. Since 0 ∈  , we observe that Jε (v0 ) →

1 2

 RN



s |(−) 2 v0 |2 + V (0)v02 dx −

i.e., (iii) is satisfied for ε sufficiently small.

RN

F (v0 ) dx < 0 as ε → 0,  

302

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Since Jε has a mountain pass geometry, for each ε ∈ (0, ε0 ], we define the mountain pass value cε = inf max Jε (γ (t))

(8.3.4)

γ ∈ε t ∈[0,1]

where ε = {γ ∈ C([0, 1], Hε ) : γ (0) = 0 and Jε (γ (1)) < 0} .

(8.3.5)

Using Lemma 8.3.1, we are able to give the following estimate for cε . Corollary 8.3.2 There exist m1 , m2 > 0 such that for any ε ∈ (0, ε0 ] m1 ≤ cε ≤ m2 . Proof For any γ ∈ ε we have γ ([0, 1]) ∩ {v ∈ Hε : v0 = ρ} = ∅. Hence, by using Lemma 8.3.1, we deduce that max Jε (γ (t)) ≥

t ∈[0,1]

inf

v0 =ρ0

Jε (v) ≥ δ0 .

Set γ0 (t) = tv0 , where v0 ∈ Cc∞ (RN ) is supplied by Lemma 8.3.1. Then we see that 

 cε = inf

γ ∈ε

max Jε (γ (t)) ≤ max Jε (γ0 (t)) ≤ sup

t ∈[0,1]



 t ∈[0,1]

ε∈(0,ε0 ]

max Jε (γ0 (t)) .

t ∈[0,1]



Putting m1 = δ0 and m2 = supε∈(0,ε0] maxt ∈[0,1] Jε (γ0 (t)) we get the desired result.   Using Lemma 8.3.1 and Theorem 2.2.15, we deduce that for all ε ∈ (0, ε0 ] there exists a Cerami sequence (vj ) ⊂ Hε such that Jε (vj ) → bε (1 + vj ε )Jε (vj )H∗ε → 0

as j → ∞.

We will show that (vj ) is bounded in Hε and has a convergent subsequence. Thus Jε has a critical point vε satisfying Jε (vε ) = bε and Jε (vε ) = 0. We also show that (vε ) is bounded

8.3 Mountain Pass Argument

303

in the sense that lim sup vε ε < ∞.

(8.3.6)

ε→0

We start by establishing this second type of boundedness. Lemma 8.3.3 Assume that f satisfies (f 1)–(f 3) and either (f 4), or (f 5). Suppose that there exists a sequence (vε )ε∈(0,ε1] , with ε1 ∈ (0, ε0 ], such that vε ∈ Hε , Jε (vε ) ∈ [m1 , m2 ] ∀ε ∈ (0, ε1 ],

(8.3.7)

(1 + vε ε )Jε (vε )H∗ε → 0

(8.3.8)

as ε → 0

with 0 < m1 < m2 . Then (8.3.6) holds. Proof First, assume that (f 4) holds. Let (vε ) be a sequence satisfying (8.3.7) and (8.3.8). Then one can see that (8.3.7) yields 1 Jε (vε ) = vε 2ε − 2

 RN



(1 − χ(εx))F (vε ) + χ(εx)F (vε ) dx ≤ m2 .

(8.3.9)

Moreover, by (8.3.8), for any ε sufficiently small we have |Jε (vε ), vε | ≤ Jε (vε )H∗ε vε ε ≤ Jε (vε )H∗ε (1 + vε ε ) ≤ 1, that is    vε 2 − ε 

RN



 (1 − χ(εx))f (vε )vε + χ(εx)f (vε )vε dx  ≤ 1.

(8.3.10)

Taking into account (8.3.9), (8.3.10) and (f 4) we get 

    1 1 1 1 − (1 − χ(εx)) F (vε ) − f (vε )vε dx + m2 + . vε 2ε ≤ 2 μ μ μ RN

Using items (i) and (iv) of Lemma 8.2.2, we know that tf (t) ≥ 0 for all t ∈ R, so we obtain    1 1 1 − (8.3.11) (1 − χ(εx))F (vε ) dx + m2 + . vε 2ε ≤ 2 μ μ RN

304

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

On the other hand, by item (ii) of Lemma 8.2.2, F (t) ≤

νt 2 2

for all t ∈ R.

Then  RN

(1 − χ(εx))F (vε ) dx ≤

1 ν νvε 2L2 (RN ) ≤ vε 2ε , 2 2V0

which together with (8.3.11) shows that 

 1 1 ν 1 − vε 2ε + m2 + . vε 2ε ≤ 2 μ 2V0 μ

In view of (8.2.2) we get vε 2ε ≤ %

1 2

m2 + μ1

− μ1 −

ν 2V0

&,

which implies that vε ε is bounded if ε is small enough. Now, let us suppose that (f 5) holds. Arguing by contradiction, we assume that lim sup vε ε = ∞. ε→0

Let εj → 0 be a subsequence such that vεj εj → ∞. For simplicity, we denote εj still vε ε ε 0 by ε. Set wε = vεvε s . Clearly, wε 0 = v vε ε ≤ vε ε = 1. Moreover, we can see that Hε

there exists C1 > 0 independent of ε such that χε wε 0 ≤ C1 ,

(8.3.12)

where χε (x) = χ(ε x). Indeed, since 0 ≤ χ ≤ 1, (|a| + |b|)2 ≤ 2(|a|2 + |b|2), ε ∈ (0, ε1 ] and s ∈ (0, 1), we obtain that  |χ(ε x)wε (x) − χ(ε y)wε (y)|2 dxdy + V0 (χε wε )2 dx N+2s 2N N |x − y| R R   2 |χ(ε x) − χ(ε y)| 2 |wε (x) − wε (y)|2 ≤2 wε (x) dxdy + 2 dxdy N+2s |x − y| |x − y|N+2s R2N R2N  + V0 wε2 dx



RN

8.3 Mountain Pass Argument

305



 ≤ 2ε

2

∇χ2L∞ (RN )

RN

 +8  +

wε2 (x) dx



RN

wε2 (x) dx

1 |z|>1

|z|N+2s

1 dz N+2s−2 |z| |z|≤1  |wε (x) − wε (y)|2 dz + 2 dxdy |x − y|N+2s R2N

V0 wε2 dx

RN

≤ (1 − s)−1 ε21 ∇χ2L∞ (RN ) ωN−1 + 4s −1 ωN−1 + V0 wε 2L2 (RN ) + 2[wε ]2s ≤ C1 wε 20 ≤ C1 . Now, (8.3.8) implies that Jε (vε ), ϕ = o(1) for any ϕ ∈ Hε , that is 

s

RN

s

[(−) 2 vε (−) 2 ϕ + V (εx)vε ϕ] dx  =

RN

[χε f (vε ) + (1 − χε )f (vε )]ϕ dx + o(1),

or equivalently 

% RN

& s s (−) 2 wε (−) 2 ϕ + V (εx)wε ϕ dx 

=

RN

  f (vε ) f (vε ) wε + (1 − χε ) wε ϕ dx + o(1). χε vε vε

(8.3.13)

Taking ϕ = wε− = min{wε , 0} in (8.3.13) and recalling that (x − y)(x − − y − ) ≥ |x − − y − |2

for all x, y ∈ R,

and that f (t) = f (t) = 0 for all t ≤ 0, we have 

%

RN

& s |(−) 2 wε− |s + V (εx)(wε− )2 dx 

 ≤

RN

 f (vε ) f (vε ) χε wε + (1 − χε ) wε wε− dx + o(1) = o(1), vε vε

so we see that wε− 2ε → 0

as ε → 0.

Now, we observe that one of the following two cases must occur.

(8.3.14)

306

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .



 Case 1: lim supε→0 supz∈RN B1 (z) |χε (x)wε |2 dx > 0;

 Case 2: lim supε→0 supz∈RN B1 (z) |χε (x)wε |2 dx = 0. Step 1

Case 1 cannot occur under assumption (f 5) with a = ∞.

Suppose, on the contrary, that Case 1 occurs. Then, up to a subsequence, there exist (xε ) ⊂ RN , d > 0 and x0 ∈  such that  |χε wε |2 dx → d > 0,

(8.3.15)

B1 (xε )

εxε → x0 ∈ .

(8.3.16)

Indeed, the existence of (yε ) satisfying (8.3.15) is clear. Moreover, (8.3.15) implies that B1 (xε )∩supp(χε ) = ∅, so there exists zε ∈ supp(χε ) such that χ(εzε ) = 0 and |zε −xε | < 1. Hence |εxε − εzε | < ε yields εxε ∈ Nε () = {z ∈ RN : dist(z, ) < ε}, and we may assume that (8.3.16) holds. Since wε 0 ≤ 1, we may assume that wε (· + xε )  w0

in H s (RN ).

(8.3.17)

In view of (8.3.16) and (8.3.17) we have (χε wε )(· + xε )  χ(x0 )w0

in H s (RN ).

To prove this, fix ϕ ∈ H s (RN ) and note that  R2N

(χε wε )(x + xε ) − (χε wε )(y + xε ) (ϕ(x) − ϕ(y)) dxdy |x − y|N+2s  (hε wε )(x + xε ) − (hε wε )(y + xε ) = (ϕ(x) − ϕ(y)) dxdy |x − y|N+2s R2N  (wε (x + xε ) − wε (y + xε )) (ϕ(x) − ϕ(y))χ(x0 ) dxdy + 2N |x − y|N+2s R = Aε + Bε ,

where hε (x) = χε (x) − χ(x0 ). In view of (8.3.17) we know that  Bε →

R2N

(w0 (x) − w0 (y)) (ϕ(x) − ϕ(y))χ(x0) dxdy. |x − y|N+2s

8.3 Mountain Pass Argument

307

Now, we observe that  Aε =  +

R2N

(hε (x + xε ) − hε (y + xε )) (ϕ(x) − ϕ(y))wε (x + xε ) dxdy |x − y|N+2s

R2N

(wε (x + xε ) − wε (y + xε )) (ϕ(x) − ϕ(y))hε (y + xε ) dxdy |x − y|N+2s

= A1ε + A2ε . Using Hölder’s inequality, (8.3.16), (8.3.17) and the dominated convergence theorem, we see that   |A2ε |

≤C

R2N

|ϕ(x) − ϕ(y)|2 |hε (y + xε )|2 dxdy |x − y|N+2s

 12

→ 0.

On the other hand   |A1ε |

≤ [ϕ]s

R2N

|hε (x + xε ) − hε (y + xε )|2 |wε (x + xε )|2 dxdy |x − y|N+2s

 12

→0

because 

|hε (x + xε ) − hε (y + xε )|2 |wε (x + xε )|2 dxdy N+2s 2N |x − y| R # $   ε2 ∇χ2L∞ (RN ) dy 4dy 2 ≤ |wε (x + xε )| dx + N+2s |x − y|N+2s−2 RN |y−x|> ε1 |x − y| |y−x|< ε1  ≤ C ε2s |wε (x + xε )|2 dx ≤ C ε2s → 0. RN

Now, let us show that χ(x0 ) = 0 and w0 ≥ 0 (≡ 0). If, by contradiction, χ(x0 ) = 0, then the dominated convergence theorem, (8.3.15), (8.3.17) and Theorem 1.1.8 imply that  0 < d = lim

ε→0 B1 (xε )

|χε wε |2 dx



= lim

ε→0 B1

|χε wε |2 (x + xε ) dx



|χ(x0 )w0 (x)|2 dx = 0,

= B1

308

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

which is impossible. For the same reason w0 ≡ 0. Using (8.3.14) and (8.3.17) we can see that w0 ≥ 0 in RN . Thus, there exists a set K ⊂ RN such that |K| > 0

(8.3.18)

wε (x + xε ) → w0 (x) > 0

for x ∈ K.

(8.3.19)

Taking ϕ = wε in (8.3.13), we get  1=

wε 2Hεs

=

RN

  f (vε ) 2 f (vε ) 2 wε + (1 − χε ) wε dx + o(1), χε vε vε

and using item (iv) of Lemma 8.2.2, we deduce that  lim sup ε→0

RN

χε

f (vε ) 2 wε dx ≤ 1, vε

(8.3.20)

that is,  lim sup ε→0

RN

χ(εx + εxε )

f (vε (x + xε )) 2 wε (x + xε ) dx ≤ 1. vε (x + xε )

In view of (8.3.18), (8.3.19) and the definition of wε , we obtain vε (x + xε ) = wε (x + xε )vε ε → w0 (x) · (∞) = ∞ ∀x ∈ K. This, together with limξ →∞

f (ξ ) ξ

 lim inf ε→0

RN

χε (x + xε ) 

f (vε (x + xε )) 2 w (x + xε ) dx wε (x + xε ) ε

χε (x + xε )

≥ lim inf ε→0

= a = ∞ and Fatou’s lemma yields

K

f (vε (x + xε )) 2 wε (x + xε ) dx = ∞, vε (x + xε )

which contradicts (8.3.20). Step 2

Case 1 cannot take place under assumption (f 5) with a < ∞.

As in Step 1, we extract a subsequence and assume that (8.3.15), (8.3.16) and (8.3.17) hold with χ(x0 ) = 0 and w0 ≥ 0 (≡ 0). We aim to prove that w0 is a weak solution to (−)s w0 + V (x0 )w0 = (χ(x0 )a + (1 − χ(x0 ))ν)w0

in RN .

(8.3.21)

This leads to a contradiction, since (−)s has no eigenvalues in H s (RN ) (this fact can be seen by using the Pohozaev identity for the fractional Laplacian).

8.3 Mountain Pass Argument

309

Fix ϕ ∈ Cc∞ (RN ). Taking into account (8.3.16), (8.3.17) and the continuity of V , we see that  % & s s (−) 2 wε (x + xε )(−) 2 ϕ(x) + V (εx + εxε )wε ϕ dx RN



→

% RN

& s s (−) 2 w0 (−) 2 ϕ + V (x0 )w0 ϕ dx.

(8.3.22)

Let us show that   g(εx + εxε , vε (x + xε )) wε ϕ dx → (χ(x0 )a + (1 − χ(x0 ))ν) w0 ϕ dx. vε (x + xε ) RN RN (8.3.23) Take R > 1 such that supp (ϕ) ⊂ BR . Then, using the fact that H s (RN ) is compactly embedded in L2loc (RN ), we see that wε −w0 2L2 (B ) → 0. Hence, there exists h ∈ L2 (BR ) R such that |wε | ≤ h a.e. in BR . ) Since a < ∞, there exists C > 0 such that | g(x,t t | ≤ C for all t > 0. Recall that

g(x, t) → χ(x)a + (1 − χ(x))ν < ∞ t

as t → ∞.

Then    g(εx + εxε , vε (x + xε ))   wε ϕ  ≤ CϕL∞ (RN ) |wε |  vε (x + xε ) ≤ CϕL∞ (RN ) h ∈ L1 (BR ),

(8.3.24)

and g(εx + εxε , vε (x + xε )) wε (x) → [χ(x0 )a + (1 − χ(x0 ))ν]w0 (x) vε (x + xε )

a.e. in BR . (8.3.25)

   ) In fact, if w0 (x) = 0, by  g(x,t t  ≤ C for all t > 0 and wε → w0 = 0 a.e. in BR , we get    g(εx + εxε , vε (x + xε ))    w ε  ≤ C|wε | → 0  vε (x + xε )

a.e. in BR .

310

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

If w0 (x) = 0, then vε (x + xε ) = wε (x + xε )vε Hεs → ∞, and since wε → w0 a.e. in BR , we have g(εx + εxε , vε (x + xε )) wε → [χ(x0 )a + (1 − χ(x0 ))ν]w0 vε (x + xε )

a.e. in BR .

Then (8.3.25) holds. Taking into account (8.3.24) and (8.3.25), we infer that (8.3.23) is true thanks to the dominated convergence theorem. Putting together Jε (vε ), ϕ = o(1), (8.3.22) and (8.3.23) we obtain (8.3.21). Step 3

Case 2 cannot take place.

Assume the contrary. Since (8.3.12) holds and  |χε wε |2 dx = 0,

lim sup

ε→0 z∈RN

B1 (z)

Lemma 1.4.4 shows that χε wε Lp+1 (RN ) → 0. Now, for any L > 1 we can see that 1 2 L − 2

Jε (Lwε ) =



' RN

( χε F (Lwε ) + (1 − χε )F (Lwε ) dx.

By item (ii) of Lemma 8.2.2 and since ν ∈ (0, V20 ), we have 

 RN

(1 − χε )F (Lwε ) dx ≤  ≤ ≤

RN

1 2 νL |wε |2 dx 2

RN

V0 2 L |wε |2 dx 4

L2 L2 wε 0 ≤ . 4 4

Consequently, Jε (Lwε ) ≥

1 2 L − 4

 RN

χε F (Lwε ) dx.

(8.3.26)

Using (8.2.1), Hölder’s inequality and the fact that χε wε Lp+1 (RN ) → 0, we see that 



 RN

χε F (Lwε ) dx ≤

RN

 δ 2 |Lwε |p+1 2 L |wε | + Cδ χε (x) dx 2 p+1 p

≤ δL2 wε 2L2 (RN ) + Cδ Lp+1 wε Lp+1 (RN ) χε wε Lp+1 (RN ) ≤

δL2 wε 2ε + o(1). V02

(8.3.27)

8.3 Mountain Pass Argument

311

Putting together (8.3.26) and (8.3.27) we have Jε (Lwε ) ≥

1 2 δL2 L − 2 wε 2ε + o(1) ∀δ > 0, 4 V0

and thanks to the arbitrariness of δ > 0, we get lim inf Jε (Lwε ) ≥ ε→0

Since vε ε → ∞, we can see that

L vε ε

1 2 L . 4

∈ (0, 1) for ε sufficiently small, and so 

max Jε (tvε ) ≥ Jε

t ∈[0,1]

L vε vε ε

 ≥

1 2 L . 4

Take L > 0 sufficiently large so that m2 < 14 L2 and recall that, by (8.3.7), Jε (vε ) ≤ m2 . Then we can find tε ∈ (0, 1) such that Jε (tε vε ) = max Jε (tvε ). t ∈[0,1]

Hence Jε (tε vε ) = max Jε (tvε ) ≥ t ∈[0,1]

1 2 L →∞ 4

as L → ∞,

that is Jε (tε vε ) → ∞

as ε → 0.

(8.3.28)

Now, since Jε (tε vε ), (tε vε ) = 0, (8.3.8) and Corollary 8.2.3-(iv) imply that 1 Jε (tε vε ) = Jε (tε vε ) − Jε (tε vε ), (tε vε ) 2   = G(εx, tε vε ) dx RN

≤ D kν

 RN

 G(εx, vε ) dx

  1

Jε (vε ) − Jε (vε ), vε  =D 2 kν

≤ D kν m2 + o(1) which contradicts (8.3.28). Then the Case 2 can not take place.

(8.3.29)

312

Step 4

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Conclusion.

Steps 1, 2 and 3, show that vε ε is bounded as ε → 0.

 

In the next lemma we prove that every Cerami sequence (vj ) ⊂ Hε at level cε is bounded and admits a convergent subsequence in Hεs . Lemma 8.3.4 Assume that f satisfies (f 1)–(f 3) and either (f 4) or (f 5). Then there exists ε1 ∈ (0, ε0 ] such that for every ε ∈ (0, ε1 ] and every sequence (vj ) ⊂ Hε satisfying Jε (vj ) → c > 0, (1 + vj ε )Jε (vj )H∗ε

(8.3.30) → 0 as j → ∞,

(8.3.31)

for some c > 0, one has that (i) vj ε is bounded as j → ∞; (ii) there exist (jk ) and v0 ∈ Hε such that vjk → v0 strongly in Hε . Proof The proof of (i) can be carried out in much the same way as the one of Lemma 8.3.3, with suitable modifications. More precisely, in Step 8.3.3, for a given  1 of Lemma N 2 sequence (vj ), there exists (xj ) ⊂ R such that B1 (xj ) |χε wj | dx → d > 0. The sequence (xj ) satisfies εxj ∈ Nε (), and we may assume that εxj → x0 ∈ Nε (), where x0 is such that χ(εx + x0 ) = 0 in B1 . In Step 2 we replace (8.3.21) by (−)s w0 + V (εx + x0 )w0 = (χ(εx + x0 )a + (1 − χ(εx + x0 ))ν)w0

in RN

(8.3.32)

where w0 ∈ H s (RN ) is non-negative and not identically zero. Indeed, by the maximum 0 ˜ = w0 ( x−x ˜ satisfies principle, w0 > 0 in RN . Set w(x) ε ). Then w ε2s (−)s w˜ + V (x)w˜ = (χ(x)a + (1 − χ(x))ν)w˜

in RN .

(8.3.33)

We aim to prove that this is impossible for ε > 0 sufficiently small. Using the Caffarelli = Ext(w) Silvestre extension technique, we know that W ˜ is a solution to the following problem ⎧ 2s 1−2s ∇ W ) = 0 ⎪ in RN+1 ⎨ −ε div(y + ,  W (·, 0) = w˜ on ∂RN+1 + , ⎪  ⎩ ∂W N+1 = −V (x) w ˜ + (χ(x)a + (1 − χ(x))ν) w ˜ on ∂R + . ∂ν 1−2s

8.3 Mountain Pass Argument

313

Take r > 0 sufficiently small such that χ(x) = 1 and V (x) < a

for x ∈ Br .

Let us recall the following notations: Br+ = {(x, y) ∈ RN+1 : y > 0, |(x, y)| < r}, + r+ = {(x, y) ∈ RN+1 : y ≥ 0, |(x, y)| = r}, + r0 = {(x, 0) ∈ ∂RN+1 : |x| < r}, + and define 1 + 1 + 1−2s H0, ) : V ≡ 0 on r+ }. + (Br ) = {V ∈ H (Br , y r

Let   μr = inf

Br+

y

1−2s

2

|∇U | dxdy : U ∈

1 + H0, + (Br ), r



 2

r0

u dx = 1 .

1 + 2 0 By the compactness of the embedding H0, + (Br )  L (r ), it is not difficult to see that r

the infimum is achieved by a function Ur ∈ H1+ (Br+ ) \ {0}. Moreover, we may assume r that Ur ≥ 0, since |U | is a minimizer whenever U is a minimizer. Then Ur is a solution, not identically zero, of ⎧ ⎪ −div(y 1−2s ∇Ur ) = 0 in Br+ , ⎪ ⎨ ∂Ur = μr u r on r0 , ⎪ ∂ν 1−2s ⎪ ⎩ on r+ . Ur = 0

(8.3.34)

By the strong maximum principle, Ur > 0 on Br+ ∪ r0 . Note μr ≥ 0 and μr is a nonincreasing function of r. Indeed, μr is decreasing in r. In fact, if by contradiction we assume that r1 < r2 and μr1 = μr2 , we can multiply the equation div(y 1−2s ∇Ur1 ) = 0 by Ur2 , and after an integration by parts, we can use the equalities satisfied by Ur1 and Ur2 , and the assumption μr1 = μr2 , to deduce that  r+1

∂Ur1 Ur dσ = 0. ∂ν 1−2s 2

This gives a contradiction, because of Ur2 > 0 and

∂Ur1 ∂ν 1−2s

< 0 on r+1 .

314

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

1−2s ). Next, extend Ur by setting Ur = 0 in RN+1 \ Br+ , so that Ur ∈ H 1 (RN+1 + + ,y Therefore,



 ε2s μr

ur w˜ dx =

r0

Br+

 · ∇Ur dxdy y 1−2s ε2s ∇ W

 =−

r0

(V (x) − a)wu ˜ r dx

that is  r0

(V (x) − a + ε2s μr )wu ˜ r dx = 0.

(8.3.35)

But this is impossible because of V (x)−a +μr ε2s < 0 in r0 for ε > 0 small and ur w˜ > 0 in r0 . In order to verify (ii), fix ε ∈ (0, ε1 ] and (vj ) satisfying (8.3.30) and (8.3.31). Using (i), we see that (vj ) is bounded in Hε . Up to a subsequence, we may assume that vj  v0 in Hε . To show that this convergence is actually strong, we follow Lemma 7.2.4 in which we observed that it suffices to show that 

s lim lim sup |(−) 2 vj |2 + V (εx)vj2 dx = 0. (8.3.36) R→∞ j →∞

|x|≥R

Let ηR ∈ C ∞ (RN ) be a cut-off function such that ⎧ ⎪ η (x) = 0, ⎪ ⎪ R ⎨ ηR (x) = 1, ⎪ 0 ≤ ηR (x) ≤ 1, ⎪ ⎪ ⎩ |∇ηR (x)| ≤ C R,

for |x| ≤ R2 , for |x| ≥ R, for x ∈ RN , for x ∈ RN .

Take R > 0 such that ε ⊂ B R . Since (vj ηR ) is bounded in Hε , we see that 2 Jε (vj ), ηR vj  = oj (1). Hence, 

s

RN

s



(−) 2 vj (−) 2 (vj ηR ) dx +  =

RN

f (vj )vj ηR dx + oj (1)



≤ν

RN

vj2 ηR dx + oj (1).

RN

V (εx)vj2 ηR dx

8.3 Mountain Pass Argument

315

By our choice of ν, we can find α ∈ (0, 1) such that 

s

RN



s

(−) 2 vj (−) 2 (vj ηR ) dx + α

RN

V (εx)vj2 ηR dx ≤ oj (1).

(8.3.37)

Now we observe that  s s (−) 2 vj (−) 2 (vj ηR ) dx RN



=

(vj (x) − vj (y))(vj (x)ηR (x) − vj (y)ηR (y)) dxdy |x − y|N+2s

R2N

 =  +

R2N

ηR (x)

|vj (x) − vj (y)|2 dxdy |x − y|N+2s

(vj (x) − vj (y))(ηR (x) − ηR (y)) vj (y) dxdy |x − y|N+2s

R2N

= AR,j + BR,j .

(8.3.38)

Clearly,  AR,j ≥

 |x|≥R RN

|vj (x) − vj (y)|2 dxdy. |x − y|N+2s

(8.3.39)

Using Hölder’s inequality, Lemma 1.4.5 and the fact that (vj ) is bounded in H s (RN ), we see that lim sup lim sup |BR,j | R→∞

j →∞

  ≤ lim sup lim sup R→∞

j →∞

R2N

|vj (x) − vj (y)|2 dxdy |x − y|N +2s

  ≤ C lim sup lim sup R→∞

j →∞

R2N

 1   2 R2N

|ηR (x) − ηR (y)|2 |vj (y)|2 dxdy |x − y|N +2s

|ηR (x) − ηR (y)|2 |vj (y)|2 dxdy |x − y|N +2s

 12

= 0.

 12

(8.3.40)

Combining (8.3.37)–(8.3.40) and using the fact that 

 |x|≥R

we infer that (8.3.36) holds.

V (εx)vj2 dx ≤

RN

V (εx)vj2 ηR dx,  

316

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Taking into account Lemma 8.3.3 and Lemma 8.3.4 we deduce the following result: Corollary 8.3.5 There exists ε1 ∈ (0, ε0 ] such that for every ε ∈ (0, ε1 ] there exists a critical point vε ∈ Hε of Jε (v) satisfying Jε (vε ) = cε , where cε ∈ [m1 , m2 ] is defined as in (8.3.4)–(8.3.5). Moreover there exists a constant M > 0 independent of ε ∈ (0, ε1 ] such that vε ε ≤ M for any ε ∈ (0, ε1 ].

8.4

Limit Equation

In the next section we will see that the sequence of critical points obtained in Corollary 8.3.5 converges, in some sense, to a sum of translated critical points of certain autonomous functionals. As proved in [39], least energy solutions for autonomous nonlinear scalar field equations admit a mountain pass characterization. This property will play a fundamental role in the proof of Theorem 8.1.1. For this reason, in this section we collect some important results on autonomous functionals associated with “limit equations". Firstly, we introduce some notations and definitions that will be useful later. For x0 ∈ RN we define the autonomous functional x0 : H s (RN ) → R by setting x0 (v) =

1 2

 RN



s |(−) 2 v|2 + V (x0 )v 2 dx −

RN

G(x0 , v) dx.

It is routine to check that x0 ∈ C 1 (H s (RN ), R) and critical points of x0 are weak solutions to the equation (−)s u + V (x0 )u = g(x0 , u) in RN .

(8.4.1)

We note that, if u is a solution to (8.2.4), then v(x) = u(ε x + x0 ) satisfies (−)s v + V (ε x + x0 )v = g(ε x + x0 , v)

in RN ,

that is, (8.4.1) can be seen as the limit equation of (8.4.2) as ε → 0. For any x0 ∈ RN and u, v ∈ H s (RN ) we denote  u, vHε =  u, vx0 = ⎪ ⎪ ⎪2 ⎪ ⎪ ⎪x 0 = ⎪v ⎪



RN

RN

s

s

s

s

(−) 2 u(−) 2 v + V (εx)uv dx, (−) 2 u(−) 2 v + V (x0 )uv dx, s

RN

|(−) 2 v|2 + V (x0 )v 2 dx.

(8.4.2)

8.4 Limit Equation

317

Finally, we define 1 H (x, t) = − V (x)t 2 + G(x, t) 2 and  

= x ∈ RN : sup H (x, t) > 0 . t >0

Remark 8.4.1 (i) ⊂  and 0 ∈ {x ∈  : V (x) = infy∈ V (y)} ⊂ . (ii) If (f 3) or (f 5) with a = ∞ holds, then = . Now, we state the following Jeanjean-Tanaka type result [234] proved in Chap. 3 (see Theorem 3.1.1) in connection with the study of nonlinear scalar field equation with fractional diffusion (−)s u = h(u) in RN ,

u ∈ H s (RN ),

(8.4.3)

where h ∈ C 1 (R, R) is an odd function satisfying the following Berestycki–Lions type assumptions [100]: (h1) −∞ < lim inft →0 h(t)/t ≤ lim supt →0 h(t)/t < 0; ) = 0; (h2) lim|t |→∞ h(t ∗ |t |2s −1 (h3) there exists t¯ > 0 such that H (t¯) > 0. We recall that the existence of a solution to (8.4.3) has been established in [39, 139]. Lemma 8.4.2 [39] Assume that h ∈ C 1 (R, R) is an odd function satisfying the Berestycki–Lions type assumptions (h1)–(h3). Let I˜ : H s (RN ) → R be the functional defined by I˜(u) =

 RN



s 1 |(−) 2 u|2 − H (u) 2

 dx.

Then I˜ has a mountain pass geometry and c = m, where m is defined as m = inf{I˜(u) : u ∈ H s (RN ) \ {0} is a solution to (8.4.3)},

(8.4.4)

318

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

and c = inf max I˜(γ (t)), γ ∈ t ∈[0,1]

where  = {γ ∈ C([0, 1], H s (RN )) : γ (0) = 0, I˜(γ (1)) < 0}. Moreover, for any least energy solution ω(x) of (8.4.3) there exists a path γ ∈  such that I˜(γ (t)) ≤ m = I˜(ω)

for all t ∈ [0, 1],

ω ∈ γ ([0, 1]).

(8.4.5) (8.4.6)

At this point, we give the proof of the following lemma which we will use in the next section to obtain a concentration-compactness type result. Lemma 8.4.3 Assume that f satisfies (f 1)–(f 3). Then we have (i) x0 (v) has non-zero critical points if and only if x0 ∈ . ⎪ ⎪ ⎪ ⎪ (ii) There exists δ1 > 0, independent of x0 ∈ RN , such that ⎪ ⎪x0 ≥ δ1 for any non-zero ⎪v ⎪ critical point v of x0 . Proof Firstly, we extend f (t) to an odd function on R. Let us consider the function h(t) = −V (x0 )t + g(x0 , t), that is h(t) = H (x0 , t). Clearly h is odd. Now we show that h satisfies assumptions (h1)–(h3). By (f 2) and (f 3), it follows that (h1) and (h2) hold. Since = {x ∈ RN : supt >0 H (x, t) > 0}, we see that (h3) holds if and only if x0 ∈ . Then (i) follows by Theorem 1 in [35] (see also Theorem 1.1 in [139]). Now let v be a non-zero critical point of x0 . Then  x0 (v), v

 = 0 $⇒

RN



s 2 2 2 |(−) v| + V (x0 )v dx −

Using item (i) of Corollary 8.2.3, we get  v20

−

RN

f (v)v dx ≤ 0,

RN

g(x0 , v)v dx = 0.

8.4 Limit Equation

319

so by (8.2.1) it follows that for any δ ∈ (0, V0 ), p+1

v20 ≤ δv2L2 (RN ) + Cδ vLp+1 (RN ) ≤

δ p+1

v20 + Cδ Cp+1 v0 . V0

Then   δ p+1

1− v20 ≤ Cδ Cp+1 v0 , V0 and we can find⎪δ1⎪> 0 such that v0 ≥ δ1 for any x0 ∈ RN and for any non-zero critical ⎪ ⎪   point v. Since ⎪ ⎪x ≥ v0 , we conclude that (ii) is verified. ⎪v ⎪ 0

For x ∈ RN , set  m(x) =

least energy level of x (v), if x ∈ , ∞, if x ∈ RN \ .

By Lemma 8.4.2, we can see that m(x) is equal to the mountain pass value for x (v) if x ∈ , that is 

 m(x) = inf

γ ∈

max x (γ (t)) ,

t ∈[0,1]

where  = {γ ∈ C([0, 1], H s (RN )) : γ (0) = 0 and x (γ (1)) < 0}. Lemma 8.4.4 m(x0 ) = inf m(x) if and only if x0 ∈  and V (x0 ) = inf V (x). x∈

x∈RN

In particular, m(0) = infx∈RN m(x). Proof Fix x0 ∈  such that V (x0 ) = infx∈ V (x). We note that x0 ∈  . Otherwise, if x0 ∈  \  , then V (x0 ) ≥

inf V (x) > inf V (x),

x∈\

x∈

which is impossible. Hence, x0 ∈  and χ(x0 ) = 1. Moreover, x0 ∈ by Remark 8.4.1. Now, using the fact that V (x) ≥ V (x0 ) in  and G(x, t) ≤ F (t) for any (x, t) ∈ RN × R,

320

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

we see that for any x¯ ∈

x¯ (v) =

s 1 1 2 (−) 2 v2L2 (RN ) + V (x)v ¯ − L2 (RN ) 2 2



s 1 1 ≥ (−) 2 v2L2 (RN ) + V (x0 )v2L2 (RN ) − 2 2

RN

G(x, ¯ v) dx



RN

F (v) dx

= x0 (v) for any v ∈ H s (RN ). This implies that m(x0 ) ≤ m(x) for all x ∈ RN , so m(x0 ) ≤ infx∈RN m(x) ≤ m(x0 ) that is m(x0 ) = infx∈RN m(x). Now fix x ∈  such that V (x ) > V (x0 ). Take γ ∈  such that (8.4.5) and (8.4.6) hold with I˜(v) = x (v). Then we deduce that m(x0 ) ≤ max x0 (γ (t)) < max x (γ (t)) = m(x ). t ∈[0,1]

t ∈[0,1]

  Finally, we note that the function m(x) is continuous (see [47]). Proposition 8.4.5 The function m(x) : RN → (−∞, ∞] is continuous in the following sense: m(xj ) → m(x0 ) m(xj ) → ∞

8.5

when xj → x0 ∈ , when xj → x0 ∈ RN \ .

ε-Dependent Concentration-Compactness Result

This section is devoted to the study of the behavior as ε → 0 of critical points (vε ) obtained in Corollary 8.3.5. More generally, we consider (vε ) such that vε ∈ Hε ,

(8.5.1)

Jε (vε ) → c ∈ R,

(8.5.2)

(1 + vε ε )Jε (vε )H∗ε

→ 0,

vε ε ≤ m, where c and m are independent of ε. We begin by proving the following concentration-compactness type result.

(8.5.3) (8.5.4)

8.5 ε-Dependent Concentration-Compactness Result

321

Lemma 8.5.1 Assume that f satisfies (f 1)–(f 3) and (vε )ε∈(0,ε1] satisfies the conditions (8.5.1)–(8.5.4). Then there exist a subsequence εj → 0, l ∈ N ∪ {0}, sequences (yεkj ) ⊂ RN , x k ∈ , ωk ∈ H s (RN ) \ {0} (k = 1, . . . , l) such that

|yεkj − yεkj | → ∞

as j → ∞, for k = k ,

(8.5.5)

εj yεkj → x k ∈

as j → ∞,

(8.5.6)

ωk ≡ 0 and  x k (ωk ) = 0,    l      k k ω (· − yεj )  vεj − ψεj  

(8.5.7)

k=1

Jεj (vεj ) →

l 

→0

as j → ∞,

(8.5.8)

εj

x k (ωk ).

(8.5.9)

k=1

Here ψε (x) = ψ(εx), and ψ ∈ Cc∞ (RN ) is such that ψ(x) = 1 for x ∈  and 0 ≤ ψ ≤ 1 on RN . When l = 0, we have vεj εj → 0 and Jεj (vεj ) → 0. Remark 8.5.2 Let us note that sup ψ(εx)V (εx) < ∞. Moreover, for all w ∈ H s (RN ), ψε w ∈ Hεs and there exists a constant C > 0, independent of ε, such that ψε wε ≤ Cw0 .

(8.5.10)

Remark 8.5.3 For any w ∈ H s (RN ) and for any sequence (yε ) ⊂ RN such that εyε → x0 ∈ , we have ψε w(· − yε )2ε  s = |(−) 2 (ψ(εx + εyε )w(x))|2 + V (εx + εyε )(ψ(εx + εyε )w(x))2 dx RN

 →

RN

⎪ s ⎪2 ⎪ ⎪ |(−) 2 w|2 + V (x0 )w2 dx = ⎪ ⎪x ⎪w⎪

0

as ε → 0.

(8.5.11)

We first prove that 

s

RN

|(−) 2 (ψ(εx + εyε )w(x))|2 dx →



s

RN

|(−) 2 w(x)|2 dx.

(8.5.12)

322

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Then, 

|ψ(εx + εyε )w(x) − ψ(εy + εyε )w(y)|2 dxdy |x − y|N+2s

R2N



R2N

|ψ(εx + εyε ) − ψ(εy + εyε )|2 |w(x)|2 dxdy |x − y|N+2s

R2N

|w(x) − w(y)|2 (ψ(ε y + ε yε ))2 dxdy |x − y|N+2s

=  +

 +2

R2N

(ψ(εx + εyε ) − ψ(εy + εyε ))(w(x) − w(y)) ω(x)ψ(ε y + ε yε ) dxdy |x − y|N+2s

= Aε + Bε + 2Cε . Now, by the dominated convergence theorem and the fact that ψ(ε · + ε yε ) → 1, we see that Bε → [w]2s . On the other hand  Aε =  +

 RN

RN

dx  dx

|x−y|≤ ε1

|ψ(εx + εyε ) − ψ(εy + εyε )|2 |w(x)|2 dy |x − y|N+2s

|x−y|> ε1

|ψ(εx + εyε ) − ψ(εy + εyε )|2 |w(x)|2 dy |x − y|N+2s 

 ≤ ε2 ∇ψ2L∞ (RN ) ωN−1  + 4ωN−1

RN



= ε ωN−1 2s

|w(x)|2 dx

|w(x)|2 dx

RN  ∞ 1 ε

∇ψ2L∞ (RN ) 2 − 2s

1 ε

0

1 z2s−1

dz

1 dz z2s+1  2 + |w(x)|2 dx → 0 s RN

as ε → 0,

(8.5.13)

and since  |Cε | ≤ [w]s Aε → 0, we infer that (8.5.12) holds. Since 

 RN

V (εx + εyε )|ψ(εx + εyε )w(x)|2 dx →

RN

V (x0 )|w(x)|2 dx,

(8.5.14)

it follows from (8.5.12) and (8.5.14) that (8.5.11). Proof We divide the proof into several steps. In what follows, we write ε instead of εj .

8.5 ε-Dependent Concentration-Compactness Result

323

Step 1 Up to a subsequence, vε  v0 in H s (RN ) and v0 is a critical point of 0 (v). Using (8.5.4) and (8.3.2), we see that vε 0 ≤ m. Thus (vε ) is bounded in H s (RN ) and we can assume that vε  v0 in H s (RN ). Let us show that v0 is a critical point of 0 (v), that is,  0 (v0 ), ϕ = 0 for all ϕ ∈ s H (RN ). Since Cc∞ (RN ) is dense in H s (RN ), it is enough to prove it for all ϕ ∈ Cc∞ (RN ). Fix ϕ ∈ Cc∞ (RN ). It follows from (8.5.3) that  RN

% & s s (−) 2 vε (−) 2 ϕ + V (εx)vε ϕ − g(εx, vε )ϕ dx → 0.

Now we show that Jε (vε ), ϕ = vε , ϕHε −



 RN

g(εx, vε )ϕ dx → v0 , ϕ0 −

RN

g(0, v0 )ϕ dx.

Note that vε , ϕHε − v0 , ϕ0   s s = (−) 2 (vε − v0 )(−) 2 ϕ dx + RN

+ V (0)

 RN

RN

[V (εx) − V (0)]vε ϕ dx

(vε − v0 )ϕ dx

= A1ε + A2ε + A3ε . Then A1ε , A3ε → 0 because vε  v0 in H s (RN ), and |A2ε | ≤ CV (ε·) − V (0)L∞ (supp(ϕ)) vε H s ϕL2 (RN ) ≤ C V (ε·) − V (0)L∞ (supp(ϕ)) → 0. q

On the other hand, using item (iii) of Corollary 8.2.3 and the fact that H s (RN )  Lloc (RN ) for any q ∈ [1, 2∗s ), we have 

 RN

g(εx, vε )ϕ dx →

RN

g(0, v0 )ϕ dx.

Hence  0 (v0 ), ϕ =



% RN

& s s (−) 2 v0 (−) 2 ϕ + V (0)v0 ϕ − g(0, v0 )ϕ dx = 0.

If v0 ≡ 0, we set yε1 = 0 and ω1 = v0 .

324

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Step 2 Suppose that there exist n ∈ N ∪ {0}, (yεk ) ⊂ RN , x k ∈ , ωk ∈ H s (RN ) (k = 1, . . . , n) such that (8.5.5), (8.5.6), (8.5.7) of Lemma 8.5.1 hold for k = 1, . . . , n and vε (· + yεk )  ωk

in H s (RN ) for k = 1, . . . , n.

(8.5.15)

Moreover, we assume that  sup y∈RN

 2 n      ωk (x − yεk ) dx → 0. vε − ψε   B1 (y)

(8.5.16)

k=1

Then  2 n      ωk (· − yεk ) → 0. vε − ψε   k=1

(8.5.17)

ε

Set ξε (x) = vε (x) − ψε (x)

n 

ωk (x − yεk ).

k=1

Inequality (8.5.10) implies that ξε ε ≤ vε ε + ψε

n 

ωk (· − yεk )ε

k=1

≤m+C

n 

ωk 0 ,

k=1

and using the fact that ξε 0 ≤ ξε ε , we deduce that (ξε ) is bounded in H s (RN ). By (8.5.16) and Lemma 1.4.4, ξε Lp+1 (RN ) → 0 as ε → 0. A direct calculation shows that ξε 2ε = vε − ψε

n 

ωk (· − yεk ), ξε Hε

k=1

= vε , ξε Hε −

n  k=1

ψε ωk (· − yεk ), ξε Hε .

(8.5.18)

8.5 ε-Dependent Concentration-Compactness Result

325

We claim that ψε ωk (· − yεk ), ξε Hε = ωk (· − yεk ), ψε ξε x k + o(1)

(8.5.19)

for all k = 1, . . . , n. Indeed ψε ωk (· − yεk ), ξε Hεs − ωk (· − yεk ), ψε ξε x k   (ψε (x) − ψε (y))(ξε (x) − ξε (y))ωk (x − yεk ) = dxdy |x − y|N+2s R2N   (ψε (x) − ψε (y))(ωk (x − yεk ) − ωk (y − yεk ))ξε (x) − dxdy |x − y|N+2s R2N  (V (εx + εyεk ) − V (x k ))ψ(εx + εyεk )ωk (x)ξε (x + yεk ) dx + RN

= Bε1 + Bε2 . Note that      (ψε (x) − ψε (y))(ξε (x) − ξε (y))ωk (x − yεk )   dxdy    R2N  |x − y|N+2s   ≤

R2N

|ξε (x) − ξε (y)|2 dxdy |x − y|N+2s

 1   2 R2N

|ψε (x) − ψε (y)|2 (ωk (x − yεk ))2 dxdy |x − y|N+2s

1 2

,

and      (ψε (x) − ψε (y))(ωk (x − yεk ) − ωk (y − yεk ))ξε (x)   dxdy    R2N  |x − y|N+2s   ≤

R2N

|ωk (x − yεk ) − ωk (y − yεk )|2 dxdy |x − y|N+2s

 1   2 R2N

|ψε (x) − ψε (y)|2 |ξε (x)|2 dxdy |x − y|N+2s

1 2

.

Hence, since ξε 0 ≤ C 1 and ωk 0 ≤ C 2 for some C¯ 1 , C¯ 2 > 0, we can argue as in the proof of (8.5.13) to see that Bε1 → 0 as ε →0. We note that the quantity (V (εx + εyεk ) − V (x k ))ψ(εx + εyεk ) is bounded in L∞ (RN ). By (8.5.5) and (8.5.15) we deduce that ξε (· + yεk )  0 ξε (· + yεk ) → 0

in H s (RN ) in L2loc (RN ).

Then Bε2 → 0 and we conclude that (8.5.19) holds.

(8.5.20)

326

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Putting together (8.5.18) and (8.5.19) we see that ξε 2ε = vε , ξε Hε −

n  ωk (· − yεk ), ψε ξε x k + o(1) k=1

=

Jε (vε ), ξε  +

 RN

g(εx, vε )ξε dx − 

 +  =

RN

RN

k=1

g(x k , ωk (x − yεk ))ψε ξε dx + o(1)

g(εx, vε )ξε dx −

= Cε1 −

n   x k (ωk (· − yεk )), ψε ξε 

n 

n   N k=1 R

g(x k , ωk (x − yεk ))ψε ξε dx + o(1)

Cε2 + o(1).

k=1

By Corollary 8.2.3-(iii),  |Cε1 | ≤ δ

 RN

|vε ξε | dx + Cδ

RN

|vε |p |ξε | dx p

≤ δvε L2 (RN ) ξε L2 (RN ) + Cδ vε Lp+1 (RN ) ξε Lp+1 (RN ) , and using that ξε Lp+1 (RN ) → 0 as ε → 0, the boundedness of vε L2 (RN ) and ξε L2 (RN ) , and the arbitrariness of δ, we get Cε1 → 0. In view of (8.5.20), we see that Cε2 → 0. Hence, ξε ε → 0 and (8.5.17) holds. Step 3 Suppose that there exist n ∈ N ∪ {0}, (yεk ) ⊂ RN , x k ∈ , ωk ∈ H s (RN ) \ {0} (k = 1, . . . , n) such that (8.5.5), (8.5.6), (8.5.7) and (8.5.15) hold. We also assume that there exists zε ∈ RN such that  2 n     k k  ω (x − yε ) dx → c > 0. vε − ψε  B1 (zε ) 



(8.5.21)

k=1

Then there exist x k+1 ∈ and ωk+1 ∈ H s (RN ) \ {0} such that |zε − yεk | → ∞

for all k = 1, . . . , n,

εzε → x k+1 ∈ , vε (· + zε )  ωk+1 ≡ 0  x k+1 (ωk+1 ) = 0.

(8.5.22) (8.5.23)

in H s (RN ),

(8.5.24) (8.5.25)

8.5 ε-Dependent Concentration-Compactness Result

327

It is routine to prove that zε satisfies (8.5.22) and that there exists ωk+1 ∈ H s (RN ) \ {0} satisfying (8.5.24). Let us show that (8.5.23) holds. First, we prove that lim supε→0 |εzε | < ∞. Assume, by contradiction, that |εzε | → ∞. Let ϕ ∈ Cc∞ (RN ) be a cut-off function such that ϕ ≥ 0, ϕ(0) = 1 and let ϕR (x) = ϕ(x/R). Since (ϕR (· − zε )vε ) is bounded in Hε , we obtain Jε (vε ), ϕR (· − zε )vε  → 0 that is 

s

as ε → 0,

s

(−) 2 vε (x + zε )(−) 2 (ϕR (x)vε (x + zε ))+V (εx + εzε )vε2 (x + zε )ϕR (x) dx

RN



−

RN

g(εx + εzε , vε (x + zε ))vε (x + zε )ϕR (x) dx → 0.

(8.5.26)

Since |εzε | → ∞, g(εx + εzε , vε (x + zε )) = f (vε (x + zε )) on supp (ϕR ) for any ε small enough. Moreover, ϕR (x) → 1 as R → ∞ and |f (ωk+1 )ωk+1 ϕR | ≤ C1 |ωk+1 |2 + C2 |ωk+1 |p+1 ∈ L1 (RN ). in view of Lemma 8.2.2-(iii) and Lemma 8.2.1-(i). Hence, by invoking the dominated convergence theorem, we infer that  lim lim

R→∞ ε→0 RN

g(εx + εzε , vε (x + zε ))vε (x + zε )ϕR (x) dx 

= lim

R→∞ RN

f (ωk+1 )ωk+1 ϕR dx



=

RN

f (ωk+1 )ωk+1 dx.

(8.5.27)

On the other hand, using (8.5.24), Hölder’s inequality and Lemma 1.4.5 (with ηR = 1 − ϕR ), we can see that  lim lim sup

R→∞

ε→0

R2N

(vε (x + zε ) − vε (y + zε ))(ϕR (x) − ϕR (y)) vε (y + zε ) dxdy = 0, |x − y|N+2s (8.5.28)

328

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

and applying Fatou’s lemma and (8.5.24) we get  lim lim inf

R→∞ ε→0

R2N

 ≥

|vε (x + zε ) − vε (y + zε )|2 ϕR (x) dxdy |x − y|N+2s

R2N

|ωk+1 (x) − ωk+1 (y)|2 dxdy. |x − y|N+2s

(8.5.29)

Taking into account (8.5.26), (8.5.27), (8.5.28) and (8.5.29), we deduce that  R2N

|ωk+1 (x) − ωk+1 (y)|2 dxdy + |x − y|N+2s

 RN

V0 (ωk+1 )2 − f (ωk+1 )ωk+1 dx ≤ 0. (8.5.30)

By Lemma 8.2.2 (i)–(ii) and (8.5.30),  R2N

|ωk+1 (x) − ωk+1 (y)|2 dxdy + |x − y|N+2s

 RN

(V0 − ν)(ωk+1 )2 dx ≤ 0.

Since V0 > ν, we infer that ωk+1 ≡ 0, which contradicts (8.5.24). Then, lim supε→0 |εzε | < ∞ and there exists x k+1 ∈ RN such that εzε → x k+1 . This and the fact that Jε (vε ), ϕ(·−zε ) → 0 for any ϕ ∈ Cc∞ (RN ) show that  x k+1 (ωk+1 ) = 0. Since ωk+1 ≡ 0, it follows that x k+1 ∈ by Lemma 8.4.3 (i). Step 4 Conclusion. Suppose first that v0 ≡ 0. Set yε1 = 0, x 1 = 0, ω1 = v0 . If vε − ψε ω1 ε → 0, then (8.5.5)–(8.5.8) are satisfied by 0 ∈ , v0 ≡ 0 and  0 (v0 ) = 0. If vε − ψε ω1 ε  0, then (8.5.16) in Step 2 does not occur, and there exists (zε ) satisfying (8.5.21) in Step 3. In view of Step 3, there exist x 2 , ω2 satisfying (8.5.22)–(8.5.25). Then set yε2 = zε . If vε −ψε (ω1 +ω2 (·−yε2 ))ε → 0, then (8.5.5)–(8.5.8) hold because |yε2 −yε1 | = |zε | → ∞, εyε2 → x 2 ∈ and  x 2 (ω2 ) = 0. Otherwise, we can use Steps 2 and 3 to continue this procedure. Now we assume that v0 ≡ 0. If vε ε → 0, we are done. Otherwise, condition (8.5.16) in Step 2 does not occur, and we can find (zε ) satisfying (8.5.21) in Step 3. Applying Step 3, there exist x 1 and ω1 satisfying (8.5.22)–(8.5.25). Thus, we set yε1 = zε . If vε − ψε (ω1 (· − yε1 ))ε → 0, we are done. Otherwise, we use Steps 2 and 3 and we continue this procedure. At this point, we aim to show that the procedure stops after a finite number of steps. First we show that, under assumptions (8.5.5)–(8.5.7) and (8.5.15), 2  n n     ⎪ ⎪ ⎪ ⎪2  k k  ⎪ ω (· − yε ) = lim vε 2ε − lim vε − ψε ⎪ωk ⎪ ⎪x k .  ε→0  ε→0 k=1

ε

k=1

(8.5.31)

8.5 ε-Dependent Concentration-Compactness Result

329

Note that  2 n     k k  ω (· − yε ) vε − ψε   k=1

= vε 2ε − 2

ε

n 

vε , ψε ωk (· − yεk )Hε +







ψε ωk (· − yεk ), ψε ωk (· − yεk )Hε .

k,k

k=1

(8.5.32) Let us verify that  vε , ψε ω

k

(· − yεk )Hε

→

RN

⎪ s ⎪ k⎪ ⎪2 |(−) 2 ωk |2 + V (x k )(ωk )2 dx = ⎪ ⎪ω ⎪ ⎪x k .

(8.5.33)

Indeed, vε , ψε ωk (· − yεk )Hε  (vε (x + yεk ) − vε (y + yεk ))(ψε (x + yεk ) − ψε (y + yεk )) k = ω (x) dxdy |x − y|N+2s R2N  (vε (x + yεk ) − vε (y + yεk ))(ωk (x) − ωk (y)) + ψε (y + yεk ) dxdy |x − y|N+2s R2N  + V (ε x + ε yεk )ψε (x + yεk )vε (x + yεk )ωk (x) dx RN

= Dε1 + Dε2 + Dε3 . Using Hölder’s inequality and the boundedness of vε (· + yεk ) we can argue as in the proof of (8.5.13) to see that Dε1 → 0. Concerning Dε2 , we observe that  Dε2

=

R2N

 +

[(vε (x + yεk ) − vε (y + yεk ))(ωk (x) − ωk (y))] dxdy |x − y|N+2s

R2N

(ψε (y + yεk ) − 1)(vε (x + yεk ) − vε (y + yεk ))(ωk (x) − ωk (y)) dxdy |x − y|N+2s

= Dε2,1 + Dε2,2 . Since vε (· + yεk )  ωk in H s (RN ), we obtain that Dε2,1 → [ωk ]2s . On the other hand, using Hölder’s inequality and the fact that vε (·+yεk ) is bounded in H s (RN ), the dominated

330

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

convergence theorem implies that   |Dε2,2 | Since Dε3 → can obtain

≤C

 RN

R2N

|(ψε (x + yεk ) − 1)(ωk (x) − ωk (y))|2 dxdy |x − y|N+2s

ψε ω

→ 0.

V (x k )(ωk )2 dx, we deduce that (8.5.33) holds. In a similar fashion, we 

k

 12



(· − yεk ), ψε ωk (· − yεk )Hε

→

0, if k = k

⎪ ⎪ ⎪ ⎪2 ⎪ ⎪ωk ⎪ ⎪x k , if k = k .

(8.5.34)

Combining (8.5.32), (8.5.33) and (8.5.34), we infer that (8.5.31) holds. Now, (8.5.31) yields that n  ⎪ ⎪ ⎪ ⎪2 ⎪ ⎪ωk ⎪ ⎪x k ≤ lim vε 2ε , ε→0

k=1

and using Lemma 8.4.3-(ii) and (8.5.4) we get δ12 n ≤ lim vε 2ε ≤ m2 . ε→0

Therefore, the procedure to find (yεk ), x k , ωk can not be iterated infinitely many times, so there exist l ∈ N ∪ {0}, (yεk ), x k , ωk such that (8.5.5)–(8.5.8) hold. Clearly, (8.5.9) follows in a standard way by (8.5.5)–(8.5.8).   In the next lemma we investigate the behavior of cε as ε → 0. Lemma 8.5.4 Let (cε )ε∈(0,ε1] be the mountain pass value of Jε defined in (8.3.4)–(8.3.5). Then cε → m(0) = inf m(x) x∈RN

as ε → 0.

Proof By Lemma 8.4.2, we can find a path γ ∈ C([0, 1], H s (RN )) such that γ (0) = 0, 0 (γ (1)) < 0, 0 (γ (t)) ≤ m(0) for all t ∈ [0, 1], and max 0 (γ (t)) = m(0).

t ∈[0,1]

8.5 ε-Dependent Concentration-Compactness Result

331

Take ϕ ∈ Cc∞ (RN ) such that ϕ(0) = 1 and ϕ ≥ 0, and set x

γR (t)(x) = ϕ γ (t)(x). R Then, it is easy to check that γR (t) ∈ C([0, 1], Hε ), γR (0) = 0 and 0 (γR (1)) < 0 for any R > 1 sufficiently large. Therefore, γR (t) ∈ ε . Now, fixed R > 0, we can see that maxt ∈[0,1] |Jε (γR (t)) − 0 (γR (t))| → 0 as ε → 0. Hence, for any R > 1 large enough, we get cε ≤ max Jε (γR (t)) → max 0 (γR (t)) t ∈[0,1]

t ∈[0,1]

as ε → 0.

On the other hand max 0 (γR (t)) → m(0)

t ∈[0,1]

as R → ∞,

so we deduce that lim supε→0 cε ≤ m(0). To complete the proof, we show that lim infε→0 cε ≥ m(0). Let vε ∈ Hε be a critical point of Jε (v) associated with cε . By Lemma 8.5.1, there exist εj → 0, l ∈ N ∪ {0}, (yεkj ) ⊂ RN , x k ∈ , ωk ∈ H s (RN ) \ {0} (k = 1, . . . , l) satisfying (8.5.5)–(8.5.9). If, by contradiction, l = 0, then (8.5.9) yields cεj = Jεj (vεj ) → 0, which contradicts Corollary 8.3.2. Consequently, l ≥ 1 and using (8.5.9) and Lemma 8.4.4, we have lim inf cεj = j →∞

l 

x k (ωk ) ≥

k=1

l 

m(x k ) ≥ lm(0) ≥ m(0).

k=1

  In view of Lemma 8.5.4, we deduce the following result. Lemma 8.5.5 For any ε ∈ (0, ε1 ], let vε denote a critical point of Jε corresponding to cε . Then for any sequence εj → 0 we can find a subsequence, still denoted by εj , and yεj , x 1 , ω1 such that εj yεj → x 1 ,

(8.5.35)

x 1 ∈  : V (x 1 ) = inf V (x),

(8.5.36)

ω1 (x) is a least energy solution of  x 1 (v) = 0,

(8.5.37)

vεj − ψεj w1 (· − yεj )εj → 0,

(8.5.38)

Jεj (vεj ) → m(x 1 ) = m(0).

(8.5.39)

x∈

332

8.6

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Proof of Theorem 8.1.1

In this last section we provide the proof of Theorem 8.1.1. Corollary 8.3.5 shows that there exists ε1 ∈ (0, ε0 ] such that for any ε ∈ (0, ε1 ], there exists a critical point vε ∈ Hε of Jε satisfying Jε (vε ) = cε . Then, by Lemma 8.5.5, we know that for any sequence εj → 0, there exists a subsequence εj and (yεj ) ⊂ RN , x 1 ∈  , ω1 ∈ H s (RN ) \ {0} 0,α satisfying (8.5.35)–(8.5.39). Note that vεj ∈ L∞ (RN )∩Cloc (RN ) (see Lemma 7.2.9), and by Theorem 1.3.5 we have vεj > 0 in RN . In view of (8.3.2) and (8.5.38) we obtain vεj − ψεj ω1 (· − yεj )H s (RN ) → 0.

(8.6.1)

We also note that (8.5.31) and (8.6.1) yield ⎪ ⎪2 ⎪ 1⎪ lim vεj 2Hεs = ⎪ ⎪x 1 = 0. ⎪ω ⎪

j →∞

(8.6.2)

j

Let v˜εj (x) = vεj (x + yεj ). Arguing as in the proof of (8.5.13), and using that ψ(x 1 ) = 1, (8.5.35), and the dominated convergence theorem, we see that [ψεj (· + yεj )ω1 − ω1 ]2s  |ψεj (x + yεj ) − ψεj (y + yεj )|2 1 ≤2 (ω (x))2 dxdy |x − y|N+2s R2N  |ψεj (y + yεj ) − 1|2 1 +2 |ω (x) − ω1 (y)|2 dxdy → 0. |x − y|N+2s R2N Clearly,  RN

|ψεj (x + yεj )ω1 − ω1 |2 dx → 0.

These two facts together with (8.6.1) imply that v˜εj − ω1 H s (RN ) → 0.

(8.6.3)

Arguing as in the proof of Lemma 7.2.9, we can find K > 0 such that v˜εj L∞ (RN ) ≤ K

for all j ∈ N,

(8.6.4)

8.6 Proof of Theorem 8.1.1

333

and (see also Remark 7.2.10) v˜εj (x) → 0

as |x| → ∞

(8.6.5)

uniformly in j ∈ N. Moreover, using interpolation in Lq spaces, we see that v˜εj → ω1 in Lq (RN ), for any q ∈ [2, ∞), hj (x) = g(εj x + εj yεj , v˜εj ) → f (ω1 ) in Lq (RN ), for any q ∈ [2, ∞).

(8.6.6)

Now let us prove that v˜εj is a solution to (8.1.1) for small εj > 0. Since εj yεj → x 1 ∈  , there exists r > 0 such that for some subsequence, still denoted by itself, we have Br (εj yεj ) ⊂ 

By setting  ε =



ε ,

for all j ∈ N.

we can see that B εr (yεj ) ⊂  εj j

for all j ∈ N

which yields RN \  εj ⊂ RN \ B εr (yεj ) j

for all j ∈ N.

In view of (8.6.5), there exists R > 0 such that v˜εj (x) < rν

for all |x| ≥ R, j ∈ N

so that vεj (x) = v˜εj (x − yεj ) < rν

for all x ∈ RN \ BR (yεj ), j ∈ N.

On the other hand, there exists j0 ∈ N such that RN \  εj ⊂ RN \ B εr (yεj ) ⊂ RN \ BR (yεj ) j

for all j ≥ j0 .

Hence, vεj (x) < rν

for all x ∈ RN \  εj , j ≥ j0 .

(8.6.7)

334

8 Fractional Schrödinger Equations with Superlinear or Asymptotically. . .

Now, up to a subsequence, we may assume that vεj L∞ (BR (yεj )) ≥ rν

for all j ≥ j0 .

(8.6.8)

Otherwise, if this is not the case, we have vεj L∞ (RN ) < rν , and taking into account the definition of g and our choice of rν , we get g(εj x, vεj )vεj = f (vεj )vεj ≤ νvε2j ≤

V0 2 v . 2 εj

Then, since Jε j (vεj ), vεj  = 0, we deduce that  vεj 2εj =

RN

f (vεj )vεj dx ≤

V0 2

 RN

vε2j dx

which implies that limj →∞ vεj 2εj = 0, and this is a contradiction in view of (8.6.2). Therefore, combining (8.6.7) and (8.6.8), we deduce that if zεj ∈ RN is a global maximum point of vεj then zεj belongs to BR (yεj ). Hence, zεj = yεj + z¯ εj for some z¯ εj ∈ BR . Recalling that the associated solution of our problem (8.1.1) is of the form uεj (x) = vεj ( εxj ), we conclude that xεj = εj yεj + εj z¯ εj is a global maximum point of uεj . Since (¯zεj ) ⊂ BR is bounded and εj yεj → x 1 ∈  , we obtain

lim V (xεj ) = V (x 1 ) = inf V (x).

j →∞

x∈

Therefore, we have proved that there exists ε0 > 0 such that for any ε ∈ (0, ε0 ], problem (8.1.1) admits a positive solution uε (x) = vε ( xε ) satisfying (1) of Theorem 8.1.1. We 0,α note that uε ∈ L∞ (RN ) ∩ Cloc (RN ). Finally, one can argue as at the end of the proof of Theorem 7.1.1 to deduce (2).  

9

Multiplicity and Concentration Results for a Fractional Choquard Equation

9.1

Introduction

This chapter deals with the following class of nonlinear fractional Choquard problems: 

ε2s (−)s u + V (x)u = εμ−N u ∈ H s (RN ),



1 |x|μ

∗ F (u) f (u) in RN ,

u > 0 in RN ,

(9.1.1)

where ε > 0 is a small parameter, s ∈ (0, 1), N > 2s and 0 < μ < 2s. The potential V : RN → R is a continuous function satisfying the following del Pino-Felmer type hypotheses [165]: (V1 ) infx∈RN V (x) = V0 > 0; (V2 ) there exists a bounded open set  ⊂ RN such that V0 < min V (x) and M = {x ∈  : V (x) = V0 } = ∅. x∈∂

Without loss of generality, we may assume that 0 ∈ M. Concerning the nonlinearity f : R → R, we assume that f is a continuous function such that f (t) = 0 for t < 0, and satisfies the following conditions: f (t) = 0; t →0 t

(f1 ) lim

(f2 ) there exists q ∈ (2,

2∗s μ 2 (2 − N ))

such that lim

f (t)

t →∞ t q−1

= 0;

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_9

335

336

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

(f3 ) the following Ambrosetti–Rabinowitz type condition [29] holds:  0 < 4F (t) = 4

t

f (τ ) dτ ≤ 2f (t) t for all t > 0;

0

(f4 ) The map t →

f (t) is increasing for every t > 0. t

We recall that the problem (9.1.1) is motivated by the search of standing wave solutions for the fractional Schrödinger equation ı



∂ψ = (−)s ψ + V (x)ψ − Is ∗ ||q ||q−2 ∂t

where Is (x) =

 π

N 2

N−s 2

(t, x) ∈ R × RN ,



2s  ( 2s )

|x|s−N is the Riesz kernel defined in Chap. 1.

When s = 1, V (x) ≡ 1, ε = 1, μ = 1 and F (u) = the so-called Choquard equation

|u|2 2 ,

equation in (9.1.1) becomes

− u + u = I2 ∗ |u|2 u in R3 ,

(9.1.2)

introduced at least as early as 1954, in a work by Pekar [289] describing the quantum mechanics of a polaron at rest. In 1976 Choquard used (9.1.2) to describe an electron trapped in its own hole, in a certain approximation to Hartree–Fock Theory of one-component plasma [248]. More recently, Penrose [290] used it as a model of self-gravitating matter. In this context (9.1.2) is usually called the nonlinear SchrödingerNewton equation. From a mathematical point of view, equation (9.1.2) and its generalizations have been widely investigated. The early existence and symmetry results are due to Lieb [248] and Lions [254]. Later, Ma and Zhao [266] classified all positive solutions to (9.1.2) and proved that they must be radially symmetric and monotonically decreasing about some fixed point. Moroz and Van Schaftingen [274] investigated existence, qualitative properties and decay asymptotics of positive ground state solutions for a generalized Choquard equation. Alves and Yang [25] studied multiplicity and concentration of positive solutions for a quasilinear Choquard equation. Further results on Choquard equations can be found in [1, 145, 275, 276, 339] and references therein. In the case s ∈ (0, 1), only few recent papers considered fractional Choquard equations like (9.1.1). In [156] d’Avenia et al. considered the fractional Choquard equation  (−)s u + u =

 1 p ∗ |u| |u|p−2 u in RN , |x|μ

9.1 Introduction

337

and proved regularity, existence and non-existence, symmetry and decay properties of solutions. Coti Zelati and Nolasco [153] established the existence of ground state solutions for a pseudo-relativistic Hartree-equation via critical point theory. Shen et al. [312] investigated the existence of ground state solutions for a fractional Choquard equation involving a nonlinearity that satisfies Berestycki–Lions type assumptions. Belchior et al. [93] dealt with existence, regularity and polynomial decay for a fractional Choquard equation involving the fractional p-Laplacian. In this chapter, we focus our attention on the multiplicity and the concentration of positive solutions of (9.1.1), involving a potential and a continuous nonlinearity satisfying the assumptions (V1 )–(V2) and (f1 )–(f4 ), respectively. In particular, we are interested in relating the number of positive solutions of (9.1.1) with the topology of the set M = {x ∈  : V (x) = V0 }. Our main result can be stated as follows: Theorem 9.1.1 ([46]) Suppose that V fulfills (V1 )–(V2 ), 0 < μ < 2s and f satisfies N : (f1 )–(f4 ) with 2 < q < 2(N−μ) N−2s . Then, for every δ > 0 such that Mδ = {x ∈ R dist(x, M) ≤ δ} ⊂ , there exists εδ > 0 such that, for any ε ∈ (0, εδ ), problem (9.1.1) has at least catMδ (M) positive solutions. Moreover, if uε denotes one of these positive solutions and xε ∈ RN is a global maximum point of uε , then lim V (xε ) = V0 .

ε→0

Firstly, we note that the restriction on the exponent q is justified by the Hardy-LittlewoodSobolev inequality (see Theorem 9.2.1 in Chap. 2). Indeed, if F (u) = |u|q , then the term  RN



 1 ∗ F (u) F (u) dx |x|μ

μ is well defined if F (u) ∈ Lt (RN ) for t > 1 such that 2t + N = 2. Hence, recalling that s N r N H (R ) is continuously embedded in L (R ) for any r ∈ [2, 2∗s ], we need to require that tq ∈ [2, 2∗s ] and this leads to assuming that

2−

2∗ μ

μ ≤q ≤ s 2− . N 2 N

Here we only consider the case q > 2. On the other hand, inspired by [25], we adapt the del Pino-Felmer penalization technique [165] by imposing further restrictions on the exponent q. Indeed, we treat the convolution |x|1μ ∗ F (u) as a bounded term and introduce the monotonicity condition on f at the same time. In this way, the assumptions 0 < μ < 2s and 2 < q < 2(N−μ) N−2s allow us to apply the penalization method. Differently from the case s = 1 considered in [25], in our setting a more accurate investigation is needed due to the presence of two nonlocal terms. Moreover, the

338

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

nonlinearity appearing in (9.1.1) is only continuous (while f ∈ C 1 in [25]), so to overcome the non-differentiability of the associated Nehari manifold, we will use the abstract critical point results in Sect. 2.4. Concerning the multiplicity result for the modified problem, our approach resembles some ideas in [95] based on the comparison between the category of some sublevel sets of the modified functional and the category of the set M. Finally, in order to prove that the solutions of the modified problem are solutions of the problem (9.1.1), we use a Moser iteration argument [278] and the boundedness of the convolution term to obtain the desired result. In Sect. 9.2 we introduce the functional setting and the modified problem. Section 9.3 is devoted to the existence of positive solutions to the autonomous problem associated to (9.1.1). In Sect. 9.4, we obtain a multiplicity result by using Lusternik–Schnirelman theory. Finally, by means of a Moser iteration scheme, we are able to prove that, for ε small enough, the solutions of the modified problem are indeed solutions of (9.1.1). We conclude this introduction by mentioning that a multiplicity result for a fractional p-Choquard equation with a global condition on the potential was established in [56].

9.2

Variational Framework

First, we recall the Hardy–Littlewood–Sobolev inequality, which will be frequently used along this section: μ + 1t = 2. Let Theorem 9.2.1 ([249]) Let r, t > 1 and 0 < μ < N such that 1r + N r N t N f ∈ L (R ) and h ∈ L (R ). Then there exists a sharp constant C(r, N, μ, t) > 0, independent of f and h, such that



 RN

RN

f (x)h(y) dxdy ≤ C(r, N, μ, t)f Lr (RN ) hLt (RN ) . |x − y|μ

For any ε > 0, we consider the fractional Sobolev space   s N Hε = u ∈ H (R ) :

 V (ε x)u dx < ∞ , 2

RN

which is a Hilbert space with the inner product  u, vε = u, vDs,2 (RN ) + and the associated norm uε =

RN

V (ε x)uv dx

√ u, uε , u ∈ Hε .

∀u, v ∈ Hε

9.2 Variational Framework

339

Using the change of variable u(x) → u(ε x), we see that (9.1.1) is equivalent to the following problem: 

(−)s u + V (ε x)u = u ∈ H s (RN ),



1 |x|μ

∗ F (u) f (u) in RN ,

(9.2.1)

u > 0 in RN .

Choose %0 > 0, to be determined later, and take a > 0 such that the functions ⎧ ⎨f (t), if t ≤ a, f˜(t) = ⎩ V0 t, if t > a,

f (a) a

=

V0 %0 .

We introduce

%0

and g(x, t) = χ (x)f (t) + (1 − χ (x))f˜(t), t where χ is the characteristic function on , and we write G(x, t) = 0 g(x, τ ) dτ . By assumptions (f1 )–(f4 ), the function g satisfies the following properties: g(x, t) = 0 uniformly in x ∈ RN ; t g(x, t) (g2 ) lim q−1 = 0 uniformly in x ∈ RN ; t →∞ t (g3 ) 0 < 4G(x, t) ≤ 2g(x, t)t for any x ∈  and t > 0, and 0 ≤ 2G(x, t) ≤ g(x, t)t ≤ V%00 t 2 for any x ∈ RN \  and t > 0;

(g1 ) lim

t →0

(g4 ) t → g(x, t) and t →

G(x,t ) t

are increasing for all x ∈ RN and t > 0.

Thus we consider the auxiliary problem 

(−)s u + V (ε x)u = u ∈ H s (RN ),



1 |x|μ

∗ G(ε x, u) g(ε x, u) in RN ,

u > 0 in RN ,

(9.2.2)

and we note that if u is a solution of (9.2.2) such that u(x) ≤ a for all x ∈ RN \ ε , where ε = {x ∈ RN : ε x ∈ }, then u solves (9.2.1).

(9.2.3)

340

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

To study weak solutions to (9.2.2), we look for critical points of the C 1 -functional Jε : Hε → R defined by Jε (u) =

1 u2ε − ε (u), 2

where ε (u) =

1 2



 RN

 1 ∗ G(ε x, u) G(ε x, u) dx. |x|μ

Lemma 9.2.2 Jε has a mountain pass geometry, that is (i) there exist α, ρ > 0 such that Jε (u) ≥ α for any u ∈ Hεs such that uε = ρ; (ii) there exists e ∈ Hε such that eε > ρ and Jε (e) < 0. Proof From (g1 ) and (g2 ) we deduce that that for every η > 0 there exists Cη > 0 such that |g(ε x, t)| ≤ η|t| + Cη |t|q−1

for (x, t) ∈ RN × R.

(9.2.4)

Using Theorem 9.2.1 and (9.2.4), we get    

 RN

   1 ∗ G(ε x, u) G(ε x, u) dx  ≤ CG(ε x, u)Lt (RN ) G(ε x, u)Lt (RN ) μ |x|  2 t 2 q t ≤C (|u| + |u| ) dx , (9.2.5) RN

where t = we have

2N 2N−μ .

Since 2 < q
2 we conclude that (i) holds. Now fix a non-negative function u0 ∈ H s (RN ) such that u ≡ 0 and supp(u0 ) ⊂ ε , and set  h(t) = ε

tu0 u0 ε

 for t > 0.

Since G(ε x, u0 ) = F (u0 ), using (f3 ) we deduce that  tu0 u0 u0 ε u0 ε       tu0 1 tu0 u0 = ∗F dx f μ N |x| u0 ε u0 ε u0 ε R       1 tu0 1 tu0 1 tu0 4 f ∗ F dx = t RN 2 |x|μ u0 ε 2 u0 ε u0 ε

h (t) = ε

>



4 h(t). t

(9.2.7)

Integrating (9.2.7) on [1, tu0 ε ] with t >

1 u0 ε ,

we obtain

h(tu0 ε ) ≥ h(1)(tu0 ε )4 , which gives  ε (tu0 ) ≥ ε

u0 u0 ε

 u0 4ε t 4 .

Therefore, Jε (tu0 ) =

t2 u0 2ε − ε (tu0 ) ≤ C1 t 2 − C2 t 4 2

for t >

Taking e = tu0 with t sufficiently large, we conclude that (ii) holds.

1 . u0 ε  

Lemma 9.2.2 and a variant of the mountain pass theorem without the (PS) condition establish the existence of a Palais–Smale sequence (un ) ⊂ Hε at the mountain pass level cε that can be also characterized as cε =

inf

max Jε (tu).

u∈Hε \{0} t >0

342

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Since supp(u0 ) ⊂ ε , there exists κ > 0 independent of ε, %, a such that cε < κ for small ε > 0. We now define B = {u ∈ H s (RN ) : u2ε ≤ 4(κ + 1)} and set 1 K˜ ε (u)(x) = ∗ G(ε x, u). |x|μ With the above notations, we are able to show the following estimate. Lemma 9.2.3 Assume that (f1 )–(f3 ) hold and 2 < q < such that supu∈B K˜ ε (u)L∞ (RN ) %0

≤

1 2

2(N−μ) N−2s . Then there exists %0

>0

for all ε > 0.

Proof We first show that there exists C0 > 0 such that sup K˜ ε (u)L∞ (RN ) ≤ C0

u∈B

for all ε > 0.

(9.2.8)

We observe that |G(ε x, u)| ≤ C(|u|2 + |u|q )

for all ε > 0.

(9.2.9)

Consequently,   ˜ |Kε (u)(x)| = 

RN

 G(ε x, u)  dy  |x − y|μ

   G(ε x, u)   G(ε x, u)  dy dy +    μ μ |x−y|≤1 |x − y| |x−y|>1 |x − y|   |u(y)|2 + |u(y)|q dy + C (|u|2 + |u|q ) dy ≤C |x − y|μ |x−y|≤1 RN  |u(y)|2 + |u(y)|q ≤C dy + C, (9.2.10) |x − y|μ |x−y|≤1

  ≤ 

where in the last line we used Theorem 1.1.8 and u2ε ≤ 4(κ + 1). Now, we take  t∈

N N , N − μ N − 2s



 and

r∈

 N 2N , . N − μ q(N − 2s)

9.2 Variational Framework

343

Hölder’s inequality, Theorem 1.1.8 and the fact that u2ε ≤ 4(κ + 1) imply that  |x−y|≤1

|u(y)|2 dy ≤ |x − y|μ

 |u| dy 2t

|x−y|≤1

 1  t |x−y|≤1

 ≤ C∗ (4(κ + 1)) because N − 1 −  |x−y|≤1

tμ t −1

1

2

tμ

|x − y| t−1  t−1 t

dρ

< ∞,

(9.2.11)

0

 |x−y|≤1

|u|rq dy

 1  r

 ≤ C∗ (4(κ + 1)) rμ r−1

t

dy

> −1. Similarly, we get

|u(y)|q dy ≤ |x − y|μ

because N − 1 −

tμ N−1− t−1

ρ

 t−1

1

|x−y|≤1 1

q

ρ

 r−1 r

1

rμ N−1− r−1

|x − y|  r−1 r

dρ

rμ r−1

dy

< ∞,

(9.2.12)

0

> −1. Combining (9.2.11) and (9.2.12) we have that

 |x−y|≤1

|u(y)|2 + |u(y)|q dy ≤ C |x − y|μ

for all x ∈ RN ,

which in view of (9.2.10) yields (9.2.8). Then there exists %0 > 0 such that supu∈B K˜ ε (u)L∞ (RN ) %0

≤

C0 1 ≤ %0 2

for all ε > 0.  

Let a > 0 be the unique number such that f (a) V0 = , a %0 where %0 is given in Lemma 9.2.3, and consider the penalized problem (9.2.2) with these choices. Let us introduce the Nehari manifold associated with (9.2.2), that is Nε = {u ∈ Hε \ {0} : Jε (u), u = 0}.

344

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Using Theorem 9.2.1 and (g1 )–(g2 ), we see that u2ε ≤ C(u4ε + u2q ε ), for all u ∈ Nε , so there exists r > 0, independent of ε > 0, such that uε ≥ r

for all u ∈ Nε .

(9.2.13)

Let Sε denote the unit sphere in Hε . Since f is only continuous, the next two results will play a fundamental role to overcome the non-differentiability of Nε . Lemma 9.2.4 Suppose that (V1 )–(V2 ) and (f1 )–(f4 ) are satisfied. Then, the following facts hold true: (a) For u ∈ Hε \ {0}, let hu : R+ → R be defined by hu (t) = Jε (tu). Then, there is a unique tu > 0 such that h u (t) > 0 in (0, tu ) and h u (t) < 0 in (tu , ∞). (b) There is a τ > 0, independent of u, such that tu ≥ τ for every u ∈ Sε . Moreover, for each compact set W ⊂ Sε , there is a CW > 0 such that tu ≤ CW for every u ∈ W. ˆ ε (u) = tu u is continuous and mε = m| ˆ Sε is (c) The map m ˆ ε : Hε \ {0} → Nε given by m u −1 a homeomorphism between Sε and Nε . Moreover, mε (u) = uε . Proof (a) From the proof of Lemma 9.2.2, we deduce that hu (0) = 0, hu (t) > 0 for t > 0 small and hu (t) < 0 for t large. Then, by the continuity of hu , it is easy to see that there exists tu > 0 such that maxt ≥0 hu (t) = hu (tu ), tu u ∈ Nε and hu v(tu ) = 0. Noting that 

 tu ∈ Nε ⇐⇒

u2ε

=

RN

 G(ε x, tu) 1 ∗ g(ε x, tu)u dx, |x|μ t

and using (g4 ), we get the uniqueness of a such tu . (b) Let u ∈ Sε . Recalling that h u (tu ) = 0, and using (g1 ), (g2 ), Theorem 9.2.1 (see estimates in Lemma 9.2.2), and Theorem 1.1.8, we get that for any small ξ > 0,  tu2 =

RN

2q K˜ ε (tu u)g(ε x, tu u)tu u dx ≤ ξ C1 tu4 + C2 Cξ tu .

Since q > 2, there exists a τ > 0, independent of u, such that tu ≥ τ . Now, by (g3 ), 1 Jε (v) = Jε (v) − Jε (v), v 4

9.2 Variational Framework

345

1 1 = v2ε − 4 4 ≥

1 v2ε 4

 RN

K˜ ε (u)[2G(ε x, u) − g(ε x, u)u] dx

for all v ∈ Nε .

(9.2.14)

Hence, if W ⊂ Sε is a compact set and (un ) ⊂ W is such that tun → ∞, it follows that un → u in Hεs and Jε (tun un ) → −∞. Taking vn = tun un ∈ Nε in (9.2.14), we get 0
0

u∈Sε t >0

In the next result we show that Jε verifies a local compactness condition. Lemma 9.2.8 Jε satisfies the (PS)c condition for all c ∈ [cε , κ]. Proof Let (un ) ⊂ Hε be a Palais–Smale sequence at the level c, that is Jε (un ) → c and Jε (un ) → 0. We divide the proof in two main steps. Step 1

For every η > 0 there exists R = Rη > 0 such that  lim sup

RN \BR

n→∞



s |(−) 2 un |2 + V (ε x)u2n dx < η.

(9.2.15)

Condition (g3 ) implies that 1 Jε (un ) − Jε (un ), un  = 4

≥



  1 1 1 − K˜ ε (un )g(ε x, un )un dx un 2Hεs + 2 4 4 RN  1 K˜ ε (un )G(ε x, un ) dx − 2 RN 1 un 2ε , 4

which shows that (un ) is bounded in Hε . Moreover, there exists n0 ∈ N such that un 2ε ≤ 4(κ + 1)

for all n ≥ n0 .

Then we may assume that un  u in Hε and un → u in Lrloc (RN ) for any r ∈ [1, 2∗s ). Moreover, by Lemma 9.2.3, we deduce that supn≥n0 K˜ ε (un )L∞ (RN ) %0

≤

1 . 2

(9.2.16)

9.2 Variational Framework

347

Fix R > 0 such that ε ⊂ BR/2 and take ψR ∈ C ∞ (RN ) such that ψR = 0 in BR/2 , ψR = 1 in BRc , 0 ≤ ψR ≤ 1 and ∇ψR L∞ (RN ) ≤ C/R for some C > 0 independent of R. Since the sequence (un ψR ) is bounded in Hε , condition (g3 ) implies that  RN

=



s s (−) 2 un (−) 2 (un ψR ) + V (ε x)ψR u2n dx

Jε (un ), un ψR  + 

≤ on (1) +

RN

 RN

K˜ ε (un )g(ε x, un )un ψR dx

K˜ ε (un ) V (ε x)u2n ψR dx, %0

whence  RN



s 2 2 2 |(−) un | ψR + V (ε x)un ψR dx ≤

RN

 −

R2N

K˜ ε (un ) V (ε x)u2n ψR dx + on (1) %0

(un (x) − un (u))(ψR (x) − ψR (y)) un (y) dxdy. |x − y|N+2s (9.2.17)

Further, the Hölder inequality, the boundedness of (un ) and Remark 1.4.6 imply that     

R2N

  (un (x) − un (u))(ψR (x) − ψR (y)) un (y) dxdy  N+2s |x − y|

  ≤

R2N

|un (x) − un (y)|2 dxdy |x − y|N+2s

  ≤C

R2N

 12   R2N

|ψR (x) − ψR (y)|2 2 un (y) dxdy |x − y|N+2s

 12

|ψR (x) − ψR (y)|2 2 un (y) dxdy |x − y|N+2s ≤

 12

C . Rs (9.2.18)

On the other hand, by (9.2.16), for all n ≥ n0  RN

 supn≥n0 K˜ ε (un )L∞ (RN ) K˜ ε (un ) V (ε x)u2n ψR dx ≤ V (ε x)u2n ψR dx N %0 % 0 R  1 ≤ V (ε x)u2n ψR dx. 2 RN (9.2.19)

348

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Putting together (9.2.17), (9.2.18), (9.2.19) and using the definition of ψR , we get  lim sup n→∞



s C |(−) 2 un |2 + V (ε x)u2n dx ≤ s , N R R \BR

and letting R→∞ we obtain (9.2.15). Step 2

Let us prove that un → u in Hε as n → ∞.

Set n = un − u2ε and observe that n = Jε (un ), un −Jε (un ), u+

 RN

K˜ ε (un )g(ε x, un )(un −u) dx +on (1).

(9.2.20)

Since Jε (un ), un  = Jε (un ), u = on (1), to infer that n → 0 as n → ∞ we only need to show that  K˜ ε (un )g(ε x, un )(un − u) dx = on (1). RN

2N

We note that G(ε x, un ) is bounded in L 2N−μ (RN ) (since 2 < q < in RN , and G(·, t) is continuous in t, so we deduce that G(ε x, un )  G(ε x, u)

2(N−μ) N−2s ),

2N

in L 2N−μ (RN ).

un → u a.e.

(9.2.21)

By virtue of Theorem 9.2.1, the convolution 2N 1 ∗ h(x) ∈ L μ (RN ), μ |x| 2N

gives a bounded linear operator from L 2N−μ (RN ) to L K˜ ε (un ) =

2N

h ∈ L 2N−μ (RN ) 2N μ

(RN ), which implies that

1 1 ∗ G(ε x, un )  ∗ G(ε x, u) = K˜ ε (u) μ |x| |x|μ

in L

2N μ

(RN ).

(9.2.22)

Since g has subcritical growth, for any fixed R > 0, the local compact embedding in Theorem 1.1.8 and (9.2.22) show that  lim

n→∞ B R

K˜ ε (un )g(ε x, un )(un − u) dx = 0.

9.2 Variational Framework

349

By (9.2.4) and the boundedness of K˜ ε (un ), we obtain  RN \BR

K˜ ε (un )|g(ε x, un )un | dx ≤ C

 

≤C

RN \BR

RN \BR

|g(ε x, un )un | dx (|un |2 + |un |q ) dx.

From Step 1, (V1 ), the interpolation inequality in Lq (RN \ BR ) with q ∈ (2, 2∗s ), and the ∗ boundedness of (un ) in L2s (RN ), we see that for every η > 0 there exists an R = Rη > 0 such that   |un |2 dx ≤ Cη and lim sup |un |q dx ≤ Cη, lim sup n→∞

RN \BR

n→∞

RN \BR

which implies that  lim sup n→∞

RN \BR

K˜ ε (un )|g(ε x, un )un | dx ≤ Cη.

On the other hand, by (9.2.4), the boundedness of K˜ ε (un ) and Hölder’s inequality we get  RN \BR

⎡  K˜ ε (un )|g(ε x, un )u| dx ≤ C ⎣

1 RN \BR

|un |2 dx

2

 +

 q−1 RN \BR

|un |q dx

q

⎤ ⎦,

whence  lim sup n→∞

K˜ ε (un )|g(ε x, un )u| dx ≤ C(η 2 + η 1

RN \BR

q−1 q

).

Taking into account the above limit inequalities, we infer that  lim

n→∞ RN

K˜ ε (un )g(ε x, un )(un − u) dx = 0.  

We also have the following result: Lemma 9.2.9 The functional ψε satisfies the (PS)c condition on Sε for any c ∈ [cε , κ]. Proof Let (un ) ⊂ Sε be a Palais–Smale sequence for ψε at the level c. Then ψε (un ) → c and ψε (un )∗ → 0, where  · ∗ denotes the norm in the dual space of (Tun Sε )∗ . By

350

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Proposition 9.2.6-(c), we infer that (mε (un )) is a Palais–Smale sequence for Jε at the level c. In view of Lemma 9.2.8, we see that, up to a subsequence, there exists u ∈ Sε such that   mε (un ) → mε (u) in Hε . By Lemma 9.2.4-(c), we conclude that un → u in Sε . Finally, we establish the existence of a ground state solution to (9.2.2). Theorem 9.2.10 For all ε > 0, problem (9.2.2) admits a positive ground state solution. Proof Combining Lemmas 9.2.2, 9.2.8, and Theorem 2.2.9, we can see that for every ε > 0 there exists uε ∈ Hε such that Jε (uε ) = cε and Jε (uε ) = 0. Since g(x, t) = 0 for t ≤ 0 and (x − y)(x − − y − ) ≥ |x − − y − |2 for all x, y ∈ R, it is readily verified N that Jε (uε ), u− ε  = 0 implies that uε ≥ 0 in R . Since uε ∈ B, we can proceed as in ˜ the proof of Lemma 9.2.3 to see that Kε (uε ) ∈ L∞ (RN ). Then, arguing as in the proof 0,α (RN ). Finally, using Theorem of Lemma 9.5.2 below, we have that uε ∈ L∞ (RN ) ∩ Cloc N   1.3.5, we deduce that uε > 0 in R .

9.3

The Autonomous Choquard Problem

In this section we deal with the limit problem associated with (9.2.1), namely 

(−)s u + V0 u =



1 |x|μ

∗ F (u) f (u) in RN ,

u ∈ H s (RN ), u > 0 in RN . The corresponding functional J0 : H0 → R associated is given by J0 (u) =

1 u20 − 0 (u), 2

where H0 is the space H s (RN ), endowed with the norm   u0 = [u]2s +

1 RN

V0 u2 dx

2

,

and 1 0 (u) = 2



 RN

 1 ∗ F (u) F (u) dx. |x|μ

Consider the corresponding Nehari manifold N0 = {u ∈ H0 \ {0} : J0 (u), u = 0}

(P0 )

9.3 The Autonomous Choquard Problem

351

and denote by S0 the unit sphere in H0 . Arguing as in the proofs of Lemma 9.2.4 and Proposition 9.2.6, we can see that the following results hold. Lemma 9.3.1 Suppose that f satisfies (f1 )–(f4 ). Then, the following assertions hold true: (a) For any u ∈ H0 \ {0}, let hu : R+ → R be defined by hu (t) = J0 (tu). Then, there is a unique tu > 0 such that h u (t) > 0 in (0, tu ) and h u (t) < 0 in (tu , ∞). (b) There is τ > 0, independent of u, such that tu ≥ τ for every u ∈ S0 . Moreover, for each compact set W ⊂ S0 , there is CW > 0 such that tu ≤ CW for every u ∈ W. ˆ 0 (u) = tu u is continuous and m0 = m| ˆ S0 is (c) The map m ˆ 0 : H0 \ {0} → N0 given by m −1 u a homeomorphism between S0 and N0 . Moreover, m0 (u) = u0 . Let us define the maps ψˆ 0 : H0 \ {0} → R by ψˆ 0 (u) = J0 (m ˆ 0 (u)), and ψ = ψˆ 0 |S0 . Proposition 9.3.2 Suppose that f satisfies (f1 )–(f4 ). Then, one has: (a) ψˆ 0 ∈ C 1 (H0 \ {0}, R) and ψˆ 0 (u), v =

m ˆ 0 (u)0

J0 (m ˆ 0 (u)), v , u0

for every u ∈ H0 \ {0} and v ∈ H0 . (b) ψ0 ∈ C 1 (S0 , R) and ψ0 (u), v = m0 (u)0 J0 (m0 (u)), v, for every v ∈ Tu S0 . (c) If (un ) is a Palais–Smale sequence for ψ0 , then (m0 (un )) is a Palais–Smale sequence for J0 . Moreover, if (un ) ⊂ N0 is a bounded Palais–Smale sequence for J0 , then (m−1 0 (un )) is a Palais–Smale sequence for ψ0 . (d) u is a critical point of ψ0 if and only if m0 (u) is a nontrivial critical point for J0 . Moreover, the corresponding critical values coincide and inf ψ0 (u) = inf J0 (u). u∈N0

u∈S0

Furthermore, we have the following characterization of the infimum of J0 on N0 : c0 = inf J0 (u) = u∈N0

inf

max J0 (tu) = inf max J0 (tu).

u∈H0 \{0} t >0

u∈S0 t >0

(9.3.1)

Arguing as in the proof of Lemma 9.2.2, we see that J0 has a mountain pass geometry. By the mountain pass theorem without the Palais–Smale condition (see Remark 2.2.10), there exists a Palais–Smale sequence (un ) ⊂ H0 such that J0 (un ) → c0

and

J0 (un ) → 0

in H0∗ .

352

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

The next lemma allows us to assume that the weak limit of a Palais–Smale sequence at the level c0 is nontrivial. Lemma 9.3.3 Let (un ) ⊂ H0 be a Palais–Smale sequence for J0 at the level c0 and such that un  0 in H0 . Then (a) either un → 0 in H0 , or (b) there exist a sequence (yn ) ⊂ RN , and constants R > 0 and γ > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ γ > 0.

Proof Suppose that (b) does not hold. Then, for all R > 0,  lim sup

n→∞

y∈RN

BR (y)

u2n dx = 0.

Since (un ) is bounded in H0 , Lemma 1.4.4 shows that un → 0 in Lr (RN ) for any r ∈ (2, 2∗s ). Then, by (f1 )–(f2 ) and applying Theorem 9.2.1, we get 

 RN

 1 ∗ F (u ) f (un )un dx = on (1). n |x|μ

Taking into account that J0 (un ), un  = on (1), we conclude that un 0 → 0 as n → ∞.   Now, we prove the following result for the autonomous problem. Lemma 9.3.4 Let (un ) ⊂ H0 be a Palais–Smale sequence for J0 at the level c0 . Then, problem (P0 ) has a positive ground state. Proof Since 1 1 J0 (un ) − J0 (un ), un  ≥ un 20 , 4 4 it is readily verified that (un ) is bounded in H0 . Moreover, arguing as in the proof of Lemma 9.3.3, we can find a sequence (yn ) ⊂ RN and constants R > 0 and γ > 0 such that  lim inf u2n dx ≥ γ > 0. n→∞

BR (yn )

9.3 The Autonomous Choquard Problem

353

Set vn = un (· − yn ). Since J0 and J0 are both invariant under translation, J0 (vn ) → c0 and J0 (vn ) → 0 in H0∗ . We observe that (vn ) is bounded in H0 , so we may assume that vn  v in H0 , for some v = 0. Let us show that v is a weak solution to (PV0 ). Fix ϕ ∈ Cc∞ (RN ). Since vn 0 ≤ C for all n ∈ N, we can argue as in the proof of Lemma 9.2.3 to deduce that    1    ∗ F (v ) n   | · |μ

L∞ (RN )

≤C

for any n ∈ N.

Then, since f is a continuous function with subcritical growth and vn → v in Lrloc (RN ) for any r ∈ [1, 2∗s ), the dominated convergence theorem gives 

 RN

    1 1 ∗ F (vn ) f (vn )ϕ dx → ∗ F (v) f (v)ϕ dx, μ |x|μ RN |x|

which combined with the weak convergence of (vn ) in H0 and the fact that J0 (vn ), ϕ = on (1), implies that J0 (v), ϕ = 0. Since Cc∞ (RN ) is dense in H0 , we get J0 (v), ϕ = 0 for all ϕ ∈ H0 . In particular, v ∈ NV0 . Using the definition of c0 together with Fatou’s lemma, we also deduce that J0 (v) = c0 . Now, recalling that f (t) = 0 for t ≤ 0 and (x − y)(x − − y − ) ≥ |x − − y − |2 for all x, y ∈ R, it is easy to deduce that J0 (v), v −  = 0 implies that v ≥ 0 in RN . Proceeding as in the proof of Lemma 9.2.3, we see that K(v)(x) = |x|1μ ∗ F (v) ∈ L∞ (RN ), and arguing as in the proof of Lemma 9.5.2 below, we have that v ∈ L∞ (RN ). Since f has subcritical growth and K(v) is bounded, we obtain that K(v)f (v) ∈ L∞ (RN ), so we can apply Proposition 1.3.2 to infer that v ∈ C 0,α (RN ) for some α ∈ (0, 1). Using Theorem 1.3.5 (or Proposition 1.3.11-(ii)), we finally conclude that   v > 0 in RN . Remark 9.3.5 As mentioned before, if u is the weak limit of a Palais–Smale sequence (un ) of J0 at the level c0 , then we can assume that u = 0. Otherwise, we would have un  0 in H0 and, assuming that un  0 in H0 , we conclude from Lemma 11.3.4 that there are (yn ) ⊂ RN and R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Set vn (x) = un (x + yn ). Then we see that (vn ) is a Palais–Smale sequence for J0 at the level c0 , (vn ) is bounded in H0 and there exists v ∈ H0 such that vn  v and v = 0.

354

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Remark 9.3.6 By using a Brezis-Lieb splitting type result for 0 (u)(see [1]), it is easy to see that if (un ) is a Palais–Smale sequence for J0 at the level c0 , and un  u in H0 , then J0 (un − u) = c0 − J0 (u) + on (1)

and

J0 (un − u) = on (1).

If one assumes that u = 0, then by Fatou’s lemma and (f3 ) we have     1

1

c0 = lim J0 (un ) − J0 (un ), un  = lim inf  (un ), un  − 0 (un ) n→∞ n→∞ 2 2 0   1

 (u), u − 0 (u) ≥ 2 0 1 = J0 (u) − J0 (u), u ≥ c0 , 2 and thus J0 (un − u) = on (1) and J0 (un − u) = on (1). Therefore,   1 1 lim sup un − u20 ≤ lim sup J0 (un − u) − J0 (un − u), un − u = 0, 4 n→∞ 4 n→∞ which implies that un →u in H0 as n→∞. In particular, we deduce that J0 satisfies the Palais–Smale condition at level c0 . The next statement is a compactness result for the autonomous problem which will be used later. Lemma 9.3.7 Let (v˜n ) ⊂ N0 be such that J0 (v˜n ) → c0 . Then (v˜n ) has a convergent subsequence in H0 . Proof Since (v˜n ) ⊂ N0 and JV0 (v˜n ) → c0 , we can apply Lemma 9.3.1-(c) and Proposition 9.3.2-(d) to infer that wn = m−1 0 (v˜n ) =

v˜n ∈ S0 v˜n 0

and ψ0 (wn ) = J0 (v˜n ) → c0 = inf ψ0 (v). v∈S0

Since ψ0 is a continuous functional bounded below, we can apply Theorem 2.2.1 to find (w˜ n ) ⊂ S0 such that (w˜ n ) is a Palais–Smale sequence for ψ0 at the level c0 and w˜ n − wn 0 = on (1). By Proposition 9.3.2-(c), (m0 (w˜ n )) is a Palais–Smale sequence for J0 at the level c0 . Taking into account Lemma 9.3.4 and Remark 9.3.6, it follows that there

9.4 Multiplicity Results

355

exists w˜ ∈ S0 such that m0 (w˜ n ) → m0 (w) ˜ in H0 . This fact together with Lemma 9.3.1-(c)   shows that v˜n → v˜ in H0 .

9.4

Multiplicity Results

In order to study the multiplicity of solutions to (9.1.1), we need to introduce some useful tools. Take δ > 0 such that Mδ ⊂ , where Mδ = {x ∈ RN : dist(x, M) ≤ δ}. Let η be a smooth nonincreasing cut-off function defined in [0, ∞), such that η = 1 in [0, 2δ ], η = 0 in [δ, ∞), 0 ≤ η ≤ 1 and |η | ≤ c for some c > 0. For any y ∈ M, set  ε,y (x) = η(| ε x − y|)w

εx −y ε

 ,

where w is a positive ground state solution for J0 (by Lemma 9.3.4). Let tε > 0 denote the unique positive number satisfying Jε (tε ε,y ) = max Jε (tε,y ). t ≥0

Finally, we consider ε (y) = tε ε,y . In the next lemma we prove an important relation between ε and the set M. Lemma 9.4.1 The functional ε has the property that lim Jε (ε (y)) = c0 ,

ε→0

uniformly in y ∈ M.

Proof Assume, by contradiction, that there exist δ0 > 0, (yn ) ⊂ M and εn → 0 such that |Jεn (εn (yn )) − c0 | ≥ δ0 .

(9.4.1)

We first show that limn→∞ tεn < ∞. Let us observe that, by using the change of variable n z = εn x−y εn , if z ∈ B εδn , it follows that ε n z ∈ Bδ and thus ε n z + yn ∈ Bδ (yn ) ⊂ Mδ ⊂ . Since G = F on  × R, we see that Jεn (εn (zn )) = +

tε2n



2 tε2n 2



s

RN

RN

|(−) 2 (η(| εn z|)w(z))|2 dz V (εn z + yn )(η(| εn z|)w(z))2 dz

− 0 (tεn η(| εn ·|)w).

356

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

In view of the dominated convergence theorem and Lemma 1.4.8, we obtain that lim εn ,yn εn = w0

n→∞

and lim 0 (εn ,yn ) = 0 (w).

n→∞

Using the fact that tεn εn ,yn ∈ Nεn and the growth assumptions on f , it is easy to prove that tεn → 1. Indeed, since  tε2n εn ,yn 2εn

=

R2N

F (tεn εn ,yn )f (tεn εn ,yn )tεn εn ,yn , |x − y|μ

(9.4.2)

we have that  w20 = lim

n→∞

R2N

F (tεn εn ,yn )f (tεn εn ,yn )tεn εn ,yn . tε2n |x − y|μ

Using that w is a ground state to (P0 ) and condition (f4 ), we conclude that tεn → 1. Consequently, lim 0 (tεn η(| εn ·|)w) = 0 (w)

n→∞

and this yields lim Jεn (εn (yn )) = J0 (w) = c0 ,

n→∞

 

which contradicts (9.4.1).

Now, take δ > 0 such that Mδ ⊂ , and choose ρ = ρ(δ) > 0 such that Mδ ⊂ Bδ . Define ϒ : RN → RN by setting ϒ(x) = x for |x| ≤ ρ and ϒ(x) = ρx |x| for |x| ≥ ρ. Then we N consider the barycenter map βε : Nε → R given by  βε (u) =

2 RNϒ(ε x)u (x) dx . 2 RN u (x) dx

Arguing as in the proof of Lemma 6.3.18 we obtain.

9.4 Multiplicity Results

357

Lemma 9.4.2 The function βε has the property that lim βε (ε (y)) = y,

ε→0

uniformly in y ∈ M.

The next compactness result will play a fundamental role in showing that the solutions of the modified problem are solutions of the original problem. Lemma 9.4.3 Let εn → 0 and (un ) = (uεn ) ⊂ Nεn be such that Jεn (un ) → c0 . Then there exists (y˜n ) = (y˜εn ) ⊂ RN such that vn (x) = un (x + y˜n ) has a convergent subsequence in H s (RN ). Moreover, up to a subsequence, yn = εn y˜n → y0 ∈ M. Proof Since Jε n (un ), un  = 0 and Jεn (un ) → c0 , we deduce that (un ) is bounded in Hεn . Note that c0 > 0, and since un εn → 0 would imply Jεn (un ) → 0, we can argue as in Lemma 9.3.3 to obtain a sequence (y˜n ) ⊂ RN and two constants R, γ > 0 such that  lim inf n→∞

BR (y˜n )

u2n dx ≥ γ > 0.

(9.4.3)

Now set vn (x) = un (x + y˜n ). Then the sequence (vn ) is bounded in H0 , and we may assume that vn  v ≡ 0 in H0 as n → ∞. Fix tn > 0 such that v˜n = tn vn ∈ N0 . Since un ∈ Nεn , we see that c0 ≤ J0 (v˜n ) = J0 (tn un ) ≤ Jεn (tn un ) ≤ Jεn (un ) = c0 + on (1), and so J0 (v˜n ) → c0 . In particular, we get v˜n  v˜ in H0 and tn → t ∗ > 0. Then, by the uniqueness of the weak limit, we have v˜ = t ∗ v ≡ 0. Using Lemma 9.3.7, we obtain that v˜n → v˜

in H0 .

(9.4.4)

To complete the proof of the lemma, we consider yn = εn y˜n and show that (yn ) admits a subsequence, still denoted by (yn ), such that yn → y0 for some y0 ∈ M. First, we prove that (yn ) is bounded in RN . We argue by contradiction and assume that, up to a subsequence, |yn | → ∞ as n → ∞. Since Jεn (un )→c0 and Jε n (un ), un  = 0, we see that un ∈ B for all n big enough. Then, in view of Lemma 9.2.3, there exists C0 ∈ (0, %20 ] such that for n large enough K˜ εn (un )L∞ (RN ) ≤ C0 .

358

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Fix R > 0 such that  ⊂ BR . Then, for n large enough, we may assume that |yn | > 2R, and for all z ∈ B R we have εn

| εn z + yn | ≥ |yn | − | εn z| > R.

(9.4.5)

Using the change of variable x → z + y˜n and (9.4.5), we deduce that for n large enough  [vn ]2s +

 RN

V0 vn2 dx ≤ C0 

RN

g(ε n z + yn , vn )vn dz f˜(vn )vn dx + C0

≤ C0 B



R εn

RN \B

f (vn )vn dx. R εn

Since vn → v in H0 as n → ∞, the dominated convergence theorem implies that  RN \B R

f (vn )vn dx = on (1),

εn

which combined with the inequality f˜(t) ≤

V0 %0 t

shows that

    1 [vn ]2s + V0 vn2 dx ≤ C0 f (vn )vn dx = on (1), 2 RN RN \B R εn

which contradicts the fact that v ≡ 0. Therefore, (yn ) is bounded, and we may assume that yn → y0 ∈ RN . Clearly, if y0 ∈ / , then we can argue as before to deduce that vn → 0 in H0 , which is impossible. Hence, y0 ∈ . Now, note that if V (y0 ) = V0 , then / ∂ in view of (V2 ), and consequently y0 ∈ M. We claim that V (y0 ) = V0 . Suppose, y0 ∈ by contradiction, that V (y0 ) > V0 . Then, using that v˜n → v˜ in H0 , Fatou’s lemma and the invariance of RN by translation, we get    1 V (y0 )v˜ 2 dx − 0 (v) ˜ [v] ˜ 2s + 2 RN    1 1 2 2 V (εn x + yn )v˜n dx − 0 (v˜n ) ≤ lim inf [v˜n ]s + n→∞ 2 2 RN

˜ < c0 = J0 (v)

≤ lim inf Jεn (tn un ) ≤ lim inf Jεn (un ) = c0 , n→∞

which gives a contradiction.

n→∞

 

9.5 Proof of Theorem 9.1.1

359

Define ε = {u ∈ Nε : Jε (u) ≤ c0 + h(ε)}, N where h(ε) = supy∈M |Jε (ε (y)) − c0 |. By Lemma 9.4.1, h(ε)→0 as ε →0. From the ε and so N ε = ∅. definition of h(ε) we deduce that, for all y ∈ M and ε > 0, ε (y) ∈ N Moreover, arguing as in the proof of Lemma 6.3.19, we obtain the following result. Lemma 9.4.4 lim sup dist(βε (u), Mδ ) = 0.

ε→0

9.5

u∈N˜ ε

Proof of Theorem 9.1.1

This last section is devoted to the proof of the main result of this chapter. First, arguing as in the proof of Theorem 6.3.22, we obtain the following multiplicity result for the modified problem (9.2.2). Theorem 9.5.1 Assume that (V1 )–(V2 ) and (f1 )–(f4 ) hold. Then, for every δ > 0 such that Mδ ⊂ , there exists ε¯ δ > 0 such that, for any ε ∈ (0, ε¯ δ ), problem (9.2.2) has at least catMδ (M) positive solutions. Now we show that the solutions obtained in Theorem 9.5.1 satisfy the estimate uε < a in cε for ε > 0 small enough. The following result will be of crucial importance in the study of the behavior of the maximum points of solutions to (9.1.1). Lemma 9.5.2 Let (vn ) be the sequence provided by Lemma 9.4.3. Then vn satisfies the following problem: 

(−)s vn + Vn (x)vn =



1 |x|μ

∗ Gn (x, vn ) gn (x, vn ) in RN ,

vn ∈ H s (RN ), vn > 0

in RN ,

(9.5.1)

where Vn (x) = V (εn x + εn y˜n ) and gn (x, vn ) = g(ε n x + εn y˜n , vn ), vn ∈ L∞ (RN ) and there exists C > 0 such that vn L∞ (RN ) ≤ C for all n ∈ N. Furthermore, lim vn (x) = 0 uniformly in n ∈ N.

|x|→∞

360

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Proof Mimicking the proof of Lemma 6.3.23 we obtain that  2  1 β−1 2(β−1) S∗ vn vL,n 2 2∗s N + Vn (x)vn2 vL,n dx L (R ) β RN  (vn (x) − vn (y)) 2(β−1) 2(β−1) ≤ ((vn vL,n )(x) − (vn vL,n )(y)) dxdy N+2s 2N |x − y| R  2(β−1) + Vn (x)vn2 vL,n dx RN

 =

RN



(9.5.2)

 1 2(β−1) ∗ Gn (x, vn ) gn (x, vn )vn vL,n dx. |x|μ

On the other hand, thanks to the boundedness of (vn ) there exists a constant C0 > 0 such that    1   (9.5.3) sup  μ ∗ Gn (·, vn )  ∞ N ≤ C0 . n∈N | · | L (R ) By assumptions (g1 ) and (g2 ), for every ξ > 0 there exists Cξ > 0 such that ∗

|gn (x, vn )| ≤ ξ |vn | + Cξ |vn |2s −1 .

(9.5.4)

Taking ξ sufficiently small and using (9.5.3) and (9.5.4), we see that (9.5.2) yields 

β−1

vn vL,n 2 2∗s

L (RN )

≤ Cβ 2

∗

RN

2(β−1)

|vn |2s vL,n

dx.

Now we can argue as in the proof of Lemma 6.3.23 to deduce the L∞ -estimate vn L∞ (RN ) ≤ C for all n ∈ N. We observe that, if wn (x, y) = Ext (vn ) = Ps (x, y) ∗ vn (x) is the s-harmonic extension of vn , then wn solves ⎧ 1−2s ∇w ) = 0 ⎪ n ⎨ − div(y wn (·, 0) = vn ⎪ ⎩ ∂w = −V (x)v + ∂ν 1−2s

n

n

in RN+1 + , on ∂RN+1 + ,

1 |x|μ

∗ Gn (x, vn ) gn (x, vn ) on ∂RN+1 + .

In view of (9.5.3), (9.5.4) and the above L∞ -estimate, we can repeat the same arguments as in Lemma 6.3.23 to show that vn (x)→0 as |x|→∞ uniformly in n ∈ N (see also Remark 7.2.10 and note that vn is a subsolution to (−)s vn +V0 vn = C0 gn (x, vn ) in RN ).   At this point, we are ready to give the proof of our main result.

9.5 Proof of Theorem 9.1.1

361

Proof of Theorem 9.1.1 Take δ > 0 such that Mδ ⊂ . We begin by proving that there ε of (9.2.2), it holds exists ε˜ δ > 0 such that for any ε ∈ (0, ε˜ δ ) and any solution uε ∈ N uε L∞ (RN \ε ) < a.

(9.5.5)

εn such that Jε (un ) = 0 Assume, by contradiction, that there exist εn → 0, un = uεn ∈ N n and un L∞ (RN \εn ) ≥ a. Since Jεn (un ) ≤ c0 + h(εn ) and h(εn ) → 0, we can argue as in the first part of the proof of Lemma 9.4.3 to deduce that Jεn (un ) → c0 . Then, by using Lemma 9.4.3, we can find (y˜n ) ⊂ RN such that vn (x) = un (x + y˜n )→v converges strongly in H s (RN ) and yn = εn y˜n → y0 ∈ M. If we take r > 0 such that Br (y0 ) ⊂ B2r (y0 ) ⊂ , then B εr ( εyn0 ) ⊂ εn . In particular, n for any y ∈ B εr (y˜n ), n

        y − y0  ≤ |y − y˜n | + y˜n − y0  < 2r    εn εn  εn

for n sufficiently large.

Therefore, for these values of n, RN \ εn ⊂ RN \ B εr (y˜n ). n

Now, by Lemma 9.5.2, we obtain that |vn (x)| → 0

as |x| → ∞, uniformly in n ∈ N,

which implies that there exists R > 0 such that vn (x) < a for |x| ≥ R and n ∈ N. Hence, un (x) < a for any x ∈ RN \ BR (y˜n ), n ∈ N. On the other hand, there exists ν ∈ N such that, for any n ≥ ν and r/ εn > R, RN \ εn ⊂ RN \ B εr (y˜n ) ⊂ RN \ BR (y˜n ), n

which gives un (x) < a for any x ∈ RN \ εn and n ≥ ν, which is impossible. Let ε¯ δ be as given in Theorem 9.5.1 and take εδ = min{˜εδ , ε¯ δ }. Fix ε ∈ (0, εδ ). By Theorem 9.5.1, problem (9.2.2) admits at least catMδ (M) nontrivial solutions uε . Since uε ∈ N˜ ε satisfies (9.5.5), by the definition of g it follows that uε is a solution of (9.2.1). Consequently, for all ε ∈ (0, εδ ), uˆ ε (x) = uε ( xε ) is a solution of (9.1.1). Now, consider a sequence εn →0 and a corresponding sequence (un ) ⊂ Hεn of solutions to (9.2.2) as above. Let us observe that (g1 ) implies that we can find γ ∈ (0, a) such that g(ε x, t)t ≤

V0 2 t %0

for any x ∈ RN , 0 ≤ t ≤ γ .

(9.5.6)

362

9 Multiplicity and Concentration Results for a Fractional Choquard Equation

Arguing as before, we can find R > 0 such that un L∞ (BRc (y˜n )) < γ .

(9.5.7)

Moreover, up to extracting a subsequence, we may assume that un L∞ (BR (y˜n )) ≥ γ .

(9.5.8)

Indeed, if (9.5.8) does not hold, then, by (9.5.7), we have un L∞ (RN ) < γ . Then, using that Jε n (un ), un  = 0, (9.5.6) and that for all n sufficiently large K˜ εn (un )L∞ (RN ) ≤ C0 (since un ∈ B for all n sufficiently large), we infer that  un 2εn

≤ C0

RN

C0 g(εn x, un )un dx ≤ %0

 RN

V0 u2n dx

which together with C%00 ≤ 12 yields un εn = 0, which is a contradiction. Consequently, (9.5.8) holds. Taking into account (9.5.7) and (9.5.8), we deduce that if pn ∈ RN is a global maximum point of un , then pn ∈ BR (y˜n ). Therefore, pn = y˜n + qn for some qn ∈ BR . Hence, ηn = εn y˜n + εn qn is a global maximum point of uˆ n (x) = un (x/ εn ). Since |qn | < R for any n ∈ N and εn y˜n → y0 ∈ M, it follows from the continuity of V that lim V (ηn ) = V (y0 ) = V0 .

n→∞

 

This ends the proof of Theorem 9.1.1.

Remark 9.5.3 Proceeding as in the proof of Theorem 7.1.2, we can show that if uε is a positive solution of (9.1.1) and xε is a global maximum point of uε , then there exists a constant C > 0 such that 0 < uε (x) ≤

C εN+2s εN+2s +|x − xε |N+2s

for all x ∈ RN .

Indeed, it is enough to observe that K˜ ε (uε )L∞ (RN ) ≤ C0 , and use (g1 ) to repeat the same arguments in the proof of Theorem 7.1.2 to deduce the desired decay estimate.

A Multiplicity Result for a Fractional Kirchhoff Equation with a General Nonlinearity

10.1

10

Introduction

In this paper we study the multiplicity of weak solutions to the nonlinear fractional Kirchhoff equation   p + q(1 − s)

R2N

 |u(x) − u(y)|2 dxdy (−)s u = g(u) |x − y|N+2s

in RN ,

(10.1.1)

where s ∈ (0, 1), N ≥ 2, p > 0, q is a small positive parameter and g is a nonlinearity which satisfies suitable assumptions. When s → 1− in (10.1.1), from the celebrated Bourgain–Brezis–Mironescu formula [108], we know that  lim (1 − s)

s→1−

R2N

ωN−1 |u(x) − u(y)|2 dxdy = |x − y|N+2s 2N

 RN

|∇u|2 dx

for all u ∈ H 1 (RN ),

and thus (10.1.1) becomes a Kirchhoff equation of the form   − p+q

RN

 |∇u(x)|2 dx u = g(u) in RN ,

(10.1.2)

which has been extensively studied in the last decade. Equation (10.1.2) is related to the stationary analogue of the Kirchhoff equation    2 ut t − p + q |∇u(x)| dx u = g(x, u)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_10

363

364

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General. . .

with ⊂ RN bounded domain, which was proposed by Kirchhoff in 1883 [240] as an extension of the classical D’Alembert’s wave equation  ρ ut t −

P0 E + h 2L



L 0

 2

|ux | dx u2xx = g(x, u)

(10.1.3)

governing the free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Here u = u(x, t) is the transverse string displacement at the space coordinate x and time t, L is the length of the string, h is the area of the cross section, E is the Young modulus of the material, ρ is the mass density and P0 is the initial tension. We also note that nonlocal boundary value problems like (10.1.2) model several physical and biological systems where u describes a process which depends on the average of itself, as for example, the population density; see [9,142]. The early classical studies devoted to Kirchhoff equations were carried out by Bernstein [103] and Pohozaev [294]. However, equation (10.1.2) received much attention only after the paper by Lions [253], where a functional analysis approach was proposed to attack it. For more recent results concerning Kirchhoff-type equations we refer to [8, 9, 79, 140, 187, 191, 291, 331]. On the other hand, great attention has been recently given to the study of nonlinear fractional Kirchhoff problems. In [196], Fiscella and Valdinoci proposed a stationary fractional Kirchhoff variational model in a bounded domain ⊂ RN with homogeneous Dirichlet boundary conditions and involving a critical nonlinearity: 

M

 RN

u=0

s ∗ |(−) 2 u|2 dx (−)s u = λf (x, u) + |u|2s −2 u in , in RN \ ,

where M is a continuous Kirchhoff function whose model case is given by M(t) = a + bt. The authors gave an interesting interpretation of Kirchhoff’s equation in the fractional scenario. In their correction of the early (one-dimensional) model, the tension on the string, which classically has a “nonlocal" nature arising from the average of the kinetic energy |ux |2 s 2 on [0, L], possesses a further nonlocal behavior governed by the H -norm (or other more general fractional norms) of the function u. Nyamoradi [283] established the existence of at least three solutions for hemivariational inequalities driven by nonlocal integro-differential operators. Pucci and Saldi in [295] studied the existence and multiplicity of nontrivial non-negative entire solutions for a Kirchhoff-type eigenvalue problem in RN involving a critical nonlinearity and the fractional Laplacian. Further results can be found in [55, 62, 72, 73, 195, 262, 273, 296].

10.2 The Truncated Problem

365

In this chapter we study the multiplicity of weak solutions to (10.1.1) with q a small parameter and g a general subcritical nonlinearity. More precisely, we assume that g : R → R satisfies Berestycki–Lions type conditions [100, 101], that is: (g1 ) g ∈ C 1,α (R, R), with α > max{0, 1 − 2s}, and odd; g(t) g(t) ≤ lim sup = −m < 0; (g2 ) −∞ < lim inf t →0 t t t →0 |g(t)| (g3 ) lim = 0; ∗ t →±∞ |t|2s −1  ζ

(g4 ) there exists ζ > 0 such that G(ζ ) =

g(t) dt > 0. 0

Let us recall that when q = 0 and p = 1 in (10.1.1), the papers [39, 139] established the existence and multiplicity of radially symmetric solutions to the fractional scalar field problem (−)s u = g(u)

in RN .

(10.1.4)

Now, we aim to study a generalization of (10.1.4), and we look for weak solutions to (10.1.1) with q > 0 sufficiently small. Our main result is the following: Theorem 10.1.1 ([72]) Assume that (g1 ), (g2 ), (g3 ) and (g4 ) hold. Then, for every h ∈ N there exists q(h) > 0 such that for any 0 < q < q(h) Eq. (10.1.1) admits at least h couples of solutions in H s (RN ) with radial symmetry. The proof of the above result is inspired by [83] and combines the mountain pass approach introduced in [218] with the truncation argument of [232].

10.2

The Truncated Problem

In this section we provide an abstract multiplicity result which allows us to prove Theorem 10.1.1. Let us introduce the following functional, defined for u ∈ H s (RN ): 1 Fq (u) = [u]2s + qR(u) − 2

 RN

G(u) dx,

where q > 0 is a small parameter and R : H s (RN ) → R. We suppose that R=

k  i=1

Ri

(10.2.1)

366

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General. . .

and, for each i = 1, . . . , k the functional Ri satisfies (R1 ) Ri ∈ C 1 (H s (RN ), R) is non-negative and even; (R2 ) there exists δi > 0 such that R i (u), u ≤ CuδHi s (RN ) for all u ∈ H s (RN ); (R3 ) if (uj ) ⊂ H s (RN ) converges weakly to u ∈ H s (RN ), then lim sup R i (uj ), u − uj  ≤ 0; j →∞

(R4 ) there exist αi , βi ≥ 0 such that if u ∈ H s (RN ), t > 0 and ut = u

· t , then

Ri (ut ) = t αi Ri (t βi u); (R5 ) Ri is invariant under the action of the N-dimensional orthogonal group, i.e., Ri (u(g·)) = Ri (u(·)) for every g ∈ O(N).  Let us observe that for any u ∈

H s (RN ),

1

Ri (u) − Ri (0) = 0

assumption (R2 ) we have Ri (u) ≤ C1 + C2 uδHi s (RN ) .

d Ri (tu) dt, so by dt

(10.2.2)

The main result of this section is Theorem 10.2.1 Suppose that (g1 )–(g4 ) and (R1 )–(R5 ) hold. Then, for every h ∈ N there exists q(h) > 0 such that for any 0 < q < q(h) the functional Fq admits at least h s (RN ). couples of critical points in Hrad Let us introduce, for any t ≥ 0, the functions g1 (t) = (g(t) + mt)+ , g2 (t) = g1 (t) − g(t), and extend them as odd functions for t ≤ 0. Observing that g1 (t) = 0, t g1 (t) lim ∗ = 0, t →∞ t 2s −1 lim

t →0

g2 (t) ≥ mt

∀t ≥ 0,

(10.2.3) (10.2.4) (10.2.5)

10.2 The Truncated Problem

367

we deduce that for any ε > 0 there exists Cε > 0 such that ∗

g1 (t) ≤ Cε t 2s −1 + εg2 (t)

∀t ≥ 0.

(10.2.6)

Setting 

t

Gi (t) =

gi (τ ) dτ

i = 1, 2,

0

(10.2.5) immediately implies that m 2 t 2

G2 (t) ≥

∀t ∈ R,

(10.2.7)

and, by (10.2.6) we can see that for every ε > 0 there exists Cε > 0 such that ∗

G1 (t) ≤ Cε |t|2s + ε G2 (t)

∀t ∈ R.

(10.2.8)

In view of (R5 ), all functionals that we will consider below are invariant under rotations, s (RN ). so, from now on we will directly define our functionals in Hrad ∞ Following [232], let χ ∈ C ([0, ∞), R) be a cut-off function such that ⎧ ⎪ χ(t) = 1, for t ∈ [0, 1], ⎪ ⎪ ⎨ 0 ≤ χ(t) ≤ 1, for t ∈ (1, 2), ⎪ χ(t) = 0, for t ∈ [2, ∞), ⎪ ⎪ ⎩

χ L∞ (0,∞) ≤ 2, and we set  ξ (u) = χ

u2H s (RN )

 .

2

s (RN ) → R defined as Then we introduce the truncated functional Fq : Hrad

Fq (u) =

1 2 [u] + q ξ (u)R(u) − 2 s

 RN

G(u) dx.

Clearly, a critical point u of Fq with uH s (RN ) ≤  is a critical point of Fq .

368

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General. . .

Our first aim is to prove that the truncated functional Fq has a symmetric mountain pass geometry: Lemma 10.2.2 There exist r0 > 0 and ρ0 > 0 such that Fq (u) ≥ 0,

for uH s (RN ) ≤ r0 ,

Fq (u) ≥ ρ0 ,

(10.2.9)

for uH s (RN ) = r0 .

Moreover, for any n ∈ N there exists an odd continuous map s γn : Sn−1 −→ Hrad (RN )

such that Fq (γn (σ )) < 0

for all σ ∈ Sn−1 ,

(10.2.10)

where Sn−1 = {σ = (σ1 , . . . , σn ) ∈ Rn : |σ | = 1}. Proof Taking ε = rem 1.1.8, we have Fq (u) =

1 2

in (10.2.8), and using (10.2.7), the positivity of R, and Theo-

1 2 [u] + 2 s



 RN

G2 (u) dx + q ξ (u)R(u) −

RN

G1 (u) dx

1 2 m 2∗ [u]s + u2L2 (RN ) − C 1 u s2∗s N L (R ) 2 2 4   1 m 2∗ , u2H s (RN ) − C 1 C ∗ uHs s (RN ) , ≥ min 2 2 4 ≥

which easily yields (10.2.9). Proceeding similarly to Theorem 10 in [101], for any n ∈ N, there exists an odd s (RN ) such that continuous map πn : Sn−1 → Hrad 0∈ / πn (Sn−1 ),  G(πn (σ )) dx ≥ 1

for all σ ∈ Sn−1 .

RN

Let us define ψnt (σ ) = πn (σ )

·

t

,

for t ≥ 1.

10.2 The Truncated Problem

369

Then, for t sufficiently large, we get t N−2s [πn (σ )]2s 2  N−2s  t [πn (σ )]2s + t N πn (σ )2L2 (RN ) +qχ R(ψnt (σ )) 2  G(πn (σ )) dx − tN

Fq (ψnt (σ )) =

RN



≤ t N−2s

 [πn (σ )]2s − t 2s < 0. 2

Therefore, we can choose t¯ such that Fq (ψnt¯ (σ )) < 0 for all σ ∈ Sn−1 , and by setting γn (σ )(x) = ψnt¯ (σ )(x), we see that γn satisfies the required properties.   s (RN ) Now we define the minimax value of Fq by using the maps γn : ∂Dn → Hrad obtained in Lemma 10.2.2. For any n ∈ N, let

bn = bn (q, ) = inf max Fq (γ (σ )), γ ∈n σ ∈Dn

where Dn = {σ = (σ1 , . . . , σn ) ∈ Rn : |σ | ≤ 1} and  n = γ ∈

s C(Dn , Hrad (RN ))

γ (−σ ) = −γ (σ ) for all σ ∈ Dn : γ (σ ) = γn (σ ) for all σ ∈ ∂Dn

Let us introduce the following modified functionals 2q (θ, u) = Fq (u(·/eθ )), F  θ 3  F q (θ, u) = Fq (u(·/e )), s (RN ), and set for (θ, u) ∈ R × Hrad

2 (θ, u) = ∂ F 2q (θ, u), F q ∂u ∂ 3 3 

(F F  (θ, u), q ) (θ, u) = ∂u q 3  (γ˜ (σ )), b˜n = b˜n (q, ) = inf max F γ˜ ∈  n σ ∈D n

q

 .

370

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General. . .

where ⎧ ⎫ ⎪ ⎪ γ˜ (σ ) = (θ(σ ), η(σ )) satisfies ⎪ ⎪ ⎨ ⎬ s N  . n = γ˜ ∈ C(Dn , R × Hrad (R )) : (θ(−σ ), η(−σ )) = (θ(σ ), −η(σ )) for all σ ∈ Dn ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (θ(σ ), η(σ )) = (0, γn (σ )) for all σ ∈ ∂Dn

By assumption (R4 ) we get (N−2s)θ  2q (θ, u) = e F [u]2s + q eαi θ Ri (eβi θ u) − eNθ 2 k



i=1

RN

G(u) dx,

and e(N−2s)θ 2 3  F [u]s + qχ q (θ, u) = 2  G(u) dx. − eNθ



e(N−2s)θ [u]2s + eNθ u2L2 (RN ) 2



k 

eαi θ Ri (eβi θ u)

i=1

RN

Proceeding as in [39, 218, 298], we can see that the following results hold. Lemma 10.2.3 We have that (1) there exists b¯ > 0 such that bn ≥ b¯ for any n ∈ N; (2) bn → ∞; (3) bn = b˜n for any n ∈ N. s (RN ) such Lemma 10.2.4 For every n ∈ N there exists a sequence ((θj , uj )) ⊂ R × Hrad that

(i) θj → 0; 3  (ii) F q (θj , uj ) → bn ; s 3 (RN ))∗ ; (iii) (Fq ) (θj , uj ) → 0 strongly in (Hrad ∂ 3 F  (θj , uj ) → 0. (iv) ∂θ q Our goal is to prove that, for a suitable choice of  and q, the sequence ((θj , uj )) given by Lemma 10.2.4 is a bounded Palais–Smale sequence for Fq . We begin by proving the boundedness of (uj ) in H s (RN ). Proposition 10.2.5 Let n ∈ N and let n > 0 be sufficiently large. There exists qn , s (RN ) is the depending on n , such that for any 0 < q < qn , if ((θj , uj )) ⊂ R × Hrad sequence given in Lemma 10.2.4, then, up to a subsequence, uj H s (RN ) ≤ n , for all j ∈ N.

10.2 The Truncated Problem

371

Proof Lemmas 10.2.3 and 10.2.4 imply that 3  NF q (θj , uj ) −

∂ 3 F  (θj , uj ) = Nbn + oj (1), ∂θ q

which can be written as se(N−2s)θj [uj ]2s   k uj (·/eθj )2H s (RN )  =qχ (αi − N)Ri (uj (·/eθj )) 2  +qχ

i=1

 k uj (·/eθj )2H s (RN )  eαi θj R i (eβi θj uj ), βi eβi θj uj  2 i=1

 +qχ



uj (·/eθj )2H s (RN )



(N − 2s)e(N−2s)θj [uj ]2s + NeNθj uj 2L2 (RN )

2

2

R(uj (·/eθj ))

+ Nbn + oj (1) = Ij + IIj + IIIj + Nbn + oj (1).

(10.2.11)

By the definition of bn , if γ ∈ n , we deduce that bn ≤ max Fq (γ (σ )) σ ∈D n

≤ max

σ ∈D n



1 [γ (σ )]2s − 2

 RN

 G(γ (σ )) dx + max (q ξ (γ (σ ))R(γ (σ ))) σ ∈D n

= A1 + A2 ().

(10.2.12)

Now, if γ (σ )2H s (RN ) ≥ 22 , then A2 () = 0: otherwise, by (10.2.2), we can find δ > 0 such that



A2 () ≤ q C1 + C2 γ (σ )δH s (RN ) ≤ q C1 + C2 δ . In addition, we have the following estimates:

|Ij | ≤ q C3 + C4 δ , |IIj | ≤ C5 q δ ,

|IIIj | ≤ q C6 + C7 δ .

(10.2.13) (10.2.14) (10.2.15)

372

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General. . .

In view of (10.2.12), (10.2.13), (10.2.14) and (10.2.15), we deduce from (10.2.11) that

[uj ]2s ≤ C + q C8 + C9 δ .

(10.2.16)

On the other hand, by item (iv) of Lemma 10.2.4 and (10.2.8), (N − 2s)e(N−2s)θj [uj ]2s + q χ 2  +qχ

uj (·/eθj )2H s (RN )



k 

2

αi Ri (uj (·/eθj ))

i=1

 k uj (·/eθj )2H s (RN )  eαi θj R i (eβi θj uj ), βi eβi θj uj  2 i=1



uj (·/eθj )2H s (RN )



+qχ





(N − 2s)e(N−2s)θj [uj ]2s + NeNθj uj 2L2 (RN )

2

2

R(uj (·/eθj ))

 + NeNθj

RN

G2 (uj ) dx

 = Ne

Nθj RN

≤ NeNθj

G1 (uj ) dx + oj (1)

  Cε



∗

RN

|uj |2s dx + ε

RN

 G2 (uj ) dx + oj (1).

(10.2.17)

Then, using (10.2.7), (10.2.14), (10.2.15), (10.2.16), (10.2.17) and Theorem 1.1.8, we infer that NeN θj m(1 − ε) 2 ≤ (1 − ε)NeN θj  ≤ Ne

N θj

 − q χ

Cε

RN

 RN

u2j dx



RN

G2 (uj ) dx  2∗s

|uj | dx − q χ

 uj (·/eθj )2H s (RN ) 2

(N

uj (·/eθj )2H s (RN ) 2

− 2s)e(N −2s)θj [u



k 

eαi θj R i (eβi θj uj ), βi eβi θj uj 

i=1 2 N θj u 2 j ]s + Ne j L2 (RN ) R(uj (·/eθj )) + oj (1) 2



2∗s

2 + q C11 + C12 δ + oj (1) ≤ C10 [uj ]2s

2∗s

≤ C10 C + q C8 + C9 δ 2 + q C11 + C12 δ + oj (1).

(10.2.18)

10.2 The Truncated Problem

373

Now, we argue by contradiction. Suppose that there is no subsequence (uj ) that is uniformly bounded by  in the H s -norm. Then we can find j0 ∈ N such that uj H s (RN ) >  for all j ≥ j0 .

(10.2.19)

Without loss of generality, we may assume that (10.2.19) is true for all uj . Consequently, using (10.2.16), (10.2.18) and (10.2.19), we deduce that 2 < uj 2H s (RN ) ≤ C13 + C14 q 

2∗ s 2 δ

which is not true for  large and q small enough. Indeed, to see this, note that one can find 2∗ s

0 such that 20 > C13 + 1 and q0 = q0 (0 ) such that C14 q 02 and this gives a contradiction.

δ

< 1, for any q < q0 ,  

At this point, we prove the following compactness result: Lemma 10.2.6 Let n ∈ N, n , qn > 0 as in Proposition 10.2.5 and ((θj , uj )) ⊂ R × s Hrad (RN ) be the sequence given in Lemma 10.2.4. Then (uj ) admits a subsequence which s (RN ) to a nontrivial critical point of Fq at the level bn . converges in Hrad Proof By Proposition 10.2.5, (uj ) is bounded, so, by using Theorem 1.1.11, we can s assume, up to a subsequence, that there exists u ∈ Hrad (RN ) such that uj  u

s in Hrad (RN ),

uj → u

in Lp (RN ), 2 < p < 2∗s ,

uj → u

a.e. in RN .

(10.2.20)

Thanks to the weak lower semicontinuity, [u]2s ≤ lim inf[uj ]2s .

(10.2.21)

j →∞

s Recalling that uj H s (RN ) ≤ n for any j ∈ N, we see that, for every v ∈ Hrad (RN ),

3 n

2 (θj , uj ), v = (F F q ) (θj , uj ), v q  (uj (x) − uj (y))(v(x) − v(y)) (N−2s)θj dxdy =e |x − y|N+2s R2N +q

k 

e(αi +βi )θj R i (eβi θj uj ), v

i=1



+ eNθj

 RN

g2 (uj )v dx − eNθj

RN

g1 (uj )v dx.

(10.2.22)

374

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General. . .

Taking into account (10.2.22) and item (iii) of Lemma 10.2.4, we have 2q (θj , uj ), u − F 2q (θj , uj ), uj  oj (1) = F  (uj (x) − uj (y)) = e(N−2s)θj [(u(x) − u(y)) − (uj (x) − uj (y))] dxdy 2N |x − y|N+2s R +q

k 

e(αi +βi )θj R i (eβi θj uj ), u − uj 

i=1



+ eNθj

 RN

g2 (uj )(u − uj ) dx − eNθj

RN

g1 (uj )(u − uj ) dx.

(10.2.23) ∗

Now, applying the first part of Lemma 1.4.2 with P (t) = gi (t), i = 1, 2, Q(t) = |t|2s −1 , vj = uj , v = gi (u), i = 1, 2 and w ∈ Cc∞ (RN ), and using (g3 ), (10.2.4) and (10.2.20), we see that, as j → ∞, 

 RN

gi (uj )w dx →

RN

gi (u)w dx

i = 1, 2,

so we obtain 

 RN

gi (uj )u dx →

RN

gi (u)u dx

i = 1, 2.

(10.2.24)

Taking X = H s (RN ), q1 = 2, q2 = 2∗s , vj = uj , v = g1 (u)u and P (t) = g1 (t)t in Lemma 1.4.3, and using (10.2.3), (10.2.4) and (10.2.20), we deduce that 

 RN

g1 (uj )uj dx →

RN

g1 (u)u dx.

(10.2.25)

On the other hand, (10.2.20) and Fatou’s lemma yield 



RN

g2 (u)u dx ≤ lim inf j →∞

RN

g2 (uj )uj dx.

(10.2.26)

Putting together (10.2.23), (10.2.24), (10.2.25), (10.2.26), and using (R3 ) we get lim sup [uj ]2s = lim sup e(N−2s)θj [uj ]2s j →∞

j →∞

  = lim sup e(N−2s)θj j →∞

+q

k  i=1

R2N

(uj (x) − uj (y))(u(x) − u(y)) |x − y|N+2s

e(αi +βi )θj R i (eβi θj uj ), u − uj 

(u(x) − u(y)) dxdy

10.2 The Truncated Problem

375

 +eNθj





RN

g2 (uj )(u − uj ) dx − eNθj

RN

g1 (uj )(u − uj ) dx

≤ [u]2s .

(10.2.27)

Therefore (10.2.21) and (10.2.27) give lim [uj ]2s = [u]2s ,

(10.2.28)

j →∞

which, in view of (10.2.23), yields 

 lim

j →∞ RN

g2 (uj )uj dx =

RN

(10.2.29)

g2 (u)u dx.

Since g2 (t)t = mt 2 + h(t), where h(t) = t (g(t) + mt)− is a non-negative and continuous function, it follows from Fatou’s lemma that 

 RN

h(u) dx ≤ lim inf j →∞



RN



2

RN

u dx ≤ lim inf j →∞

RN

(10.2.30)

h(uj ) dx,

u2j dx.

(10.2.31)

Using (10.2.29) and (10.2.30), we have that  lim sup j →∞

 RN

mu2j dx = lim sup j →∞

 =

RN

RN

(g2 (uj )uj − h(uj )) dx

RN

j →∞



h(uj ) dx



2

=

RN

(mu + h(u)) dx − lim inf

 =  ≤



  g2 (u)u dx + lim sup −

RN

RN

j →∞

 mu2 dx +

RN

RN

h(uj ) dx

h(u) dx − lim inf j →∞

 RN

h(uj ) dx

mu2 dx

which in conjunction with (10.2.31), implies that uj → u strongly in L2 (RN ). We s conclude that uj → u strongly in Hrad (RN ), and since bn > 0, u is a nontrivial critical   point of Fq at the level bn .

376

10 A Multiplicity Result for a Fractional Kirchhoff Equation with a General. . .

Now we are ready to prove the main result of this section: Proof of Theorem 10.2.1 Let h ≥ 1. Since bn → ∞ (see item (2) of Lemma 10.2.3), up to a subsequence, we can consider that b1 < b2 < · · · < bh . Then, in view of Lemma 10.2.6, we define q(h) = qh > 0 and we get the desired conclusion.  

10.3

A Multiplicity Result

In this section we give the proof of Theorem 10.1.1. Let us introduce the functional

q 1 p + (1 − s)[u]2H s (RN ) [u]2s − Fq (u) = 2 2

 RN

G(u) dx.

In view of Theorem 10.2.1, it is enough to verify that R(u) =

1−s 4 [u]s 4

(10.3.1)

satisfies the assumptions (R1 )–(R5 ). Clearly, R is an even and nonnegative C 1 -functional on H s (RN ). Since [u]2s ≤ u2H s (RN ) , we can see that assumptions (R1 ) and (R2 ) are satisfied. s (RN ) and [u ]2 → % ≥ 0. If Regarding (R3 ), suppose that uj  u weakly in Hrad j s % = 0, then we are done. Assume that % > 0. By the weak lower semicontinuity, [u]2s ≤ lim inf[uj ]2s . j →∞

(10.3.2)

Using the following properties of lim inf and lim sup for sequences of real numbers: lim sup aj bj = a lim sup bj if aj , bj ≥ 0, aj → a, j →∞

j →∞

lim sup (aj + bj ) = a + lim sup bj if aj → a, j →∞

j →∞

lim sup (−aj ) = − lim inf aj , j →∞

j →∞

and applying (10.3.2), we obtain lim sup R (uj ), u − uj  = j →∞



= (1 − s) lim sup j →∞

 [uj ]2s

R2N

(uj (x) − uj (y)) |x − y|N+2s

 [(u(x) − u(y)) − (uj (x) − uj (y))] dx dy

10.3 A Multiplicity Result

377

 = (1 − s)% lim sup j →∞

 = (1 − s)%

(uj (x) − uj (y))

R2N



|x − y|N+2s

[(u(x) − u(y)) − (uj (x) − uj (y))] dx dy

(uj (x) − uj (y))(u(x) − u(y))

lim

j →∞ |x − y|N+2s R2N   = (1 − s)% [u]2s − lim inf[uj ]2s ≤ 0,

 dx dy − lim inf[uj ]2s j →∞

j →∞

which gives (R3 ). Now, recalling the definition of ut and using (10.3.1), we conclude that (R4 ) is verified, because 1−s R(ut ) = 4 =

  R2N

|u( xt ) − u( yt )|2 dx dy |x − y|N+2s

(1 − s) t 2(N−2s) 4

  R2N

2

|u(x) − u(y)|2 dx dy |x − y|N+2s

4

= t 2(N−2s) R(u). Finally, we prove (R5 ). Using a change of variable, we see that, for any g ∈ O(N), R(u(g·)) =

1−s 4 1−s [u(g·)]4s = [u]s = R(u). 4 4

Then, by applying Theorem 10.2.1, we infer that for every h ∈ N there exists q(h) > 0 such that for any 0 < q < q(h) the functional Fq admits at least h couples of critical points in H s (RN ) with radial symmetry. This means that (10.1.1) admits at least h couples s (RN ). of weak solutions in Hrad

Multiplicity and Concentration of Positive Solutions for a Fractional Kirchhoff Equation

11.1

11

Introduction

In this chapter we study the multiplicity and concentration of positive solutions for the following fractional Schrödinger–Kirchhoff type problem ⎧  ⎪ ⎨M

 ' 2s ( 1  |u(x) − u(y)|2 2 dx dxdy + V (x)u ε (−)s u + V (x)u = f (u) in R3 , 3 R ε3−2s |x − y|3+2s ε3 ⎪ ⎩ u ∈ H s (R3 ), u > 0 in R3 , 1



R6

(11.1.1) where ε > 0 is a small parameter and s ∈ ( 34 , 1). We assume that M : [0, ∞) → [0, ∞) is a continuous function such that: (M1 ) there exists m0 > 0 such that M(t) ≥ m0 for any t ≥ 0; (M2 ) M(t) is nondecreasing; (M3 ) for any t1 ≥ t2 > 0, M(t1 ) M(t2 ) − ≤ m0 t1 t2



1 1 − t1 t2

 .

 As a model for M, we can take M(t) = m0 + bt + ki=1 bi t γi with bi ≥ 0 and γi ∈ (0, 1) for all i ∈ {1, . . . , k}. Concerning the potential V : R3 → R, we suppose that V ∈ C(R3 , R) and fulfills the following hypotheses: (V1 ) there exists V0 > 0 such that V0 = inf V (x); x∈R3

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_11

379

380

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

(V2 ) there is a bounded set ⊂ R3 such that V0 < min V , ∂

and  = {x ∈ : V (x) = V0 } = ∅.

Without loss of generality, we may assume that 0 ∈ . Finally, concerning the nonlinear term in (11.1.1), we assume that f : R → R is a continuous function satisfying the following conditions: (f1 ) lim

t →0

f (t) = 0; t3

6 (f2 ) there is q ∈ (4, 3−2s ) such that lim

f (t)

t →∞ t q−1

= 0;

(f3 ) there is ϑ ∈ (4, q) such that 0 < ϑF (t) ≤ f (t)t for all t > 0; f (t) (f4 ) the function t → 3 is increasing in (0, ∞). t A typical example is f (t) =

k 

ai (t + )qi −1

i=1 6 with ai ≥ 0 not all identically zero and qi ∈ [ϑ, 3−2s ) for all i ∈ {1, . . . , k}. Since we are interested in positive solutions, we assume that f vanishes in (−∞, 0). Now, we are ready to state our main result.

Theorem 11.1.1 Let s ∈ ( 34 , 1) and assume that (M1 )–(M3 ), (V1 )–(V2 ) and (f1 )–(f4 ) hold. Then, for every δ > 0 such that δ = {x ∈ R3 : dist(x, ) ≤ δ} ⊂ , there exists an εδ > 0 such that, for any ε ∈ (0, εδ ), problem (11.1.1) has at least catδ () positive solutions. Moreover, if uε denotes one of these positive solutions and ηε ∈ R3 is a global maximum point of uε , then lim V (ηε ) = V0 .

ε→0

The proof of Theorem 11.1.1 relies on variational methods inspired by [191, 321, 322]. In particular, Theorem 11.1.1 corresponds to the fractional counterpart of Theorem 1.1 in [191]. Clearly, the presence of the fractional Laplacian makes our analysis more delicate and intriguing compared to the case s = 1, and the results obtained in Chaps. 6 and 7 about fractional Schrödinger equations, will play a fundamental role to overcome our difficulties.

11.2 The Modified Kirchhoff Problem

381

In what follows, we give a sketch of the proof. The lack of informations on the behavior of V at infinity suggest us to use the penalization method introduced by del Pino and Felmer [165]. Since f and M are only continuous, the Nehari manifold associated with the modified problem is not differentiable, so we will use some abstract results obtained by Szulkin and Weth in [321, 322]. After a careful study of the autonomous problem associated to (11.1.1), we deal with the multiplicity of solutions of the modified problem, by invoking the Lusternik–Schnirelman theory. Then, in order to prove that the solutions uε of the truncated problem are also solutions to (11.1.1) when ε > 0 is sufficiently small, we argue as in Chap. 7. We point out that the restriction s ∈ ( 34 , 1) is essential in our technical approach in order to guarantee the continuous embedding of the space H s (R3 ) 6 (see conditions (f1 )–(f3 )). into the Lebesgue spaces Lr (RN ) with 4 ≤ r < 3−2s

11.2

The Modified Kirchhoff Problem

This section is devoted to the existence of positive solutions to (11.1.1). After a change of variable, problem (11.1.1) reduces to 

M



u∈

 |u(x)−u(y)|2 R6 |x−y|3+2s dxdy + R3 H s (R3 ), u > 0 in R3 .

V (ε x)u2 dx [(−)s u + V (ε x)u] = f (u) in R3 , (11.2.1)

Take K >

2 m0

and a > 0 such that f (a) =  f˜(t) =

V0 K a.

Let us define

f (t), if t ≤ a, V0 K t, if t > a,

and g(x, t) = χ (x)f (t) + (1 − χ (x))f˜(t). In view of the assumptions on f we deduce that g is a Carathéodory function and satisfies g(x, t) = 0 uniformly in x ∈ R3 ; t →0 t3 g(x, t) (g2 ) lim q−1 = 0 uniformly in x ∈ R3 ; t →∞ t (g3 ) (i) 0< ϑG(x, t) ≤ g(x, t)t for any x ∈ and for any t > 0, V0 2 t for any x ∈ R3 \ and for any t > 0; (ii) 0 ≤ 2G(x, t) ≤ g(x, t)t ≤ K (g1 ) lim

382

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

g(x, t) is increasing in (0, ∞) and for each t3 g(x, t)  is increasing in (0, a). fixed x ∈ R3 \ the function t → t3

(g4 ) for each fixed x ∈ the function t →

By the definition of g it follows that g(x, t) ≤ f (t) for all t > 0, for all x ∈ R3 , g(x, t) = 0 for all t < 0, for all x ∈ R3 . In what follows, we consider the auxiliary problem   M

R6

|u(x) − u(y)|2 dxdy + |x − y|3+2s

 R3

 V (ε x)u2 dx [(−)s u + V (ε x)u] = g(ε x, u) in R3 .

(11.2.2)

Moreover, we focus our attention on positive solutions to (11.2.2) having the property u(x) ≤ a for each x ∈ R3 \ . Therefore, solutions of (11.2.2) can be found as critical points of the following energy functional Jε (u) =

1 M 2

  R6

|u(x) − u(y)|2 dxdy + |x − y|3+2s

 R3

  V (ε x)u2 dx −

R3

G(ε x, u) dx

where  = M(t)





t

M(τ ) dτ

t

G(ε x, t) =

and

0

g(ε x, τ ) dτ, 0

which is well defined on the Hilbert space   Hε = u ∈ H s (R3 ) :

 R3

V (ε x)u2 dx < ∞

endowed with the inner product  u, ϕε = u, ϕDs,2 (R3 ) +

R3

V (ε x) uϕ dx

and the corresponding norm   uε = [u]2s +

1 2

R3

V (ε x) u dx

2

.

11.2 The Modified Kirchhoff Problem

383

By the assumptions on M and f , and using Theorem 1.1.8, it is easy to check that Jε is well defined, that Jε ∈ C 1 (Hε , R) and that its differential Jε is given by Jε (u), ϕ

 =

M(u2ε )u, ϕε

−

R3

g(ε x, u)ϕ dx,

for any u, ϕ ∈ Hε . Let us introduce the Nehari manifold associated with Jε , that is Nε = {u ∈ Hε \ {0} : Jε (u), u = 0}. The main result of this section reads: Theorem 11.2.1 Under assumptions (M1 )–(M3 ), (V1 )–(V2) and (f1 )–(f4 ), the auxiliary problem (11.2.2) has a positive ground state solution for all ε > 0. Let ε = {x ∈ R3 : ε x ∈ } and Hε+ = {u ∈ Hε : |supp(u+ ) ∩ ε | > 0} ⊂ Hε . + + Let Sε be the unit sphere of Hε and denote S+ ε = Sε ∩ Hε . We observe that Hε is open in Hε . Indeed, consider a sequence (un ) ⊂ Hε \ Hε+ such that un → u in Hε and assume, by contradiction, that u ∈ Hε+ . Now, by the definition of Hε+ it follows that + + |supp(u+ n ) ∩ ε | = 0 for all n ∈ N and un (x) → u (x) a.e. x ∈ ε . So,

u+ (x) = lim u+ n (x) = 0 n→∞

a.e. x ∈ ε ,

and this contradicts the fact that u ∈ Hε+ . Therefore, Hε+ is open. + + By the definition of S+ ε and the fact that Hε is open in Hε , it follows that Sε is a 1,1 incomplete C -manifold of codimension 1, modeled on Hε and contained in the open + Hε+ . Then, Hε = Tu S+ ε ⊕ Ru for each u ∈ Sε , where Tu S+ ε = {v ∈ Hε : (u, ϕ)ε = 0}. In the next lemma we prove that Jε has a mountain pass geometry [29]. Lemma 11.2.2 The functional Jε satisfies the following conditions: (a) there exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ; (b) there exists e ∈ Hε such that eε > ρ and Jε (e) < 0.

384

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

Proof (a) By (M1 ), (g1 ), (g2 ) and Theorem 1.1.8, for every ξ > 0 there exists Cξ > 0 such that Jε (u) =

1 M(u2ε ) − 2

 R3

G(ε x, u) dx ≥

m0 u2ε − ξ Cu4ε − Cξ Cuqε . 2

Then, we can find α, ρ > 0 such that Jε (u) ≥ α with uε = ρ. (b) In view of (M3 ), there exists a positive constant γ such that M(t) ≤ γ (1 + t) for all t ≥ 0.

(11.2.3)

Then, by (g3 )-(i), we can see that, for any u ∈ Hε+ and t > 0 Jε (tu) =

1 M(tu2ε ) − 2

 R3

G(ε x, tu) dx

γ γ ≤ t 2 u2ε + t 4 u4ε − 2 4 ≤

 G(ε x, tu) dx

ε

γ 2 γ t u2ε + t 4 u4ε − C1 t ϑ 2 4



(u+ )ϑ dx + C2 |supp(u+ ) ∩ ε |,

ε

(11.2.4) for some positive constants C1 and C2 . 6 ), we conclude that Jε (tu) → −∞ as t → Taking into account that ϑ ∈ (4, 3−2s ∞.   Since f and M are continuous functions, the next results will play a fundamental role in overcoming the non-differentiability of Nε and the incompleteness of S+ ε. Lemma 11.2.3 Assume that (M1 )–(M3 ), (V1 )–(V2) and (f1 )–(f4 ) hold. Then, (i) For each u ∈ Hε+ , let h : R+ → R be defined by hu (t) = Jε (tu). Then, there is a unique tu > 0 such that h u (t) > 0 in (0, tu ), h u (t) < 0 in (tu , ∞). (ii) There exists τ > 0 independent of u such that tu ≥ τ for all u ∈ S+ ε . Moreover, for there is a positive constant C such that tu ≤ CK for any each compact set K ⊂ S+ K ε u ∈ K.

11.2 The Modified Kirchhoff Problem

385

(iii) The map m ˆ ε : Hε+ → Nε given by m ˆ ε (u) = tu u is continuous and mε = m ˆ ε |S+ε is a u + −1 homeomorphism between Sε and Nε . Moreover, mε (u) = uε . + (iv) If there is a sequence (un ) ⊂ S+ ε such that dist(un , ∂Sε ) → 0, then mε (un )ε → ∞ and Jε (mε (un )) → ∞. Proof (i) We know that hu ∈ C 1 (R+ , R) and, by Lemma 11.2.2, we have that hu (0) = 0, hu (t) > 0 for t > 0 small enough and hu (t) < 0 for t > 0 sufficiently large. Hence, there exists tu > 0 such that h u (tu ) = 0, and tu is a global maximum for hu . Then, 0 = h u (tu ) = Jε (tu u), u =

1

J (tu u), tu u, tu ε

which implies that tu u ∈ Nε . Next, let us prove the uniqueness of such a tu . Assume, by contradiction, that there exist t1 > t2 > 0 such that h u (t1 ) = h u (t2 ) = 0, or equivalently,  t1 M(t1 u2ε )u2ε =  t2 M(t2 u2ε )u2ε

=

R3

R3

g(ε x, t1 u)u dx,

(11.2.5)

g(ε x, t2 u)u dx.

(11.2.6)

Dividing both members of (11.2.5) by t13 u4ε we get M(t1 u2ε ) 1 = 2 t1 uε u4ε

 R3

g(ε x, t1 u) 4 u dx; (t1 u)3

similarly, dividing both members of (11.2.6) by t23 u4ε we obtain 1 M(t2 u2ε ) = t2 u2ε u4ε

 R3

g(ε x, t2 u) 4 u dx. (t2 u)3

Taking the difference of the last two identities and using (M3 ) and (g4 ), we can see that m0 u2ε ≥



1 1 − 2 t12 t2



M(t1 u2ε ) M(t2 u2ε ) − 2 t1 uε t2 u2ε

386

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .  g(ε x, t1 u) g(ε x, t2 u) 4 − u dx (t1 u)3 (t2 u)3 R3       1 g(ε x, t1 u) g(ε x, t2 u) 4 1 g(ε x, t1 u) g(ε x, t2 u) 4 = − u dx + − u dx u4ε R3 \ ε (t1 u)3 (t2 u)3 u4ε ε (t1 u)3 (t2 u)3    1 g(ε x, t2 u) 4 g(ε x, t1 u) ≥ − u dx u4ε R3 \ ε (t1 u)3 (t2 u)3    1 g(ε x, t2 u) 4 g(ε x, t1 u) = − u dx u4ε (R3 \ ε )∩{t2 u>a} (t1 u)3 (t2 u)3    1 g(ε x, t2 u) 4 g(ε x, t1 u) + − u dx u4ε (R3 \ ε )∩{t2 u≤aa}



1 1 = u4ε K



1 1 − 2 t12 t2

 V0 1 V0 1 − u4 dx K (t1 u)2 K (t2 u)2

(R3 \ ε )∩{t2 u>a}

V0 u2 dx.

Regarding I I , using again the definition of g, we infer that II ≥

1 u4ε



 (R3 \ ε )∩{t2 u≤aa}    1 f (t2 u) 4 V0 1 + − u dx. u4ε (R3 \ ε )∩{t2 u≤aa}

(R3 \ ε )∩{t2 u>a}

V0 u2 dx +

V0 u2 dx

< 0, we have t12 t22

t22 − t12

 (R3 \ ε )∩{t2 u≤a 0 there exists a positive constant Cξ such that |g(x, t)| ≤ ξ |t|3 + Cξ |t|q−1 ,

for all (x, t) ∈ R3 × R,

which in conjunction with (M1 ) and Theorem 1.1.8 yields  m0 tu ≤ M(tu2 )tu =

R3

g(ε x, tu u) u dx q−1

≤ ξ C1 tu3 + Cξ C2 tu

.

We conclude that there exists τ > 0, independent of u, such that tu ≥ τ . Now, let K ⊂ S+ ε be a compact set. Let us show that tu can be estimated from above by a constant depending on K. Assume, by contradiction, that there exists a sequence (un ) ⊂ K such that tn = tun → ∞. Therefore, there exists u ∈ K such that un → u in Hε . From (11.2.4) we get Jε (tn un ) → −∞.

(11.2.7)

Fix v ∈ Nε . Then, using the fact that Jε (v), v = 0, and assumptions (g3 )-(i) and (g3 )-(ii), we see that 1

J (v), v ϑ ε  1 1 1 2 2 2 = M(v M(v ) − )v + [g(ε x, v)v − ϑG(ε x, v)] dx ε ε ε 2 ϑ ϑ R3  1 1 1 2 2 2 M(v = M(v ) − )v + [g(ε x, v)v − ϑG(ε x, v)] dx ε ε ε 2 ϑ ϑ R3 \ ε

Jε (v) = Jε (v) −

388

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

1 + ϑ

 [g(ε x, v)v − ϑG(ε x, v)] dx

ε

 1 1 1 [g(ε x, v)v − ϑG(ε x, v)] dx M(v2ε ) − M(v2ε )v2ε + 2 ϑ ϑ R3 \ ε    1 ϑ −2 1 1 2 2 2 M(v ) − )v − V (ε x)v 2 dx ≥ M(v ε ε ε 2 ϑ 2ϑ K R3 \ ε   1 ϑ −2 1 1 2 2 2 M(v v2ε . ) − )v − ≥ M(v ε ε ε 2 ϑ 2ϑ K ≥

Now, by (M3 ), we know that  ≥ M(t) + m0 t M(t) 2

for all t ≥ 0.

(11.2.8)

This together with (M1 ) implies that   & 1% 1 ϑ−2 1 2 2 2 2 Jε (v) ≥ M(vε ) + m0 vε − M(vε )vε − v2ε 4 ϑ 2ϑ K   ϑ −4 1 ϑ −2 1 2 2 2 M(vε )vε + m0 vε − v2ε = 4ϑ 4 2ϑ K     ϑ −4 1 ϑ −2 1 + v2ε ≥ m0 v2ε − 4ϑ 4 2ϑ K    ϑ −2 1 m0 − v2ε . = (11.2.9) 2ϑ K Taking into account that (tun un ) ⊂ Nε and K > m20 , from (11.2.9) we deduce that (11.2.7) cannot hold. + (iii) First, note that m ˆ ε , mε and m−1 ε are well defined. Indeed, by (i), for each u ∈ Hε + there exists a unique mε (u) ∈ Nε . On the other hand, if u ∈ Nε , then u ∈ Hε . Otherwise, if u ∈ / Hε+ , then |supp(u+ ) ∩ ε | = 0, which together with (g3)-(ii) gives  0
0. Keeping in mind the definitions of mε (un ) and M(t), and by using (11.2.11) and assumption (M1 ), we have lim inf Jε (mε (un )) ≥ lim inf Jε (mε (un )) n→∞ n→∞    1 2 = lim inf M(tun ε ) − G(ε x, tun ) dx n→∞ 2 R3 1 2 t2 M(t ) − 2 K   1 1 m0 − t 2. ≥ 2 K ≥

Recalling that K > 2/m0 , we get lim Jε (mε (un )) = ∞.

n→∞

11.2 The Modified Kirchhoff Problem

391

Moreover, the definition of Jε (mε (un )) and (11.2.3) imply that Jε (mε (un )) ≤

1 2 M(tun ) ≤ C(tu2n + tu4n ) 2

for all n ∈ N,

and so mε (un )ε → ∞ as n → ∞.   Let us define the maps ψˆ ε : Hε+ → R

and

ψε : S+ ε → R,

by ψˆ ε (u) = Jε (m ˆ ε (u)) and ψε = ψˆ ε |S+ε . The next result is a direct consequence of Lemma 11.2.3 and Corollary 2.4.4 (see also [191, 321, 322]). Proposition 11.2.4 Assume that (M1 )–(M3 ), (V1 )–(V2) and (f1 )–(f4 ) hold. Then, (a) ψˆ ε ∈ C 1 (Hε+ , R) and ψˆ ε (u), v =

m ˆ ε (u)ε

Jε (m ˆ ε (u)), v, uε

for every u ∈ Hε+ and v ∈ Hε . (b) ψε ∈ C 1 (S+ ε , R) and ψε (u), v = mε (u)ε Jε (mε (u)), v, for every v ∈ Tu S+ ε . (c) If (un ) is a Palais–Smale sequence for ψε , then (mε (un )) is a Palais–Smale sequence for Jε . If (un ) ⊂ Nε is a bounded Palais–Smale sequence for Jε , then (m−1 ε (un )) is a Palais–Smale sequence for ψε . (d) u is a critical point of ψε if and only if mε (u) is a nontrivial critical point for Jε . Moreover, the corresponding critical values coincide and inf ψε (u) = inf Jε (u).

u∈S+ ε

u∈Nε

Remark 11.2.5 As in [322], we can see that, thanks to (M1 )–(M3 ), cε = inf Jε (u) = inf max Jε (tu) = inf max Jε (tu). u∈Nε

u∈H+ ε t >0

u∈S+ ε t >0

It is also easy to check that cε coincides with the mountain pass level of Jε . Let us verify that the functional Jε satisfies the Palais–Smale condition.

392

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

Lemma 11.2.6 Let (un ) ⊂ Hε be a Palais–Smale sequence for Jε at the level d ∈ R. Then (un ) is bounded in Hε . Proof Since (un ) is a Palais–Smale sequence at the level d, we know that Jε (un ) → d

and

Jε (un ) → 0 in Hε∗ .

Then, arguing as in the proof of Lemma 11.2.3-(ii) (see formula (11.2.9)), we see that 1 C(1 + un ε ) ≥ Jε (un ) − Jε (un ), un  ϑ    1 ϑ −2 ≥ m0 − un 2ε . 2ϑ K Since ϑ > 4 and K > 2/m0 , we conclude that (un ) is bounded in Hε .

 

The next two lemmas are fundamental to obtain compactness of bounded Palais–Smale sequences. Lemma 11.2.7 Let (un ) ⊂ Hε be a Palais–Smale sequence for Jε at the level d. Then, for each ζ > 0, there exists R = R(ζ ) > 0 such that  lim sup n→∞

 R3 \BR

dx

R3

|un (x) − un (y)|2 dy + |x − y|3+2s

 R3 \BR

 V (ε x)u2n dx < ζ.

Proof For any R > 0, let ηR ∈ C ∞ (R3 ) be such that ηR = 0 in B R and ηR = 1 in 2

R3 \ BR , with 0 ≤ ηR ≤ 1 and ∇ηR L∞ (R3 ) ≤ C R , where C is a constant independent of R. Since (ηR un ) is bounded in Hε , it follows that Jε (un ), ηR un  = on (1), that is  

  |un (x) − un (y)|2 2 η (x) dxdy + V (ε x)u η dx R n R |x − y|3+2s R6 R3   (ηR (x) − ηR (y))(un (x) − un (y)) g(ε x, un )un ηR dx − M(un 2ε ) un (y) dxdy. = on (1) + |x − y|3+2s R3 R6

M(un 2ε )

Take R > 0 such that Ωε ⊂ B R . Then, using (M1 ) and (g3 )-(ii), we have 2

  m0  ≤

R3

  |un (x) − un (y)|2 2 η (x) dxdy + V (ε x)u η dx R R n |x − y|3+2s R6 R3  1 (ηR (x) − ηR (y))(un (x) − un (y)) un (y) dxdy + on (1), V (ε x)u2n ηR dx − M(un 2ε ) 6 K |x − y|3+2s R

11.2 The Modified Kirchhoff Problem

393

whence       |un (x) − un (y)|2 1 2 η (x) dxdy + V (ε x)u η dx m0 − R n R K |x − y|3+2s R6 R3  (ηR (x) − ηR (y))(un (x) − un (y)) ≤ −M(un 2ε ) un (y) dxdy + on (1). |x − y|3+2s R6 (11.2.12) Now note that the boundedness of (un ) in Hε and assumption (M2 ) imply that M(un 2ε ) ≤ C

for any n ∈ N.

(11.2.13)

On the other hand, using Hölder’s inequality, the boundedness of (un ) and Lemma 1.4.5 we get  lim lim sup

R→∞ n→∞

R6

(ηR (x) − ηR (y))(un (x) − un (y)) un (y) dxdy = 0. |x − y|3+2s

(11.2.14)

Finally, (11.2.12), (11.2.13), (11.2.14) and the definition of ηR show that 

 lim lim sup

R→∞ n→∞

R3 \BR

dx

R3

|un (x) − un (y)|2 dy + |x − y|3+2s

 R3 \BR

V (ε x)u2n dx = 0,  

which completes the proof of lemma.

Lemma 11.2.8 Let (un ) ⊂ Hε be a Palais–Smale sequence for Jε at the level d and such that un  u. Then, for all R > 0 

 lim

n→∞ B R

dx



=

R3



dx BR

|un (x) − un (y)|2 dy + |x − y|3+2s R3



|u(x) − u(y)|2 dy + |x − y|3+2s

BR

V (ε x)u2n dx



V (ε x)u2 dx. BR

Proof By Lemma 11.2.6, (un ) is bounded in Hε , so we may assume that un  u in Hε and un ε → t0 . Then, by the weak lower semicontinuity uε ≤ t0 . Moreover, by the continuity of M, we know that M(un 2ε ) → M(t0 ). c , with 0 ≤ η ≤ 1. Fix Let ηρ ∈ C ∞ (R3 ) be such that ηρ = 1 in Bρ and ηρ = 0 in B2ρ ρ R > 0 and choose ρ > R. Then we have  0 ≤M(un 2ε )

 dx BR

R3

|(un (x) − un (y)) − (u(x) − u(y))|2 dy + |x − y|3+2s



 V (ε x)(un − u)2 dx BR

394

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .  

  |(un (x) − un (y)) − (u(x) − u(y))|2 ηρ (x) dxdy + V (ε x)(un − u)2 ηρ dx 3+2s |x − y| R6 R3     2 |un (x) − un (y)| ηρ (x) dxdy + V (ε x)u2n ηρ dx ≤ M(un 2ε ) 6 |x − y|3+2s R R3     |u(x) − u(y)|2 2 + M(un 2ε ) η (x) dxdy + V (ε x)u η dx ρ ρ 3+2s R6 |x − y| R3     (un (x) − un (y))(u(x) − u(y)) − 2M(un 2ε ) ηρ (x) dxdy + V (ε x)un uηρ dx 3+2s |x − y| R6 R3 ≤ M(un 2ε )

= In,ρ − I In,ρ + I I In,ρ + I Vn,ρ ≤ |In,ρ | + |I In,ρ | + |I I In,ρ | + |I Vn,ρ |

(11.2.15)

where   In,ρ =M(un 2ε )

R6

  I In,ρ=M(un 2ε )  −

R3

|un (x) − un (y)|2 ηρ (x) dxdy + |x − y|3+2s

R6

 R3

  V (ε x)u2n ηρ dx −

(un (x) − un (y))(u(x) − u(y)) ηρ (x) dxdy + |x − y|3+2s

 R3

R3

g(ε x, un )un ηρ dx, 

V (ε x)un uηρ dx

g(ε x, un )uηρ dx,  

  (un (x) − un (y))(u(x) − u(y)) η (x)dxdy + V (ε x)u uη dx ρ n ρ |x − y|3+2s R6 R3     |u(x) − u(y)|2 ηρ (x) dxdy + V (ε x)u2 ηρ dx , + M(un 2ε ) 3+2s R6 |x − y| R3   = g(ε x, un )un ηρ dx − g(ε x, un )uηρ dx.

I I In,ρ = −M(un 2ε )

I Vn,ρ

R3

R3

Let us prove that lim lim sup |In,ρ | = 0.

ρ→∞ n→∞

(11.2.16)

First, In,ρ can be written as In,ρ = Jε (un ), un ηρ  − M(un 2ε )

 R6

(un (x) − un (y))(ηρ (x) − ηρ (y)) un (y) dxdy. |x − y|3+2s

Since (un ηρ ) is bounded in Hε , we have Jε (un ), un ηρ  = on (1), so  In,ρ = on (1) − M(un 2ε )

R6

(un (x) − un (y))(ηρ (x) − ηρ (y)) un (y) dxdy. |x − y|3+2s (11.2.17)

11.2 The Modified Kirchhoff Problem

395

By Lemma 1.4.5,    2  lim lim sup M(un ε ) ρ→∞ n→∞ 

R6

  (un (x) − un (y))(ηρ (x) − ηρ (y))  = 0, u (x) dxdy n  |x − y|3+2s

which together with (11.2.17) implies that (11.2.16) holds. Next, note that I In,ρ =

Jε (un ), uηρ  − M(un 2ε )

 R6

(un (x) − un (y))(ηρ (x) − ηρ (y)) u(x) dxdy. |x − y|3+2s

Similar calculations to the proof of Lemma 1.4.5 show that    2  lim lim sup M(un ε ) ρ→∞ n→∞ 

R6

  (un (x) − un (y))(ηρ (x) − ηρ (y)) u(x) dxdy  = 0, |x − y|3+2s

and since Jε (un ), uηρ  = on (1), we obtain lim lim sup |I In,ρ | = 0.

(11.2.18)

lim lim sup |I I In,ρ | = 0.

(11.2.19)

ρ→∞ n→∞

Now let us prove that ρ→∞ n→∞

By the weak convergence,  

[(un (x) − u(x)) − (un (y) − u(y))] (u(x) − u(y))ηρ (x) dxdy |x − y|3+2s  V (ε x)(un − u)uηρ dx

I I In,ρ = −M(un 2ε )  +

R3

R6

 

[(un (x) − u(x)) − (un (y) − u(y))] (u(x)ηρ (x) − u(y)ηρ (y)) dxdy |x − y|3+2s  V (ε x)(un − u)uηρ dx

= −M(un 2ε )  +

R3

R6



[(un (x) − u(x)) − (un (y) − u(y))] (ηρ (x) − ηρ (y))u(y) dxdy |x − y|3+2s  [(un (x) − u(x)) − (un (y) − u(y))] (ηρ (x) − ηρ (y))u(y) dxdy. = on (1) − M(un 2ε ) |x − y|3+2s R6 − M(un 2ε )

R6

Using Hölder’s inequality and the boundedness of (un ) and M(un 2ε ) we see that    2  sup M(un ε )

n∈N

R6

  [(un (x) − u(x)) − (un (y) − u(y))] (ηρ (x) − ηρ (y))u(y) dxdy  3+2s |x − y|

396

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

  ≤C

R6

|ηρ (x) − ηρ (y)|2 |u(y)|2 dxdy |x − y|3+2s

1/2 →0

as ρ → ∞,

so (11.2.19) holds true. Finally, we deal with the fourth term. By Theorem 1.1.8, un → u p 6 in Lloc (R3 ) for all 1 ≤ p < 3−2s . Hence, in view of (g1 ) and (g2 ), we deduce that, for any ρ > R, lim |I Vn,ρ | = 0.

(11.2.20)

n→∞

Putting together (11.2.15), (11.2.16), (11.2.18), (11.2.19) and (11.2.20), and recalling that un ε → t0 we complete the proof.   The previous lemmas enable us to prove the following result. Proposition 11.2.9 The functional Jε satisfies the (PS)d condition in Hε at any level d ∈ R. Proof Let (un ) ⊂ Hε be a Palais–Smale sequence for Jε at the level d. By Lemma 11.2.6, (un ) is bounded in Hε . Thus, up to a subsequence, in Hε .

un  u

(11.2.21)

Lemma 11.2.8 implies that 

 lim

n→∞ B R

dx

R3





=

dx BR

|un (x) − un (y)|2 dy + |x − y|3+2s |u(x) − u(y)|2 |x − y|3+2s

R3

 BR

V (ε x)u2n dx (11.2.22)



2

dy +

V (ε x)u dx. BR

Moreover, by Lemma 11.2.7, for each ζ > 0 there exists R = R(ζ ) > independent of ζ , such that  lim sup n→∞

 R3 \BR

dx

R3

|un (x) − un (y)|2 dy + |x − y|3+2s

 R3 \BR

C ζ,

with C > 0

 V (ε x)u2n dx < ζ. (11.2.23)

Putting together (11.2.21), (11.2.22) and (11.2.23) we infer that u2ε ≤ lim inf un 2ε ≤ lim sup un 2ε n→∞

n→∞





= lim sup n→∞

dx BR

R3

|un (x) − un (y)|2 dy + |x − y|3+2s

 BR

V (ε x) u2n dx

11.2 The Modified Kirchhoff Problem

397

  |un (x) − un (y)|2 2 dy + V (ε x) u dx n |x − y|3+2s R3 \BR R3 R3 \BR    |u(x) − u(y)|2 ≤ dx dy + V (ε x)u2 dx + ζ. 3+2s 3 |x − y| BR R BR 

+



dx

Taking the limit as ζ → 0, we have R → ∞, therefore u2ε ≤ lim inf un 2ε ≤ lim sup un 2ε ≤ u2ε , n→∞

n→∞

which implies un ε → uε . Since Hε is a Hilbert space, we deduce that un → u in Hε .   Corollary 11.2.10 The functional ψε satisfies the (PS)d condition on S+ ε at any level d ∈ R. Proof Let (un ) ⊂ S+ ε be a Palais–Smale sequence for ψε at the level d. Then ψε (un ) → d

and

ψε (un ) → 0

∗ in (Tun S+ ε) .

It follows from Proposition 11.2.4-(c) that (mε (un )) is a Palais–Smale sequence for Jε in Hε at the level d. Then, using Proposition 11.2.9, we see that Jε fulfills the (PS)d condition in Hε , so there exists u ∈ S+ ε such that, up to a subsequence, mε (un ) → mε (u)

in Hε .

Applying Lemma 11.2.3-(iii) we conclude that un → u in S+ ε .

 

At this point, we are able to prove the main result of this section. Proof of Theorem 11.2.1 In view of Lemma 11.2.2 and Proposition 11.2.9, we can apply Theorem 2.2.9, and thus obtain the existence of a nontrivial critical point uε of Jε . Now, − we show that uε ≥ 0 in R3 . Since Jε (uε ), u− ε  = 0, where x = min{x, 0}, we see that   M(uε 2ε )  =

R3

R6

− (uε (x) − uε (y))(u− ε (x) − uε (y)) dxdy + |x − y|3−2s

g(ε x, uε )u− ε dx.

 R3

V (ε x)uε u− ε dx



398

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

Recalling that (x − y)(x − − y − ) ≥ |x − − y − |2 and g(x, t) = 0 for t ≤ 0, we deduce that 2 0 ≤ M(uε 2ε )u− ε ε ≤ 0. 2 3 Using (M1 ), we have u− ε ε = 0, that is, uε ≥ 0 in R . Finally, arguing as in the proof of 0,α 3 (R ), and by Theorem 1.3.5 Lemma 11.5.1 below, we can see that uε ∈ L∞ (R3 ) ∩ Cloc we deduce that uε > 0 in R3 .  

11.3

The Autonomous Kirchhoff Problem

In this section we deal with the limit problem associated with (11.2.1), namely, 

M



u∈

 |u(x)−u(y)|2 dxdy + R3 |x−y|3+2s H s (R3 ), u > 0 in R3 . R6

V0 u2 dx [(−)s u + V0 u] = f (u) in R3 , (11.3.1)

The Euler–Lagrange functional associated with (11.3.1) is 1 J0 (u) = M 2

  R6

|u(x) − u(y)|2 dxdy + |x − y|3+2s





V0 u dx −



2

R3

R3

F (u) dx,

and so is well defined on the Hilbert space H0 = H s (R3 ) endowed with the inner product  u, ϕ0 = u, ϕDs,2 (R3 ) +

R3

V0 uϕ dx.

The norm induced by this inner product is

1 2 u0 = [u]2s + V0 u2L2 (R3 ) . The Nehari manifold associated with J0 is given by N0 = {u ∈ H0 \ {0} : J0 (u), u = 0}. Let H0+ denote the open subset of H0 defined as H0+ = {u ∈ H0 : |supp(u+ )| > 0}, + + and let S+ 0 = S0 ∩H0 , where S0 is the unit sphere of H0 . We note that S0 is an incomplete + 1,1 C -manifold of codimension 1 modeled on H0 and contained in H0 . Thus H0 = Tu S+ 0 ⊕

11.3 The Autonomous Kirchhoff Problem

399

+ Ru for each u ∈ S+ 0 , where Tu S0 = {u ∈ H0 : (u, v)0 = 0}. Arguing as in the previous subsection, we can see that the following results hold.

Lemma 11.3.1 Assume that (M1 )–(M3 ) and (f1 )–(f4 ) hold. Then, (i) For each u ∈ H0+ , let h : R+ → R be defined by hu (t) = J0 (tu). Then, there is a unique tu > 0 such that h u (t) > 0 in (0, tu ), h u (t) < 0 in (tu , ∞). (ii) There exists τ > 0 independent of u such that tu ≥ τ for any u ∈ S+ 0 . Moreover, for there is a positive constant C such that tu ≤ CK for any each compact set K ⊂ S+ K 0 u ∈ K. ˆ 0 (u) = tu u is continuous and m0 = m ˆ 0 |S+ is a (iii) The map m ˆ 0 : H0+ → N0 given by m 0

−1 u homeomorphism between S+ 0 and N0 . Moreover m0 (u) = u0 . + (iv) If there is a sequence (un ) ⊂ S+ 0 such that dist(un , ∂S0 ) → 0 then m0 (un )0 → ∞ and J0 (m0 (un )) → ∞.

Let us define the maps ψˆ 0 : H0+ → R

and

ψ0 : S+ 0 → R,

by ψˆ 0 (u) = J0 (m ˆ 0 (u)) and ψ0 = ψˆ 0 |S+ . 0

Proposition 11.3.2 Assume that (M1 )–(M3 ) and (f1 )–(f4 ) hold. Then, (a) ψˆ 0 ∈ C 1 (H0+ , R) and ψˆ 0 (u), v =

m ˆ 0 (u)0

J0 (m ˆ 0 (u)), v u0

for every u ∈ H0+ and v ∈ H0 . (b) ψ0 ∈ C 1 (S+ 0 , R) and ψ0 (u), v = m0 (u)0 J0 (m0 (u)), v, for every v ∈ Tu S+ 0. (c) If (un ) is a Palais–Smale sequence for ψ0 , then (m0 (un )) is a Palais–Smale sequence for J0 . If (un ) ⊂ N0 is a bounded Palais–Smale sequence for J0 , then (m−1 0 (un )) is a Palais–Smale sequence for ψ0 .

400

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

(d) u is a critical point of ψ0 if and only if m0 (u) is a nontrivial critical point for J0 . Moreover, the corresponding critical values coincide and inf ψ0 (u) = inf J0 (u).

u∈S+ 0

u∈N0

Remark 11.3.3 We have the following equalities: c0 = inf J0 (u) = inf max J0 (tu) = inf max J0 (tu). t >0 u∈H+ 0

u∈N0

t >0 u∈S+ 0

Arguing as in the proof of Lemma 11.2.2, it is easy to check that J0 has a mountain pass geometry. Then we can apply a variant of the mountain pass theorem without the Palais– Smale condition (see Remark 2.2.10) to find a Palais–Smale sequence of J0 at level c0 (which coincides with the mountain pass level of J0 ). The next lemma is very important because it allows us to deduce that the weak limit of a Palais–Smale sequence at the level d is nontrivial. Lemma 11.3.4 Let (un ) ⊂ H0 be a Palais–Smale sequence for J0 at the level d with un  0 in H0 . Then (a) either un → 0 in H0 , or (b) there are a sequence (yn ) ⊂ R3 and constants R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Proof Assume that (b) does not hold. Then, for every R > 0,  lim sup

n→∞

y∈R3

BR (y)

u2n dx = 0.

Since (un ) is bounded in H0 , Lemma 1.4.4 shows that un → 0

in Lp (R3 )

for any 2 < p < 2∗s .

Moreover, by conditions (f1 ) and (f2 ), we can see that  lim

n→∞ R3

f (un )un dx = 0.

11.3 The Autonomous Kirchhoff Problem

401

Then, using J0 (un ), un  = on (1) and (M1 ), we get  0≤

m0 un 20

≤

M(un 20 )un 20

=

R3

f (un )un dx + on (1) = on (1).  

Therefore, (a) holds true.

Remark 11.3.5 Let us observe that, if u is the weak limit of a Palais–Smale sequence (un ) of J0 at the level c0 , then we can assume that u = 0. Otherwise, we would have un  0 and, if un  0 in H0 , we conclude from Lemma 11.3.4 that there are (yn ) ⊂ R3 and R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Set vn (x) = un (x + yn ). Then we see that (vn ) is a Palais–Smale sequence for J0 at the level c0 , (vn ) is bounded in H0 and there exists v ∈ H0 such that vn  v and v = 0. Theorem 11.3.6 Problem (11.3.1) admits a positive ground state solution. Proof Arguing as in the proof of Lemma 11.2.2, it is easy to check that J0 has a mountain pass geometry. Then there exists a Palais–Smale sequence (un ) ⊂ H0 for J0 at the level c0 , that is, J0 (un ) → c0

and

J0 (un ) → 0

in H0∗ .

We claim that (un ) is a bounded sequence in H0 . Indeed, using (11.2.8) and assumptions (f3 ) and (M1 ), we have 1

J (un ), un  ϑ 0  1 1 1 2 2 2 = M(u [f (u)u − F (u)] dx n 0 ) − M(un 0 )un 0 + 2 ϑ ϑ R3   1 m0 1 ≥ − M(un 20 )un 20 + un 20 4 ϑ 4

C(1 + un 0 ) ≥ J0 (un ) −

≥

ϑ −2 m0 un 20 , 2ϑ

which yields the boundedness of (un ) because ϑ > 4. Therefore, in view of Theorem 1.1.8 and Remark 11.3.5, we may assume that there exists u ∈ H0 , u = 0, such that un  u

in H0 ,

(11.3.2)

402

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

un → u

p

for all 2 ≤ p
0, we have 1 c0 ≤ J0 (tu) = J0 (tu) − J0 (tu), tu 4    1 1 1 2 2 2 M(tu f (tu)tu − F (tu) dx ) − )tu + = M(tu 0 0 0 2 4 R3 4

11.3 The Autonomous Kirchhoff Problem

403

   1 1 1 2 2 2 f (u)u − F (u) dx < M(u0 ) − M(u0 )u0 + 2 4 R3 4      1 1 1 2 2 2 f (un )un − F (un ) dx ≤ lim inf M(un 0 ) − M(un 0 )un 0 + n→∞ 2 4 R3 4   1 = lim inf J0 (un ) − J0 (un ), un  = c0 , n→∞ 4 so we reached a contradiction. Therefore, combining (11.3.5), (11.3.6), (11.3.7) and (11.3.8) we obtain  M(u20 )u, ϕ0 =

R3

for any ϕ ∈ Cc∞ (R3 ).

f (u)ϕ dx

Since Cc∞ (R3 ) is dense in H0 , we deduce that J0 (u) = 0. Moreover, a similar calculation as done above with t = 1 yields u ∈ N0 . Let us show that u > 0 in R3 . First we prove that u ≥ 0. Indeed, observing that J0 (u), u−  = 0 and using the inequality (x − y)(x − − y − ) ≥ |x − − y − |2 and the fact that f (t) = 0 for t ≤ 0, we have 0 ≤ M(u0 20 )u− 20 ≤ 0, which together with (M1 ) implies that u ≥ 0 and u ≡ 0. Next, we claim that u ∈ C 1,α (R3 ) for some α ∈ (0, 1). Let v=

u 1

.

M(u20 ) q−2

Then v is a nonnegative solution to in R3

(−)s v + V0 v = h(v) where h is the continuous function 1

h(t) =

f (tM(u20 ) q−2 ) q−1

.

M(u20 ) q−2

Using the conditions (f1 ) and (f3 ), we can see that 1

1

f (tM(u20 ) q−2 ) (tM(u20 ) q−2 )3 h(t) = lim lim =0 1 q−1 t →0 t t →0 (tM(u2 ) q−2 )3 M(u2 ) q−2 0

0

404

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

and 1

f (tM(u20 ) q−2 ) h(t) lim q−1 = lim = 0, 1 q−1 t →∞ t t →∞ 2 q−2 tM(u0 ) so we deduce that h(t) ≤ C(|t| + |t|q−1 ) for any t ∈ R. Using Proposition 3.2.14 we deduce that v ∈ L∞ (R3 ). Since s > 34 > 12 , Proposition 1.3.2 shows that v ∈ C 1,α (R3 ) for any α < 2s − 1. From Proposition 1.3.11-(ii) (or Theorem 1.3.5), we get v > 0 in R3 . Therefore, u ∈ C 1,α (R3 ) is a positive solution to (11.3.1) and this ends the proof of theorem.   The next lemma is a compactness result for the autonomous problem that will be used later. Lemma 11.3.7 Let (un ) ⊂ N0 be a sequence such that J0 (un ) → c0 . Then (un ) has a convergent subsequence in H s (R3 ). Proof Since (un ) ⊂ N0 and J0 (un ) → c0 , we can apply Lemma 11.3.1-(iii), Proposition 11.3.2-(d) and Remark 11.3.3 to infer that vn = m−1 (un ) =

un ∈ S+ 0 un 0

and ψ0 (vn ) = J0 (un ) → c0 = inf ψ0 (v). v∈S+ 0

+

Let us introduce the map F : S0 → R ∪ {∞} defined by ⎧ ⎨ψ (u), 0 F (u) = ⎩∞,

if u ∈ S+ 0, if u ∈ ∂S+ 0.

We note that +

• (S0 , d0 ), where d(u, v) = u − v0 , is a complete metric space; + • F ∈ C(S0 , R ∪ {∞}), by Lemma 11.3.1-(iii); • F is bounded below, by Proposition 11.3.2-(d). Hence, by applying Theorem 2.2.1 to F , we can find (vˆn ) ⊂ S+ 0 such that (vˆn ) is a Palais– Smale sequence for ψ0 at the level c0 and vˆn − vn 0 = on (1). Then, using Proposition

11.4 Multiple Solutions for (11.2.2)

405

11.3.2, Theorem 11.3.6 and arguing as in the proof of Corollary 11.2.10 we obtain the desired conclusion.   Lemma 11.3.8 lim supε→0 cε = c0 . Proof Arguing as in the proof of Lemma 7.2.7 and using the continuity of M, we see that lim sup cε ≤ c0 . ε→0

To complete the proof, note that, by (V1 ), lim infε→0 cε ≥ c0 .

11.4

 

Multiple Solutions for (11.2.2)

In this section, our main purpose is to apply the Lusternik–Schnirelman category theory to prove a multiplicity result for the problem (11.2.2). We begin with several technical results. Lemma 11.4.1 Let εn → 0 and (un ) = (uεn ) ⊂ Nεn be such that Jεn (un ) → c0 . Then there exists (y˜n ) = (y˜εn ) ⊂ R3 such that the translated sequence u˜ n (x) = un (x + y˜n ) has a subsequence which converges in H s (R3 ). Moreover, up to a subsequence, (yn ) = (εn y˜n ) is such that yn → y0 ∈ . Proof Since Jε n (un ), un  = 0 and Jεn (un ) → c0 , it is easy to see that (un ) is bounded in Hεn . Let us observe that un εn  0, since c0 > 0. Therefore, arguing as in Remark 11.3.5, we can find a sequence (y˜n ) ⊂ R3 and constants R, α > 0 such that  lim inf n→∞

BR (y˜n )

u2n dx ≥ α.

Set u˜ n (x) = un (x + y˜n ). Then it is clear that (u˜ n ) is bounded in H s (R3 ), and we may assume that u˜ n  u˜

in H s (R3 ),

406

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

for some u˜ = 0. Let (tn ) ⊂ (0, ∞) be such that v˜n = tn u˜ n ∈ N0 and set yn = εn y˜n . Then, using (M2 ) and the fact that g(x, t) ≤ f (t), we have 1 2 2 c0 ≤ J0 (v˜n ) = M(t n un 0 ) − 2 ≤

 R3

1 2 M(tn un 2εn ) − 2

F (tn un ) dx



R3

G(ε x, tn un ) dx

= Jεn (tn un ) ≤ Jεn (un ) = c0 + on (1), which shows that J0 (v˜n ) → c0 and (v˜n ) ⊂ N0 .

(11.4.1)

In particular, (11.4.1) implies that (v˜n ) is bounded in H s (R3 ), so we may assume that ˜ Obviously, (tn ) is bounded and tn → t0 ≥ 0. If t0 = 0, then the boundedness v˜n  v. of (u˜ n ) implies that v˜n 0 = tn u˜ n 0 → 0, that is J0 (v˜n ) → 0, in contrast with the fact c0 > 0. Then, t0 > 0. From the uniqueness of the weak limit, we have v˜ = t0 u˜ and u˜ = 0. By (11.4.1) and Lemma 11.3.7 we deduce that v˜n → v˜

in H s (R3 ),

(11.4.2)

v˜n v˜ → = u˜ in H s (R3 ) and tn t0

which implies that u˜ n =

˜ = c0 and J0 (v), ˜ v ˜ = 0. J0 (v) Let us show that (yn ) has a subsequence such that yn → y0 ∈ . Assume, by contradiction, that (yn ) is not bounded, that is, there exists a subsequence, still denoted by (yn ), such that |yn | → ∞. Since un ∈ Nεn , we see that  m0 u˜ n 20

≤

R3

g(ε n x + yn , u˜ n )u˜ n dx.

Take R > 0 such that ⊂ BR . We may assume that |yn | > 2R, so, for any x ∈ BR/ εn we get | εn x + yn | ≥ |yn | − | εn x| > R. Then,  m0 u˜ n 20

f˜(u˜ n )u˜ n dx +

≤ BR/ εn

 R3 \BR/ εn

f (u˜ n )u˜ n dx.

11.4 Multiple Solutions for (11.2.2)

407

Since u˜ n → u˜ in H s (R3 ), the dominated convergence theorem shows that  R3 \BR/ εn

Recalling that f˜(u˜ n ) ≤

V0 ˜ n, Ku

f (u˜ n )u˜ n dx = on (1).

we get

m0 u˜ n 20 ≤

1 K

 BR/ εn

V0 u˜ 2n dx + on (1),

which yields   1 m0 − u˜ n 20 ≤ on (1). K Since u˜ n → u˜ = 0 in H s (R3 ), we have a contradiction. Thus (yn ) is bounded and, up / , then there exists r > 0 to a subsequence, we may assume that yn → y0 . If y0 ∈ such that yn ∈ Br/2 (y0 ) ⊂ R3 \ for any n large enough. Reasoning as before, we get a contradiction. Hence, y0 ∈ . Now, we prove that V (y0 ) = V0 . Assume, by contradiction, that V (y0 ) > V0 . Taking into account (11.4.2), Fatou’s lemma and the translation invariance of R3 , we have 

1 c0 < lim inf M n→∞ 2

 R3



  s 2 2 2 |(−) v˜n | + V (εn z + yn )v˜n dz −

R3

 F (v˜n ) dx

≤ lim inf Jεn (tn un ) ≤ lim inf Jεn (un ) = c0 , n→∞

n→∞

which gives a contradiction. Therefore, V (y0 ) = V0 and using (V2 ) we deduce that y0 ∈ .   Our next objective is to relate the number of positive solutions of (11.2.2) to the topology of the set . Take δ > 0 such that δ = {x ∈ R3 : dist(x, ) ≤ δ} ⊂ . Let η be a smooth nonincreasing cut-off function defined in [0, ∞), such that η = 1 in [0, 2δ ], η = 0 in [δ, ∞), 0 ≤ η ≤ 1 and |η | ≤ c for some c > 0. For any y ∈ , we define  ε,y (x) = η(| ε x − y|)w

εx −y ε

 ,

where w ∈ H s (R3 ) is a positive ground state solution to the autonomous problem (11.3.1) (such a solution exists in view of Theorem 11.3.6).

408

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

Let tε > 0 be the unique number such that max Jε (tε,y ) = Jε (tε ε,y ). t ≥0

Finally, we consider ε :  → Nε defined by ε (y) = tε ε,y . Lemma 11.4.2 The functional ε has the property that lim Jε (ε (y)) = c0 ,

uniformly in y ∈ .

ε→0

Proof Assume, by contradiction, that there exist δ0 > 0, (yn ) ⊂  and εn → 0 such that |Jεn (εn (yn )) − c0 | ≥ δ0 .

(11.4.3)

ε n x − yn , if z ∈ B δ , then εn z ∈ Bδ and thus εn z+yn ∈ εn εn Bδ (yn ) ⊂ δ ⊂ . Since G = F in × R we have

With the change of variable z =

  s 1 2 |(−) 2 (η(| εn z|)w(z))|2 dz M tεn 2 R3   2 + V (εn z + yn )(η(| εn z|)w(z)) dz

Jε (εn (yn )) =

 −

R3

R3

F (tεn η(| εn z|)w(z)) dz.

(11.4.4)

Let us to show that the sequence (tεn ) is such that tεn → 1 as n → ∞. By the definition of tεn , it follows that Jε n (εn (yn )), εn (yn ) = 0, which gives M(tε2n A2n ) tε2n A2n

1 = 4 An



 R3

 f (tεn η(| εn z|)w(z)) (η(| εn z|)w(z))4 dz (tεn η(| εn z|)w(z))3

(11.4.5)

where  A2n

=



s 2

R3

|(−) (η(| εn z|)w(z))| dz + 2

Since η(|x|) = 1 for x ∈ B δ and B δ ⊂ B 2

M(tε2n A2n ) tε2n A2n

2

≥

1 A4n

δ 2 εn



 Bδ

2

R3

V (εn z + yn )(η(| εn z|)w(z))2 dz.

for all n large enough, (11.4.5) shows that  f (tεn w(z)) w4 (z) dz. (tεn w(z))3

11.4 Multiple Solutions for (11.2.2)

409

By the continuity of w we can find a vector zˆ ∈ R3 such that w(ˆz) = min w(z) > 0. z∈B δ

2

Then, using (f4 ), we deduce that M(tε2n A2n ) tε2n A2n

≥

    1 f (tεn w(ˆz)) 1 f (tεn w(ˆz)) 4 w (z) dz ≥ w4 (ˆz)|B δ |. 4 3 2 A4n (tεn w(ˆz))3 A (t w(ˆ z )) ε Bδ n n 2

(11.4.6) Now, assume by contradiction that tεn → ∞. Let us observe that Lemma 1.4.8 and the dominated convergence theorem yield εn ,yn 2εn = A2n → w20 ∈ (0, ∞).

(11.4.7)

Using that tεn → ∞, (M3 ) and (11.4.7), we see that lim

n→∞

M(tε2n A2n ) tε2n A2n

≤ lim

γ (1 + tε2n A2n )

n→∞

tε2n A2n

≤ C < ∞.

(11.4.8)

On the other hand, assumption (f3 ) implies that f (tεn w(ˆz)) = ∞. n→∞ (tε n w(ˆ z))3 lim

(11.4.9)

Putting together (11.4.6), (11.4.8) and (11.4.9) we get a contradiction. Therefore, (tεn ) is bounded and, up to subsequence, we may assume that tεn → t0 for some t0 ≥ 0. Let us prove that t0 > 0. Suppose, by contradiction, that t0 = 0. Then, taking into account (11.4.7), (M1 ), (f1 ) and (f2 ), we see that (11.4.5) yields 0 < m0 w20 ≤ lim

n→∞

  Ctε2n

 R3

w4 dz + Ctεq−2 n

R3

 wq dz = 0,

which is impossible. Hence t0 > 0. Passing to the limit as n → ∞ in (11.4.5), it follows from (11.4.7), the continuity of M and the dominated convergence theorem that  M(t02 w20 ) t0 w20 =

R3

f (t0 w)w dx.

410

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

Since w ∈ N0 , we obtain that M(t02 w20 ) t02 w20

−

M(w20 ) w20

1 = w40



 R3

 f (t0 w) f (w) w4 dx. − (t0 w)3 w3

(11.4.10)

If t0 > 1, then by (M3 ) and (f4 ), the left-hand side of (11.4.10) is negative and the righthand side is positive. A similar argument works when t0 < 1. Therefore, t0 = 1. Then, taking the limit as n → ∞ in (11.4.4) and using that tεn → 1, that 

 R3

F (η(| εn z|)w(z)) dz →

R3

F (w) dz,

and (11.4.7), we obtain lim Jεn (εn ,yn ) = J0 (w) = c0 ,

n→∞

 

which contradicts (11.4.3).

For δ > 0, take ρ = ρ(δ) > 0 such that δ ⊂ Bρ , and consider the map ϒ : R3 → R3 given by  ϒ(x) =

x, ρx |x| ,

if |x| < ρ, if |x| ≥ ρ.

Now define the barycenter map βε : Nε → R3 by  ϒ(ε x)u2 (x) dx

βε (u) =

R3 

. 2

R3

u (x) dx

Arguing as in the proof of Lemma 6.3.18 we can prove the next result. Lemma 11.4.3 The function βε has the property that lim βε (ε (y)) = y,

ε→0

uniformly in y ∈ .

ε of Nε by taking a function h1 : R+ → R+ such At this point, we introduce a subset N that h1 (ε) → 0 as ε → 0, and setting ε = {u ∈ Nε : Jε (u) ≤ c0 + h1 (ε)} . N

11.4 Multiple Solutions for (11.2.2)

411

ε = ∅ for In view of Lemma 11.4.2, h1 (ε) = supy∈ |Jε (ε (y))−c0| → 0 as ε → 0, so N all ε > 0. Furthermore, arguing as in the proof of Lemma 6.3.19, we deduce the following assertion. Lemma 11.4.4 lim sup dist(βε (u), δ ) = 0.

ε→0

ε u∈N

Now we are in the position to prove the following multiplicity result. Theorem 11.4.5 Assume that (M1 )–(M3 ), (V1 )–(V2 ) and (f1 )–(f4 ) hold. Then, given δ > 0 such that δ ⊂ , there exists ε¯ δ > 0 such that, for any ε ∈ (0, ε¯ δ ), problem (11.2.2) has at least catδ () positive solutions. −1 Proof Let ε > 0, and consider the map αε :  → S+ ε defined as αε (y) = mε (ε (y)). Using Lemma 11.4.2, we see that

lim ψε (αε (y)) = lim Jε (ε (y)) = c0

ε→0

ε→0

uniformly in y ∈ .

(11.4.11)

Set Sε+ = {w ∈ S+ ε : ψε (w) ≤ c0 + h1 (ε)}, where h1 (ε) = supy∈ |ψε (αε (y)) − c0 |. It follows from (11.4.11) that h1 (ε) → 0 as ε → 0. Moreover, αε (y) ∈ Sε+ for all y ∈  and this shows that Sε+ = ∅ for all ε > 0. In the light of Lemmas 11.4.2, 11.2.3-(iii), 11.4.4, and 11.4.3, we can find ε¯ = ε¯ δ > 0 such that the following diagram ε

m−1 ε

mε

βε

 −→ ε () −→ αε () −→ ε () −→ δ is well defined for any ε ∈ (0, ε¯ ). Thanks to Lemma 11.4.3, and decreasing ε¯ if necessary, we see that βε (ε (y)) = y + θ (ε, y) for all y ∈ , for some function θ (ε, y) satisfying |θ (ε, y)| < 2δ uniformly in y ∈  and for all ε ∈ (0, ε¯ ). Define H (t, y) = y + (1 − t)θ (ε, y). Then H : [0, 1] ×  → δ is continuous. Clearly, H (0, y) = βε (ε (y)) and H (1, y) = y for all y ∈ . Consequently, H (t, y) is a homotopy between βε ◦ ε = (βε ◦ mε ) ◦ (m−1 ε ◦ ε ) and the inclusion map id:  → δ . This fact together with Lemma 6.3.21 yields catαε () αε () ≥ catδ ().

(11.4.12)

412

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

Applying Corollary 11.2.10, Lemma 11.3.8, and Theorem 2.4.6 with c = cε ≤ c0 + h1 (ε) = d and K = αε (), we infer that ψε has at least catαε () αε () critical points on Sε+ . Taking into account Proposition 11.2.4-(d) and (11.4.12), we conclude that Jε admits ε . at least catδ () critical points in N  

11.5

Proof of Theorem 11.1.1

In this last section, we provide the proof of Theorem 11.1.1. First, we establish the following useful L∞ -estimate for the solutions of the modified problem (11.2.2). εn be a solution to (11.2.2). Then, up to a Lemma 11.5.1 Let εn → 0 and un ∈ N ∞ N subsequence, u˜ n = un (· + y˜n ) ∈ L (R ), and there exists C > 0 such that u˜ n L∞ (R3 ) ≤ C

for all n ∈ N.

Moreover, u˜ n (x) → 0 as |x| → ∞ uniformly in n ∈ N. Proof Since Jεn (un ) ≤ c0 + h(εn ) with h(εn ) → 0 as n → ∞, we can proceed as in the proof of (11.4.1) to deduce that Jεn (un ) → c0 . Thus, we may invoke Lemma 11.4.1 to find a sequence (y˜n ) ⊂ R3 such that εn y˜n → y0 ∈  and u˜ n = un (· + y˜n ) admits a convergent subsequence in H s (R3 ). Now, arguing as in the proof of Lemma 6.3.23, and using (M1 ), (g1 ) and (g2 ), we have  2  1 1 β−1 2(β−1) S∗−1 u˜ n u˜ n,L 2 2¯ ∗s 3 ≤ [g(εn x + εn y˜n , u˜ n ) − m0 V (εn x + εn y˜n )]u˜ n u˜ n,L dx L (R ) β m0 R3  2∗ 2(β−1) u˜ ns u˜ n,L dx. ≤C R3

Hence, we can argue as in the proof of Lemma 6.3.23 to obtain the required L∞ -estimate. We note that the s-harmonic extension w˜ n (x, y) = Ext(u˜ n ) = Ps (x, y) ∗ u˜ n (x) of u˜ n solves ⎧ 1−2s ∇ w ⎪ ˜ n) = 0 in R3+1 + , ⎨ − div(y on ∂R3+1 w˜ n (·, 0) = u˜ n + , ⎪ ⎩ ∂wn = 3+1 1 [−V (ε x + ε y ˜ ) u ˜ + g(ε x + ε y ˜ , u ˜ )] on ∂R n n n n n n n n + , M(un ε ) ∂ν 1−2s n

and that m0 ≤ M(un εn ) ≤ C

for all n ∈ N.

(11.5.1)

11.5 Proof of Theorem 11.1.1

413

0,α Applying Proposition 1.3.11-(iii), we see that u˜ n ∈ L∞ (R3 ) ∩ Cloc (R3 ). Next, using ∞ (11.5.1) and the L -estimate, we argue as in the proof of Lemma 6.3.23 to conclude that u˜ n (x) → 0 as |x| → ∞ uniformly in n ∈ N (see also Remark 7.2.10 and note that u˜ n is a subsolution to (−)s u˜ n + V0 u˜ n = m10 g(ε n x + εn y˜n , u˜ n ) in R3 ).  

Now, we are able to give the proof of our main result. Proof of Theorem 11.1.1 Take δ > 0 such that δ ⊂ . We begin by proving that there ε of (11.2.2), exists ε˜ δ > 0 such that for any ε ∈ (0, ε˜ δ ) and any solution uε ∈ N uε L∞ (R3 \ ε ) < a.

(11.5.2)

Suppose, by contradiction, that for some subsequence (εn ) such that εn → 0, we can find εn such that Jε (un ) = 0 and un = uε n ∈ N n un L∞ (R3 \ εn ) ≥ a.

(11.5.3)

Since Jεn (un ) ≤ c0 + h1 (εn ) and h1 (εn ) → 0, we can proceed as in the first part of the proof of Lemma 11.4.1 to deduce that Jεn (un ) → c0 . Then, by Lemma 11.4.1, there exists (y˜n ) ⊂ R3 such that u˜ n = un (· + y˜n ) → u˜ in H s (R3 ) and εn y˜n → y0 ∈ . Now, if we choose r > 0 such that Br (y0 ) ⊂ B2r (y0 ) ⊂ , we see that B εr ( εyn0 ) ⊂ εn . n Moreover, for any y ∈ B εr (y˜n ), n

        y − y0  ≤ |y − y˜n | + y˜n − y0  < 1 (r + on (1)) < 2r    εn εn  εn εn

for n sufficiently large.

Therefore, R3 \ εn ⊂ R3 \ B εr (y˜n ) n

for n sufficiently large. On the other hand, by Lemma 11.5.1, u˜ n (x) → 0

as |x| → ∞

uniformly in n ∈ N. Therefore, there exists R > 0 such that u˜ n (x) < a

for all |x| ≥ R and all n ∈ N.

(11.5.4)

414

11 Multiplicity and Concentration of Positive Solutions for a Fractional. . .

Hence, un (x) < a for all x ∈ R3 \ BR (y˜n ) and n ∈ N. On the other hand, there exists ν ∈ N such that for any n ≥ ν and r/ εn > R, we have R3 \ εn ⊂ R3 \ B εr (y˜n ) ⊂ R3 \ BR (y˜n ), n

which implies that un (x) < a for all x ∈ R3 \ εn and n ≥ ν. This contradicts (11.5.3). Let ε¯ δ > 0 be given by Theorem 11.4.5, and fix ε ∈ (0, εδ ), where εδ = min{˜εδ , ε¯ δ }. In view of Theorem 11.4.5, problem (11.2.2) admits at least catδ () nontrivial solutions. ε satisfies (11.5.2), by the definition of g it Let uε be one of these solutions. Since uε ∈ N follows that uε is a solution of (11.2.1). Then, uˆ ε (x) = u(x/ ε) is a solution to (11.1.1), and we deduce that (11.1.1) has at least catδ () solutions. Finally, we study the behavior of the maximum points of solutions to (11.2.1). Take εn → 0 and consider a sequence (un ) ⊂ Hεn of solutions to (11.2.1) as above. Condition (g1 ) ensurers that we can find γ > 0 such that g(ε x, t)t ≤

V0 2 t K

for all x ∈ R3 and 0 ≤ t ≤ γ .

(11.5.5)

Arguing as before, we can find R > 0 such that un L∞ (BRc (y˜n )) < γ .

(11.5.6)

Moreover, up to extracting a subsequence, we may assume that un L∞ (BR (y˜n )) ≥ γ .

(11.5.7)

Indeed, if (11.5.7) does not hold, it follows from (11.5.6) that un L∞ (R3 ) < γ . Then, using Jε n (un ), un  = 0 and (11.5.5) we can infer that 

m0 un 2εn

V0 ≤ g(εn x, un )un dx ≤ 3 K R

 R3

u2n dx

which yields un εn = 0, and this is impossible. Consequently, (11.5.7) holds. Taking into account (11.5.6) and (11.5.7), we deduce that if pn ∈ R3 is a global maximum point of un , then pn ∈ BR (y˜n ). Therefore, pn = y˜n + qn for some qn ∈ BR . Hence, ηn = εn y˜n + εn qn is a global maximum point of uˆ n (x) = un (x/ εn ). Since |qn | < R for all n ∈ N and since εn y˜n → y0 ∈  (in view of Lemma 11.4.1), it follows from the continuity of V that lim V (ηn ) = V (y0 ) = V0 .

n→∞

 

11.5 Proof of Theorem 11.1.1

415

Remark 11.5.2 In [62] we considered the problem 

ε2s M(ε2s−N [u]2s )(−)s u + V (x)u = f (u) in RN , in RN , u ∈ H s (RN ), u > 0

where s ∈ (0, 1), N ≥ 2, V : RN → R is a continuous potential satisfying the local conditions (V1 )–(V2 ) introduced in Chap. 7, the nonlinearity f : R → R is a continuous function such that f (t) = 0 for t ≤ 0 and f fulfills the following Berestycki-Lions type assumptions [100]: (f1 ) (f2 ) (f3 ) (f4 )

0,α f ∈ Cloc (R) for some α ∈ (1 − 2s, 1) if s ∈ (0, 12 ], limt →0 f (tt ) = 0, lim supt →∞ ft(tp ) < ∞ for some p ∈ (1, 2∗s − 1), t there exists T > 0 such that F (T ) > V20 T 2 , where F (t) = 0 F (τ ) dτ ,

and the Kirchhoff term M : [0, ∞) → R+ is a continuous function such that: (M1) there exists %m0 > 0 such that M(t) ≥ & m0 for all t ≥ 0, 2s  (M2) lim inft →∞ M(t) − (1 − N )M(t)t = ∞, 2s

(M3) M(t)/t N−2s → 0 as t → ∞, (M4) M is nondecreasing in [0, ∞), 2s (M5) t → M(t)/t N−2s is nonincreasing in (0, ∞). Clearly, M(t) = m0 + bt, with b ≥ 0, satisfies (M1)–(M5) when b = 0, N ≥ 2, s ∈ (0, 1), and when b > 0, N = 3, s ∈ ( 34 , 1). By combining the penalization approaches in [165] and [121], we established the existence of a family of positive solutions (uε ) which concentrates at a local minimum of V as ε → 0.

12

Concentrating Solutions for a Fractional Kirchhoff Equation with Critical Growth

12.1

Introduction

This chapter is devoted to the existence and concentration of positive solutions for the following fractional Kirchhoff type equation with critical nonlinearity:

  s ∗ ε2s a + ε4s−3 b R3 |(−) 2 u|2 dx (−)s u + V (x)u = f (u) + |u|2s −2 u in R3 , u ∈ H s (R3 ),

u > 0 in R3 , (12.1.1)

6 where ε > 0 is a small parameter, a, b > 0 are constants, s ∈ ( 34 , 1) is fixed, 2∗s = 3−2s 3 is the fractional critical exponent. The potential V : R → R is a continuous function satisfying the following conditions:

(V1 ) (V2 )

V1 = infx∈R3 V (x) > 0; there exists a bounded open set  ⊂ R3 such that 0 < V0 = inf V < min V . 

∂

The nonlinearity f : R → R is a continuous function which fulfills the following hypotheses: (f1 ) (f2 )

f (t) = o(t 3 ) as t → 0; there exist q, σ ∈ (4, 2∗s ), C0 > 0 such that f (t) ≥ C0 t q−1

∀t > 0,

lim

f (t)

t →∞ t σ −1

= 0;

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_12

417

418

(f3 ) (f4 )

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

there exists ϑ ∈ (4, σ ) such that 0 < ϑF (t) ≤ tf (t) for all t > 0; the function t → ft(t3 ) is increasing in (0, ∞).

Since we will look for positive solutions to (12.1.1), we assume that f (t) = 0 for t ≤ 0. We note that when a = 1, b = 0 and R3 is replaced by RN , then (12.1.1) reduces to the fractional Schrödinger equation ε2s (−)s u + V (x)u = h(x, u)

in RN

(12.1.2)

studied in Chaps. 6, 7, and 8. On the other hand, if we set s = ε = 1 and we replace ∗ f (u) + |u|2s −2 u by a more general nonlinearity h(x, u), then (12.1.1) becomes the wellknown classical Kirchhoff equation    2 |∇u| dx u + V (x)u = h(x, u) in R3 . (12.1.3) − a+b R3

We recall that He and Zou [215] obtained existence and multiplicity results for small ε > 0 for the perturbed Kirchhoff equation   2 − aε + bε

 2

R3

|∇u| dx u + V (x)u = g(u)

in R3 ,

(12.1.4)

where the potential V satisfies condition (V ) and g is a subcritical nonlinearity. Wang et al. [331] studied the multiplicity and concentration phenomenon for (12.1.4) when g(u) = λf (u) + |u|4 u, f is a continuous subcritical nonlinearity and λ is large. Figueiredo and Santos Junior [191] used the generalized Nehari manifold method to obtain a multiplicity result for a subcritical Kirchhoff equation under conditions (V1 )–(V2). He et al. [214] dealt with the existence and multiplicity of solutions to (12.1.4), where g(u) = f (u) + u5 , f ∈ C 1 is a subcritical nonlinearity which does not satisfies the Ambrosetti–Rabinowitz condition [29] and V fulfills (V1 )–(V2). In this chapter we study the existence and concentration behavior of solutions to (12.1.1) under assumptions (V1 )–(V2 ) and (f1 )–(f4 ). More precisely, our main result can be stated as follows: Theorem 12.1.1 ([50]) Assume that (V1 )–(V2) and (f1 )–(f4 ) hold. Then, there exists ε0 > 0 such that, for each ε ∈ (0, ε0 ), problem (12.1.1) has a positive solution uε . Moreover, if ηε denotes a global maximum point of uε , then lim V (ηε ) = V0 ,

ε→0

and there exists a constant C > 0 such that 0 < uε (x) ≤

ε3+2s

Cε3+2s + |x − ηε |3+2s

for all x ∈ R3 .

12.2 The Modified Critical Problem

419

The proof of Theorem 12.1.1 will be carried out by means of appropriate variational arguments. After considering the ε-rescaled problem associated with (12.1.1), we use a variant of the penalization technique introduced in [165] (see also [12,190]). The solutions of the modified problem will be obtained as critical points of the modified energy functional Jε which, in view of the growth assumptions on f and the auxiliary nonlinearity, possesses a mountain pass geometry [29]. In order to recover some compactness properties for Jε , we have to circumvent several difficulties which make our study rather delicate. The first one is related to the presence of the Kirchhoff term in (12.1.1) which does not permit to verify in a standard way that if u is the weak limit of a Palais–Smale sequence (un ) for Jε , then u is a weak solution for the modified problem. The second one is due to the lack of compactness caused by the unboundedness of the domain R3 and the critical Sobolev exponent. Anyway, we will be able to overcome these problems looking for critical points of a suitable functional whose quadratic part involves the limit term of (a + b[un ]2s ), and showing that the mountain pass level cε of Jε is strictly less than a threshold value related ∗ to the best constant of the embedding H s (R3 ) in L2s (R3 ). Then, applying the mountain pass lemma, we will deduce the existence of a positive solution for the modified problem. Finally, combining a compactness argument with a Moser iteration procedure [278], we prove that the solution of the modified problem is also a solution to the original one for ε > 0 small enough, and that it decays to zero at infinity with polynomial rate.

12.2

The Modified Critical Problem

In order to study (12.1.1), we use the change of variable x → ε x and look for solutions to  ∗ (a + b[u]2s )(−)s u + V (ε x)u = f (u) + |u|2s −2 u in R3 , (12.2.1) u ∈ H s (R3 ), u > 0 in R3 . Now, we introduce a penalization method in the spirit of [165] which will be crucial for obtaining our main result. First of all, without loss of generality we may assume that 0∈ Let K >

ϑ ϑ−2

and

V (0) = V0 = inf V . 

and a0 > 0 be such that 2∗ −1

f (a0 ) + a0 s

=



∗

V1 a0 K

and we define f˜(t) =

f (t) + (t + )2s −1 , if t ≤ a0 , V1 if t > a0 , K t,

(12.2.2)

420

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

and  g(x, t) =

∗ χ (x)(f (t) + (t + )2s −1 ) + (1 − χ (x))f˜(t), if t > 0, 0, if t ≤ 0.

It is easy to check that g enjoys the following properties: ) (g1 ) limt →0 g(x,t = 0 uniformly with respect to x ∈ R3 ; t3 ∗ (g2 ) g(x, t) ≤ f (t) + t 2s −1 for all x ∈ R3 , t > 0; (g3 ) (i) 0 < ϑG(x, t) ≤ g(x, t)t for all x ∈  and t > 0, (ii) 0 ≤ 2G(x, t) ≤ g(x, t)t ≤ VK1 t 2 for all x ∈ R3 \  and t > 0; ) (g4 ) for each x ∈  the function t → g(x,t is increasing in (0, ∞), and for each x ∈ t3

R3 \  the function t →

g(x,t ) t3

is increasing in (0, a0 ).

Then we consider the following modified problem: 

(a + b[u]2s )(−)s u + V (ε x)u = g(ε x, u) in R3 , u ∈ H s (R3 ), u > 0 in R3 .

(12.2.3)

The corresponding energy functional is given by Jε (u) =

1 b u2ε + [u]4s − 2 4

 R3

G(ε x, u) dx,

and is well defined on the space   Hε = u ∈ H s (R3 ) :

 R3

V (ε x)u2 dx < ∞

endowed with the norm  uε =

1

 a[u]2s

+

2

2

R3

V (ε x)u dx

.

Clearly, Hε is a Hilbert space with the inner product  u, vε = au, vDs,2 (R3 ) +

R3

V (ε x)uv dx.

It is standard to show that Jε ∈ C 1 (Hε , R) and its differential is given by Jε (u), v = u, vε + b[u]2s u, vDs,2 (R3 ) −

 R3

g(ε x, u)v dx

12.2 The Modified Critical Problem

421

for all u, v ∈ Hε . Let us introduce the Nehari manifold associated with (12.2.3), that is, ! " Nε = u ∈ Hε \ {0} : Jε (u), u = 0 . We begin by proving that Jε possesses a nice geometric structure: Lemma 12.2.1 The functional Jε has a mountain pass geometry: (a) there exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ; (b) there exists e ∈ Hε such that eε > ρ and Jε (e) < 0. Proof (a) By assumptions (g1 ) and (g2 ), for every ξ > 0 there exists Cξ > 0 such that 1 Jε (u) ≥ u2ε − 2

 R3

G(ε x, u) dx ≥

1 2∗ u2ε − ξ Cu2ε − Cξ Cuε s . 2

Then, there exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ. (b) Using (g3 )-(i), we deduce that for any u ∈ Cc∞ (R3 ) such that u ≥ 0, u ≡ 0 and supp(u) ⊂ ε , and τ2 τ4 Jε (τ u) = u2ε + b [u]4s − 2 4

 G(ε x, τ u) dx ε

τ2 τ4 ≤ u2ε + b [u]4s − C1 τ ϑ 2 4

 uϑ dx + C2 ,

(12.2.4)

ε

for all τ > 0, with some constants C1 , C2 > 0. Recalling that ϑ ∈ (4, 2∗s ), we conclude that Jε (τ u) → −∞ as τ → ∞.   In view of Lemma 12.2.1, we can use a variant of the mountain pass theorem without the Palais–Smale condition (see Remark 2.2.10) to deduce the existence of a Palais–Smale sequence (un ) ⊂ Hε such that Jε (un ) → cε

and

Jε (un ) → 0 in Hε∗ ,

(12.2.5)

where cε = inf max Jε (γ (t)) γ ∈ε t ∈[0,1]

and

ε = {γ ∈ C([0, 1], Hε ) : γ (0) = 0, Jε (γ (1)) < 0} . (12.2.6)

422

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

As in [299], we use the following equivalent characterization of cε that is more appropriate for our aims: cε =

max Jε (tu).

inf

u∈Hε \{0} t ≥0

Moreover, the monotonicity of g readily implies that for every u ∈ Hε \ {0} there exists a unique t0 = t0 (u) > 0 such that Jε (t0 u) = max Jε (tu). t ≥0

In the next lemma, we will see that cε is less then a threshold value involving the best ∗ constant S∗ of Sobolev embedding of Ds,2 (R3 ) in L2s (R3 ). More precisely: Lemma 12.2.2 There exists T > 0 such that 0 < cε
0 is such that B2ρ ⊂ . For simplicity, we assume that ρ = 1. We know (see Remark 1.1.9) that S∗ is achieved by U (x) = 3−2s κ(μ2 + |x − x0 |2 )− 2 , with κ ∈ R, μ > 0 and x0 ∈ R3 . Taking x0 = 0, as in [310], we define vδ (x) = η(εx)uδ (x) ∀δ > 0, where 1

uδ (x) = δ

− 3−2s ∗ 2

u (x/δ)

and

U (x/S∗2s ) u (x) = . U L2∗s (R3 ) ∗

2∗

∗

3

Then (−)s uδ = |uδ |2s −2 uδ in R3 and [uδ ]2s = uδ  s2∗s 3 = S∗2s . We also recall the L (R ) following useful estimates (see [310]): 3

Aδ = [vδ ]2s = S∗2s + O(δ 3−2s ),

(12.2.7)

Bδ = vδ 2L2 (R3 ) = O(δ 3−2s ),

(12.2.8)

12.2 The Modified Critical Problem

Cδ =

q vδ Lq (R3 )

2∗s

Dδ = vδ 

423

⎧ 3− (3−2s)q ⎪ 2 ⎪ ), if q > ⎨ O(δ (3−2s)q 1 3− ≥ O(log( δ )δ 2 ), if q = ⎪ ⎪ ⎩ O(δ (3−2s)q 2 ), if q
0 there exists t0 > 0 such that Jε (γδ (t0 )) < 0, where γδ (t) = vδ (·/t). Indeed, setting V2 = maxx∈ V (x), by (f2 ) we have a 3−2s V2 3 b t3 t3 2∗ q t t vδ 2L2 (R3 ) + t 6−4s [vδ ]4s − ∗ vδ  s2∗s 3 − vδ Lq (R3 ) C0 [vδ ]2s + L (R ) 2 2 4 2s q   Dδ a b C0 Cδ 3 Bδ − ∗ − t . = t 3−2s Aδ + A2δ t 6−4s + V2 (12.2.11) 2 4 2 2s q

Jε (γδ (t )) ≤

Since 0 < 6 − 4s < 3, we use (12.2.8) to deduce that V2

Dδ Bδ 1 3 − ∗ → − ∗ S∗2s 2 2s 2s

as δ → 0. Hence, using (12.2.7), we see that for every δ > 0 sufficiently small, Jε (γδ (t)) → −∞ as t → ∞, that is, there exists t0 > 0 such that Jε (γδ (t0 )) < 0. Now, as t → 0, we have [γδ (t)]2s + γδ (t)2L2 (R3 ) = t 3−2s Aδ + t 3 Bδ → 0

uniformly for δ > 0 small.

We set γδ (0) = 0. Then γδ (t0 ·) ∈ ε , where ε is defined as in (12.2.6) and we infer that cε ≤ sup Jε (γδ (t)). t ≥0

Since cε > 0, inequality (12.2.11) shows that exists tδ > 0 such that sup Jε (γδ (t)) = Jε (γδ (tδ )). t ≥0

In the light of (12.2.7), (12.2.9) and (12.2.11), we deduce that Jε (γδ (t)) → 0 as t → 0 and Jε (γδ (t)) → −∞ as t → ∞ uniformly for δ > 0 small. Then there exist t1 , t2 > 0 (independent of δ > 0) satisfying t1 ≤ tδ ≤ t2 . Setting Hδ (t) =

aAδ 3−2s bA2δ 6−4s Dδ 3 t t + − ∗t , 2 4 2s

424

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

we thus have,  cε ≤ sup Hδ (t) + t ≥0

C0 Cδ V2 Bδ − 2 q

 tδ3 .

Next, by (12.2.9), for every q ∈ (2, 2∗s ), we have Cδ ≥ O(δ 3− we infer cε ≤ sup Hδ (t) + O(δ 3−2s ) − O(C0 δ 3−

(3−2s)q 2

(3−2s)q 2

t ≥0

Since 3 − 2s > 0 and 3 −

(3−2s)q 2

). Then, by (12.2.8),

).

> 0, we obtain

sup Hδ (t) ≥ t ≥0

cε 2

uniformly for δ > 0 small.

Arguing as above, there exist t3 , t4 > 0 (independent of δ > 0) such that sup Hδ (t) = sup Hδ (t). t ≥0

t ∈[t3 ,t4 ]

By (12.2.7) we deduce 1

cε ≤ sup K(S∗2s t) + O(δ 3−2s ) − O(C0 δ 3−

(3−2s)q 2

t ≥0

),

(12.2.12)

where K(t) =

aS∗ 3−2s bS∗2 6−4s 1 t t + − ∗ t 3. 2 4 2s

Let us note that for t > 0, 3 − 2s 3 − 2s 2 5−4s 3 − 2s 2 aS∗ t 2−2s + bS∗ t t − 2 2 2

(3 − 2s)t 2−2s (3 − 2s)t 2−2s ˜ aS∗ + bS∗2 t 3−2s − t 2s = K(t). = 2 2

K (t) =

Moreover, K˜ (t) = bS∗ (3 − 2s)t 2−2s − 2st 2s−1 = t 2−2s [bS∗2 (3 − 2s) − 2st 4s−3 ]. ˜ ˜ Since 4s > 3, there exists a unique T > 0 such that K(t) > 0 for t ∈ (0, T ) and K(t) T . Then, T is the unique maximum point of K(t). By virtue of (12.2.12), we have, cε ≤ K(T ) + O(δ 3−2s ) − O(C0 δ 3−

(3−2s)q 2

).

(12.2.13)

12.2 The Modified Critical Problem

425

4s If q > 3−2s , then 0 < 3 − (3−2s)q < 3 − 2s, and by (12.2.13), for any fixed C0 > 0, 2 4s , then, for δ > 0 small and it holds that cε < K(T ) for δ > 0 small. If 2 < q < 3−2s

C0 > δ

(3−2s)q −2s−1 2

 

, we also have cε < K(T ).

Lemma 12.2.3 Every sequence (un ) satisfying (12.2.5) is bounded in Hε . Proof In view of (g3 ), we deduce that for all n ∈ N 1 (12.2.14) C(1 + un ε ) ≥ Jε (un ) − Jε (un ), un  ϑ      ϑ −2 ϑ −4 1 = [g(ε x, un )un − ϑG(ε x, un )] dx un 2ε + b [un ]4 + 2ϑ 4ϑ ϑ R3 \ε  1 + [g(ε x, un )un − ϑG(ε x, un )] dx ϑ ε    ϑ −2 1 2 ≥ [g(ε x, un )un − ϑG(ε x, un )] dx un ε + 2ϑ ϑ R3 \ε      ϑ −2 1 ϑ −2 2 V (ε x)u2n dx ≥ un ε − 2ϑ 2ϑ K R3 \ε    ϑ −2 1 (12.2.15) ≥ 1− un 2ε . 2ϑ K Since ϑ > 4 and K > 2, we conclude that (un ) is bounded in Hε .

 

Remark 12.2.4 Arguing as in Remark 5.2.8, we may always suppose that the Palais– Smale sequence (un ) is non-negative in R3 . Lemma 12.2.5 There exist a sequence (zn ) ⊂ R3 and constants R, β > 0 such that  BR (zn )

u2n dx ≥ β.

Moreover, (zn ) is bounded in R3 . Proof Assume, by contradiction, that the first conclusion of lemma is not true. Then, by Lemma 1.4.4, we have un → 0

in Lq (R3 )

∀q ∈ (2, 2∗s ),

426

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

which together with (f1 ) and (f2 ) shows that 

 R3

F (un ) dx =

R3

f (un )un dx = on (1)

as n → ∞.

Since (un ) is bounded in Hε , we may assume that un  u in Hε . Now, we observe that  R3

G(ε x, un ) dx ≤

1 2∗s



∗

ε ∪{un ≤a0 }

2s (u+ n ) dx +

V1 2K

 (R3 \ε )∩{un >a0 }

u2n dx + on (1) (12.2.16)

and 

 R3

g(ε x, un )un dx =

ε ∪{un ≤a0 }

2∗s (u+ n)

V1 dx + K

 (R3 \ε )∩{un >a0 }

u2n dx + on (1). (12.2.17)

Using the fact that Jε (un ), un  = on (1) and (12.2.17), we have un 2ε

V1 − K



 (R3 \ε )∩{un >a0 }

u2n dx

+ b[un ]4s

=

∗

ε ∪{un ≤a0 }

2s (u+ n ) dx + on (1).

(12.2.18) Assume that 

∗

ε ∪{un ≤a0 }

2s 3 (u+ n ) dx → % ≥ 0

and [un ]2s → B 2 . Note that % > 0: otherwise, (12.2.18) yields un ε → 0 as n → ∞ which implies that Jε (un ) → 0, and this is impossible because cε > 0. Then, by (12.2.18) and (1.1.1) we obtain  aS∗

∗

ε ∪{un ≤a0 }

2s (u+ n ) dx

 2∗ 2s

 + bS∗2  ≤

∗

ε ∪{un ≤a0 }

2s (u+ n ) dx ∗

ε ∪{un ≤a0 }



4 2∗ s

2s (u+ n ) dx + on (1).

(12.2.19)

12.2 The Modified Critical Problem

427

Since % > 0, it follows from (12.2.19) that K (%) =

3 − 2s −1 % (aS∗ %3−2s + bS∗2 %6−4s − %3 ) ≤ 0, 2

so we deduce that % ≥ T , where T is the unique maximum of K defined in Lemma 12.2.2. Let us consider the following functional: Iε (u) =

(a + bB 2 ) 2 1 [u]s + 2 2



 R3

V (ε x)u2 dx −

R3

G(ε x, u) dx

b b = Jε (u) − [u]4s + B 2 [u]2s , 4 2

(12.2.20)

and note that (un ) is a Palais–Smale sequence for Iε at the level cε + b4 B 4 , that is b Iε (un ) = cε + B 4 + on (1), 4

Iε (un ) = on (1).

(12.2.21)

Then, since % ≥ T , using (12.2.16), (12.2.21) and (1.1.1) we infer that b cε = Iε (un ) − B 4 + on (1) 4

  bB 2 b 1 V1 a [un ]2s + [un ]2s − B 4 + V (ε x)u2n dx − u2 dx 2 2 4 2 R3 2K (R3 \ε )∩{un >a0 } n  1 2∗s − ∗ (u+ n ) dx + on (1) 2s ε ∪{un ≤a0 }  b 1 a ∗ 2 4 (u+ )2s dx + on (1) ≥ [un ]s + [un ]s − ∗ 2 4 2s ε ∪{un ≤a0 } n  2∗  4∗   2s 2s a b 2 + 2∗s + 2∗s ≥ S∗ (un ) dx + S∗ (un ) dx 2 4 ε ∪{un ≤a0 } ε ∪{un ≤a0 }  1 ∗ − ∗ (u+ )2s dx + on (1) 2s ε ∪{un ≤a0 } n ≥

=

b 1 a S∗ %3−2s + S∗2 %6−4s − ∗ %3 2 4 2s

≥

a b 1 S∗ T 3−2s + S∗2 T 6−4s − ∗ T 3 = c∗ , 2 4 2s

and this gives a contradiction by Lemma 12.2.2. Let us show that (zn ) is bounded in R3 . For any ρ > 0, let ψρ ∈ C ∞ (R3 ) be such that ψρ = 0 in Bρ and ψρ = 1 in R3 \ B2ρ , with 0 ≤ ψρ ≤ 1 and ∇ψρ L∞ (R3 ) ≤ Cρ ,

428

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

where C is a constant independent of ρ. Since (ψρ un ) is bounded in Hε , it follows that Jε (un ), ψρ un  = on (1), that is  (a

+ b[un ]2s )

R6

 = on (1) + − (a

R3

|un (x) − un (y)|2 ψρ (x) dxdy + |x − y|3+2s

 R3

V (ε x)u2n ψρ dx

g(ε x, un )un ψρ dx 

+ b[un ]2s )

R6

(ψρ (x) − ψρ (y))(un (x) − un (y)) un (y) dxdy. |x − y|3+2s

Take ρ > 0 sufficiently large such that ε ⊂ Bρ . Then, using (g3 )-(ii), we get  R6

a



|un (x) − un (y)|2 ψρ (x) dxdy + |x − y|3+2s

 R3

V (ε x)u2n ψρ dx

1 V (ε x)u2n ψρ dx K  (ψρ (x) − ψρ (y))(un (x) − un (y)) − (a + b[un ]2s ) un (y) dxdy + on (1) |x − y|3+2s R6 ≤

R3

which implies that    1 V1 1− u2n ψρ dx 3 K R  (ψρ (x) − ψρ (y))(un (x) − un (y)) 2 un (y) dxdy + on (1). ≤ −(a + b[un ]s ) |x − y|3+2s R6 (12.2.22) Now, by the Hölder inequality and the boundedness on (un ) in Hε ,     

R6

  (un (x) − un (y))(ψρ (x) − ψρ (y)) un (y) dxdy  3+2s |x − y|

  ≤C

R6

|ψρ (x) − ψρ (y)|2 |un (y)|2 dxdy |x − y|3+2s

 12 .

(12.2.23)

On the other hand, recalling that 0 ≤ ψρ ≤ 1 and ∇ψρ L∞ (R3 ) ≤ C/ρ, and using polar coordinates, we obtain 

|ψρ (x) − ψρ (y)|2 |un (x)|2 dxdy |x − y|3+2s R6     |ψρ (x) − ψρ (y)|2 |ψρ (x) − ψρ (y)|2 2 = |u (x)| dxdy + |un (x)|2 dxdy n |x − y|3+2s |x − y|3+2s R3 |y−x|>ρ R3 |y−x|≤ρ

12.2 The Modified Critical Problem

429



    C dy dy 2 dx + dx |u (x)| n 3+2s 3+2s−2 ρ 2 R3 R3 |y−x|>ρ |x − y| |y−x|≤ρ |x − y|       C dz dz 2 ≤C dx + dx |un (x)|2 |u (x)| n 3+2s 1+2s ρ 2 R3 R3 |z|>ρ |z| |z|≤ρ |z|    ∞  ρ   C dρ dρ 2 + |un (x)|2 dx |u (x)| dx ≤C n 2s−1 ρ 2s+1 ρ 2 R3 R3 ρ 0 ρ   C C ≤ 2s |un (x)|2 dx + 2 ρ −2s+2 |un (x)|2 dx ρ ρ R3 R3  C C ≤ 2s |un (x)|2 dx ≤ 2s 3 ρ ρ R 

≤C

|un (x)|2

where in the last step we used the boundedness of (un ) in Hε . Taking into account (12.2.22), (12.2.23) and the above estimate, we infer that       1 C 1 2 V1 V1 un dx ≤ 1 − u2n ψρ dx ≤ s + on (1). 1− c K K ρ B2ρ R3 Now, if by contradiction |zn | → ∞, then |zn | ≥ R + 2ρ for n large and we obtain 



β ≤ lim sup n→∞

that is, 0 < β ≤

C ρs ,

BR (zn )

u2n dx

≤ lim sup n→∞

c B2ρ

u2n dx ≤

C , ρs

which is not true for ρ large. Hence, (zn ) is bounded in R3 .

 

We conclude this section with the proof of the main result of this section: Theorem 12.2.6 Assume that (V1 )–(V2 ) and (f 1)–(f 4) hold. Then, for all ε > 0, problem (12.2.3) admits a positive ground state. Proof Using Lemma 12.2.1 and a variant of the mountain pass theorem without the Palais–Smale condition (see Remark 2.2.10), we know that there exists a Palais–Smale sequence (un ) for Jε at the level cε , where 0 < cε < c∗ by Lemma 12.2.2. By Lemma 12.2.3, (un ) is bounded in Hε , so we may assume that un  u in Hε and q un → u in Lloc (R3 ) for all q ∈ [1, 2∗s ). By Lemma 12.2.5, u is nontrivial. Since Jε (un ), ϕ = on (1) for all ϕ ∈ Hε , we see that  u, ϕε + bB 2

s

R3

s

(−) 2 u(−) 2 ϕ dx



 =

R3

g(ε x, u)ϕ dx,

(12.2.24)

where B 2 = limn→∞ [un ]2s . Note that B 2 ≥ [u]2s by Fatou’s lemma. If, by contradiction, B 2 > [u]2s , we may use (12.2.24) to deduce that Jε (u), u < 0. Moreover, conditions

430

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

(g1 )–(g2 ) imply that Jε (τ u), τ u > 0 for some 0 < τ 0 in R3 . Finally, arguing as in (12.2.25) with t0 = 1, we conclude that u is a ground state solution to (12.2.3).  

12.3

The Autonomous Critical Fractional Kirchhoff Problem

Let us consider the following limit problem related to (12.2.3), that is, for μ > 0: 

∗

(a + b[u]2s )(−)s u + μu = f (u) + |u|2s −2 u in R3 , u ∈ H s (R3 ), u > 0 in R3 .

(12.3.1)

The corresponding Euler–Lagrange functional is given by Iμ (u) =

b 1 a[u]2s + μu2L2 (R3 ) + [u]4s − 2 4



  1 ∗ F (u) + ∗ (u+ )2s dx 2s R3

which is well defined on the Hilbert space Xμ = H s (R3 ) endowed with the inner product  u, ϕμ = au, vDs,2 (R3 ) + μ

R3

uϕ dx

and the induced norm

1 2 uμ = a[u]2s + μu2L2 (R3 ) . We denote by Mμ the Nehari manifold associated with Iμ , that is, ! " Mμ = u ∈ Xμ \ {0} : Iμ (u), u = 0 ,

12.3 The Autonomous Critical Fractional Kirchhoff Problem

431

and dμ = inf Iμ (u), u∈Mμ

or equivalently dμ =

inf

max Iμ (tu).

u∈Xμ \{0} t ≥0

Arguing as in the proof of Theorem 12.2.6 it is easy to deduce that: Theorem 12.3.1 For all μ > 0, problem (12.3.1) admits a positive ground state solution. The following useful relation holds between cε and dV0 : Lemma 12.3.2 lim supε→0 cε ≤ dV0 . Proof For any ε > 0 we set ωε (x) = ψε (x)ω(x), where ω is a positive ground state provided by Theorem 12.3.1 with μ = V0 , and ψε (x) = ψ(ε x) with ψ ∈ Cc∞ (R3 ), ψ ∈ [0, 1], ψ(x) = 1 if |x| ≤ 12 and ψ(x) = 0 if |x| ≥ 1. Here we assume that supp(ψ) ⊂ B1 ⊂ . Using Lemma 1.4.8 and the dominated convergence theorem we can see that ωε → ω in H s (R3 ) and IV0 (ωε ) → IV0 (ω) = dV0 as ε → 0. For every ε > 0 there exists tε > 0 such that Jε (tε ωε ) = max Jε (tωε ). t ≥0

Then,

d dt [Jε (tωε )]t =tε

1 tε2 =



= 0 and this implies that s 2

R3



R3

 2

a|(−) ωε | +

V (ε x)ωε2 dx

f (tε ωε ) 4 2∗s −4 ω dx + t ε ε (tε ωε )3



+b

s 2

R3

|(−) ωε | dx

2∗

R3

2 2

ωε s dx.

(12.3.2)

By (f1 )–(f4 ), ω ∈ MV0 and (12.3.2), it follows that tε → 1 as ε → 0. On the other hand, cε ≤ max Jε (tωε ) = Jε (tε ωε ) = IV0 (tε ωε ) + t ≥0

tε2 2

 R3

(V (ε x) − V0 )ωε2 dx.

Since V (ε x) is bounded on the support of ωε , the dominated convergence theorem and the above inequality complete the proof.  

432

12.4

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

Proof of Theorem 12.1.1

This last section is devoted to the proof of the main result of this chapter. First, we prove the following compactness result which will play a fundamental role in showing that the solutions of (12.2.3) are also solutions to (12.2.1) for ε > 0 small enough. Lemma 12.4.1 Let εn → 0 and (un ) = (uεn ) ⊂ Hεn be such that Jεn (un ) = cεn and Jε n (un ) = 0. Then there exists (y˜n ) = (y˜εn ) ⊂ R3 such that u˜ n (x) = un (x + y˜n ) has a convergent subsequence in H s (R3 ). Moreover, up to a subsequence, yn = εn y˜n → y0 for some y0 ∈  such that V (y0 ) = V0 . Proof Using that Jε n (un ), un  = 0 and the conditions (g1 ), (g2 ), it is easy to see that there is γ > 0 (independent of εn ) such that un εn ≥ γ > 0

∀n ∈ N.

Since Jεn (un ) = cεn and Jε n (un ), un  = 0, Lemma 12.3.2 shows that we can argue as in the proof of Lemma 12.2.3 to deduce that (un ) is bounded in Hεn . Therefore, proceeding as in the proof of Lemma 12.2.5, we can find a sequence (y˜n ) ⊂ R3 and constants R, α > 0 such that  u2n dx ≥ α. lim inf n→∞

BR (y˜n )

Set u˜ n (x) = un (x + y˜n ). Then, (u˜ n ) is bounded in H s (R3 ), and we may assume that u˜ n  u˜ in H s (R3 ),

(12.4.1)

and [u˜ n ]2s → B 2 as n → ∞. Moreover, u˜ = 0 because  u˜ 2 dx ≥ α.

(12.4.2)

BR

Next, set yn = εn y˜n . First, we show that (yn ) is bounded in R3 . Claim 1 limn→∞ dist(yn , ) = 0. If, by contradiction, this is not true, then we can find δ > 0 and a subsequence of (yn ), still denoted (yn ), such that dist(yn , ) ≥ δ

for all n ∈ N.

12.4 Proof of Theorem 12.1.1

433

Therefore, there is r > 0 such that Br (yn ) ⊂ R3 \  for all n ∈ N. Since u˜ ≥ 0 and Cc∞ (R3 ) is dense in H s (R3 ), we can approximate u˜ by a sequence (ψj ) ⊂ Cc∞ (R3 ) such that ψj ≥ 0 in R3 and ψj → u˜ in H s (R3 ). Fix j ∈ N and use ψ = ψj as test function in Jε n (un ), ψ = 0. We have  (a + b[u˜ n ]2s )

R3

|x − y|3+2s

R6

 =

(u˜ n (x) − u˜ n (y))(ψj (x) − ψj (y))

 dxdy +

R3

V (εn x + εn y˜n )u˜ n ψj dx

g(εn x + εn y˜n , u˜ n )ψj dx.

(12.4.3)

Since un , ψj ≥ 0, the definition of g implies that 

 R3

g(εn x + εn y˜n , u˜ n )ψj dx =  + ≤

Br/ ε n

g(εn x + εn y˜n , u˜ n )ψj dx

R3 \Br/ ε n

g(εn x + εn y˜n , u˜ n )ψj dx

 

V1 2∗ −1 f (u˜ n )ψj + u˜ ns ψj dx. u˜ n ψj dx + K Br/ ε n R3 \Br/ ε n

This fact together with (12.4.3) gives 

(a

+ b[u˜ n ]2s ) 

≤

R3 \Br/ εn

(u˜ n (x) − u˜ n (y))(ψj (x) − ψj (y)) dxdy + A 6 |x − y|3+2s R

2∗ −1 f (u˜ n )ψj + u˜ ns ψj dx,

 R3

u˜ n ψj dx (12.4.4)

where A = V1 (1 − K1 ). Taking into account (12.4.1), that ψj has compact support in R3 and that εn → 0, we infer that as n → ∞ 

(u˜ n (x) − u˜ n (y))(ψj (x) − ψj (y)) dxdy |x − y|3+2s  (u(x) ˜ − u(y))(ψ ˜ j (x) − ψj (y)) dxdy → |x − y|3+2s R6 R6

and  R3 \Br/ ε n



2∗ −1 f (u˜ n )ψj + u˜ ns ψj dx → 0.

The above limits, (12.4.4) and the fact that [u˜ n ]2s → B 2 imply that  2

(a + bB )

R6

(u(x) ˜ − u(y))(ψ ˜ j (x) − ψj (y)) dxdy + A |x − y|3+2s

 R3

uψ ˜ j dx ≤ 0

434

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

and letting j → ∞ we obtain (a + bB 2 )[u] ˜ 2s + Au ˜ 2L2 (R3 ) ≤ 0, which contradicts (12.4.2). Hence, there exists a subsequence of (yn ) such that yn → y0 ∈ . Claim 2 y0 ∈ . In view of (g2 ) and (12.4.3), (a + b[u˜ n ]2s )



|x − y|3+2s

R6

 ≤

(u˜ n (x) − u˜ n (y))(ψj (x) − ψj (y)) 2∗ −1

R3

(f (u˜ n ) + u˜ ns

 dxdy +

R3

V (εn x + εn y˜n )u˜ n ψj dx

)ψj dx.

Letting n → ∞ we find that  (a + bB 2 )  ≤

R6

(u(x) ˜ − u(y))(ψ ˜ j (x) − ψj (y)) dxdy + |x − y|3+2s

 R3

V (y0 )uψ ˜ j dx

∗

R3

(f (u) ˜ + u˜ 2s −1 )ψj dx,

and then passing to the limit as j → ∞ we obtain  (a + bB

2

)[u] ˜ 2s

+ V (y0 )u ˜ 2L2 (R3 )

≤

∗

R3

(f (u) ˜ + u˜ 2s −1 )u˜ dx.

˜ 2s (by Fatou’s lemma), this inequality yields Since B 2 ≥ [u]  (a + b[u] ˜ 2s )[u] ˜ 2s + V (y0 )u ˜ 2L2 (R3 ) ≤

∗

R3

(f (u) ˜ + u˜ 2s −1 )u˜ dx.

Therefore, we can find τ ∈ (0, 1) such that τ u˜ ∈ MV (y0 ) . Then, by Lemma 12.3.2, ˜ ≤ lim inf Jεn (un ) = lim inf cεn ≤ dV0 , dV (y0 ) ≤ IV (y0 ) (τ u) n→∞

n→∞

which implies that V (y0 ) ≤ V (0) = V0 . Since V0 = min¯ V , we deduce that V (y0 ) = V0 . / ∂. Consequently, y0 ∈ . This fact together with (V2 ) yields y0 ∈ Claim 3 u˜ n → u˜ in H s (R3 ) as n → ∞.

12.4 Proof of Theorem 12.1.1

435

Let ˜ n =  − εn y˜n  εn and  χ˜ n1 (x)

=

˜ n, 1, if x ∈  ˜ n, 0, if x ∈ R3 \ 

χ˜ n2 (x) = 1 − χ˜n1 (x). Let us also consider the following functions for x ∈ R3 :  h1n (x) = 

1 1 − 2 ϑ

 V (εn x + εn y˜n )u˜ 2n (x)χ˜ n1 (x),

 1 1 − V (y0 )u˜ 2 (x), 2 ϑ   1 1 1 2 − V (εn x + εn y˜n )u˜ 2n (x) + g(εn x + εn y˜n , u˜ n (x))u˜ n (x) hn (x) = 2 ϑ ϑ  −G(εn x + εn y˜n , u˜ n (x)) χ˜ n2 (x) h1 (x) =

 ≥ 

1 1 − 2 ϑ

 −

1 2K

 V (εn x + εn y˜n )u˜ 2n (x)χ˜ n2 (x),

 1 g(εn x + εn y˜n , u˜ n (x))u˜ n (x) − G(εn x + ε n y˜n , u˜ n (x)) χ˜ n1 (x) ϑ  

 1 1 2∗s 2∗s − F (u˜ n (x)) + ∗ (u˜ n (x)) f (u˜ n (x))u˜ n (x) + (u˜ n (x)) = χ˜ n1 (x), ϑ 2s 

 1 1 3 2∗s 2∗s h (x) = − F (u(x)) ˜ + ∗ (u(x)) f (u(x)) ˜ u(x) ˜ + (u(x)) ˜ ˜ . ϑ 2s

h3n (x) =

In view of (f3 ) and (g3 ), all these functions are non-negative. Moreover, by (12.4.1) and Claim 2, ˜ u˜ n (x) → u(x)

a.e. x ∈ R3 ,

yn = εn y˜n → y0 ∈ , which implies that χ˜ n1 (x) → 1, h1n (x) → h1 (x), h2n (x) → 0 and h3n (x) → h3 (x)

a.e. x ∈ R3 .

436

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

Hence, applying Fatou’s lemma and using the translation invariance of R3 , we see that  1

≥ lim sup cεn = lim sup Jεn (un ) − Jεn (un ), un  ϑ n→∞ n→∞        1 1 1 1 2 4 1 2 3 ≥ lim sup a − [u˜ n ]s + − b[u˜ n ]s + (hn + hn + hn ) dx 2 ϑ 4 ϑ n→∞ R3        1 1 1 1 − − ≥ lim inf a (h1n + h2n + h3n ) dx [u˜ n ]2s + b[u˜ n ]4s + n→∞ 2 ϑ 4 ϑ R3      1 1 1 1 ≥a − [u] ˜ 2s + − b[u] ˜ 4s + (h1 + h3 ) dx ≥ dV0 . 2 ϑ 4 ϑ R3 

dV0

Accordingly, lim [u˜ n ]2s = [u] ˜ 2s

(12.4.5)

n→∞

and h1n → h1 , h2n → 0 and h3n → h3 in L1 (R3 ). Then 

 lim

n→∞ R3

V (εn x + εn y˜n )u˜ 2n dx =

R3

V (y0 )u˜ 2 dx,

and we deduce that lim u˜ n 2L2 (R3 ) = u ˜ 2L2 (R3 ) .

n→∞

(12.4.6)

Putting together (12.4.1), (12.4.5) and (12.4.6) and using the fact that H s (R3 ) is a Hilbert space we conclude that u˜ n − u ˜ V0 → 0 which completes the proof of lemma.

as n → ∞,  

The next lemma provides a very useful L∞ -estimate for the solutions of the modified problem (12.2.3).

12.4 Proof of Theorem 12.1.1

437

Lemma 12.4.2 Let (u˜ n ) be the sequence given in Lemma 12.4.1. Then, u˜ n ∈ L∞ (R3 ) and there exists C > 0 such that u˜ n L∞ (R3 ) ≤ C

for all n ∈ N.

Moreover, u˜ n (x) → 0 as |x| → ∞, uniformly in n ∈ N. Proof Arguing as in the proof of Lemma 6.3.23 we can see that  2  1 β−1 2(β−1) S∗ u˜ n u˜ L,n 2 2∗s 3 + Vn (x)u˜ 2n u˜ L,n dx L (R ) 3 β R  ( u ˜ (x) − u ˜ n n (y)) 2(β−1) 2(β−1) ≤ (a + b[u˜ n ]2s ) ((u˜ n u˜ L,n )(x) − (u˜ n u˜ L,n )(y)) dxdy N+2s |x − y| R6  2(β−1) + Vn (x)u˜ 2n u˜ L,n dx

a

R3

 ≤

2(β−1)

R3

gn (x, u˜ n )u˜ n u˜ L,n

(12.4.7)

dx,

where Vn (x) = V (εn x + εn y˜n ) and gn (x, u˜ n ) = g(ε n x + εn y˜n , u˜ n ). By assumptions (g1 ) and (g2 ), for every ξ > 0 there exists Cξ > 0 such that ∗

|gn (x, u˜ n )| ≤ ξ |u˜ n | + Cξ |u˜ n |2s −1 .

(12.4.8)

Taking ξ ∈ (0, V1 ), and using (12.4.8) and (12.4.7) we get β−1 u˜ n u˜ L,n 2 2∗s 3 L (R )

 ≤ Cβ

∗

2 R3

2(β−1)

|u˜ n |2s u˜ L,n

dx.

Then we can proceed as in the proof of Lemma 11.5.1 to complete the proof (see also V1 Remark 7.2.10 and note that u˜ n is a subsolution to (−)s u˜ n + bD u˜ n = a1 gn (x, u˜ n ) in R3 , 2   where D > 0 is such that a ≤ a + b[u˜ n ]s ≤ D for all n ∈ N). Now, we give the proof of Theorem 12.1.1. Proof of Theorem 12.1.1 Firstly, we prove that there exists ε˜ 0 > 0 such that for any ε ∈ (0, ε˜ 0 ) and any mountain pass solution uε ∈ Hε of (12.2.3), uε L∞ (R3 \ε ) < a0 .

(12.4.9)

438

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

Suppose, by contradiction, that for some subsequence (εn ) so that εn → 0, we can find un = uεn ∈ Hεn such that Jεn (un ) = cεn , Jε n (un ) = 0 and un L∞ (R3 \εn ) ≥ a0 .

(12.4.10)

By Lemma 12.4.1, there exists (y˜n ) ⊂ R3 such that u˜ n = un (· + y˜n ) → u˜ in H s (R3 ) and εn y˜n → y0 for some y0 ∈  such that V (y0 ) = V0 . Now, if we choose r > 0 so that Br (y0 ) ⊂ B2r (y0 ) ⊂ , we have B εr ( εyn0 ) ⊂ εn . Then, for any y ∈ B εr (y˜n ), n

n

        y − y0  ≤ |y − y˜n | + y˜n − y0  < 1 (r + on (1)) < 2r    εn εn  εn εn

for n sufficiently large.

Hence, for these values of n, we get R3 \ εn ⊂ R3 \ B εr (y˜n ). n

(12.4.11)

By Lemma 12.4.2, we deduce that u˜ n (x) → 0

as |x| → ∞, uniformly in n ∈ N.

(12.4.12)

Therefore, we can find R > 0 such that u˜ n (x) < a0

for all |x| ≥ R, n ∈ N,

which yields un (x) < a0 for any x ∈ R3 \ BR (y˜n ) and n ∈ N. On the other hand, there exists ν ∈ N such that for any n ≥ ν and r/ εn > R, it holds R3 \ εn ⊂ R3 \ B εr (y˜n ) ⊂ R3 \ BR (y˜n ), n

which gives un (x) < a0

for all x ∈ R3 \ εn .

contradicting (12.4.10). Thus, (12.4.9) is verified. Now, let uε be a solution to (12.2.3). Since uε satisfies (12.4.9) for any ε ∈ (0, ε˜ 0 ), it follows from the definition of g that uε is a solution to (12.2.1), and then uˆ ε (x) = u(x/ ε) is a solution to (12.1.1) for any ε ∈ (0, ε˜ 0 ). Finally, we study the behavior of the maximum points of solutions to problem (12.2.1). Take εn → 0 and consider a sequence (un ) ⊂ Hεn of solutions to (12.2.1). We first notice

12.4 Proof of Theorem 12.1.1

439

that, by (g1 ), there exists γ ∈ (0, a0 ) such that ∗

g(ε n x, t)t = f (t)t + t 2s ≤

V1 2 t K

for any x ∈ R3 , 0 ≤ t ≤ γ .

(12.4.13)

The same argument as before shows that, for some R > 0, un L∞ (R3 \BR (y˜n )) < γ .

(12.4.14)

Moreover, up to extracting a subsequence, we may assume that un L∞ (BR (y˜n )) ≥ γ .

(12.4.15)

Indeed, if (12.4.15) does not hold, then (12.4.14) implies that un L∞ (R3 ) < γ , and so in view of Jε n (un ), un  = 0 and (12.4.13),  un 2εn ≤ un 2εn + b[un ]4s =

R3

g(ε n x, un )un dx ≤

V1 K

 R3

u2n dx.

This implies that un εn = 0, a contradiction. Hence, (12.4.15) holds true. Let pn ∈ R3 be a global maximum point of un . In the light of (12.4.14) and (12.4.15), we deduce that pn ∈ BR (y˜n ). Thus pn = y˜n + qn for some qn ∈ BR . Recalling that the solution to (12.1.1) is of the form uˆ n (x) = un (x/ εn ), we conclude that ηn = εn y˜n + εn qn is a global maximum point of uˆ n . Since (qn ) ⊂ BR is bounded and εn y˜n → y0 with V (y0 ) = V0 , it follows from the continuity of V that lim V (ηn ) = V (y0 ) = V0 .

n→∞

Next, we give a decay estimate for uˆ n . Invoking Lemma 3.2.17, we know that there exists a positive function w such that 0 < w(x) ≤

C 1 + |x|3+2s

for all x ∈ R3

(12.4.16)

and (−)s w +

V1 w=0 2(a + bA21 )

in R3 \ B R1 ,

for some suitable R1 > 0, with A1 > 0 such that a ≤ a + b[un ]2s ≤ a + bA21

for all n ∈ N.

(12.4.17)

440

12 Concentrating Solutions for a Fractional Kirchhoff Equation with Critical. . .

Using (g1 ) and (12.4.12), we can find R2 > 0 sufficiently large such that (−)s u˜ n +

V1 V1 u˜ n u˜ n ≤ (−)s u˜ n + 2(a + b[u˜ n ]2 ) 2(a + bA21 )     1 V1 u˜ n gn (x, u˜ n ) − Vn − = 2 a + b[u˜ n ]2s   V1 1 3 u ˜ g (x, u ˜ ) − ≤ n n n ≤ 0 in R \ B R2 . 2 a + b[u˜ n ]2s (12.4.18)

Set R3 = max{R1 , R2 } > 0 and c = min w > 0

w˜ n = (d + 1)w − cu˜ n ,

and

B R3

(12.4.19)

where d = supn∈N u˜ n L∞ (R3 ) < ∞. We claim that w˜ n ≥ 0 in R3 .

(12.4.20)

First observe that (12.4.17), (12.4.18) and (12.4.19) yield w˜ n ≥ cd + w − cd > 0 in B R3 , (−)s w˜ n +

V1 w˜ n ≥ 0 in R3 \ B R3 . 2(a + bA21)

Applying Lemma 1.3.8 we deduce that (12.4.20) is satisfied. Combining (12.4.16) and (12.4.20) we obtain 0 < u˜ n (x) ≤

C˜ 1 + |x|3+2s

for all x ∈ R3 , n ∈ N,

(12.4.21)

for some constant C˜ > 0. Since uˆ n (x) = un ( εxn ) = u˜ n ( εxn − y˜n ) and ηn = εn y˜n + εn qn , we can use (12.4.21) to deduce that  0 < uˆ n (x) = un ≤

x εn

1 + | εxn



 = u˜ n

x − y˜n εn

C˜ − y˜n |3+2s



12.4 Proof of Theorem 12.1.1

441

= ≤

C˜ εn3+2s ε3+2s n

+|x − εn y˜n |3+2s C˜ ε3+2s n

ε3+2s +|x − ηn |3+2s n

This ends the proof of Theorem 12.1.1.

for all x ∈ R3 .  

Remark 12.4.3 In [62] we extended Theorem 12.1.1 for more general nonlinearities with critical growth.

Multiplicity and Concentration Results for a Fractional Schrödinger-Poisson System with Critical Growth

13.1

13

Introduction

In this chapter we focus our attention on the multiplicity and concentration of positive solutions for the following critical fractional nonlinear Schrödinger-Poisson system: ⎧ 2s s 2∗ −2 3 ⎪ ⎨ ε (−) u + V (x)u + φu = f (u) + |u| s u in R , 2t t 2 in R3 , ε (−) φ = u ⎪ ⎩ u ∈ H s (R3 ), u > 0 in R3 ,

(13.1.1)

6 is the fractional critical where ε > 0 is a small parameter, s ∈ ( 34 , 1), t ∈ (0, 1), 2∗s = 3−2s 3 Sobolev exponent. Here, the potential V : R → R is a continuous function satisfying the following del Pino-Felmer hypotheses [165]:

(V1 ) (V2 )

there exists V0 > 0 such that V0 = infx∈R3 V (x); there exists a bounded open set  ⊂ R3 such that V0 < min V ∂

and

M = {x ∈  : V (x) = V0 } = ∅.

Without loss of generality, we may assume that 0 ∈ M. The nonlinearity f : R → R is assumed to be a continuous function such that f (t) = 0 for t ≤ 0 and to satisfy the following conditions: (f1 ) (f2 )

f (t) = o(t 3 ) as t → 0; there exist q, σ ∈ (4, 2∗s ), C0 > 0 such that f (t) ≥ C0 t q−1

∀t > 0,

lim

f (t)

t →∞ t σ −1

= 0;

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_13

443

444

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

there exists ϑ ∈ (4, σ ) such that 0 < ϑF (t) ≤ tf (t) for all t > 0; f (t) (f4 ) the function t → 3 is increasing in (0, ∞). t We note that when φ = 0, then (13.1.1) reduces to a fractional Schrödinger equation of the type (f3 )

ε2s (−)s u + V (x)u = h(x, u)

in R3 ,

(13.1.2)

which was investigated in depth in Chaps. 6–8. It s = t = 1, then (13.1.1) becomes the classical Schrödinger-Poisson system 

− ε2 u + V (x)u + μφu = g(u) in R3 , in R3 , − ε2 φ = u2

(13.1.3)

which describes systems of identical charged particles interacting with each other in the case that magnetic field effects can be ignored, and its solution represents, in particular, a standing wave for such a system. For a more detailed physical description of this system we refer to [96]. Concerning some classical existence and multiplicity results for Schrödinger-Poisson systems we refer to [82, 212, 213, 303, 331, 348]. For instance, Ruiz [303] obtained existence and nonexistence results to (13.1.3) when g(u) = up , p ∈ (1, 5) and μ > 0. Azzollini et al. [82] investigated the existence of nontrivial solutions when g satisfies Berestycki-Lions type assumptions. Wang et al. [331] considered the existence and concentration of positive solutions to (13.1.3) involving subcritical nonlinearities. He and Li [213] obtained an existence result for a critical Schrödinger-Poisson system, assuming that the potential V satisfies the conditions (V1 )-(V2 ). By contrast, only few results for fractional Schrödinger-Poisson systems are available in literature. Giammetta [204] studied the local and global well-posedness of a fractional Schrödinger-Poisson system in which the fractional diffusion term appears only in the second equation in (13.1.1). Teng [325] analyzed the existence of ground state solutions for (13.1.1) with critical Sobolev exponent, by combining the method of the PohozaevNehari manifold, arguments of Brezis-Nirenberg type [114], the monotonicity trick and a global compactness lemma. In [342] Zhang et al. used a perturbation approach to prove the existence of positive solutions to (13.1.1) when V (x) = μ > 0 and g is a general nonlinearity with subcritical or critical growth. They also investigated the asymptotic behavior of solutions as μ → 0. Liu and Zhang [264] studied the multiplicity and concentration of solutions to (13.1.1) when the potential V satisfies the global condition due to Rabinowitz [299]. Murcia and Siciliano [281] showed that, for suitably small ε, the number of positive solutions to (13.1.1) is estimated below by the Lusternik–Schnirelman category of the set of minima of the potential. In this chapter, we study the multiplicity and concentration of solutions to (13.1.1) under the local conditions (V1 )-(V2 ) on the potential V and the conditions (f1 )–(f4 ) for the nonlinearity f . Then we are able to prove the following result:

13.1 Introduction

445

Theorem 13.1.1 ([48]) Assume that (V1 )-(V2 ) and (f1 )–(f4 ) hold. Then, for any δ > 0 such that Mδ = {x ∈ R3 : dist(x, M) ≤ δ} ⊂ , there exists εδ > 0 such that, for any ε ∈ (0, εδ ), problem (13.1.1) admits at least catMδ (M) positive solutions in Hε × D t,2 (R3 ). Moreover, if (uε , φε ) denotes one of these solutions and xε ∈ R3 is a global maximum point of uε , then lim V (xε ) = V0 ,

ε→0

and there exists C > 0 such that 0 < uε (x) ≤

ε3+2s

Cε3+2s + |x − xε |3+2s

for all x ∈ R3 .

The proof of Theorem 13.1.1 is obtained by applying a penalization argument [165], a concentration-compactness lemma and the generalized Nehari method [321, 322]. We emphasize that the presence of two fractional Laplacian operators and the critical Sobolev exponent make our task more complicated and intriguing compared to the ones in the previous chapters and a more careful analysis will be needed. We conclude this introduction by giving some requisite preliminary results. Let s, t ∈ (0, 1) be such that 4s + 2t ≥ 3. Using Theorem 1.1.8 we can see that 12

H s (R3 ) ⊂ L 3+2t (R3 ).

(13.1.4)

For any u ∈ H s (R3 ), the linear functional Lu : Dt,2 (R3 ) → R given by  Lu (v) =

R3

u2 v dx

is well defined and continuous thanks to the Hölder inequality and (13.1.4). Indeed,  |Lu (v)| ≤

R3

|u|

12 3+2t

 3+2t  6

dx

2∗t

R3

|v| dx

 1∗ 2t

≤ Cu2 vDt,2 ,

where  vDt,2 = 2

R6

|v(x) − v(y)|2 dxdy. |x − y|3+2t

446

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

Then, by the Lax-Milgram Theorem, there exists exactly one φut ∈ Dt,2 (R3 ) such that (−)t φut = u2 in R3 . Therefore, the following t-Riesz formula holds:  φut (x)

= ct

R3

u2 (y) dy |x − y|3−2t

3

ct = π − 2 2−2t

(x ∈ R3 ),

( 3−2t 2 ) . (t)

(13.1.5)

In the sequel, we will omit the constant ct in order to lighten the notation. Finally, we collect some useful properties of φut which will be used later. Lemma 13.1.2 If s, t ∈ (0, 1) and 4s + 2t ≥ 3, then, for all u ∈ H s (R3 ) we have: (1) φut Dt,2 ≤ Cu2

12 L 3+2t

(2) (3) (4) (5) (6)

(R3 )

≤ Cu2H s (R3 ) and



φut u2 dx ≤ Ct u4

R3

12

.

L 3+2t (R3 )

Moreover, the mapping u ∈ H s (R3 ) → φut ∈ Dt,2 (R3 ) is continuous and sends bounded sets into bounded sets; φut ≥ 0 in R3 , and φut is radial if u is radial;  ¯ = u(x + y), then φut¯ (x) = φut (x + y) and R3 φut¯ u¯ 2 dx = if y ∈ R3 and u(x)  t 2 R3 φu u dx; t = r 2 φ t for all r ∈ R, φ t (x) = θ 2s φ t ( x ) for any θ > 0, where u (x) = u( x ); φru θ u uθ u θ θ if un  u in H s (R3 ), then φut n  φut in Dt,2 (R3 ); if un  u in H s (R3 ), then 

 R3

φut n u2n dx =

 R3

t φ(u (un − u)2 dx + n −u)

(7) if un → u in H s (R3 ), then φut n → φut in Dt,2 (R3 ) and

R3

φut u2 dx + on (1);

 R3

φut n u2n dx →

 12

R3

φut u2 dx.

s 3 3 3+2t (8) Let  4st + 2t > 3. If tun  u in H (Rs ) 3and un →t u2 in L  (R t), 2then R3 φun un v dx → R3 φu uv dx for all v ∈ H (R ) and R3 φun un dx → R3 φu u dx.

Proof The proofs of properties (1)–(7) are similar to those in [303,348], so we omit them (see [264, 325, 342] for details). Here we only prove (8). Note that 4s + 2t > 3 and 12 6 t ∈ (0, 1) imply that 2 < 3+2t < 3−2s . Take v ∈ H s (R3 ). From (1) and un → u in 12

L 3+2t (R3 ), we see that φut n → φut in Dt,2 (R3 ). Then, by using the Hölder inequality, the 6

embedding Dt,2 (R3 ) ⊂ L 3−2t (R3 ) and the boundedness of (un ) in H s (R3 ), we see that    

 R3

φut n un v dx −

R3

     φut uv dx  =  (φut n − φut )un v dx + R3

≤ φut n − φut 

6

L 3−2t (R3 )

un 

R3 12

  φut (un − u)v dx 

L 3+2t (R3 )

v

12

L 3+2t (R3 )

13.2 Functional Setting

447

+ φut 

6

L 3−2t (R3 )

un − u

12

L 3+2t (R3 )

≤ Cφut n − φut Dt,2 + Cun − u

v

12

L 3+2t (R3 )

→ 0.

12

L 3+2t (R3 )

In much the same way,    

 R3

φut n u2n dx

−

R3

 

φut u2 dx 

   t t 2  =  (φun − φu )un dx + 3 R

≤ φut n − φut 

6

L 3−2t (R3 )

+ φut 

6

L 3−2t (R3 )

R3

un 2

φut (un

  − u)(un + u) dx 

12

L 3+2t (R3 )

un − u

12

L 3+2t (R3 )

≤ Cφut n − φut Dt,2 + Cun − u

un + u

12

L 3+2t (R3 )

12

L 3+2t (R3 )

→ 0.  

s (R3 ), then Remark 13.1.3 As in [303], we can see that if 4s + 2t > 3 and un  u in Hrad   t t t,2 3 t 2 t 2 φun → φu in D (R ) and R3 φun un dx → R3 φu u dx.

13.2

Functional Setting

In order to study (13.1.1), we use the change of variable x → ε x and we look for solutions to  ∗ (−)s u + V (ε x)u + φut u = f (u) + |u|2s −2 u in R3 , (13.2.1) u ∈ H s (R3 ), u > 0 in R3 , where φut is given by (13.1.5). In what follows we introduce a useful penalization function [165]. ∗ Let K > 2 and a > 0 such that f (a) + a 2s −1 = VK0 a, and define  f˜(t) =

∗

f (t) + (t + )2s −1 , if t ≤ a, V0 if t > a, K t,

and ∗

g(x, t) = χ (x)(f (t) + (t + )2s −1 ) + (1 − χ (x))f˜(t).

448

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

It is easy to check that g satisfies the following properties: ) limt →0 g(x,t = 0 uniformly with respect to x ∈ R3 ; t3 ∗ g(x, t) ≤ f (t) + t 2s −1 for all x ∈ R3 , t > 0; (i) 0 < ϑG(x, t) ≤ g(x, t)t for all x ∈  and t > 0, (ii) 0 ≤ 2G(x, t) ≤ g(x, t)t ≤ VK0 t 2 for all x ∈ R3 \  and t > 0; ) (g4 ) for each fixed x ∈  the function g(x,t is increasing in (0, ∞), and for each fixed t3

(g1 ) (g2 ) (g3 )

x ∈ R3 \  the function

g(x,t ) t3

is increasing in (0, a).

Let us consider the following modified problem: 

(−)s u + V (ε x)u + φut u = g(ε x, u) in R3 , u ∈ H s (R3 ), u > 0 in R3 .

(13.2.2)

It is clear that weak solutions to (13.2.2) are critical points of the following functional Jε (u) =

1 1 u2ε + 2 4



 R3

φut u2 dx −

R3

G(ε x, u) dx,

defined for all u ∈ Hε , where   Hε = u ∈ H s (R3 ) :

 R3

V (ε x)u2 dx < ∞

is endowed with the norm   2 uε = [u]s +

1 2

R3

V (ε x)u dx

2

.

Obviously, Hε is a Hilbert space with the inner product  u, vε = u, vDs,2 (R3 ) +

R3

V (ε x)uv dx.

We also note that Jε ∈ C 1 (Hε , R) and its differential is given by Jε (u), v = u, vε +

 R3

 φut uv dx −

R3

g(ε x, u)v dx

∀u, v ∈ Hε .

Let us introduce the Nehari manifold associated with (13.2.2), that is, Nε = {u ∈ Hε \ {0} : Jε (u), u = 0},

13.2 Functional Setting

449

and denote Hε+ = {u ∈ Hε : |supp(u+ ) ∩ ε | > 0} + + and S+ ε = Sε ∩ Hε , where Sε is the unit sphere in Hε . Then Hε = Tu Sε ⊕ Ru and + Tu Sε = {v ∈ Hε : u, vε = 0}.

Lemma 13.2.1 The functional Jε has a mountain pass geometry: (a) there exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ; (b) there exists e ∈ Hε such that eε > ρ and Jε (e) < 0. Proof (a) Indeed, by assumptions (g1 ), (g2 ), (f2 ), for every ξ > 0 we can find Cξ > 0 such that Jε (u) ≥

1 u2ε − 2

 R3

G(ε x, u) dx ≥

1 2∗ u2ε − ξ Cu4ε − Cξ Cuε s . 2

Then there exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ. (b) In view of (g3 )-(i) and Lemma 13.1.2-(4), for any u ∈ Hε+ and τ > 0 τ2 τ4 Jε (τ u) ≤ u2ε + 2 4 τ4 τ2 ≤ u2ε + 2 4 ≤

τ2 τ4 u2ε + 2 4



  

R3

R3

R3

φut u2 dx

− 

φut u2 dx

R3

−

G(ε x, τ u) dx G(ε x, τ u) dx

ε

φut u2 dx − C1 τ ϑ



(u+ )ϑ dx + C2 |supp(u+ ) ∩ ε |, ε

(13.2.3) for some positive constants C1 and C2 . Since ϑ ∈ (4, 2∗s ), we see that Jε (τ u) → −∞ as τ → ∞.   Since f is only continuous, the next results will be crucial for overcoming the nondifferentiability of Nε and the incompleteness of S+ ε. Lemma 13.2.2 Assume that (V1 )-(V2 ) and (f1 )–(f4 ) hold true. Then,

450

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

(i) For each u ∈ Hε+ , let hu : R+ → R be defined by hu (t) = Jε (tu). Then, there is a unique tu > 0 such that h u (t) > 0

in (0, tu ),

h u (t) < 0

in (tu , ∞).

(ii) There exists τ > 0, independent of u, such that tu ≥ τ for all u ∈ S+ ε . Moreover, for there is a positive constant C such that tu ≤ CK for all each compact set K ⊂ S+ K ε u ∈ K. ˆ ε (u) = tu u is continuous and mε = m ˆ ε |S+ε is a (iii) The map m ˆ ε : Hε+ → Nε given by m −1 (u) = u . and N . Moreover, m homeomorphism between S+ ε ε ε uε + (iv) If there is a sequence (un ) ⊂ S+ ε such that dist(un , ∂Sε ) → 0, then mε (un )ε → ∞ and Jε (mε (un )) → ∞. Proof (i) We note that hu ∈ C 1 (R+ , R), and in view of Lemma 13.2.1, we can see that hu (0) = 0, hu (t) > 0 for t > 0 small enough and hu (t) < 0 for t > 0 sufficiently large. Then there exists tu > 0 such that h u (tu ) = 0 and tu is a global maximum for hu . This implies that tu u ∈ Nε . Let us show that there is exactly one tu . Suppose, by contradiction, that there exist t1 > t2 > 0 such that h u (t1 ) = h u (t2 ) = 0, that is, 

 t1 u2ε + t13  t2 u2ε + t23

R3

R3

φut u2 dx =  φut u2 dx =

R3

R3

g(ε x, t1 u)u dx

(13.2.4)

g(ε x, t2 u)u dx.

(13.2.5)

Using (13.2.4), (13.2.5) and (g4 ) we see that  u2ε

1 1 − 2 2 t1 t2



 g(ε x, t1 u) g(ε x, t2 u) 4 u dx − (t1 u)3 (t2 u)3 R3    g(ε x, t1 u) g(ε x, t2 u) 4 u dx = − (t1 u)3 (t2 u)3 R3 \ε    g(ε x, t1 u) g(ε x, t2 u) 4 u dx − + (t1 u)3 (t2 u)3 ε    g(ε x, t1 u) g(ε x, t2 u) 4 − ≥ u dx (t1 u)3 (t2 u)3 R3 \ε    g(ε x, t1 u) g(ε x, t2 u) 4 − = u dx (t1 u)3 (t2 u)3 (R3 \ε )∩{t2 u>a} 

=



13.2 Functional Setting

451



 g(ε x, t1 u) g(ε x, t2 u) 4 + − u dx (t1 u)3 (t2 u)3 (R3 \ε )∩{t2 u≤aa} K (t1 u)   1 1 1 = − 2 V0 u2 dx. K t12 t2 (R3 \ε )∩{t2 u>a} Concerning I I , using again the definition of g, we get #

 II ≥

(R3 \ε )∩{t2 u≤aa}

#

 +

(R3 \ε )∩{t2 u≤aa}

t12 t22



V0 u2 dx #

$ ∗ V0 1 f (t2 u) + (t2 u+ )2s −1 u4 dx − K (t1 u)2 (t2 u)3

t22 − t12 (R3 \ε )∩{t2 u≤aa} t1 − t2 (R3 \ε )∩{t2 u≤a 0 such that hu (tu ) = 0, or equivalently 

 tu + tu3

R3

φut u2 dx

=

R3

g(ε x, tu u)u dx.

(13.2.6)

In the light of (g1 ) and (g2 ), given ξ > 0 there exists a positive constant Cξ such that ∗

|g(x, t)| ≤ ξ |t|3 + Cξ |t|2s −1 ,

for all (x, t) ∈ R3 × R.

From (13.2.6) and applying Theorem 1.1.8 we have that 2∗ −1

tu ≤ ξ tu3 C1 + Cξ tu s

C2 ,

which implies that there exists τ > 0, independent of u, such that tu ≥ τ . Now, let K ⊂ S+ ε be a compact set. Let us show that tu can be estimated from above by a constant depending on K. Assume, by contradiction, that there exists a sequence (un ) ⊂ K such that tn = tun → ∞. Therefore, there exists u ∈ K such that un → u in Hε . In view of (13.2.3), Jε (tn un ) → −∞.

(13.2.7)

Fix v ∈ Nε . Then, using the fact that Jε (v), v = 0, and assumptions (g3 )-(i) and (g3 )(ii), we can infer that 1

J (v), v ϑ ε      ϑ −4 ϑ −2 1 2 φvt v 2 dx + [g(ε x, v)v − ϑG(ε x, v)] dx = vε + 2ϑ 4ϑ ϑ R3 \ε R3  1 + [g(ε x, v)v − ϑG(ε x, v)] dx ϑ ε    ϑ −2 1 [g(ε x, v)v − ϑG(ε x, v)] dx v2ε + ≥ 2ϑ ϑ R3 \ε      ϑ −2 ϑ −2 1 2 ≥ V (ε x)v 2 dx vε − 2ϑ 2ϑ K R3 \ε    ϑ −2 1 ≥ (13.2.8) 1− v2ε . 2ϑ K

Jε (v) = Jε (v) −

Since (tun un ) ⊂ Nε and K > 2, (13.2.8) implies that (13.2.7) does not hold, a contradiction.

13.2 Functional Setting

453

(iii) First of all, we observe that m ˆ ε , mε and m−1 ε are well defined. In fact, by (i), for each + u ∈ Hε there exists a unique m ˆ ε (u) ∈ Nε . On the other hand, if u ∈ Nε then u ∈ Hε+ . + Otherwise, if u ∈ / Hε , then |supp(u+ ) ∩ ε | = 0, which together with (g3)-(ii) gives  u2ε +

 R3

φut u2 dx = ≤

R3 \ε

1 K

g(ε x, u+ ) u+ dx



R3 \ε

V (ε x)u2 dx ≤

1 u2ε . K

(13.2.9)

Using that φut ≥ 0 and (13.2.9), we get 0 < u2ε ≤

1 u2ε K

which is not possible, because K > 2. Accordingly, m−1 ε (u) = defined and it is a continuous mapping. Now, take u ∈ S+ ε ; then −1 m−1 ε (mε (u)) = mε (tu u) =

u uε

−1 ∈ S+ ε , mε is well

tu u u = = u, tu uε uε

which shows that mε is a bijection. Next, we show that m ˆ ε is continuous. Let (un ) ⊂ Hε+ + and u ∈ Hε be such that un → u in Hε . Since m ˆ ε (tu) = m ˆ ε (u) for all t > 0, we may assume that un ε = uε = 1 for all n ∈ N. Then, in view of (ii), we can find t0 > 0 such that tn = tun → t0 . Since tn un ∈ Nε , we obtain 

 tn2 un 2ε

+

tn4

R3

φut n u2n dx

=

R3

g(ε x, tn un ) tn un dx,

and letting n → ∞ we get 

 t02 u2ε

+

t04

R3

φut u2 dx

=

R3

g(ε x, t0 u) t0 u dx,

which shows that t0 u ∈ Nε and tu = t0 . Therefore, m ˆ ε (un ) → m ˆ ε (u) and m ˆ ε and mε are continuous maps.

in Hε ,

454

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

+ ∗ (iv) Let (un ) ⊂ S+ ε be such that dist(un , ∂Sε ) → 0. Note that for each p ∈ [2, 2s ] and n ∈ N,

u+ n Lp (ε ) ≤ inf un − vLp (ε ) v∈∂S+ ε

≤ Cp inf un − vε , v∈∂S+ ε

Hence, by (g1 ), (g2 ), and (g3 )-(ii), we can infer that for all t > 0 

 R3

G(ε x, tun ) dx = ≤ ≤ ≤

 R3 \ε

t2 K t2 K t2 K

G(ε x, tun ) dx +

G(ε x, tun ) dx ε





R3 \ε

un 2ε

V (ε x)u2n dx + 

+ C1 t

4 ε

∗

F (tun ) + ε

4 (u+ n ) dx

+ C2 t

2∗s



t 2s + 2∗s (u ) dx 2∗s n ∗

ε

2s (u+ n ) dx

∗

∗

4

2s + 2s + C1 t 4 dist(un , ∂S+ ε ) + C2 t dist(un , ∂Sε ) .

Therefore, for all t > 0,  lim sup n→∞

R3

G(ε x, tun ) dx ≤

t2 . K

(13.2.10)

Recalling the definition of mε (un ) and using (13.2.10) we see that 

1 1 lim inf mε (un )2ε + n→∞ 2 4



 R3

t φm (mε (un ))2 dx ε (un )

≥ lim inf Jε (tun ) n→∞

≥ lim inf Jε (mε (un )) n→∞



t2 t4 un 2ε + n→∞ 2 4   1 1 − ≥ t 2. 2 K

≥ lim inf





 R3

φut n u2n dx −

R3

G(ε x, tun ) dx

Since t > 0 is arbitrary and K > 2, we conclude that Jε (mε (un )) → ∞ and   mε (un )ε → ∞ as n → ∞, and this ends the proof of Lemma 13.2.2.

13.2 Functional Setting

455

Let us introduce the maps ψˆ ε : Hε+ → R

and

ψε : S+ ε → R,

by ψˆ ε (u) = Jε (m ˆ ε (u)) and ψε = ψˆ ε |S+ε . Using Lemma 13.2.2 and arguing as in Chap. 11, we obtain the following result. Proposition 13.2.3 Assume that hypotheses (V1 )-(V2 ) and (f1 )–(f4 ) hold. Then, (a) ψˆ ε ∈ C 1 (Hε+ , R) and ψˆ ε (u), v =

m ˆ ε (u)ε

Jε (m ˆ ε (u)), v uε

for every u ∈ Hε+ and v ∈ Hε . (b) ψε ∈ C 1 (S+ ε , R) and ψε (u), v = mε (u)ε Jε (mε (u)), v, for every v ∈ Tu S+ ε . (c) If (un ) is a Palais-Smale sequence for ψε , then (mε (un )) is a Palais-Smale sequence for Jε . If (un ) ⊂ Nε is a bounded Palais-Smale sequence for Jε , then (m−1 ε (un )) is a Palais-Smale sequence for ψε . (d) The point u is a critical point of ψε if and only if mε (u) is a nontrivial critical point for Jε . Moreover, the corresponding critical values coincide and inf ψε (u) = inf Jε (u).

u∈S+ ε

u∈Nε

Remark 13.2.4 As in [322], we see that cε = inf Jε (u) = inf max Jε (tu) = inf max Jε (tu). u∈Nε

u∈H+ ε t >0

u∈S+ ε t >0

It is also easy to check that cε coincides with the mountain pass level of Jε . Remark 13.2.5 Let us note that if u ∈ Nε , then using (g1 ), (g2 ) and taking ξ ∈ (0, 12 ) we obtain that   φu u2 dx − g(ε x, u)u dx 0 = u2ε + R3

≥

1 2∗ u2ε − Cuε s , 2

and so uε ≥ α > 0 for some α independent of u.

R3

456

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

The next lemma gives an upper bound of the minimax level cε . 3

Lemma 13.2.6 It holds, 0 < cε < 3s S∗2s . Proof We follow [264]. We know (see Remark 1.1.9) that S∗ is achieved by zδ (x) =

κδ − (μ2 + |

3−2s 2

3−2s x 2 2 1/2s | ) δS∗

for any fixed δ > 0, where κ ∈ R and μ > 0 are fixed constants. Let η ∈ Cc∞ (R3 ) be a cut-off function such that η = 1 in Bρ , supp(η) ⊂ B2ρ and 0 ≤ η ≤ 1, where B2ρ ⊂ . For simplicity, we assume that ρ = 1. We define Zδ (x) = η(εx)zδ (x), and let vδ =

Zδ Zδ  2∗

be such that

L s (R3 )

[vδ ]2s ≤ S∗ + O(δ 3−2s ).

(13.2.11)

Moreover, vδ 2L2 (R3 ) = O(δ 3−2s ), ⎧ 3− (3−2s)p ⎪ 2 ), if p > ⎪ ⎨ O(δ (3−2s)p q 1 3− vδ Lp (R3 ) = O(log( δ )δ 2 ), if p = ⎪ ⎪ ⎩ O(δ (3−2s)p 2 ), if p
0. We begin by showing that the functional Jε satisfies the Palais-Smale 3

condition at any level 0 < d < 3s S∗2s , where S∗ is the best constant of the Sobolev ∗ embedding Ds,2 (R3 ) into L2s (R3 ). We recall that the existence of Palais-Smale sequences of Jε is justified by Lemma 13.2.1 and a variant of the mountain pass theorem without the Palais-Smale condition (see Remark 2.2.10). Firstly, we note that every Palais-Smale sequence is bounded. 3

Lemma 13.3.1 Let 0 < d < 3s S∗2s and let (un ) ⊂ Hε be a Palais-Smale sequence for Jε at the level d. Then (un ) is bounded in Hε . Proof Let (un ) ⊂ Hε be a Palais-Smale sequence at the level d, that is Jε (un ) → d

and

Jε (un ) → 0 in Hε∗ .

Arguing as in the proof of Lemma 13.2.2-(ii) (see formula (13.2.8) there), we deduce that 1 C(1 + un ε ) ≥ Jε (un ) − Jε (un ), un  ϑ    ϑ −2 1 ≥ 1− un 2ε . 2ϑ K Since ϑ > 4 and K > 2, we conclude that (un ) is bounded in Hε .

 

Remark 13.3.2 Arguing as in Remark 5.2.8, we may always assume that the Palais-Smale sequence (un ) is non-negative in RN . The next result will play a crucial role in establishing compactness of bounded PalaisSmale sequences. 3

Lemma 13.3.3 Let 0 < d < 3s S∗2s and let (un ) ⊂ Hε be a Palais-Smale sequence for Jε at the level d. Then, for each ζ > 0, there exists R = R(ζ ) > 0 such that  lim sup n→∞

 R3 \BR

dx

R3

|un (x) − un (y)|2 dy + |x − y|3+2s



 R3 \BR

V (ε x)u2n dx

< ζ.

13.3 An Existence Result for the Modified Problem

459

Proof For R > 0, let ηR ∈ C ∞ (R3 ) be such that ηR = 0 in B R and ηR = 1 in BRc , 2

with 0 ≤ ηR ≤ 1 and ∇ηR L∞ (R3 ) ≤ C R , where C is a constant independent of R. Since (ηR un ) is bounded in Hε , it follows that Jε (un ), ηR un  = on (1), that is 

  |un (x) − un (y)|2 2 ηR (x) dxdy + V (ε x)un ηR dx + φut n u2n ηR dx |x − y|3+2s R6 R3 R3   (ηR (x) − ηR (y))(un (x) − un (y)) = on (1) + g(ε x, un )un ηR dx − un (y) dxdy. 3 6 |x − y|3+2s R R

Take R > 0 such that ε ⊂ B R . Then, using (g3 )-(ii), we see that 2



 |un (x) − un (y)|2 η (x) dxdy + V (ε x)u2n ηR dx R |x − y|3+2s R6 R3   1 (ηR (x) − ηR (y))(un (x) − un (y)) 2 V (ε x)un ηR dx − un (y) dxdy + on (1), ≤ 3 6 K |x − y|3+2s R R

which implies that   |un (x) − un (y)|2 1 ηR (x) dxdy + 1 − V (ε x)u2n ηR dx |x − y|3+2s K R6 R3  (ηR (x) − ηR (y))(un (x) − un (y)) ≤− un (y) dxdy + on (1). 6 |x − y|3+2s R



(13.3.1)

Using Hölder’s inequality, the boundedness of (un ) and Remark 1.4.6, we have     

R6

    12  (ηR (x) − ηR (y))(un (x) − un (y)) |ηR (x) − ηR (y)|2  un (y) dxdy  ≤ C dxdy |x − y|3+2s |x − y|3+2s R6 ≤

C . Rs

(13.3.2)

Using (13.3.1), (13.3.2) and the definition of ηR , we obtain that   1 |un (x) − un (y)|2 dx dy + 1 − V (ε x)u2n dx |x − y|3+2s K R3 \BR R3 R3 \BR    |un (x) − un (y)|2 1 ≤ η (x) dxdy + 1 − V (ε x)u2n ηR dx R |x − y|3+2s K R6 R3



≤



(13.3.3)

C + on (1), Rs

from which we deduce the thesis.

 

460

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

Proposition 13.3.4 The functional Jε satisfies the (PS)d condition in Hε at any level 3

0 < d < 3s S∗2s . Proof Let (un ) ⊂ Hε be a Palais-Smale sequence for Jε at the level d. By Lemma 13.3.1, we know that (un ) is bounded in Hε , and, up to a subsequence, we may assume that un  u

in Hε .

(13.3.4)

In view of Lemma 13.3.3, for each ζ > 0 there exists R = Rζ > 0 such that ε ⊂ B R 2 and  lim sup n→∞

 R3 \BR

dx

R3

|un (x) − un (y)|2 dy + |x − y|3+2s

 R3 \BR

 V (ε x)u2n dx < ζ.

(13.3.5)

Using (13.3.5) and the fact that Hε is compactly embedded in L2loc (R3 ), it is easy to deduce that un → u in L2 (R3 ). By interpolation, un → u in Lr (R3 ) for all r ∈ [2, 2∗s ). In particular, un → u

12

in L 3+2t (R3 ).

(13.3.6)

Then, in view of (13.3.4), (13.3.6), the fact that 4s + 2t > 3 and Lemma 13.1.2-(8), 

 φut n u2n dx →

R3

R3

φut u2 dx,

(13.3.7)

and 

 R3

φut n un ψ dx →

R3

φut uψ dx

for all ψ ∈ Cc∞ (R3 ).

Now, we know that Jε (un ), ψ = on (1) for all ψ ∈ Cc∞ (R3 ). Since it is clear that (13.3.4) and (f1 )-(f2 ) ensure that  (un , ψ)ε → (u, ψ)ε

and

 R3

g(ε x, un )ψ dx →

R3

g(ε x, u)ψ dx,

for all ψ ∈ Cc∞ (R3 ), we can use the denseness of Cc∞ (R3 ) in Hε to deduce that u is a critical point of Jε . In particular,  u2ε +

R3

 φut u2 dx =

R3

g(ε x, u)u dx.

(13.3.8)

13.3 An Existence Result for the Modified Problem

461

Let us show that 

 lim

n→∞ R3

g(ε x, un )un dx =

R3

(13.3.9)

g(ε x, u)u dx.

For this purpose, we argue as in the proof of Lemma 7.3.2. By the fractional Sobolev inequality (1.1.1) in Theorem 1.1.8,  R3 \BR

∗

|un |2s dx

 2∗ 2s

 ≤

∗

R3

|un ηR |2s dx



2 2∗ s

≤ C[un ηR ]2s .

Since 0 ≤ ηR ≤ 1, (13.3.3) and Remark 1.4.6 imply that 

[un ηR ]2s

|(un (x) − un (y))ηR (x) + (ηR (x) − ηR (y))un(y)|2 dxdy |x − y|N +2s R2N     |un (x) − un (y)|2 2 |ηR (x) − ηR (y)|2 2 η (x) dxdy + |u (y)| dxdy ≤C n R |x − y|N +2s |x − y|N +2s R2N R2N    C |un (x) − un (y)|2 ηR (x) dxdy + 2s ≤C N +2s 2N |x − y| R R =

≤

C C + on (1) + 2s , s R R

and so  lim lim sup

R→∞ n→∞

∗

R3 \B

|un |2s dx = 0.

(13.3.10)

|un |2 dx = 0,

(13.3.11)

R

Now, note that by (13.3.5) and (V1 ),  lim lim sup

R→∞ n→∞

R3 \BR

∗

and using interpolation on Lp -spaces and the boundedness of (un ) in L2s (R3 ), we also deduce that for every p ∈ (2, 2∗s )  lim lim sup

R→∞ n→∞

R3 \BR

|un |p dx = 0.

(13.3.12)

By (f1 ), (f2 ), (g2 ), (13.3.10), (13.3.11) and (13.3.12), we see that for every ζ > 0 there exists an R = Rζ > 0 such that  lim sup n→∞

R3 \BR

g(ε x, un )un dx < Cζ.

(13.3.13)

462

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

On the other hand, choosing R larger if necessary, we may assume that  R3 \BR

(13.3.14)

g(ε x, u)u dx < ζ.

Combining (13.3.13) and (13.3.14) we get   lim sup  n→∞

 R3 \BR

g(ε x, un )un dx −

R3 \BR

  g(ε x, u)u dx  < Cζ

for all ζ > 0,

and consequently 

 lim

n→∞ R3 \B R

g(ε x, un )un dx =

R3 \BR

g(ε x, u)u dx.

(13.3.15)

Next, note that, by the definition of g, ∗

g(ε x, un )un ≤ f (un )un + a 2s +

V0 2 u K n

in R3 \ ε .

Since the set BR ∩ (R3 \ ε ) is bounded, we can use (f1 )-(f2 ), the dominated convergence theorem and the strong convergence in Lrloc (R3 ) for all r ∈ [1, 2∗s ), to deduce that 

 lim

n→∞ B ∩(R3 \ ) ε R

g(ε x, un )un dx =

BR ∩(R3 \ε )

g(ε x, u)u dx

(13.3.16)

as n → ∞. We claim that  lim

n→∞  ε

∗

2s (u+ n ) dx =



∗

(u+ )2s dx.

(13.3.17)

ε

If we can prove this, then from Theorem 1.1.8, (g2 ), (f1 )-(f2 ), (13.3.4) and the dominated convergence theorem it will follow that 

 lim

n→∞  ∩B ε R

g(ε x, un )un dx =

ε ∩BR

g(ε x, u)u dx.

(13.3.18)

Putting together (13.3.15), (13.3.16) and (13.3.18), we conclude that (13.3.9) holds. Hence, in view of Jε (un ), un  = on (1), we see that (13.3.7), (13.3.8) and (13.3.9) imply that un ε → uε , and then un → u in Hε (since Hε is a Hilbert space). Thus, it remains to prove (13.3.17). Since (un ) is bounded in Hε , we may assume s 2 + 2∗s  ν, where μ and ν are two bounded non-negative that |(−) 2 u+ n |  μ and (un )

13.3 An Existence Result for the Modified Problem

463

measures on R3 . Invoking Lemma 1.5.1 we can find an at most countable index set I and sequences (xi )i∈I ⊂ R3 , (μi )i∈I , (νi )i∈I ⊂ (0, ∞), such that s

μ ≥ |(−) 2 u+ |2 +



μi δ x i ,

i∈I ∗

ν = (u+ )2s +



νi δxi

and

(13.3.19)

2 2∗

S∗ νi s ≤ μi

i∈I

for any i ∈ I , where δxi is the Dirac mass at the point xi . Let us show that (xi )i∈I ∩ε = ∅. Assume, by contradiction, that xi ∈ ε for some i ∈ I . For any ρ > 0, we define ∞ 3 3 i ψρ (x) = ψ( x−x ρ ) where ψ ∈ Cc (R , [0, 1]) is such that ψ = 1 in B1 , ψ = 0 in R \ B2 and ∇ψL∞ (R3 ) ≤ 2. We suppose that ρ > 0 is such that supp(ψρ ) ⊂ ε . Since (ψρ u+ n)  = o (1), whence is bounded, we have Jε (un ), ψρ u+ n n  R6

ψρ (y)

 ≤

R6

ψρ (y)

 ≤−  +

R3

+ 2 |u+ n (x) − un (y)| dxdy 3+2s |x − y| + 2 |u+ n (x) − un (y)| dxdy + |x − y|3+2s

 R3

2 φut n (u+ n ) ψρ dx +

 R3

2 V (ε x)(u+ n ) ψρ dx

(un (x) − un (y)(ψρ (x) − ψρ (y))) dxdy |x − y|3+2s  2∗s u+ ψ f (u ) dx + ψρ (u+ ρ n n n ) dx + on (1). R6

u+ n (x)

R3

(13.3.20)

Since f has subcritical growth and ψρ has compact support, we obtain that  lim lim

ρ→0 n→∞ R3

f (un )u+ n ψρ dx = lim



ρ→0 R3

ψρ f (u)u+ dx = 0.

On the other hand, by Hölder’s inequality, the boundedness of (un ) in Hε and Lemma 1.4.7 we get  lim lim sup

ρ→0 n→∞

R6

u+ n (x)

(un (x) − un (y)(ψρ (x) − ψρ (y))) dxdy = 0. |x − y|3+2s

Then, by (13.3.19) and taking the limit as ρ → 0 and n → ∞ in (13.3.20), we deduce that 3 νi ≥ μi . In view of the last statement in (13.3.19), we have νi ≥ S 2s , and using (g3 ), (V1 ) and the fact that K > 2 we obtain that 1 d = Jε (un ) − Jε (un ), un  + on (1) 4

464

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

    1 4s − 3 1 2 2∗s un g(ε x, un ) − G(ε x, un ) dx + ≥ un ε + (u+ n ) dx + on (1) 3 4 4 12 R \ε ε     s 1 1 ψρ |(−) 2 un |2 dx + V (ε x)u2n dx − V0 u2n dx ≥ 3 4 ε 4K R \ε R3 \ε  4s − 3 2∗s + (u+ n ) dx + on (1) 12 ε  s 1 ψρ |(−) 2 un |2 dx ≥ 4 ε    1 4s − 3 1 2 2∗s − + V (ε x)un dx + (u+ n ) dx + on (1) 3 4 4K 12 R \ε ε   s 1 4s − 3 2 2∗s ≥ ψρ |(−) 2 u+ ψρ (u+ n | dx + n ) dx + on (1), 4 ε 12 ε

where in the last inequality we used that K > 1, and that |x + − y + | ≤ |x − y| for all x, y ∈ R yields + 2 2 |u+ n (x) − un (y)| ≤ |un (x) − un (y)| . 3

Putting together (13.3.19) and the fact that νi ≥ S 2s , and letting n → ∞, we see that d≥ ≥ ≥

1 4 1 4



ψρ (xi )μi +

{i∈I :xi ∈ε }



4s − 3 12

2/2∗s

ψρ (xi )S∗ νi

+

{i∈I : xi ∈ε }



ψρ (xi )νi

{i∈I :xi ∈ε }

4s − 3 12



ψρ (xi )νi

{i∈I : xi ∈ε }

1 2s3 4s − 3 2s3 s 3 S∗ + S∗ = S∗2s , 4 12 3

which gives a contradiction. This means that (13.3.17) holds, which completes the proof.   Corollary 13.3.5 The functional ψε satisfies the (PS)d condition on S+ ε at any level 0 < 3

d < 3s S∗2s . Proof Let (un ) ⊂ S+ ε be a Palais-Smale sequence for ψε at the level d, that is ψε (un ) → d

and

∗ ψε (un ) → 0 in (Tun S+ ε ) .

13.3 An Existence Result for the Modified Problem

465

Using Proposition 13.2.3-(c), we have that (mε (un )) is a Palais-Smale sequence for Jε at the level d. Then, by Proposition 13.3.4, Jε satisfies the (PS)d condition in Hε , so there exists u ∈ S+ ε such that, up to a subsequence, mε (un ) → mε (u)

in Hε .

Finally, by Lemma 13.2.2-(iii), we infer that un → u in S+ ε .

 

We conclude with the proof of the main result of this section: Theorem 13.3.6 Assume that (V1 )-(V2 ) and (f 1)–(f 4) hold. Then, for all ε > 0, problem (13.2.2) admits a positive ground state. 3

Proof By Lemma 13.2.6, 0 < cε < 3s S∗2s . Then, taking into account Lemma 13.2.1, Lemma 13.3.1, Proposition 13.3.4, and applying Theorem 2.2.9, we see that Jε admits a nontrivial critical point u ∈ Hε . Since Jε (u), u−  = 0, where u− = min{u, 0}, it is easy to check that u ≥ 0 in R3 . Proceeding as in the proof of Lemma 13.6.1 below, we see that u ∈ L∞ (R3 ). In particular, 

u2 (y) dy + 3−2t |y−x|≥1 |x − y|  ≤ u2L2 (R3 ) + u2L∞ (R3 )

φut (x) =

 |y−x| 0 in R3 .

(13.4.1)

The Euler-Lagrange functional associated with (13.4.1) is given by J0 (u) =

1 1 u20 + 2 4



 R3

φut u2 dx −

R3

F (u) dx −

1 2∗s



∗

R3

(u+ )2s dx,

which is well defined on H0 = H s (R3 ) endowed with the norm

1 2 u0 = [u]2s + V0 u2L2 (R3 ) . Clearly, H0 is a Hilbert space with the inner product  u, ϕ0 = u, ϕDs,2 (R3 ) +

R3

V0 uϕ dx.

The Nehari manifold associated with J0 is given by N0 = {u ∈ H0 \ {0} : J0 (u), u = 0}. Let H0+ denote the open subset of H0 defined as H0+ = {u ∈ H0 : |supp(u+ )| > 0}, + + and S+ 0 = S0 ∩ H0 , where S0 is the unit sphere of H0 . We note that S0 is a incomplete 1,1 C -manifold of codimension 1 modeled on H0 and contained in H0+ . Thus, H0 = + + Tu S+ 0 ⊕ Ru for each u ∈ S0 , where Tu S0 = {u ∈ H0 : (u, v)0 = 0}. As in the previous section, we can verify that the following results hold.

Lemma 13.4.1 Assume that (f1 )–(f4 ) hold. Then: (i) For each u ∈ H0+ , let hu : R+ → R be defined by hu (t) = J0 (tu). Then, there is a unique tu > 0 such that h u (t) > 0 in (0, tu ), h u (t) < 0 in (tu , ∞).

13.4 The Autonomous Schrödinger-Poisson Equation

467

(ii) There exists τ > 0 independent of u such that tu ≥ τ for any u ∈ S+ 0 . Moreover, for + each compact set K ⊂ S0 there is a positive constant CK such that tu ≤ CK for any u ∈ K. ˆ 0 (u) = tu u is continuous and m0 = m ˆ 0 |S+ is a (iii) The map m ˆ 0 : H0+ → N0 given by m 0

−1 u homeomorphism between S+ 0 and N0 . Moreover m0 (u) = u0 . + (iv) If there is a sequence (un ) ⊂ S+ 0 such that dist(un , ∂S0 ) → 0, then m0 (un )0 → ∞ and J0 (m0 (un )) → ∞.

Let us define the maps ψˆ 0 : H0+ → R

and

ψ0 : S+ 0 → R,

ˆ 0 (u)) and ψ0 = ψˆ 0 |S+ . by ψˆ 0 (u) = J0 (m 0

Proposition 13.4.2 Assume that assumptions (f1 )–(f4 ) hold. Then: (a) ψˆ 0 ∈ C 1 (H0+ , R) and ψˆ 0 (u), v =

m ˆ 0 (u)0

J0 (m ˆ 0 (u)), v u0

for every u ∈ H0+ and v ∈ H0 . (b) ψ0 ∈ C 1 (S+ 0 , R) and ψ0 (u), v = m0 (u)0 J0 (m0 (u)), v, for every v ∈ Tu S+ 0. (c) If (un ) is a Palais-Smale sequence for ψ0 , then (m0 (un )) is a Palais-Smale sequence for J0 . If (un ) ⊂ N0 is a bounded Palais-Smale sequence for J0 , then (m−1 0 (un )) is a Palais-Smale sequence for ψ0 . (d) u is a critical point of ψ0 if and only if m0 (u) is a nontrivial critical point for J0 . Moreover, the corresponding critical values coincide and inf ψ0 (u) = inf J0 (u).

u∈S+ 0

u∈N0

Remark 13.4.3 The following equalities hold: c0 = inf J0 (u) = inf max J0 (tu) = inf max J0 (tu). u∈N0

t >0 u∈H+ 0

t >0 u∈S+ 0 3

Moreover, J0 has a mountain pass geometry and 0 < c0 < 3s S∗2s .

468

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

Now we prove the following useful lemma. 3

Lemma 13.4.4 Let 0 < d < 3s S∗2s . Let (un ) ⊂ H0 be a Palais-Smale sequence for J0 at the level d and un  0 in H0 . Then (a) either un → 0 in H0 , or (b) there exist a sequence (yn ) ⊂ R3 and constants R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Proof Assume that (b) does not occur. Then, by Lemma 1.4.4, un → 0 in Lp (R3 ) for all  p ∈ (2, 2∗s ). Moreover, by Lemma 13.1.2-(8), R3 φut n u2n dx → 0. Now we can argue as in the proof of Lemma 6.4.9 to get the thesis.   Remark 13.4.5 Let us observe that, if (un ) is a Palais-Smale sequence at the level c0 for the functional J0 such that un  u, then we may assume that u = 0. Otherwise, if un  0 and, if un  0 in H0 , then in view of Lemma 13.4.4, there exist a sequence (yn ) ⊂ R3 and R, β > 0 such that  lim inf n→∞

BR (yn )

u2n dx ≥ β > 0.

Set vn (x) = un (x + yn ). Then we see that (vn ) is a Palais-Smale sequence for J0 at the level c0 , (vn ) is bounded in H0 , and there exists v ∈ H0 such that vn  v with v = 0. As a consequence of Lemma 13.4.4, we can prove the following existence result: Theorem 13.4.6 Problem (13.4.1) admits a positive ground state solution. Proof To establish the existence of a ground state solution u to (13.4.1), it is enough to argue as in the proof of Lemma 6.4.10. We only have to observe that 1 c0 + on (1) = J0 (un ) − J0 (un ), un  4      1 1 1 1 2∗s f (un )un − F (un ) dx + − ∗ = un 20 + (u+ n ) dx 3 3 4 4 4 2 R R s

13.4 The Autonomous Schrödinger-Poisson Equation

469

and use assumption (f3 ). From J0 (u), u−  = 0 we get u ≥ 0 in R3 . Arguing as in the proof of Lemma 13.6.1, we see that u ∈ L∞ (R3 ). In particular, 

φut (x)

u2 (y) = dy + 3−2t |y−x|≥1 |x − y|  ≤ u2L2 (R3 ) + u2L∞ (R3 )

 |y−x| 0 in R3 . Now we give a compactness result for the autonomous problem which we will use later. Lemma 13.4.7 Let (un ) ⊂ N0 be a sequence such that J0 (un ) → c0 . Then (un ) has a convergent subsequence in H s (R3 ). Proof Since (un ) ⊂ N0 and J0 (un ) → c0 , we can apply Lemma 13.4.1-(iii), Proposition 13.4.2-(d) and the definition of c0 to infer that vn = m−1 (un ) =

un ∈ S+ 0 un 0

and ψ0 (vn ) = J0 (un ) → c0 = inf ψ0 (v). v∈S+ 0

+

Let us introduce the map F : S0 → R ∪ {∞} defined by ⎧ ⎨ψ (u), 0 F (u) = ⎩∞,

if u ∈ S+ 0, if u ∈ ∂S+ 0.

We note that +

• (S0 , d0 ), where d0 (u, v) = u − v0 , is a complete metric space; + • F ∈ C(S0 , R ∪ {∞}), by Lemma 13.4.1-(iv); • F is bounded below, by Proposition 13.4.2-(d). Hence, applying Theorem 2.2.1 to F , we can find (vˆn ) ⊂ S+ 0 such that (vˆn ) is a Palais-Smale sequence for ψ0 at the level c0 and vˆn − vn 0 = on (1). Then, using

470

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

Proposition 13.4.2, and Theorem 13.4.6 and arguing as in the proof of Corollary 13.3.5, we complete the proof.   Finally, arguing as in the proof of Lemma 12.3.2 and using the fact that V (ε ·) ≥ V0 , we obtain the following useful relation between cε and c0 : Lemma 13.4.8 limε→0 cε = c0 .

13.5

Barycenter Map and Multiplicity of Solutions to (13.2.2)

In this section, our main purpose is to apply the Lusternik-Schnirelman category theory to prove a multiplicity result for problem (13.2.2). We begin with the following technical result. Lemma 13.5.1 Let εn → 0 and (un ) = (uεn ) ⊂ Nεn be such that Jεn (un ) → c0 . Then there exists (y˜n ) = (y˜εn ) ⊂ R3 such that the translated sequence u˜ n (x) = un (x + y˜n ) has a subsequence which converges in H s (R3 ). Moreover, up to a subsequence, (yn ) = (εn y˜n ) is such that yn → y0 ∈ M. Proof Since Jε n (un ), un  = 0 and Jεn (un ) → c0 , it is easy to see that (un ) is bounded in Hεn . Let us observe that un εn  0, since c0 > 0. Therefore, arguing as in the first part of Lemma 7.3.7, we can find a sequence (y˜n ) ⊂ R3 and constants R, α > 0 such that  lim inf n→∞

BR (y˜n )

u2n dx ≥ α.

Set u˜ n (x) = un (x + y˜n ). Then, (u˜ n ) is bounded in H s (R3 ), and we may assume that u˜ n  u˜

in H s (R3 ),

for some u˜ = 0. Let (tn ) ⊂ (0, ∞) be such that v˜n = tn u˜ n ∈ N0 (see Lemma 13.4.1-(i)), and set yn = εn y˜n . Then, using (g2 ), Lemma 13.1.2-(4) and the translation invariance of R3 , we see that c0 ≤ J0 (v˜n )   s 1 1 2 2 2 ≤ |(−) v˜n | + V (εn x + yn )v˜n dx + φ t v˜ 2 dx 2 R3 4 R3 v˜n n

13.5 Barycenter Map and Multiplicity of Solutions to (13.2.2)

471

  1 + 2∗s − F (v˜n ) + ∗ (v˜n ) dx 2s R3    s t4 t2 |(−) 2 un |2 + V (εn x)u2n dx + n φut n u2n dx − G(εn z, tn un ) dx ≤ n 2 R3 4 R3 R3 

= Jεn (tn un ) ≤ Jεn (un ) = c0 + on (1), and so J0 (v˜n ) → c0

(v˜n ) ⊂ N0 .

and

(13.5.1)

In particular, (13.5.1) yields that (v˜n ) is bounded in H s (R3 ), so we may assume that v˜n  v. ˜ Obviously, (tn ) is bounded and we may assume that tn → t0 ≥ 0. If t0 = 0, then from the boundedness of (u˜ n ) it follows that v˜n 0 = tn u˜ n 0 → 0, that is, J0 (v˜n ) → 0, in contrast with the fact c0 > 0. Hence, t0 > 0. By the uniqueness of the weak limit, v˜ = t0 u˜ and v˜ = 0. Using (13.5.1) and Lemma 13.4.7 we deduce that v˜n → v˜

H s (R3 ),

in

(13.5.2)

which implies that u˜ n → u˜ in H s (R3 ) and J0 (v) ˜ = c0

J0 (v), ˜ v ˜ = 0.

and

Now let us show that (yn ) admits a subsequence, still denoted by (yn ), such that yn → y0 ∈ M. Assume, by contradiction, that (yn ) is not bounded. Then there exists a subsequence, still denoted by (yn ), such that |yn | → ∞. Since un ∈ Nεn , we see that  u˜ n 20

≤

[u˜ n ]2s

+

 R3

V (εn x

+ yn )u˜ 2n dx

+



R3

φut˜ n u˜ 2n dx

=

R3

g(ε n x + yn , u˜ n )u˜ n dx.

Take R > 0 such that  ⊂ BR , and assume that |yn | > 2R for n large. Thus, for any x ∈ BR/ εn , we get | εn x + yn | ≥ |yn | − | εn x| > R for all n large enough. Hence, by the definition of g, we deduce that  u˜ n 20 ≤

f˜(u˜ n )u˜ n dx + BR/ ε n



2∗

R3 \BR/ εn

f (u˜ n )u˜ n + u˜ ns dx.

Since u˜ n → u˜ in H s (R3 ), it follows from the dominated convergence theorem that  R3 \BR/ εn

f (u˜ n )u˜ n dx = on (1).

472

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

Therefore, u˜ n 20 ≤

1 K

 BR/ ε n

V0 u˜ 2n dx + on (1),

which yields   1 1− u˜ n 20 ≤ on (1). K But u˜ n → u˜ = 0 and K > 2, so we get a contradiction. Thus (yn ) is bounded and, up / , then there exists r > 0 to a subsequence, we may assume that yn → y0 . If y0 ∈ such that yn ∈ Br/2 (y0 ) ⊂ R3 \  for any n large enough. Reasoning as before, we reach a contradiction. Hence, y0 ∈ . Now, we show that V (y0 ) = V0 . Assume, by contradiction, that V (y0 ) > V0 . Taking into account (13.5.2), Fatou’s lemma and the translation invariance of R3 , we have    s 1 |(−) 2 v˜n |2 + V (εn x + yn )v˜n2 dx n→∞ 2 R3      1 1 + 2∗s t 2 φ v˜ dx − + F (v˜n ) + ∗ (v˜n ) dx 4 R3 v˜n n 2s R3

c0 < lim inf

≤ lim inf Jεn (tn un ) ≤ lim inf Jεn (un ) = c0 , n→∞

n→∞

which is impossible. Therefore, in view of (V2 ), we conclude that y0 ∈ M.

 

Now, we aim to relate the number of positive solutions of (13.2.2) to the topology of the set . For this reason, we take δ > 0 such that Mδ = {x ∈ R3 : dist(x, M) ≤ δ} ⊂ , and let η be a smooth nonincreasing cut-off function defined in [0, ∞) such that η = 1 in [0, 2δ ], η = 0 in [δ, ∞), 0 ≤ η ≤ 1 and |η | ≤ c for some c > 0. For y ∈ , define  ε,y (x) = η(| ε x − y|)w

εx −y ε



where w ∈ H s (R3 ) is a positive ground state solution to problem (13.4.1) (see Theorem 13.4.6). Let tε > 0 be the unique number such that max Jε (tε,y ) = Jε (tε ε,y ). t ≥0

13.5 Barycenter Map and Multiplicity of Solutions to (13.2.2)

473

Finally, we introduce ε : M → Nε given by ε (y) = tε ε,y . Lemma 13.5.2 The functional ε has the property that lim Jε (ε (y)) = c0 ,

ε→0

uniformly in y ∈ M.

Proof Suppose this is not the case, i.e., there exist δ0 > 0, (yn ) ⊂ M and εn → 0 such that |Jεn (εn (yn )) − c0 | ≥ δ0 .

(13.5.3)

ε n x − yn , if z ∈ B δ , we have that εn z ∈ Bδ εn εn ∗ and εn z + yn ∈ Bδ (yn ) ⊂ Mδ ⊂ . Then, recalling that G(x, t) = F (t) + 21∗ (t + )2s for s (x, t) ∈  × R, we have Then, using the change of variable z =

Jε (εn (yn )) =

+

tε2n 2 tε4n 4

R3

tε ns 2∗s





s

R3



2∗

−



|(−) 2 (η(| εn z|)w(z))|2 dz + 

t 2 φη(| εn z|) (η(| ε n z|)w(z)) dz −



R3

R3

V (εn z + yn )(η(| ε n z|)w(z))2 dz

F (tεn η(| εn z|)w(z)) dz

∗

R3

(η(| εn z|)w(z))2s dz.

(13.5.4)

Let us show that the sequence (tεn ) satisfies tεn → 1 as εn → 0. By the definition of tεn , it follows that Jε n (εn (yn )), εn (yn ) = 0, which gives  tε2n



s

R3

|(−) 2 (η(| εn z|)w(z))|2 + V (εn z + yn )(η(| εn z|)w(z))2 dz



+ tε4n

R3

 =

R3

t 2 φη(| εn z|) (η(| εn z|)w(z)) dz

g(ε n z + yn , tεn η(| εn z|)w(z))tεn η(| εn z|)w(z) dz.

Since η(|x|) = 1 for x ∈ B δ and B δ ⊂ B 2

1 tε2n



s

R3

2

δ 2 εn

(13.5.5)

for all n sufficiently large, (13.5.5) yields

|(−) 2 εn ,yn |2 + V (εn x)ε2n ,yn dx +

 R3

t φ ε2n ,yn dx εn ,yn

474

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .



∗

f (tεn εn ,yn ) + (tεn εn ,yn )2s −1 4 εn ,yn dx (tεn εn ,yn )3 R3  ∗ 2∗ −4 ≥ tεns |w(z)|2s dz.

=

Bδ

2

By the continuity of w, there exists a vector zˆ ∈ R3 such that w(ˆz) = min w(z) > 0, z∈B δ

2

which implies that   s 1 t 2 ε ,y |2 + V (ε n x) 2 |(−) dx + φ ε2n ,yn dx εn ,yn n n εn ,yn tε2n R3 R3 2∗ −4

≥ tεns

∗

w2s (ˆz)|B δ |.

(13.5.6)

2

Now, assume by contradiction that tεn → ∞. Lemma 1.4.8, Lemma 13.1.2-(7) and the dominated convergence theorem imply that   2 2 t 2 t 2 φεn ,yn εn ,yn dx → φw w dx, εn ,yn εn → w0 ∈ (0, ∞), R3

 εn ,yn L2∗s (R3 ) → wL2∗s (R3 ) ,

R3

f (tεn εn ,yn ) 4  dx → (tεn εn ,yn )3 εn ,yn

R3

 R3

f (t0 w) 4 w dx. (t0 w)3 (13.5.7)

Hence, using that tεn → ∞, (13.5.6) and (13.5.7), we obtain  t 2 φw w dx = ∞, R3

which is a contradiction. Therefore, the sequence (tεn ) is bounded and, up to subsequence, we may assume that tεn → t0 for some t0 ≥ 0. Let us prove that t0 > 0. Suppose this is not the case, i.e., t0 = 0. Then, taking into account (13.5.7) and the growth assumptions on g, we see that (13.5.5) gives tεn εn ,yn 2εn → 0 which is impossible because tεn εn ,yn ∈ Nεn and because of Remark 13.2.5. Hence, t0 > 0. Then, letting n → ∞ in (13.5.5), we deduce from (13.5.7) and the dominated convergence theorem that 1 w20 + t02



 R3

t 2 φw w dx =

∗

R3

f (t0 w) + (t0 w)2s −1 4 w dx. (t0 w)3

13.5 Barycenter Map and Multiplicity of Solutions to (13.2.2)

475

Since w ∈ N0 and (f5 ) holds, we infer that t0 = 1. Then, passing to the limit as n → ∞ in (13.5.4), by tεn → 1 and (13.5.7) we obtain lim Jεn (εn (yn )) = J0 (w) = c0 ,

n→∞

 

which contradicts (13.5.3).

It is time to define the barycenter map. Take ρ = ρ(δ) > 0 such that Mδ ⊂ Bρ , and consider the map ϒ : R3 → R3 given by  ϒ(x) =

x, ρx |x| ,

if |x| < ρ, if |x| ≥ ρ.

Define the barycenter map βε : Nε → R3 as follows:  ϒ(ε x)u2 (x) dx

βε (u) =

R3 

R3

. u2 (x) dx

Arguing as in the proof of Lemma 6.3.18, we see that the function βε has the following property: Lemma 13.5.3 lim βε (ε (y)) = y,

ε→0

uniformly in y ∈ M.

ε of Nε by taking a function h1 : R+ → R+ such that Next, we introduce a subset N h1 (ε) → 0 as ε → 0 and setting ε = {u ∈ Nε : Jε (u) ≤ c0 + h1 (ε)} . N ε = ∅ By Lemma 13.5.2, h1 (ε) = supy∈M |Jε (ε (y)) − c0 | → 0 as ε → 0. Therefore, N for any ε > 0. Additionally, as in the proof of Lemma 6.3.19, we have: Lemma 13.5.4 For every δ > 0, lim sup dist(βε (u), Mδ ) = 0.

ε→0

ε u∈N

Now we prove a multiplicity result for (13.2.2).

476

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

Theorem 13.5.5 Assume that (V1 )-(V2 ) and (f1 )–(f4 ) hold. Then, for any δ > 0 such that Mδ ⊂ , there exists ε¯ δ > 0 such that, for any ε ∈ (0, ε¯ δ ), problem (13.2.2) has at least catMδ (M) positive solutions. −1 Proof For any ε > 0, define the map αε : M → S+ ε by αε (y) = mε (ε (y)). By Lemma 13.5.2,

lim ψε (αε (y)) = lim Jε (ε (y)) = c0 ,

ε→0

ε→0

uniformly in y ∈ M.

(13.5.8)

Set Sε+ = {w ∈ S+ ε : ψε (w) ≤ c0 + h1 (ε)}, where h1 (ε) = supy∈M |ψε (αε (y)) − c0 |. It follows from (13.5.8) that h1 (ε) → 0 as ε → 0. By the definition of h1 (ε), we see that, for all y ∈ M and ε > 0, αε (y) ∈ Sε+ . Consequently, Sε+ = ∅ for all ε > 0. From the above considerations, together with Lemma 13.2.2-(ii), Lemma 13.5.2, Lemma 13.5.3 and Lemma 13.5.4, we see that there exists ε¯ = ε¯ δ > 0 such that the following diagram ε

m−1 ε

mε

βε

M −→ ε (M) −→ αε (M) −→ ε (M) −→ Mδ is well defined for any ε ∈ (0, ε¯ ). Thanks to Lemma 13.5.3, and decreasing ε¯ if necessary, we have that βε (ε (y)) = y + θ (ε, y) for all y ∈ M, for some function θ (ε, y) such that |θ (ε, y)| < 2δ uniformly in y ∈ M and for all ε ∈ (0, ε¯ ). Then, it is easy to check that the map H : [0, 1] × M → Mδ defined by H (t, y) = y + (1 − t)θ (ε, y) is a homotopy between βε ◦ ε = (βε ◦ mε ) ◦ (m−1 ε ◦ ε ) and the inclusion map id: M → Mδ . This fact together with Lemma 6.3.21 implies that catαε (M) αε (M) ≥ catMδ (M).

(13.5.9)

Applying Corollary 13.3.5, Lemma 13.4.8 and Theorem 2.4.6 with c = cε ≤ c0 + h1 (ε) = d and K = αε (M), we deduce that ψε has at least catαε (M) αε (M) critical points on Sε+ . Therefore, by Proposition 13.2.3-(d) and (13.5.9), we infer that (13.2.2) has at least   catMδ (M) solutions.

13.6 Proof of Theorem 13.1.1

13.6

477

Proof of Theorem 13.1.1

This last section is devoted to the proof of Theorem 13.1.1, namely, that the solutions of (13.2.2) are indeed solutions of the original problem (13.1.1). Firstly, we prove the following useful L∞ -estimate for the solutions of the modified problem (13.2.2). εn be a solution to (13.2.2). Then, up to a Lemma 13.6.1 Let εn → 0 and un ∈ N ∞ N subsequence, vn = un (· + y˜n ) ∈ L (R ) and there exists C > 0 such that vn L∞ (R3 ) ≤ C

for all n ∈ N,

where (y˜n ) is given in Lemma 13.5.1. Moreover, vn (x) → 0 as |x| → ∞ uniformly in n ∈ N. Proof Recalling that Jεn (un ) ≤ c0 +h1 (εn ), with h1 (εn ) → 0 as n → ∞, we can proceed as in the proof of (13.5.1) to deduce that Jεn (un ) → c0 . Invoking Lemma 13.5.1, we find a sequence (y˜n ) ⊂ R3 such that εn y˜n → y0 ∈ M and vn = un (· + y˜n ) has a convergent subsequence in H s (R3 ). To obtain the L∞ -estimate, we can proceed as in the proof of Lemma 6.3.23. Indeed,  2  1 β−1 2(β−1) S∗ vn vL,n 2 2∗s 3 + Vn (x)vn2 vL,n dx L (R ) β R3    (vn (x) − vn (y)) 2(β−1) 2(β−1) ≤ ((vn vL,n )(x) − (vn vL,n )(y)) dxdy |x − y|N+2s R6  2(β−1) Vn (x)vn2 vL,n dx + R3

 ≤

2(β−1)

R3

gn (x, vn )vn vL,n

(13.6.1)

dx,

where we used the notations Vn (x) = V (εn x +εn y˜n ) and gn (x, vn ) = g(εn x +εn y˜n , vn ). Assumptions (g1 ) and (g2 ) imply that for every ξ > 0 there exists Cξ > 0 such that ∗

|gn (x, vn )| ≤ ξ |vn | + Cξ |vn |2s −1 . Taking ξ ∈ (0, V0 ), and using (13.6.1) we obtain 

β−1

vn vL,n 2 2∗s

L (R3 )

≤ Cβ 2

∗

R3

2(β−1)

|vn |2s vL,n

dx.

478

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

Therefore, we deduce that vn L∞ (R3 ) ≤ C for all n ∈ N. Let now wn (x, y) = Ext(vn ) = Ps (x, y) ∗ vn (x) the s-harmonic extension of vn and we note that it solves ⎧ 1−2s ∇w ) = 0 ⎪ in R3+1 n ⎨ − div(y + , on ∂R3+1 wn (·, 0) = vn + , ⎪ ⎩ ∂w = −V (x)v − φ t v + g (x, v ) on ∂R3+1 . n n n n + vn n ∂ν 1−2s From the uniform boundedness of (vn ) in L2 (R3 ) ∩ L∞ (R3 ), we deduce that 0 ≤ φvt n ≤ C for all n ∈ N (see the estimate in Theorem 13.4.6). Then we can argue as in the proof of Lemma 6.3.23 to conclude that vn (x) → 0 as |x| → ∞ uniformly in n ∈ N (see also Remark 7.2.10 and note that vn is a subsolution to (−)s vn + V0 vn = gn (x, vn ) in R3 ).   Now, we are ready to give the proof of the main result of this chapter. Proof of Theorem 13.1.1 Take δ > 0 such that Mδ ⊂ . We begin by proving that there ε of (13.2.2), exists ε˜ δ > 0 such that for every ε ∈ (0, ε˜ δ ) and every solution uε ∈ N uε L∞ (R3 \ε ) < a.

(13.6.2)

Assume, by contradiction, that for some subsequence (εn ) such that εn → 0, there exists εn such that Jε (un ) = 0 and un = uε n ∈ N n un L∞ (R3 \εn ) ≥ a.

(13.6.3)

Since Jεn (un ) ≤ c0 + h1 (εn ) and h1 (εn ) → 0, we can argue as in the first part of the proof of Lemma 13.5.1 to deduce that Jεn (un ) → c0 . In view of Lemma 13.5.1, there exists a sequence (y˜n ) ⊂ R3 such that u˜ n = un (· + y˜n ) → u˜ in H s (R3 ) and εn y˜n → y0 ∈ M. Now, if we choose r > 0 such that Br (y0 ) ⊂ B2r (y0 ) ⊂ , we see that B εr ( εyn0 ) ⊂ εn . n Then, for any y ∈ B εr (y˜n ), we have n

        y − y0  ≤ |y − y˜n | + y˜n − y0  < 1 (r + on (1)) < 2r    εn εn  εn εn

for n sufficiently large.

Therefore, for these values of n, we get R3 \ εn ⊂ R3 \ B εr (y˜n ). n

(13.6.4)

On the other hand, using Lemma 13.6.1 we can verify that u˜ n (x) → 0

as |x| → ∞

(13.6.5)

13.6 Proof of Theorem 13.1.1

479

uniformly in n ∈ N. Consequently, there exists R > 0 such that u˜ n (x) < a

for all |x| ≥ R, n ∈ N.

Hence, un (x) < a for any x ∈ R3 \ BR (y˜n ) and n ∈ N. On the other hand, there exists ν ∈ N such that for any n ≥ ν and r/ εn > R we have R3 \ εn ⊂ R3 \ B εr (y˜n ) ⊂ R3 \ BR (y˜n ), n

which implies that un (x) < a for any x ∈ R3 \ εn and n ≥ ν. This is impossible in view of (13.6.3). Let ε¯ δ > 0 be given by Theorem 13.5.5, and fix ε ∈ (0, εδ ), where εδ = min{˜εδ , ε¯ δ }. By Theorem 13.5.5, problem (13.2.2) admits at least catMδ (M) ε satisfies (13.6.2), by nontrivial solutions. Let uε be one of these solutions. Since uε ∈ N the definition of g it follows that uε is a solution of (13.2.1). Then uˆ ε (x) = uε (x/ ε) is a solution to (13.1.1). Therefore, (13.1.1) has at least catMδ (M) nontrivial solutions. Finally, let us analyze the behavior of the maximum points of solutions to problem (13.1.1). Take εn → 0 and consider a sequence (un ) ⊂ Hεn of solutions to (13.2.1) as above. Note that (g1 ) implies that we can find γ ∈ (0, a) sufficiently small such that g(ε x, t)t ≤

V0 2 t K

for any x ∈ R3 , 0 ≤ t ≤ γ .

(13.6.6)

Arguing as before, we can find R > 0 such that un L∞ (BRc (y˜n )) < γ .

(13.6.7)

Moreover, up to extracting a subsequence, we may assume that un L∞ (BR (y˜n )) ≥ γ .

(13.6.8)

Indeed, if (13.6.8) does not hold, then, in view of (13.6.7), we have un L∞ (R3 ) < γ . Further, using that Jε n (un ), un  = 0 and (13.6.6) we infer that  un 2εn ≤ un 2εn +

 R3

φut n u2n dx =

R3

g(εn x, un )un dx ≤

V0 K

 R3

u2n dx

which yields un εn = 0, which is impossible. Hence, (13.6.8) holds. Taking into account (13.6.7) and (13.6.8), we deduce that if pn ∈ R3 is a global maximum point of un , then it belongs to BR (y˜n ). Therefore, pn = y˜n + qn for some qn ∈ BR . Consequently, ηn = εn y˜n + εn qn is a global maximum point of uˆ n (x) = un (x/ εn ). Since |qn | < R for any n ∈ N and εn y˜n → y0 ∈ M (in view of Lemma 13.5.1), the continuity of V ensures that lim V (ηn ) = V (y0 ) = V0 .

n→∞

480

13 Multiplicity and Concentration Results for a Fractional Schrödinger-. . .

To conclude the proof of Theorem 13.1.1, we estimate the decay of solutions to (13.1.1). By Lemma 3.2.17, we can find a positive function w such that 0 < w(x) ≤

C 1 + |x|3+2s

for all x ∈ R3 ,

(13.6.9)

and (−)s w +

V0 w = 0 in R3 \ B R1 , 2

(13.6.10)

in the classical sense, for some suitable R1 > 0. By (g1 ) and (13.6.5), we can find R2 > 0 sufficiently large such that   V0 V0 (−) u˜ n + u˜ n = gn (x, u˜ n ) − Vn − u˜ n − φut˜ n u˜ n 2 2 s

≤ gn (x, u˜ n ) −

V0 u˜ n ≤ 0 in R3 \ B R2 . 2

(13.6.11)

Choose R3 = max{R1 , R2 }, and set a = min w > 0

w˜ n = (b + 1)w − a u˜ n ,

and

B R3

(13.6.12)

where b = supn∈N u˜ n L∞ (R3 ) < ∞. We claim that w˜ n ≥ 0 in R3 .

(13.6.13)

Indeed, note first that (13.6.10), (13.6.11) and (13.6.12) yield w˜ n ≥ ba + w − ba > 0 (−)s w˜ n +

V0 w˜ n ≥ 0 2

in BR3 ,

(13.6.14)

in R3 \ B R3 .

(13.6.15)

Then we can use Lemma 1.3.8 to verify that (13.6.13) holds. Using (13.6.9), we see that u˜ n (x) ≤

C˜ 1 + |x|3+2s

for all x ∈ R3 , n ∈ N,

(13.6.16)

13.6 Proof of Theorem 13.1.1

481

for some C˜ > 0. Since uˆ n (x) = un ( εxn ) = u˜ n ( εxn − y˜n ) and ηn = εn y˜n + εn qn , inequality (13.6.16) implies that  0 < uˆ n (x) = un ≤ = ≤

x εn

1 + | εxn



 = u˜ n

x − y˜εn εn



C˜ − y˜n |3+2s C˜ εn3+2s

ε3+2s n

+|x − εn y˜n |3+2s C˜ ε3+2s n

ε3+2s +|x − ηn |3+2s n

This completes the proof of Theorem 13.1.1.

for all x ∈ R3 .  

Remark 13.6.2 The approach used in this chapter can be easily adapted to deal with the subcritical case (in this situation the Palais-Smale condition holds for all d ∈ R).

An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

14.1

14

Introduction

In this chapter we deal with the following nonlinear fractional Kirchhoff–Schrödinger– Poisson system

  2 p + q R6 |u(x)−u(y)| dxdy (−)s u + μφu = g(u) in R3 , N+2s |x−y| (14.1.1) (−)t φ = μu2 in R3 , where s ∈ ( 34 , 1), t ∈ (0, 1), p > 0, q ≥ 0, μ > 0 is a parameter, and g : R → R is an odd C 1,α (R) nonlinearity, with α > max{0, 1 − 2s}, satisfying the following Berestycki–Lions type assumptions [100]: (g1 ) (g2 ) (g3 )

g(τ ) ) −∞ < lim infτ →0 g(τ τ ≤ lim supτ →0 τ = −m < 0; g(τ ) 6 −∞ ≤ lim supτ →∞ 2∗s −1 ≤ 0, where 2∗s = 3−2s ; τ ζ there exists ζ > 0 such that G(ζ ) = 0 g(r) dr > 0.

We note that if μ = q = 0 and p = 1, problem (14.1.1) reduces to the fractional Schrödinger equation (−)s u + V (x)u = g(u)

in R3 .

(14.1.2)

On the other hand, when μ = 0, problem (14.1.1) reduces to the fractional Kirchhoff equation   p+q

R6

 |u(x) − u(y)|2 dxdy (−)s u = g(u) in R3 , |x − y|N+2s

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_14

(14.1.3)

483

484

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

Finally, when p = 1 and q = 0, (14.1.1) becomes the fractional Schrödinger-Poisson system 

(−)s u + μφu = g(u) in R3 , in R3 . (−)t φ = μu2

(14.1.4)

In the present chapter, we prove an existence result for (14.1.1) when μ > 0 is small enough. More precisely, we are able to show that: Theorem 14.1.1 Let us suppose that (g1 ), (g2 ), and (g3 ) are satisfied. Then, there exists μ0 > 0 such that for any 0 < μ < μ0 , the system (14.1.1) admits a positive solution (u, φ) ∈ H s (R3 ) × Dt,2 (R3 ). The proof of our main result is obtained by exploiting suitable variational methods based on a truncation argument and the Struwe’s monotonicity trick developed by Jeanjean [231]. It is worth pointing out that our approach for attacking the problem has to take care of the presence of general nonlinearities and combined effects of different nonlocal terms, so an accurate and delicate analysis is required.

14.2

Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional

Since we are dealing with general nonlinearities and we seek positive solutions of (14.2.7), similarly to [100] (see also [82, 139, 342]), we modify the nonlinearity g in a convenient way. Without loss of generality, we assume that 0 < ζ = inf{τ ∈ (0, ∞) : G(τ ) > 0}, where ζ > 0 is given in (g3 ). Define g˜ : R → R by  g(τ ˜ )=

g(τ ) for τ ∈ [0, τ0 ], 0 for τ ∈ R \ [0, τ0 ],

where τ0 = min{τ ∈ (ζ, ∞) : g(τ ) = 0} (τ0 = ∞ if g(τ ) > 0 for all τ ≥ ζ ). It is easy to check that if u is a nontrivial solution of (14.1.1) with g˜ in the place of g, then 0 ≤ u ≤ τ0 in R3 , that is, u is a non-negative solution of (14.1.1) with nonlinearity g. Hence, we assume that g is extended as g. ˜ In particular, g satisfies the assumptions (g1 ), (g3 ) and lim

t →∞

g(τ ) = 0. ∗ τ 2s −1

(g2 )

14.2 Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional

485

Now set  g1 (τ ) =

(g(τ ) + mτ )+ for τ ≥ 0, 0 for τ < 0,

g2 (τ ) = g1 (τ ) − g(τ )

for τ ∈ R.

Note that g1 , g2 ≥ 0 in [0, ∞) and g1 (τ ) = 0, τ g1 (τ ) lim = 0, ∗ τ →∞ τ 2s −1 lim

(14.2.1)

τ →0

g2 (τ ) ≥ mτ

(14.2.2)

for all τ ≥ 0.

(14.2.3)

Consequently, for every ε > 0 there exists Cε > 0 such that ∗

g1 (τ ) ≤ Cε τ 2s −1 + εg2 (τ )

for all τ ≥ 0.

(14.2.4)

Set 

τ

Gi (τ ) =

gi (r) dr,

i = 1, 2.

0

Then, by (14.2.1)–(14.2.4), we have G2 (τ ) ≥

m 2 τ 2

for all τ ∈ R,

(14.2.5)

and for any ε > 0 there exists Cε > 0 such that ∗

G1 (τ ) ≤ ε G2 (τ ) + Cε |τ |2s

for all τ ∈ R.

(14.2.6)

As observed in Chap. 13, the system (14.1.1) can be reduced to a single equation. Substituting the expression (13.1.5) of φut in (14.1.1), we obtain the following fractional equation: (p + q[u]2s )(−)s u + μφut u = g(u) in R3 ,

(14.2.7)

whose solutions can be found by looking for critical points of the functional Jμ : H s (R3 ) → R defined by Jμ (u) =

p 2 q 4 μ [u] + [u]s + 2 s 4 4



 R3

φut u2 dx −

R3

G(u) dx.

(14.2.8)

486

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

From the growth assumptions on g and Lemma 13.1.2-(1), it is easy to check that Jμ ∈ C 1 (H s (R3 ), R) and that the critical points of Jμ are the weak solutions of (14.2.7). Definition 14.2.1 (i) We say that (u, φ) ∈ H s (R3 ) × Dt,2 (R3 ) is a weak solution to (14.1.1) if u is a weak solution to (14.2.7). (ii) We say that u ∈ H s (R3 ) is a weak solution to (14.2.7) if 

% R3

 & s s (p + q[u]2s )(−) 2 u(−) 2 v + μφut uv dx =

R3

g(u)v dx

for any v ∈ H s (R3 ). s (R3 ), which is a natural constraint (we We will look for critical points of Jμ on Hrad remark that, by Lemma 13.1.2-(2), if u is a radial function, then so is φut ). Due to the presence of different nonlocal terms and the general nonlinearity, it is not easy to verify the geometric assumptions of the mountain pass theorem and the boundedness of Palais– Smale sequences for Jμ . Therefore, inspired by [232, 239], we introduce the cut-off function χ ∈ C ∞ ([0, ∞), R) by

⎧ χ(τ ) = 1 for τ ∈ [0, 1], ⎪ ⎪ ⎪ ⎨ 0 ≤ χ(τ ) ≤ 1 for τ ∈ (1, 2), ⎪ χ(τ ) = 0 for τ ∈ [2, ∞), ⎪ ⎪ ⎩

χ L∞ (0,∞) ≤ 2, and then  ξk (u) = χ

uαLα (R3 )



kα

with α =

12 . 3 + 2t

s (R3 ) → R given by Now consider the truncated functional Jμk : Hrad

Jμk (u) =

p 2 q 4 μ [u] + [u] + ξk (u) 2 s 4 s 4



 R3

φut u2 dx −

R3

G(u) dx.

Clearly, if u is a critical point of Jμk with uLα (R3 ) ≤ k, then u is also a critical point of Jμ . The C 1 -functional Jμk satisfies the geometric assumptions of the mountain pass theorem (note that s > 34 ) but, since g is a general nonlinearity, we are not able to prove the boundedness of the Palais–Smale sequences. For this reason, we use a slightly different version of Theorem 5.2.2.

14.2 Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional

487

Theorem 14.2.2 ([82, 231]) Let (X,  · ) be a Banach space and let  ⊂ R+ be an interval. Consider a family (Jλ )λ∈ of C 1 -functionals on X of the form Jλ (u) = A(u) − λB(u),

for λ ∈ ,

with B nonnegative and either A(u) → ∞ or B(u) → ∞ as u → ∞, and such that Jλ (0) = 0. For λ ∈ , set λ = {γ ∈ C([0, 1], X) : γ (0) = 0, Jλ (γ (1)) < 0}. If for any λ ∈  the set λ is nonempty and cλ = inf max Jλ (γ (t)) > 0, γ ∈λ t ∈[0,1]

then, for almost every λ ∈ , there is a sequence (uj ) ⊂ X such that (i) (uj ) is bounded; (ii) Jλ (uj ) → cλ ; (iii) Jλ (uj ) → 0 on X∗ . Moreover, the function λ → cλ is non-increasing and continuous from the left. In order to apply Theorem 14.2.2, we consider the following parametrized family of C 1 -functionals: k Jμ,λ (u)

p q μ = [u]2s + [u]4s + ξk (u) 2 4 4

 R3

 φut u2 dx

+

 R3

G2 (u) dx − λ

R3

G1 (u) dx.

s (R3 ), with λ ∈ [δ, ¯ 1], where δ¯ ∈ (0, 1) will be chosen later in a suitable way. for u ∈ Hrad More precisely, we have the following result.

Lemma 14.2.3 Under assumptions (g1 ), (g2 ), (g3 ), the conclusions of Theorem 14.2.2 hold. s (R3 ) such that Proof First, we show that there exists w ∈ Hrad R > 1, we define

 R3

⎧ ⎪ for |x| ≤ R, ⎨ζ wR (x) = ζ(R + 1 − |x|) for |x| ∈ [R, R + 1], ⎪ ⎩ 0 for |x| ≥ R + 1.

G(w) dx > 0. For

488

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

s (R3 ). By the definitions of w and g, It is clear that wR ∈ Hrad R



 R3

G(wR )dx = G(ζ )|BR | +

{R≤|x|≤R+1}

G(ζ (R + 1 − |x|)) dx

≥ G(ζ )|BR | − |BR+1 − BR | max |G(t)| t ∈[0,ζ ]

3

=

π2 ( 32 + 1)

[G(ζ )R 3 − max |G(t)|((R + 1)3 − R 3 )] t ∈[0,ζ ]

 $  1 3 ≥ − 1 R3 , G(ζ ) − max |G(t)| 1 + 3 t ∈[0,ζ ] R ( 2 + 1) 3

π2

#

 ¯ Then we take w = wR so there exists R¯ > 0 such that R3 G(wR ) dx > 0 for all R ≥ R. ¯ with R large enough. Moreover, we can find δ ∈ (0, 1) such that δ¯



 R3

G1 (w) dx −

R3

G2 (w) dx > 0.

Therefore, if we consider the path γ (t) = w( ¯ t· ) if t ∈ (0, 1] and γ (0) = 0, where w¯ = w( θ·¯ ) and θ¯ > 0, we can use the definition of χ, λ ≥ δ¯ and the above inequality to see that k Jμ,λ (γ (1))

  3 θ¯ wαLα (R3 )  pθ¯ 3−2s q θ¯ 2(3−2s) θ¯ 3+2t 2 4 t 2 [w]s + [w]s + μχ ≤ φw w dx 2 4 4 kα R3    3 ¯ ¯ +θ G2 (w) dx − δ G1 (w) dx → −∞ as θ¯ → ∞, R3

R3

due to the fact that 6 − 4s < 3. Now, by using (14.2.5), (14.2.6) with ε = ¯ 1] Theorem 1.1.8, we have for any λ ∈ [δ,

1 2

and

   p 2 q 4 μ [u]s + [u]s + ξk (u) φut u2 dx + G2 (u) dx − G1 (u) dx 2 4 4 R3 R3 R3   C1/2 p 2 m ∗ 2 u dx − ∗ |u|2s dx ≥ [u]s + 3 3 2 4 R 2s R p m 2∗ , u2H s (R3 ) − C ∗ uHs s (R3 ) . ≥ min 2 4

k (u) ≥ Jμ,λ

¯ 1] and u ∈ H s (R3 ) with u = 0 and Then there exists ρ > 0 such that for any λ ∈ [δ, rad k (u) > 0. In particular, for any u uH s (R3 ) ≤ ρ, we get Jμ,λ H s (R3 ) = ρ, we have that k ¯ 1] and γ ∈ λ . Since γ (0) = 0 and Jμ,λ (u) ≥ c¯ > 0, for some c¯ > 0. Fix λ ∈ [δ,

14.2 Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional

489

k (γ (1)) < 0, it is clear that γ (1) Jμ,λ H s (R3 ) > ρ. By continuity, there exists tγ ∈ (0, 1) such that γ (tγ )H s (R3 ) = ρ. Consequently, k cμ,λ ≥ inf Jμ,λ (γ (tγ )) ≥ c¯ > 0 λ∈λ

¯ 1]. for any λ ∈ [δ,

s (R3 ),  = [δ, ¯ 1], Finally, we are in the position to apply Theorem 14.2.2 with X = Hrad

A(u) =

p 2 q 4 μ [u] + [u] + ξk (u) 2 s 4 s 4



 R3

φut u2 dx +

R3

G2 (u) dx

and  B(u) =

R3

G1 (u) dx.

¯ 1], there exists a bounded sequence (uλ ) ⊂ H s (R3 ) such Hence, for almost every λ ∈ [δ, j rad that k k λ (uλj ) → cμ,λ and (Jμ,λ ) (uj ) → 0 Jμ,λ

s in (Hrad (R3 ))∗ .

  Next we prove a compactness result. ¯ 1], every bounded Palais–Smale sequence for J k admits Lemma 14.2.4 For any λ ∈ [δ, μ,λ a convergent subsequence. k Proof Let (uj ) be a bounded Palais–Smale sequence for Jμ,λ , that is, k Jμ,λ (uj )

is bounded and

k

(Jμ,λ ) (uj ) → 0

s in (Hrad (R3 ))∗ .

(14.2.9)

Then, using Theorem 1.1.11, we may suppose that, up to a subsequence, there exists u ∈ s Hrad (R3 ) such that [uj ]2s → L ≥ 0, uj  u

s in Hrad (R3 ),

uj → u

in Lp (R3 ), 2 < p < 2∗s ,

uj → u

a.e. in R3 .

(14.2.10)

490

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

By the weak lower semicontinuity of the fractional Sobolev seminorm, [u]2s ≤ lim inf[uj ]2s .

(14.2.11)

j →∞

∗

Applying the first part of Lemma 1.4.2 with P (t) = gi (t), i = 1, 2, Q(t) = |t|2s −1 , vj = uj , v = gi (u), i = 1, 2 and w ∈ Cc∞ (R3 ), and using (g2 ), (14.2.2) and (14.2.10), we have that   gi (uj )w dx → gi (u)w dx i = 1, 2 R3

R3

as j → ∞. Now (14.2.10) and Remark 13.1.3 imply that, as j → ∞, 

 φut j uj w dx → ξk (u)

ξk (uj )

R3   uj α α 3  L (R )

χ kα

R3

 R3

φut j u2j

R3

φut uw dx 

uα α

L (R3 ) kα

|uj |α−2 uj w → χ





R3

φut u2

R3

|u|α−2 uw.

It follows that u satisfies  (p + qL)u, wD s,2 (R3 ) + μξk (u) 

 +

R3

g2 (u)w = λ

R3

μα φut uw + α χ

4k

R3



uα α

L (R3 ) kα



 R3

φut u2

R3

|u|α−2 α uw

(14.2.12)

g1 (u)w

s (R3 ), we get for all w ∈ Cc∞ (R3 ). Since Cc∞ (R3 ) is dense in Hrad

 (p

+ qL)[u]2s

+ μξk (u)

R3

φut u2 dx

    α μα uLα (R3 ) α t 2 u + αχ φ u dx + g2 (u)u dx u Lα (R3 ) 4k kα R3 R3  g1 (u)u dx. (14.2.13) =λ R3

Again, from (14.2.10) and Remark 13.1.3, we obtain that, as j → ∞,  ξk (uj )  χ



 R3

φut j u2j dx → ξk (u)

uj αLα (R3 ) kα



 uj αLα (R3 )

R3

R3

φut u2 dx,  φut j u2j

dx → χ



uαLα (R3 ) kα



 uαLα (R3 )

R3

φut u2 dx. (14.2.14)

14.2 Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional

491

s (R3 ), q = 2, q = 2∗ , v = u , v = g (u)u and P (t) = g (t)t in Taking X = Hrad 1 2 j j 1 1 s Lemma 1.4.3, by (14.2.1), (14.2.2) and (14.2.10) we deduce that



 R3

g1 (uj )uj dx →

R3

(14.2.15)

g1 (u)u dx.

On the other hand, (14.2.10) and Fatou’s lemma yield 

 R3

g2 (u)u dx ≤ lim inf j →∞

R3

(14.2.16)

g2 (uj )uj dx.

Putting together (14.2.13), (14.2.14), (14.2.15), (14.2.16), and using the fact that k

(Jμ,λ ) (uj ), uj  → 0, we see that lim sup (p + qL)[uj ]2s j →∞

  = lim sup λ j →∞

 R3

g1 (uj )uj dx −

 R3

g2 (uj )uj dx − μξk (uj )

$   α  μα uj Lα (R3 ) α t 2 − αχ φuj uj dx uj Lα (R3 ) 4k kα R3    g1 (u)u dx − g2 (u)u dx − μξk (u) φut u2 dx ≤λ R3

−

μα

χ 4k α



R3  α  uLα (R3 ) α u Lα (R3 ) kα

R3

φut j u2j dx

R3

R3

φut u2 dx

= (p + qL)[u]2s .

(14.2.17)

Now, (14.2.11) and (14.2.17) imply that lim [uj ]2s = [u]2s

(14.2.18)

j →∞

and thus 

 lim

j →∞ R3

g2 (uj )uj dx =

R3

g2 (u)u dx.

(14.2.19)

Since g2 (τ )τ = mτ 2 + h(τ ), where h(τ ) = τ (g(τ ) + mτ )− is a non-negative and continuous function, we can apply Fatou’s lemma to infer that 

 R3

h(u) dx ≤ lim inf j →∞

R3

h(uj ) dx

492

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

and 

 u dx ≤ lim inf 2

R3

j →∞

R3

u2j dx.

These two inequalities and (14.2.19) imply that 

 mu dx ≤ lim inf



2

R3

j →∞

R3

mu2j

dx ≤ lim sup j →∞

R3



= lim sup j →∞

 =

 =  =

R3

R3

mu2j dx

R3

(g2 (uj )uj − h(uj )) dx

 ≤

R3

R3

j →∞

j →∞

 mu dx +

R3

h(uj ) dx



(mu2 + h(u)) dx − lim inf 2

R3



  g2 (u)u dx + lim sup −

R3

h(uj ) dx 

h(u) dx − lim inf j →∞

R3

h(uj ) dx

mu2 dx,

that is, uj → u in L2 (R3 ), which combined with (14.2.18) implies that uj → u strongly s (R3 ).   in Hrad Lemma 14.2.3, Lemma 14.2.4 and Theorem 14.2.2 lead to the following result. ¯ 1], there exists uλ ∈ H s (R3 ), uλ = 0, such that Lemma 14.2.5 For almost every λ ∈ [δ, rad k k λ (uλ ) = cμ,λ and (Jμ,λ ) (u ) = 0. Jμ,λ ¯ 1] we can find a Proof Applying Theorem 5.2.2, we know that for almost every λ ∈ [δ, s λ 3 bounded sequence (uj ) ⊂ Hrad (R ) such that k (uλj ) → cμ,λ Jμ,λ

and

k λ s (Jμ,λ ) (uj ) → 0 in (Hrad (R3 ))∗ .

(14.2.20)

s Up to a subsequence, by Lemma 14.2.4, we may assume that there exists uλ ∈ Hrad (R3 ) s (R3 ). By Lemma 14.2.3, we know that cμ,λ ≥ c¯ > 0, which in such that uλj → uλ in Hrad view of the first relation in (14.2.20) yields uλ = 0.  

¯ 1], λj → 1 and (uj ) ⊂ H s (R3 ) such that Therefore, we can find (λj ) ⊂ [δ, rad k Jμ,λ (uj ) = cμ,λj j

and

k s (Jμ,λ ) (uj ) = 0 in (Hrad (R3 ))∗ . j

14.2 Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional

493

k Lemma 14.2.6 Let uj be a critical point of Jμ,λ at the level cμ,λj . Then, for every j sufficiently large k > 0, there exists μ0 = μ0 (k) > 0 such that for any μ ∈ (0, μ0 ), up to a subsequence, uj Lα (R3 ) ≤ k for any j ∈ N. k ) (u ) = 0, we can see that u is a weak solution to the problem Proof Since (Jμ,λ j j j

⎧

2 s ⎪ ⎪ ⎨ p + q[u]s (−) u + μξk (u)φu +

α

kα χ



uα α

L (R3 ) kα

 |u|α−2u

 R3

φu dx

= λg1 (u) − g2 (u) inR3 ⎪ ⎪ ⎩ (−)t φ = μu2 in R3 . Therefore, arguing as in the proof of Theorem 3.5.1 (see also [325]), we see that uj satisfies the following Pohozaev identity:  3 − 2s 3 + 2t 2 2 (p + q[uj ]s )[uj ]s + μξk (uj ) φut j u2j dx 2 4 R3   α  3μ uj Lα (R3 ) α u + αχ  φut j u2j dx j α 3 L (R ) k kα R3   G1 (uj ) dx − 3 G2 (uj ) dx. (14.2.21) = 3λj R3

R3

k (uj ) = cμ,λj , (14.2.21), and Lemma 13.1.2-(1) imply that Since Jμ,λ j

q s p + [uj ]2s [uj ]2s 2 = 3cμ,λj ≤ 3cμ,λj

   α 3μ uj Lα (R3 ) α uj Lα (R3 ) dx + α χ φut j u2j dx k kα R3 R3   uj αLα (R3 ) μ2 2 4

+ C1 μ ξk (uj )uj Lα (R3 ) + C2 χ uj 4+α . Lα (R3 ) kα kα t + μξk (uj ) 2



φut j u2j

(14.2.22) Let us estimate the right-hand side of this inequality. Using the min–max definition of cμ,λj , we have ·

k cμ,λj ≤ max Jμ,λ w j θ>0 θ   3−2s     θ qθ 3−2s 2 2 3 ¯ [w]s [w]s + θ (G2 (w) dx − δ (G2 (w) dx ≤ max p+ θ>0 2 2 R3 R3   3+2t  μθ t 2 + max ξk (γ (σ )) φw w dx = A1 + A2 (k). (14.2.23) θ>0 4 R3

494

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

Now, if θ 3 ≥ 2k α /wαLα (R3 ) then A2 (k) = 0. Otherwise, if θ 3 < 2k α /wαLα (R3 ) , then by the definition of χ and Lemma 13.1.2-(1), there is C > 0 such that μ A2 (k) ≤ 4 μ ≤ 4





2k α wαLα (R3 ) 2k α wαLα (R3 )

 3+2t  3 R3

t 2 φw w dx

 3+2t 3

Ct μw4Lα (R3 ) ≤ C3 μ2 k 4 .

In a similar fashion we can prove the following estimates: C1 μ2 ξk (uj )uj 4Lα (R3 ) ≤ C4 μ4 k 4 ,   uj αLα (R3 ) μ2 C2 χ

uj 4+α ≤ C5 μ2 k 4 . Lα (R3 ) kα kα

(14.2.24) (14.2.25)

Combining (14.2.23), (14.2.24), (14.2.25) and (14.2.22), we obtain

q sp[uj ]2s ≤ s p + [uj ]2s [uj ]2s ≤ 3A1 + C6 μ2 k 4 . 2

(14.2.26)

k ) (u ), u  = 0 and (14.2.4), we deduce that On the other hand, by using that (Jμ,λ j j j

 (p

+ q[uj ]2s )[uj ]2s

+ μξk (uj )

R3

φut j u2j dx

   α  μα uj Lα (R3 ) α t 2 + αχ  φ u dx + g2 (uj )uj dx u j α 3 uj j L (R ) 4k kα R3 R3   2∗ g1 (uj )uj dx ≤ Cε uj  s2∗s 3 + ε g2 (uj )uj dx. (14.2.27) = λj L (R )

R3

R3

Hence, using (14.2.3), (14.2.26), (14.2.27), the definition of χ, Lemma 13.1.2 and Theorem 1.1.8, we infer that  m(1 − ε)uj 2L2 (R3 ) ≤ (1 − ε) 2∗s

μα ≤ Cε uj  2∗s 3 − α χ

L (R ) 4k



R3

g2 (uj )uj dx

uj αLα (R3 ) kα



 uj αLα (R3 )

R3

φut j u2j dx

2∗

≤ C∗ Cε [uj ]s s + Cμ2 k 4 2 4 ˆ ¯ 2k4. ≤ C(3A 1 + C6 μ k ) 3−2s + Cμ 3

(14.2.28)

14.2 Struwe–Jeanjean Monotonicity Trick for a Perturbed Functional

495

Now suppose, by contradiction, that (uj ) admits no subsequence that is uniformly bounded by k in the Lα (R3 )-norm. Then, there exists ν ∈ N such that uj Lα (R3 ) > k

for any j ≥ ν.

(14.2.29)

Without loss of generality, we can assume that (14.2.29) holds for all uj . Taking into account (14.2.26), (14.2.28) and (14.2.29), we get 6

12

k 2 < uj 2Lα (R3 ) ≤ Cuj 2H s (R3 ) ≤ C7 + C8 μ 3−2s k 3−2s + C9 μ2 k 4 , which gives a contradiction for k large and μ sufficiently small, because we can find k0 > 0 12

6

¯ 0 ) such that C8 μ 3−2s k03−2s + C9 μ2 k04 < 1 for all such that k02 > C7 + 1 and μ¯ = μ(k μ ∈ (0, μ). ¯   Now, we are ready to give the proof of the main result of this chapter. Proof of Theorem 14.1.1 Let k and μ0 be as in Lemma 14.2.6, and fix μ ∈ (0, μ0 ). Let uj k be a critical point for Jμ,λ at the level cμ,λj . We claim that (uj ) is a bounded Palais–Smale j sequence for Jμ at the level cμ,1 . Since by Lemma 14.2.6 we know that uj Lα (R3 ) ≤ k, arguments similar to those used to prove (14.2.26) and (14.2.28) imply that (uj ) is bounded s (R3 ). Up to a subsequence, we may assume that u  u in H s (R3 ). By the in Hrad j rad definition of χ, it follows that k Jμ,λ (uj ) j

=

q μ + [uj ]2s [uj ]2s + 2 4 4

p



 R3

φuj u2j

dx+



R3

G2 (uj ) dx−λj

R3

G1 (uj ) dx. (14.2.30)

s (R3 ))∗ , Lemma 1.4.2 and the fact that Since (g1 (uj )) is bounded in (Hrad



 R3

g1 (u)ψ dx =

R3

g1 (uj )ψ dx + oj (1)

for all ψ ∈ Cc∞ (R3 ), we obtain that k (Jμ ) (uj ), ψ = (Jμ,λ ) (uj ), ψ + (λj − 1) j

 R3

g1 (uj )ψ dx → 0.

s s (R3 ), we see that Jμ (uj ) → 0 in (Hrad (R3 ))∗ . Further, Since Cc∞ (R3 ) is dense in Hrad since the function λ → cμ,λ is left continuous,

 k Jμ (uj ) = Jμ,λ (uj ) + (λj − 1) j

R3

g1 (uj ) dx = cμ,λj + oj (1) = cμ,1 + oj (1).

496

14 An Existence Result for a Fractional Kirchhoff–Schrödinger–Poisson System

Consequently, (uj ) is a bounded Palais–Smale sequence for Jμ . By Lemma 14.2.4, we s (R3 ) and thus J (u) = c

deduce that uj → u in Hrad μ μ,1 and Jμ (u) = 0. From Jμ (u), u−  = 0 we see that u ≥ 0 in R3 . In light of Lemma 14.2.3, we know that cμ,1 > 0, which implies that u ≡ 0. Since φut ≥ 0, we can argue as in the proof of Lemma 3.2.14 to see that u ∈ L∞ (R3 ). Therefore, φut ∈ L∞ (R3 ). Note that, by (14.2.7), u satisfies (−)s u =

1 [−μφut u + g(u)] (p + q[u]2s )

in R3 ,

and applying Proposition 1.3.2 we obtain that u ∈ C 1,α (R3 ) for any α < 2s − 1 (we remark that s > 34 > 12 ). By using Proposition 1.3.11-(ii) (or Theorem 1.3.5), we have that   u > 0 in R3 . This completes the proof of Theorem 14.1.1.

Multiple Positive Solutions for a Non-homogeneous Fractional Schrödinger Equation

15.1

15

Introduction

In this chapter we deal with the existence of positive solutions for the nonlinear fractional equation 

(−)s u + u = k(x)f (u) + h(x) u ∈ H s (RN ), u > 0 in RN ,

in RN ,

(15.1.1)

where s ∈ (0, 1), N ≥ 2, k is a bounded positive function, h ∈ L2 (RN ), h ≥ 0, h ≡ 0, and the nonlinearity f : R → R is a smooth function which can be either asymptotically linear or superlinear at infinity. Our purpose is to investigate the existence and the multiplicity of positive solutions for the nonhomogeneous equation (15.1.1), subject to a small perturbation h ∈ L2 (RN ) and suitable assumptions on the nonlinearity f . More precisely, we assume that f satisfies the following conditions: (f 1) f ∈ C 1 (R, R+ ), f (0) = 0 and f (t) = 0 for t ≤ 0; f (t) = 0; (f 2) lim t →0 t f (t) (f 3) there exists p ∈ (1, 2∗s − 1) such that lim p = 0; t →∞ t f (t) (f 4) there exists l ∈ (0, ∞] such that lim = l. t →∞ t

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_15

497

498

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

Note that (f 1)–(f 3), yield that for any ε > 0 there exists Cε > 0 such that |F (t)| ≤

1 ε 2 t + Cε |t|p+1 2 p+1

for all t ∈ R,

(15.1.2)

while (f 4) implies that f is asymptotically linear if l < ∞, or superlinear when l = ∞. Due to the presence of the fractional Laplacian, which is a nonlocal operator, we analyze (15.1.1) by using the s-harmonic extension method [127]. This approach allows us to write a given nonlocal equation in a local way and to apply some known variational techniques to these kind of problems. Hence, instead of (15.1.1), we consider the following degenerate elliptic equation with a nonlinear Neumann boundary condition: 

in RN+1 − div(y 1−2s ∇U ) = 0 + , N+1 ∂U = κ [−u + k(x)f (u) + h(x)] on ∂R s + , ∂ν 1−2s

(15.1.3)

where u denotes the trace of U , that is u = U (·, 0). For simplicity, we will assume that κs = 1. Taking into account this fact, we are able to enlist some variational techniques developed in the papers [233, 263, 320, 333], dealing with asymptotically or superlinear classical problems, by introducing the following functional I (U ) =

1 U 2 s N+1 − X (R+ ) 2





RN

k(x)F (u) dx −

RN

h(x)u dx

N+1 ∞ where the weighted Sobolev space X s (RN+1 + ) is defined as the completion of Cc (R+ ) with respect to the norm

  U X s (RN+1 ) = +

RN+1 +

1

 y

1−2s

2

|∇U | dxdy +

2

2

RN

u dx

< ∞.

Clearly, this functional simplification comes at the price of some additional technical difficulties. For instance, some weighted embedding result will be needed (see Lemma 1.3.9) to obtain convergence results (see Lemma 15.2.1). Moreover, the arguments used in [233, 263] to prove the non-existence of solutions for certain eigenvalue problems have to be handled carefully in order to take care of the trace of the involved functions (see Lemma 15.2.6). Now, we state our first main result concerning the existence of positive solutions to (15.1.1) in the asymptotically linear case, that is l < ∞. Theorem 15.1.1 ([70]) Let s ∈ (0, 1) and N ≥ 2. Assume that h ∈ L2 (RN ), h(x) ≥ 0, h(x) ≡ 0 and k ∈ L∞ (RN , R+ ) satisfies the following condition:

15.1 Introduction

499

(K) there exists R0 > 0 such that  sup

   1 f (t) : t > 0 < inf : |x| ≥ R0 . t k(x)

(15.1.4)

Suppose that f satisfies the conditions (f 1)–(f 4) and μ∗ ∈ (l, ∞), where ∗



μ = inf

s 2

RN





(|(−) u| + u ) dx : u ∈ H (R ), 2

2

s

k(x)u dx = 1 . 2

N

RN

(15.1.5)

Assume that  hL2 (RN ) < m = max t ≥0

  1 ε Cε p+1 p − kL∞ (RN ) t − S∗ t kL∞ (RN ) , 2 2 p+1 (15.1.6)

where ε ∈ (0, k−1 ) is fixed and S∗ is the best Sobolev constant of the embedding L∞ (RN ) ∗

H s (RN ) ⊂ L2s (RN ). Let E : H s (RN ) → R be the energy functional associated with (15.1.1), that is E(u) =

1 2



s

RN

(|(−) 2 u|2 + u2 ) dx −



 RN

k(x)F (u) dx −

RN

h(x)u dx.

Then problem (15.1.1) possesses at least two positive solutions u1 , u2 ∈ H s (RN ) with the property that E(u1 ) < 0 < E(u2 ). Remark 15.1.2 The assumption on the size of h is necessary for (15.1.1) to admit a solution. In fact, proceeding as in [130], one can obtain a non-existence result to (15.1.1) when hL2 (RN ) is sufficiently large. The proof of the above theorem goes as follows: under the assumption l < ∞, we first use Ekeland’s variational principle to prove that for hL2 (RN ) small enough, there exists a positive solution to (15.1.3) such that I (U0 ) < 0. Then, we use a variant of the mountain pass theorem [177] to find a Cerami sequence that converges strongly in X s (RN+1 + ) to a solution U1 of (15.1.3) with I (U1 ) > 0. Clearly, these two solutions U0 and U1 are different. Our second result deals with the existence of positive solutions to (15.1.1) in the superlinear case l = ∞. Theorem 15.1.3 ([70]) Let s ∈ (0, 1) and N ≥ 2. Assume that f fulfills (f 1)–(f 4) with l = ∞. Let k(x) ≡ 1, and let h ∈ C 1 (RN ) ∩ L2 (RN ) be a radial function such that h(x) ≥ 0, h(x) ≡ 0 and

500

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

(H ) x · ∇h(x) ∈ L1 (RN ) ∩ L∞ (RN ) and x · ∇h(x) ≥ 0 for all x ∈ RN . Assume that  hL2 (RN ) < m1 = max t ≥0

  Cε p+1 p 1 ε − t− S∗ t , 2 2 p+1

where ε ∈ (0, 1) is fixed. Then, (15.1.1) admits two positive solutions u3 , u4 ∈ Hrs (RN ) such that E(u3 ) < 0 < E(u4 ). Due to the presence of radial functions k(x) = 1 and h = h(|x|), we work in the N+1 s s (RN+1 subspace Xrad + ) of the weighted space X (R+ ) that consists of the functions that are radial with respect to x ∈ RN . We point out that the methods used to study the asymptotically linear case do not work any more. Indeed, to prove that a Palais-Smale sequence converges to a second solution different from the first one, we have to use the concentration-compactness principle, which seems very hard to apply without requiring s (RN+1 further assumptions on k(x) and f (t). This time we use the compactness of Xrad + ) q N ∗ in L (R ) for any q ∈ (2, 2s ), and the Ekeland variational principle, to get a first solution to (15.1.3) with negative energy, provided that hL2 (RN ) is sufficiently small. The existence of a second solution with positive energy is obtained by combining the StruweJeanjean monotonicity trick in [231], which allows us to prove the existence of bounded Palais-Smale sequences for parametrized functionals, with the Pohozaev identity for the fractional Laplacian and assumption (H ), which guarantee the existence of a bounded Palais-Smale sequence for I that converges to a radial positive solution to (15.1.3). We point out that in the current literature there are only few papers concerning the existence and the multiplicity of solutions for nonhomogeneous problems in a nonlocal setting [150, 227, 296, 309]; this is rather surprising given that in the classical framework such type of problems have been extensively investigated by many authors [85, 130, 229, 319, 349, 350]. Now, we provide some examples of functions f , k and h for which our main results are applicable. Example 15.1.4 Let R0 > 0 and let  k(x) =

1 1+|x| , 1 1+R0 ,

if |x| < R0 , if |x| ≥ R0 ,

 and f (t) =

R0 t 2 1+t ,

0,

if t > 0, if t ≤ 0.

15.1 Introduction

501

It is clear that kL∞ (RN ) = 1, and f satisfies (f 1)–(f 3) and (f 4) with l = R0 . Moreover, (K) holds because    f (t) 1 : t > 0 = R0 < R0 + 1 = inf : |x| ≥ R0 . sup t k(x) 

To verify that l > μ∗ , we have to choose a special R0 > 0. For R > 0, we take φ ∈ Cc∞ (RN ) such that φ(x) = 1 if |x| ≤ R, φ(x) = 0 if |x| ≥ 2R, and |∇φ(x)| ≤ C R for all x ∈ RN . Since φ ∈ H 1 (RN ) ⊂ H s (RN ), we see that φH s (RN ) ≤ CφH 1 (RN ) . On the other hand, for any R0 > 2R, we have  

φ 2 dx ≤ k(x)φ 2 dx



RN

RN

RN 1 1+2R



φ 2 dx

RN

φ 2 dx

= 1 + 2R

and  C2 2 |B2R | N |∇φ| dx R2 R  ≤ ≤ 2 RN k(x)φ dx BR k(x) dx

C2 |B | R 2 2R 1 1+R |BR |

= C1

(1 + R) . R2

Therefore 

φ2H s (RN )

RN

Cφ2H 1 (RN ) (1 + R)  + C3 (1 + 2R), ≤ ≤ C2 2 2 k(x)φ dx R2 RN k(x)φ dx

where C2 , C3 > 0 are constants independent of R. ≤ C3 , we can infer that μ∗ ≤ 2C3 (R + 1). Then, Choosing R > 0 such that C2 (1+R) R2 taking R0 = 2C3 (R + 1) + 2R, we have lim

t →∞

f (t) = l = R0 > μ∗ . t

Now, fix ε ∈ (0, 1), and let h ∈ L2 (RN ) be such that  hL2 (RN ) < m = max t ≥0

  1 ε Cε p+1 p − S∗ t . t− 2 2 p+1

Then, all assumptions of Theorem 15.1.1 are satisfied, and we can find at least two positive solutions to (15.1.1).

502

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

Example 15.1.5 Fix ε ∈ (0, 1), and consider the following functions  h(x) =

√ 0, if |x| < 3 ∨ |x| > 2, √ C(|x|2 − 2)2 (|x|2 − 3)2 (|x|2 − 4)2 , if 3 ≤ |x| ≤ 2,

and  f (t) =

t log(1 + t), if t > 0, 0, if t ≤ 0,

where C > 0 is a constant such that  hL2 (RN ) < m = max t ≥0

  1 ε Cε p+1 p − S∗ t . t− 2 2 p+1

1 N 2 N It is clear that f satisfies √ (f 1)–(f 3) and (f 4) with l = ∞, and h ∈ C (R ) ∩ L (R ). Moreover, for any 3 < |x| < 2,

% & x · ∇h = 4C |x|2 (|x|2 − 2)(|x|2 − 3)(|x|2 − 4)(3|x|4 − 22|x|2 + 26) ≥ 0, so x · ∇h ≥ 0 on RN . In particular, x · ∇h ∈ Lq (RN ) for any q ∈ [1, ∞]. Then, we can apply Theorem 15.1.3 to deduce that the problem (15.1.1) admits at least two positive solutions.

15.2

The Asymptotically Linear Case

In this section we discuss the existence of positive solutions to (15.1.1) under the assumption that f is asymptotically linear. We consider the following degenerate elliptic problem 

in RN+1 − div(y 1−2s ∇U ) = 0 + , ∂U N, = −u + k(x)f (u) + h(x) on R ∂ν 1−2s

(15.2.1)

where k(x) is a bounded positive function, h ∈ L2 (RN ), h ≥ 0 (h ≡ 0) and f satisfies (f 1)-(f 4) with l < ∞. Since the proof of Theorem 15.1.1 consists of several steps, we first collect some useful lemmas.

15.2 The Asymptotically Linear Case

503

Lemma 15.2.1 Suppose that (f 1)–(f 4) with l < ∞ hold. Let h ∈ L2 (RN ), let k satisfy (15.1.4), and let (Un ) ⊂ X s (RN+1 + ) be a bounded Palais-Smale sequence for I . Then (Un ) has a strongly convergent subsequence in X s (RN+1 + ). Proof First, we show that for every ε > 0 there exist an R(ε) > R0 (where R0 is given by (K)) and an n(ε) > 0 such that 

 + RN+1 + \BR

Let R ∈

y 1−2s |∇Un |2 dxdy +

RN \BR

u2n dx ≤ ε,

∀R ≥ R(ε) and n ≥ n(ε). (15.2.2)

C ∞ (RN+1 + )

be a smooth function such that 0 ≤ R ≤ 1,  R (x, y) =

0 for (x, y) ∈ B + R,

(15.2.3)

2

1 for (x, y) ∈ / BR+ ,

and C R

|∇R (x, y)| ≤

for all (x, y) ∈ RN+1 +

(15.2.4)

for some positive constant C independent of R. Then, for any U ∈ X s (RN+1 + ) and all R ≥ 1, there exists a constant C1 > 0 such that R U X s (RN+1 ) ≤ C1 U X s (RN+1 ) . +

+

Indeed, using Young’s inequality and Lemma 1.3.9-(i), we see that 

 RN+1 +

y 1−2s |∇(U R )|2 dxdy +



≤2  ≤2

RN+1 +

RN+1 +

y

1−2s

2C |∇U | dxdy + 2 R



 RN+1 +

y 1−2s |∇R |2 U 2 dxdy +

2

 RN+1 +



y 1−2s |∇U |2 R2 dxdy + 2

 ≤2

(uψR )2 dx

RN

y 1−2s |∇U |2 dxdy +

RN

BR+ \B + R 2

 y

1−2s

U dxdy + 2

RN

u2 dx

u2 dx

⎛ ⎞1 ⎛ ⎞ γ −1 γ γ   2C ⎝ 1−2s 2γ 1−2s ⎝ ⎠ ⎠ y |∇U | dxdy y dxdy + 2 R BR+ \B + BR+ \B + R R 2

≤

2 (1 + C) U 2 s N+1 X (R+ )

2

≤

C1 U 2 s N+1 , X (R+ )

RN

u2 dx

504

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

where we used the facts  y 1−2s dxdy ≤ CR N+2−2s BR+ \B + R 2

and

2 γ −1 = . γ N + 2 − 2s

Since I (Un ) → 0 as n → ∞ and (Un ) is bounded in X s (RN+1 + ), we know that, for any ε > 0, there exists n(ε) > 0 such that I (Un ), R Un  ≤ C1 I (Un )∗ Un X s (RN+1 ) ≤ +

ε 4

for n ≥ n(ε).

Equivalently, for all n ≥ n(ε), 

 RN+1 +

y 1−2s |∇Un |2 R dxdy +

 ≤

RN

RN

u2n ψR dx 

(k(x)f (un ) + h(x))un ψR dx −

RN+1 +

ε y 1−2s ∇Un · ∇ψR Un dxdy + . 4 (15.2.5)

Now, by (f 1) and (15.1.4), we obtain that there exists 0 < θ < 1 such that k(x)f (un )un ≤ θ u2n

for |x| ≥ R0 .

(15.2.6)

Since h ∈ L2 (RN ) and Un X s (RN+1 ) ≤ C for some constant C > 0, it follows + from (15.2.3) that there exists R(ε) > R0 such that  RN

h(x)un ψR dx ≤ hψR L2 (RN ) un L2 (RN ) ≤

ε , 4

for R ≥ R(ε).

(15.2.7)

Thanks to the boundedness of (Un ) in X s (RN+1 + ), we may assume that, up to a subsequence, there exists U ∈ X s (RN+1 ) such that Un  U in X s (RN+1 + + ), un → u q N ∗ in Lloc (R ) for any q ∈ [1, 2s ), and un → u a.e. in RN . Therefore, by (15.2.4), the bound Un X s (RN+1 ) ≤ C, Hölder’s inequality and Lemma 1.3.9-(i), +

       1−2s lim lim sup  y ∇Un · ∇R Un dxdy   R→∞ n→∞  RN+1 + ⎛ ⎞1 ⎛ ⎞1 2 2   C⎝ 1−2s 2 1−2s 2 ⎝ ⎠ ⎠ ≤ lim lim sup y |∇Un | dxdy y |Un | dxdy R→∞ n→∞ R BR+ \B + BR+ \B + R R 2

2

15.2 The Asymptotically Linear Case

505

⎛ ⎞1 2  C ⎝ ≤ lim y 1−2s |U |2 dxdy ⎠ R→∞ R BR+ \B + R 2

⎛ ⎞1 ⎛ ⎞ γ −1 2γ 2γ   C⎝ 1−2s 2γ 1−2s ⎝ ⎠ ⎠ ≤ lim y |U | dxdy y dxdy R→∞ R BR+ \B + BR+ \B + R R 2

⎛  ⎝ ≤ C lim R→∞

2

⎞1

2γ

y

BR+ \B + R

1−2s

|U |

2γ

dxdy ⎠

= 0.

(15.2.8)

2

Then, putting together (15.2.5), (15.2.6), (15.2.7) and (15.2.8), we have for any R ≥ R(ε) and n ≥ n(ε) sufficiently large 

 RN+1 +

y

1−2s

|∇Un | R dxdy + 2

RN

(1 − θ )u2n ψR dx ≤ ε .

(15.2.9)

From θ ∈ (0, 1) and (15.2.3), we deduce that (15.2.9) implies (15.2.2). Now, we exploit the relation (15.2.2) in order to prove the existence of a convergent subsequence for (Un ). Using the fact that I (Un ) = 0 and (Un ) is bounded in X s (RN+1 + ), we see that  

1−2s 2 y |∇Un | dxdy + u2n dx I (Un ), Un  = N+1 R+

−

RN

RN





k(x)f (un )un dx −

h(x)un dx = o(1)

RN

(15.2.10)

and I (Un ), U  =





RN+1 +

y 1−2s ∇Un · ∇U dxdy +

 −

RN



k(x)f (un )u dx −

RN

RN

un u dx

h(x)u dx = o(1).

(15.2.11)

Hence, in order to prove our lemma, it suffices to show that Un X s (RN+1 ) → U X s (RN+1 ) +

+

as n → ∞.

In view of (15.2.10) and (15.2.11), this is equivalent to showing that 

 RN

k(x)f (un )(un − u) dx +

RN

h(x)(un − u) dx = o(1).

(15.2.12)

506

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

Clearly, since k ∈ L∞ (RN ), h ∈ L2 (RN ) and un → u in L2 (BR ) for any R > 0, we have that   k(x)f (un )(un − u) dx + h(x)(un − u) dx = o(1). (15.2.13) BR

BR

On the other hand, by (15.2.2), we know that for any ε > 0 there exists R(ε) > 0 such that   k(x)f (un )(un − u) dx + h(x)(un − u) dx |x|≥R(ε)

RN

 1 

 ≤

|x|≥R(ε)

 +

|x|≥R(ε)

k(x)|un − u|2 dx

 1 

|x|≥R(ε)

|h(x)|2 dx

 ≤C

1

2

k(x)|f (un )|2 dx

1

2

|x|≥R(ε)

|un − u|2 dx

 1  |x|≥R(ε)

|un |2 dx

≤Cε

2

1

2

|x|≥R(ε)

|un − u|2 dx

2

1

 + hL2 (RN )

2

|x|≥R(ε)

|un − u|2 dx

2

(15.2.14)

for n large enough. Combining (15.2.13) with (15.2.14) we obtain (15.2.12), which completes the proof of lemma.   In the next lemma we show that I is positive on the boundary of a some ball in X s (RN+1 + ), as long as hL2 (RN ) is sufficiently small. This property will allow us to apply Ekeland’s variational principle. Lemma 15.2.2 Assume that (f 1)–(f 3) hold, h ∈ L2 (RN ) is such that (15.1.6) is satisfied, and k ∈ L∞ (RN ). Then there exist ρ, α, m > 0 such that I (U )|U  s N+1 =ρ ≥ α for hL2 (RN ) < m.

X (R+

)

15.2 The Asymptotically Linear Case

507

Proof Fix ε ∈ (0, k−1 ). Then, in view of (15.1.2) and (1.2.9), L∞ (RN ) I (U ) ≥

1 ε C(ε) p+1 p+1 U 2 s N +1 − kL∞ (RN ) U 2 s N +1 − kL∞ (RN ) S∗ U  s N +1 X (R+ ) X (R+ ) X (R+ ) 2 2 p+1

− hL2 (RN ) U X s (RN +1 ) +    1 p = U X s (RN +1 ) − C1 ε U X s (RN +1 ) − C2 (ε)U  s N +1 − hL2 (RN ) , X (R+ ) + + 2

(15.2.15) where C1 =

1 kL∞ (RN ) 2

and

C2 (ε) =

C(ε) p+1 kL∞ (RN ) S∗ . p+1

Using (15.1.6) and (15.2.15), we can find ρ, α > 0 such that I (U )|U 

X s (RN+1 + )

provided that hL2 (RN ) < m.

=ρ

≥ α,  

For ρ given by Lemma 15.2.2, we denote by Bρ = {U ∈ X s (RN+1 + ) : U Xs (RN+1 ) < ρ} +

the ball in X s (RN+1 + ) with center in 0 and radius ρ. By means of the Ekeland variational principle and Lemma 15.2.1, we can verify that I has a local minimum if hL2 (RN ) is small enough. Theorem 15.2.3 Assume that (f 1)–(f 4) with l < ∞ hold, h ∈ L2 (RN ), h ≥ 0 (h ≡ 0) and k satisfies (15.1.4). If hL2 (RN ) < m, where m is given by Lemma 15.2.2, then there exists U0 ∈ X s (RN+1 + ) such that I (U0 ) = inf{I (U ) : U ∈ B ρ } < 0 and U0 is a nontrivial nonnegative solution of problem (15.2.1). Proof Since h(x) ∈ L2 (RN ), h ≥ 0 and h ≡ 0, we can choose a function V ∈ X s (RN+1 + ) such that  h(x)v dx > 0. (15.2.16) RN

508

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

Next, note that t2 I (tV ) = 2 ≤

# +1 RN +

$

 y

1−2s

|∇V | dxdy + 2

t2 V 2 s N +1 − t X (R+ ) 2

v dx −





2

RN

RN

k(x)F (tv) dx − t

RN

h(x)v dx

 RN

h(x)v dx < 0

for t > 0 small enough.

Then c0 = inf{I (U ) : U ∈ B ρ } < 0. Note that B ρ is a complete metric space with the distance dist(U, V ) = U − V X s (RN+1 ) , +

for any U, V ∈ X s (RN+1 + ).

Applying Theorem 2.2.1, there exists (Un ) ⊂ B ρ such that (i) c0 ≤ I (Un ) < c0 + n1 , (ii) I (W ) ≥ I (Un ) − n1 W − Un X s (RN+1 ) for all W ∈ B ρ . +

Let us prove that (Un ) is a bounded Palais-Smale sequence of I . First, we show that Un X s (RN+1 ) < ρ for a n large enough. If Un X s (RN+1 ) = ρ for infinitely many n, + + then we may assume that Un X s (RN+1 ) = ρ for all n ≥ 1. Hence, by Lemma 15.2.2, + I (Un ) ≥ α > 0. Taking the limit as n → ∞ and by using (i), we deduce that 0 > c0 ≥ α > 0, which is a contradiction. = 1, Now, we show that I (Un ) → 0. Indeed, for any U ∈ X s (RN+1 + ) with U X s (RN+1 + ) let Wn = Un + tU . For a fixed n, we have Wn X s (RN+1 ) ≤ Un X s (RN+1 ) + t < ρ when t + + is small enough. Using (ii), we deduce that I (Wn ) ≥ I (Un ) −

t U X s (RN+1 ) , + n

that is, U X s (RN+1 ) I (Wn ) − I (Un ) 1 + ≥− =− . t n n Letting t → 0, we deduce that I (Un ), U  ≥ − n1 , which means that |I (Un ), U | ≤ n1 for any U ∈ X s (RN+1 + ) with U X s (RN+1 ) = 1. This shows that (Un ) is indeed a bounded +

Palais-Smale sequence of I . Then, by Lemma 15.2.1, we can find U0 ∈ X s (RN+1 + ) such

15.2 The Asymptotically Linear Case

509

that I (U0 ) = 0 and I (U0 ) = c0 < 0. Since I (U0 ), U0−  = 0 and h ≥ 0, we get U0 ≥ 0 in RN+1   + , U0 ≡ 0. In what follows, we show that problem (15.2.1) has a mountain pass type solution. In order to do this, we use Theorem 2.2.15, which furnishes a Cerami sequence (Un ). Since this type of Palais-Smale sequence enjoys some useful properties, we are able to prove its boundedness in the asymptotically linear case. The lemma below shows that I possesses a mountain pass geometry. Lemma 15.2.4 Suppose that (f 1)–(f 4) hold and μ∗ ∈ (l, ∞) with μ∗ given by (15.1.5). Then there exists V ∈ X s (RN+1 > ρ, ρ is given by Lemma 15.2.2, + ) with V X s (RN+1 + ) such that I (V ) < 0. Proof Since l > μ∗ , we can find a non-negative function W ∈ X s (RN+1 + ) such that 

 k(x)w dx = 1 such that



2

RN

RN+1 +

y

1−2s

|∇W | dxdy + 2

RN

w2 dx < l.

Using (f 4) and Fatou’s lemma, we see that I (tW ) 1 = W 2 s N+1 − lim 2 X (R+ ) t →∞ t →∞ t 2 lim

≤

 RN

k(x)

1 F (tw) dx − lim 2 t →∞ t t

 RN

h(x)w dx

1 (W 2 s N+1 − l) < 0. X (R+ ) 2

It remains to take V = t0 W with t0 large enough.   Putting together Lemmas 15.2.2 and 15.2.4, we see that the assumptions of Theorem 2.2.15 are satisfied. Then, we can find a sequence (Un ) ⊂ X s (RN+1 + ) such that I (Un ) → c > 0

and

I (Un )∗ (1 + Un X s (RN+1 ) ) → 0. +

Let Wn =

Un . Un X s (RN+1 ) +

(15.2.17)

510

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

N+1 s Obviously, (Wn ) is bounded in X s (RN+1 + ), so there exists a W ∈ X (R+ ) such that, up to a subsequence, we have

Wn  W

in X s (RN+1 + ),

wn → w

a.e. in RN ,

wn → w

in L2loc (RN ).

(15.2.18)

With the notation introduced above, we state. Lemma 15.2.5 Assume that (f 1)–(f 4) and (K) hold. Let h ∈ L2 (RN ) and μ∗ ∈ (l, ∞) for μ∗ given by (15.1.5). If Un X s (RN+1 ) → ∞, then W given by (15.2.18) is a nontrivial + nonnegative solution of 

− div(y 1−2s ∇W ) = 0 in RN+1 + , ∂W = −w + lk(x)w on RN . ∂ν 1−2s

(15.2.19)

Proof First, we show that W ≡ 0. We argue by contradiction and assume that W ≡ 0. Then Theorem 1.1.8 shows that wn → 0 strongly in L2 (BR0 ), where R0 is given by condition (K). On the other hand, by (f 1), (f 4) and l < ∞, there is C > 0 such that f (t) ≤ C, for all t ∈ R. t

(15.2.20)

Therefore,  |x| 0} and 0 = {x ∈ RN : w(x) = 0}. In view of (15.2.18), it is clear that un (x) → ∞ a.e. x ∈ + . Then, by (f 4), f (un ) wn (x) → lw(x) un

a.e. x ∈ + .

(15.2.29)

Since wn → 0 a.e. in 0 , it follows from (15.2.20) that f (un ) wn (x) → 0 ≡ lw(x) un

a.e. x ∈ 0 .

(15.2.30)

in L2 (RN ).

(15.2.31)

Now (15.2.29) and (15.2.30) imply that f (un ) wn  lw un

Since φ ∈ Cc∞ (RN ) and k ∈ L∞ (RN ), we see that z = kφ ∈ L2 (RN ), which combined with (15.2.31) implies that  RN

that is, (15.2.28) holds.

f (un ) wn z dx → un

 RN

lwz dx

as n → ∞,  

15.2 The Asymptotically Linear Case

513

Lemma 15.2.6 Let k ∈ L∞ (RN , R+ ) and let μ∗ be defined by (15.1.5) with l ∈ (μ∗ , ∞). Then, (15.2.19) has no nontrivial non-negative solution. Proof Since l > μ∗ , there is a constant δ > 0such that μ∗ < μ∗ +δ < l. By the definition 2 of μ∗ , there exists Vδ ∈ X s (RN+1 + ) such that RN k(x)vδ dx = 1 and μ∗ ≤ Vδ 2 s

X (RN+1 + )

< μ∗ + δ.

N+1 N+1 s ∞ Since Cc∞ (RN+1 + ) is dense in X (R+ ), we may assume that Vδ ∈ Cc (R+ ). Let R > 0 + be such that supp(Vδ ) ⊂ BR and define

  μR = inf

BR+

 y

1−2s

|∇U | dxdy +





2

u dx : 2

0 R

k(x)u dx = 1, U ∈ 2

0 R

H1+ (BR+ ) R

,

where we used the notation H1+ (BR+ ) = {V ∈ H 1 (BR+ , y 1−2s ) : V ≡ 0 on R+ }. R

Since Vδ ≡ 0 on R+ , we infer that Vδ ∈ H1+ (BR+ ) and R

μR ≤ Vδ 2 s

X (RN+1 + )

< μ∗ + δ < l.

(15.2.32)

By the compactness of the embedding H1+ (BR+ ) ⊂ L2 (R0 ), it is not difficult to see that R  2 dx = 1 such that there exists WR ∈ H1+ (BR+ ) \ {0} with WR ≥ 0 and  0 k(x)wR R

R

⎧ 1−2s ∇W ) = 0 ⎪ in BR+ , R ⎨ − div(y ∂WR = −wR + μR k(x)wR on R0 , ∂ν 1−2s ⎪ ⎩ R W =0 on R+ .

(15.2.33)

It follows from the strong maximum principle that WR > 0 on BR+ . Extend WR by setting N+1 s WR = 0 in RN+1 \ BR+ , so that WR ∈ X s (RN+1 + + ). Therefore, if U ≡ 0, U ∈ X (R+ ) is a non-negative solution of (15.2.19). Then, 

 μR

0 R

k(x)wR u dx =  =l

 BR+

0 R

y 1−2s ∇WR · ∇U dxdy + k(x)uwR dx.

0 R

wR u dx (15.2.34)

514

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

 Sonce u ≥ 0 and u ≡ 0, we can choose R > 0 large enough so that  0 K(x)uwR dx > 0. R   Hence, (15.2.34) implies that μR = l, which is a contradiction because (15.2.32). Proof of Theorem 15.1.1 In view of Lemmas 15.2.5 and 15.2.6, it is obvious that the situation Un  → ∞ cannot occur. Therefore, the sequence (Un ) is bounded in X s (RN+1 + ). Taking into account Lemma 15.2.1, we deduce that problem (15.2.1) admits a nontrivial non-negative solution U2 ∈ X s (RN+1 + ) with I (U2 ) > 0. On the other hand, by Theorem 15.2.3, there exists a nontrivial non-negative solution U1 ∈ X s (RN+1 + ) with I (U1 ) < 0. Applying Theorem 1.3.4, we obtain that u1 , u2 > 0 in RN .  

15.3

The Superlinear Case

This section is devoted to the proof of Theorem 15.1.3. We consider the following problem: 

− div(y 1−2s ∇U ) = 0 in RN+1 + , ∂U = −u + f (u) + h(x) on RN , ∂ν 1−2s

(15.3.1)

where h(x) = h(|x|) ∈ C 1 (RN ) ∩ L2 (RN ), h(x) ≥ 0, h(x) ≡ 0 and f satisfies the conditions (f 1)–(f 4) with l = ∞. Since we assume that k(x) ≡ 1 and h(x) is radial, it is natural to work on the space of the function belonging to X s (RN+1 + ) which are radial with respect to x, that is  s Xrad (RN+1 ) = U ∈ X s (RN+1 + + ) : U (x, y) = U (|x|, y) . We begin by proving the following preliminary result. Theorem 15.3.1 Suppose that h(x) = h(|x|) ∈ L2 (RN ), h(x) ≥ 0, h(x) ≡ 0, and s (RN+1 ) such that conditions (f 1)–(f 3) hold. Then there exist m1 > 0 and U˜ 0 ∈ Xrad +

˜ ˜ I (U0 ) = 0 and I (U0 ) < 0 if hL2 (RN ) < m1 . Proof Arguing as in the proof of Theorem 15.2.3, it follows from Theorem 2.2.1 that there s (RN+1 ) such that exists a bounded Palais-Smale sequence (U˜ n ) ⊂ Xrad + s I (U˜ n ) → c˜0 = inf{I (U ) : U ∈ Xrad (RN+1 + ) and U X s (RN+1 ) = ρ} < 0, +

where ρ is given by Lemma 15.2.2. We claim that this infimum is achieved. s (RN+1 ) such that Using Theorem 1.1.11, we may assume that there exists U˜ 0 ∈ Xrad + N+1 s p+1 N (R ). U˜ n  U˜ 0 in Xrad (R+ ), u˜ n → u˜ 0 in L

15.3 The Superlinear Case

515

Taking into account (f 1)-(f 3), Theorem 1.1.8, and by exploiting the fact that (Un ) is s (RN+1 ), we see that bounded in Xrad +    

RN

  f (u˜ n )(u˜ n − u˜ 0 ) dx  ≤ ε u˜ n L2 (RN ) u˜ n − u˜ 0 L2 (RN ) p

+ Cε u˜ n Lp+1 (RN ) u˜ n − u˜ 0 Lp+1 (RN ) ≤ C ε +Cε Cu˜ n − u˜ 0 Lp+1 (RN ) . Hence,   lim  n→∞ 

RN

  f (u˜ n )(u˜ n − u˜ 0 ) dx  ≤ C ε

and so, by the arbitrariness of ε,  RN

f (u˜ n )(u˜ n − u˜ 0 ) dx → 0.

Using the assumptions (f 1)–(f 3) and Lemma 1.4.3, we obtain that 



RN

f (u˜ n )u˜ n dx →

RN

f (u˜ 0 )u˜ 0 dx.

Then, 

 RN

(f (u˜ n )−f (u˜ 0 ))u˜ 0 dx =

 RN

(f (u˜ n )u˜ n −f (u˜ 0 )u˜ 0 ) dx −

RN

f (u˜ n )(u˜ n − u˜ 0 ) dx → 0.

On the other hand, u˜ n  u˜ 0 in L2 (RN ), and using the fact that h ∈ L2 (RN ), we also have 

 RN

h(x)u˜ n dx →

RN

h(x)u˜ 0 dx.

Since I (U˜ n ), U˜ n  → 0, and I (U˜ n ), U˜ 0  → 0, the above relations imply that U˜ n → U˜ 0 s strongly in Xrad (RN+1 + ). Therefore, I (U˜ 0 ) = c˜0 < 0

and

I (U˜ 0 ) = 0.  

516

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

s (RN+1 ) → R For any λ ∈ [ 12 , 1], we introduce the following family of functionals Iλ : Xrad + defined by

Iλ (U ) =

1 U 2 s N+1 − λ X (R+ ) 2

 RN

(F (u) + h(x)u) dx.

s (RN+1 ). Our purpose is to show that I satisfies the assumptions of for any U ∈ Xrad λ + Theorem 5.2.2.

Lemma 15.3.2 Assume that (f 1)-(f 4) with l = ∞ hold. Then, 1 s ¯ (RN+1 (i) There exists V¯ ∈ Xrad + ) \ {0} such that Iλ (V ) < 0 for all λ ∈ [ 2 , 1]. (ii) For m1 > 0 given in Theorem 15.3.1, if hL2 (RN ) < m1 , then

cλ = inf max Iλ (γ (t)) > max{Iλ (0), Iλ (V¯ )} γ ∈ t ∈[0,1]

 1 ,1 , ∀λ ∈ 2 

¯ where  = {γ ∈ C([0, 1], Xrs (RN+1 + ))) : γ (0) = 0, γ (1) = V }. Proof s (RN+1 ) \ {0} and V ≥ 0 such that (i) For every δ > 0, we can find V ∈ Xrad +



 RN+1 +

y

1−2s

|∇V | dxdy < δ 2

RN

v 2 dx.

This is lawful due to the fact that   inf

RN+1 +



s y 1−2s |∇U |2 dxdy : U ∈ Xrad (RN+1 + ) and uL2 (RN ) = 1 = 0

(using the Pohozaev identity, one can see that (−)s has no eigenvalues in H s (RN )). By (f 4) with l = ∞, and applying Fatou’s lemma, we deduce that  lim

t →∞ RN

F (tv) dx ≥ (1 + δ) t2

 RN

v 2 dx.

Hence, for any λ ∈ [ 12 , 1], we get I 1 (tV ) Iλ (tV ) 1 2 lim ≤ lim ≤ 2 2 t →∞ t →∞ t t 2

 



 RN+1 +

y

1−2s

|∇V | dxdy − δ 2

2

RN

v dx

< 0.

15.3 The Superlinear Case

517

Take t1 > 0 large enough such that I 1 (t1 V ) < 0, and set V¯ = t1 V . Then, Iλ (V¯ ) ≤ 2 I 1 (V¯ ) < 0, i.e., (i) holds. 2

s (RN+1 ), we have (ii) It is clear that, for any λ ∈ [ 12 , 1] and U ∈ Xrad +

Iλ (U ) ≥

1 U 2 s N+1 − X (R+ ) 2

 RN

F (u) dx − hL2 (RN ) uL2 (RN ) = J (U ).

Proceeding as in the proof of Lemma 15.2.2, we deduce that inf max J (γ (t)) > 0,

γ ∈ t ∈[0,1]

provided that hL2 (RN ) < m1 , with m1 given by Theorem 15.3.1. Then, for every % & λ ∈ 12 , 1 , it holds cλ = inf max Iλ (γ (t)) ≥ inf max J (γ (t)) > max{Iλ (0), Iλ (V¯ )}. γ ∈ t ∈[0,1]

γ ∈ t ∈[0,1]

This ends the proof of the lemma.   It follows from Lemma 15.3.2 and Theorem 5.2.2 that there exists (λj ) ⊂ [ 12 , 1] such that (i) λj → 1 as j → ∞; j (ii) Iλj has a bounded Palais-Smale sequence (Un ) at the level cλj . s (RN+1 ) In view of Theorem 1.1.11, we deduce that for each j ∈ N, there exists Uj ∈ Xrad + j s (RN+1 ) and U is a positive solution of such that Un → Un strongly in Xrad j +



− div(y 1−2s ∇Uj ) = 0 in RN+1 + , ∂Uj N. = −u + λ [f (u ) + h(x)] on R j j j ∂ν 1−2s

Arguing as in Theorem 3.5.1 (see also [307]), it is easy to see that each Uj satisfies the following Pohozaev identity:  N y |∇Uj | dxdy + u2j dx N+1 N 2 R+ R   (F (uj ) + huj ) dx + λj ∇h(x) · xuj dx. = Nλj

N − 2s 2



1−2s

RN

2

RN

(15.3.2)

In the next lemma, we use condition (H ) to prove the boundedness of the sequence (Uj ).

518

15 Multiple Positive Solutions for a Non-homogeneous Fractional. . .

Lemma 15.3.3 Assume that (f 1)–(f 4) with l = ∞ hold, and h satisfies (H ) and s (RN+1 ) is bounded. hL2 (RN ) < m1 , for m1 given in Theorem 15.3.1. Then, (Uj ) ⊂ Xrad + Proof Using Theorem 5.2.2, we know that the function λ → cλ is continuous from the left. Then, by Lemma 15.3.2-(ii), we deduce that Iλj (Uj ) = cλj → c1 > 0 as λj → 1. Hence, we can find a constant K > 0 such that Iλj (Uj ) ≤ K for all j ∈ N. Combining this with (15.3.2), the fact that uj > 0 and (H ), we see that  RN+1 +

y 1−2s |∇Uj |2 dxdy ≤

λj KN − s s

 RN

∇h(x) · x uj dx ≤

KN , s

which together with the Sobolev inequality (1.2.9) implies that 1

  uj L2∗s (RN ) ≤ C∗

2

RN+1 +

y 1−2s |∇Uj |2 dxdy

≤ C.

(15.3.3)

Now, it follows Iλj (Uj ) ≤ K for all j ∈ N that 1 Uj 2 s N+1 − λj X (R+ ) 2

 RN

(F (uj ) + h(x)uj ) dx ≤ K.

(15.3.4)

On the other hand, by (f 2), (f 3), there exists a constant C > 0 such that  RN

1 F (uj ) dx ≤ 4



 RN

u2j

dx + C

∗

RN

|uj |2s dx.

Substituting this inequality into (15.3.4) and using (15.3.3), (1.2.9), we deduce that 1 2



 RN

u2j

dx ≤ λj

RN

(F (uj ) + h(x)uj ) dx + K

1 2∗ uj 2L2 (RN ) + Cuj  s2∗s N + hL2 (RN ) uj L2 (RN ) + K L (R ) 4 1 ≤ uj 2L2 (RN ) + C¯ + hL2 (RN ) uj L2 (RN ) + K. 4

≤

Then, 1 uj 2L2 (RN ) ≤ C˜ + hL2 (RN ) uj L2 (RN ) , 4 that is uj L2 (RN ) ≤ C for all j ∈ N,

(15.3.5)

15.3 The Superlinear Case

519

for some positive constant C independent of j . Putting together (15.3.3) and (15.3.5), we complete the proof.   Lemma 15.3.4 Under the assumptions of Lemma 15.3.3, the above sequence (Uj ) is also a Palais-Smale sequence of I . Proof By the definitions of I and Iλj we deduce that  I (Uj ) = Iλj (Uj ) + (λj − 1)

RN

(F (uj ) + h(x)uj ) dx.

(15.3.6)

Using Theorem 5.2.2, we obtain Iλj (Uj ) = cλj → c1 > 0

as λj → 1.

Hence, applying Lemma 15.3.3 and (15.3.6), we see that I (Uj ) → c1 > 0. Since Iλ j (Uj ) = 0, we infer that, for any  ∈ Cc∞ (RN+1 + ), I (Uj ),  = Iλ j (Uj ),  + (λj − 1)

 RN

(f (uj ) + h(x))ψ dx → 0,

s that is, I (Uj ) → 0 as j → ∞ in the dual space of Xrad (RN+1 + ).

 

Finally, we give the proof of the main result of this section: Proof of Theorem 15.1.3 It follows from Theorem 15.3.1 that (15.3.1) admits a nontrivial s non-negative solution U3 ∈ Xrad (RN+1 + ) such that I (U3 ) < 0. On the other hand, by Lemma 15.3.4 and Theorem 1.1.11, we know that problem (15.3.1) has a nontrivial nons (RN+1 ) with I (U ) = c > 0. Consequently, U ˜ 0 ≡ U˜ 1 . By negative solution U4 ∈ Xrad 4 1 + N using Theorem 1.3.4, we get u3 , u4 > 0 in R and this ends the proof of Theorem 15.1.3.  

Sign-Changing Solutions for a Fractional Schrödinger Equation with Vanishing Potential

16.1

16

Introduction

In this chapter we study the existence of least energy sign-changing (or nodal) solutions for the following nonlinear problem involving the fractional Laplacian: 

(−)s u + V (x)u = K(x)f (u) in RN , u ∈ Ds,2 (RN ),

(16.1.1)

with s ∈ (0, 1), N > 2s, V and K are continuous potentials which satisfy suitable assumptions, and f is a continuous nonlinearity. When s = 1, equation in (16.1.1) becomes the classical nonlinear Schrödinger equation − u + V (x)u = K(x)f (u)

in RN ,

(16.1.2)

which is extensively studied for at least 20 years. We do not intend to review the huge bibliography devoted to equations like (16.1.2); we just emphasize that the potential V : RN → R has a crucial role concerning the existence and behavior of solutions. For instance, when V is a positive constant, or V is radially symmetric, it is natural to look for radially symmetric solutions, see [318,340]. On the other hand, after the seminal paper of Rabinowitz [299], where the potential V is assumed to be coercive, several different assumptions were adopted in order to obtain existence and multiplicity results, see [88,89]. An important class of problems associated with (16.1.2) is the zero mass case, which occurs when lim V (x) = 0.

|x|→∞

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_16

521

522

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

To study these problems, many authors used several variational methods; see [21,26,30,96, 97] and [100, 107, 192] for problems posed in RN , as well as [22, 91, 92, 133] for problems in bounded domains with homogeneous boundary conditions. Motivated by the above papers, our goal is to prove the existence of sign-changing solutions to problem (16.1.1). Before stating our main result, we introduce the basic assumptions on V , K and f . More precisely, we suppose that the functions V , K : RN → R are continuous on RN , and we say that (V , K) ∈ K if: (h1 ) V (x), K(x) > 0 for all x ∈ RN and K ∈ L∞ (RN ); (h2 ) If (An ) ⊂ RN is a sequence of Borel sets such that the Lebesgue measure m(An ) ≤ R, for all n ∈ N and some R > 0, then  lim

r→∞ A ∩B c n r

K(x) dx = 0,

for every n ∈ N, where Brc = RN \ Br . Furthermore, one of the following conditions is in force: (h3 ) K/V ∈ L∞ (RN ) or (h4 ) there exists m ∈ (2, 2∗s ) such that K(x) 2∗ s −m ∗

→0

as |x| → ∞.

V (x) 2s −2

We recall that assumptions (h1 )–(h4 ) were introduced for the first time by Alves and Souto in [21]. It is very important to observe that (h2 ) is weaker than any one of the conditions listed below that in the above mentioned papers to study zero mass problems: (a) there are r ≥ 1 and ρ ≥ 0 such that K ∈ Lr (RN \ Bρ ); (b) K(x) → 0 as |x| → ∞; (c) K = H1 + H2 , with H1 and H2 satisfying (a) and (b), respectively. Now, we provide some examples of functions V and K satisfying (h1 )–(h4 ). Let (Bn ) be a sequence of disjoint open balls in RN centered in ξn = (n, 0, . . . , 0) and consider a non-negative function H3 such that H3 = 0 in R \ N

∞ 7 n=1

 Bn ,

H3 (x) dx = 2−n .

H3 (ξn ) = 1 and Bn

16.1 Introduction

523

Then, the pairs (V , K) given by K(x) = V (x) = H3 (x) +

1 log(2 + |x|)

and 

1 K(x) = H3 (x) + log(2 + |x|)

and

V (x) = H3 (x) +

1 log(2 + |x|)

 2∗∗s −2

2s −m

for some m ∈ (2, 2∗s ), belong to the class K. On the nonlinearity f : R → R we assume that it is a continuous function and satisfies the following growth conditions at the origin and at infinity: f (t) = 0 if (h3 ) holds; |t |→0 |t|

(f1 ) lim or

f (t) < ∞ if (h4 ) holds for some m ∈ (2, 2∗s ); |t |→0 |t|m−1 (f2 ) f has a quasicritical growth at infinity, namely

(f˜1 ) lim

f (t) = 0; ∗ |t |→∞ |t|2s −1 lim

(f3 ) F has a superquadratic growth at infinity, that is F (t) = ∞, |t |→∞ |t|2 lim



t

where, as usual, we set F (t) =

f (τ ) dτ ; 0

(f4 ) the function t →

f (t) is increasing for every t ∈ R \ {0}. |t|

As models for f we can take, for instance, the following nonlinearities  + m

f (t) = (t ) for some m ∈ (2, 2∗s ).

and

f (t) =

log 2(t + )m , if t ≤ 1, t log(1 + t), if t > 1,

524

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

Remark 16.1.1 It follows from (f4 ) that the function t →

1 f (t)t − F (t) 2

is increasing for every t > 0

(16.1.3)

and decreasing for every t < 0. Now, we are ready to state the main result of this chapter. Theorem 16.1.2 ([69]) Suppose that (V , K) ∈ K and f ∈ C(R) satisfies either condition (f1 ), or conditions (f˜1 ) and (f2 )–(f4 ). Then, problem (16.1.1) possesses a least energy nodal weak solution. In addition, if the nonlinear term f is odd, then problem (16.1.1) has infinitely many nontrivial weak solutions not necessarily nodals. The proof of Theorem 16.1.2 is obtained by enlisting variational arguments. We note that the Euler–Lagrange functional associated with (16.1.1), that is,     1 2 2 J (u) = [u]s + V (x)u dx − K(x)F (u) dx, 2 RN RN does not satisfy the following decompositions J (u), u±  = J (u± ), u±  J (u) = J (u+ ) + J (u− ), which were fundamental in the application of variational methods to study (16.1.2); see [86, 92, 322, 340]. Anyway, we prove that the geometry of the classical minimization theorem is respected in the nonlocal framework: more precisely, we develop a functional analytical setting that is inspired by (but not equivalent to) the fractional Sobolev spaces, in order to correctly encode the variational formulation of problem (16.1.1). Secondly, the nonlinearity f is only continuous, so to overcome the nondifferentiability of the Nehari manifold associated with J , we use the generalized Nehari manifold method developed by Szulkin and Weth. Of course, also the compactness properties (see Proposition 16.3.2) required by these abstract theorems are satisfied in the nonlocal case, thanks to our functional setting. Then, to obtain nodal solutions, we look for critical points of J (tu+ + su− ), and, due to the fact that f is only continuous, we do not apply the Miranda’s Theorem [270] as in [22,86], but we use an iterative procedure and the properties of J to prove the existence of a sequence which converges to a critical point of J (tu+ + su− ) (see Lemma 16.4.1). Finally, we emphasize that Theorem 16.1.2 improves the recent result established in [71], in which the existence of a least energy nodal solution to problem (16.1.1) has been proved under the stronger assumption that f ∈ C 1 and satisfies the Ambrosetti-Rabinowitz condition.

16.2 Preliminary Lemmas

16.2

525

Preliminary Lemmas

In order to give the weak formulation of problem (16.1.1), we need to work in a special functional space. Indeed, one of the difficulties in treating problem (16.1.1) is related to his variational formulation. With this respect the standard fractional Sobolev spaces are not sufficient in order to study the problem. We overcome this difficulty by working in a suitable functional space, whose definition and basic analytical properties are recalled here. Let us introduce the following functional space 





X= u∈D

s,2

2

(R ) : N

RN

V (x)u dx < ∞

endowed with the norm   u = [u]2s +

1

RN

2

V (x)u2 dx

.

q

For q ∈ R with q ≥ 1, we define the Lebesgue space LK (RN ) as   q LK (RN ) = u : RN → R measurable and

RN

 K(x)|u|q dx < ∞ ,

endowed with the norm  uLq (RN ) = K

1 q

RN

q

K(x)|u| dx

.

We start by giving the following useful results established in [71]. q

Lemma 16.2.1 Assume that (V , K) ∈ K. Then X is continuously embedded in LK (RN ) N for every q ∈ [2, 2∗s ] if (h3 ) holds. Moreover, X is continuously embedded in Lm K (R ) if (h4 ) holds. Proof Assume that (h3 ) is true. The proof is trivial if q = 2 or q = 2∗s . Fix q ∈ (2, 2∗s ) 2∗ − q . We observe that q can be written as q = 2λ + (1 − λ)2∗s . Then we and let λ = s∗ 2s − 2 have   ∗ q K(x)|u| dx = K(x)|u|2λ|u|(1−λ)2s dx RN

RN

 ≤

1 λ

RN

λ 

|K(x)| |u| dx

2∗s

2

RN

|u| dx

1−λ

526

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .



  λ  1−λ |K(x)| 2 2∗s ≤ sup V (x)|u| dx |u| dx λ RN RN x∈RN |V (x)|    λ |K(x)| (1−λ)2∗s 2 ≤ C sup V (x)|u| dx [u]s λ N R x∈RN |V (x)|  

(1−λ)2∗ |K(x)| 2 λ+ 2 s u ≤ C sup λ x∈RN |V (x)|   |K(x)| uq . = C sup λ |V (x)| N x∈R Since K ∈ L∞ (RN ) and (h3 ) hold, we conclude that uLq (RN ) ≤ Cu. K

Now, suppose that (h4 ) holds. Setting λ0 = m = 2λ0 + (1 − λ0 )2∗s . As above, we have 

2∗s − m , we see that m can be written as 2∗s − 2

 RN

K(x)|u|m dx =

∗

RN

K(x)|u|2λ0 |u|(1−λ0 )2s dx

 ≤ 

1

RN

|K(x)| λ0 |u|2 dx

λ0 

 

∗

RN

|u|2s dx

|K(x)| ≤ sup V (x)|u|2 dx |V (x)|λ0 RN x∈RN   |K(x)| ≤ C sup um . λ x∈RN |V (x)| Since

K(x) V (x)

2∗ s −m 2∗ s −2

1−λ0

λ0 

2∗s

RN

|u| dx

1−λ0

∈ L∞ (RN ), we infer that uLm (RN ) ≤ Cu. K

This complete the proof of the lemma. Lemma 16.2.2 Assume that (V , K) ∈ K. Then (1) X is compactly embedded into LK (RN ) for all q ∈ (2, 2∗s ) if (h3 ) holds; q

 

16.2 Preliminary Lemmas

527

N (2) X is compactly embedded into Lm K (R ) if (h4 ) holds.

Proof (1) Assume that (h3 ) holds. Fix q ∈ (2, 2∗s ) and let ε > 0. Then there exist numbers 0 < t0 < t1 and a positive constant C such that & % ∗ ∗ K(x)|t|q ≤ ε C V (x)|t|2 + |t|2s + C K(x)χ[t0,t1 ] (|t|)|t|2s , for all t ∈ R. Integrating over Brc we have, for all u ∈ X and r > 0, 

 K(x)|u| dx ≤ ε C

%

q

Brc

=ε

Brc

V (x)|u| + |u| 2

2∗ CQ(u) + Ct1 s

2∗s

& dx

2∗ + Ct1 s



 K(x) dx A∩Brc

K(x) dx, A∩Brc

(16.2.1) where we set  Q(u) =

% Brc

& ∗ V (x)|u|2 + |u|2s dx

 and A = x ∈ RN : t0 ≤ |u(x)| ≤ t1 .

Now, if (un ) ⊂ X is a sequence such that un  u in X, then there is M > 0 such that  un 2 ≤ M and

∗

RN

|un |2s dx ≤ M

∀n ∈ N.

(16.2.2)

from above by a positive constant. This implies that the!sequence (Q(un )) is bounded " Let us denote An = x ∈ RN : t0 ≤ |un | ≤ t1 . By (16.2.2), 2∗ t0 s m(An )



∗

≤

|un |2s dx ≤ M

for all n ∈ N,

An

which implies that supn∈N |m(An )| < ∞. Therefore, by (h2 ), there exists a positive radius r large enough such that  An ∩Brc

K(x) dx
0 small enough and (16.2.5), we have 

 K(x)|un | dx = q

lim

n→∞ RN

RN

K(x)|u|q dx

from which we conclude that un → u in LK (RN ), for every q ∈ (2, 2∗s ). q

(2) Suppose that (h4 ) holds. Then we see that for each fixed x ∈ RN , the function ∗

g(t) = V (x)t 2−m + t 2s −m , t > 0, 

2∗ s −m ∗

has Cm V (x) 2s −2 as its minimum value, where Cm = Hence 2∗ s −m ∗

∗

Cm V (x) 2s −2 ≤ V (x)t 2−m + t 2s −m ,

2∗s − 2 2∗s − m



m−2 2∗s − 2

 2−m ∗

2s −2

.

for every x ∈ RN and t > 0.

Combining this inequality with (h4 ), we see that for every ε > 0 we can find a sufficiently large radius r such that % & ∗

K(x)|t|m ≤ εCm V (x)|t|2 + |t|2s , for every t ∈ R and |x| ≥ r,

is the inverse of C , and integrating over B c we get where Cm m r

 Brc

% 2∗

u2 + u s2∗s K(x)|u|m dx ≤ εCm L

& (RN )

,

for all u ∈ X.

(16.2.6)

16.2 Preliminary Lemmas

529

If (un ) ⊂ X is a sequence such that un  u in X, then by (16.2.6) we deduce that  Brc



K(x)|u|m dx ≤ εCm ,

for all n ∈ N.

(16.2.7)

Since m ∈ (2, 2∗s ) and K is a continuous function, it follows from Theorem 1.1.8 that 

 lim

n→∞ B r

K(x)|un |m dx =

K(x)|u|m dx.

(16.2.8)

Br

Then, (16.2.7) and (16.2.8) yield 

 lim

n→∞ RN

K(x)|un |m dx =

RN

K(x)|u|m dx,

from which we deduce that N ∗ un → u in Lm K (R ), for every m ∈ (2, 2s ).

 

The next lemma is a compactness result related to the nonlinear term (see [71]). Lemma 16.2.3 Assume that (V , K) ∈ K and f satisfies either (f1 )–(f2 ) or (f˜1 )–(f2 ). Let (un ) be a sequence such that un  u in X. Then, up to a subsequence, one has 

 lim

n→∞ RN

K(x)F (un ) dx =

RN

K(x)F (u) dx

and 

 lim

n→∞ RN

K(x)f (un )un dx =

RN

K(x)f (u)u dx.

Proof Assume that (h3 ) holds. By (f1 )–(f2 ), for fixed q ∈ (2, 2∗s ) and given ε > 0, there exists C > 0 such that & % ∗ |K(x)f (t)t| ≤ εC V (x)|t|2 + |t|2s + CK(x)|t|q ,

for all t ∈ R.

By Proposition 16.2.2, 

 lim

n→∞ RN

K(x)|un | dx = q

RN

K(x)|u|q dx,

(16.2.9)

530

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

so there exists r > 0 such that  K(x)|un |q dx < ε,

for all n ∈ N.

Brc

(16.2.10)

Since (un ) ⊂ X is bounded, there exists a positive constant C such that  RN

V (x)|un |2 dx ≤ C

 and

∗

RN

|un |2s dx ≤ C ,

for all n ∈ N,

(16.2.11)

Next, (16.2.9), (16.2.10), and (16.2.11) imply that  Brc

K(x)|un |q dx < (2CC + 1)ε, for all n ∈ N.

Assume that (h4 ) holds. Similarly to the second part of Proposition 16.2.2, given ε > 0 sufficiently small, there exists r > 0 large enough such that % & ∗

V (x)|t|2−m + |t|2s −m , K(x) ≤ εCm

for all |t| > 0 and |x| > r.

Consequently, % & ∗

V (x)|f (t)t||t|2−m + |f (t)t||t|2s −m K(x)|f (t)t| ≤ εCm

for all |t| > 0 and |x| > r.

From (f˜1 ) and (f2 ), there exist C, t0 , t1 > 0 satisfying & % ∗ K(x)|f (t)t| ≤ εC V (x)t 2 + |t|2s , for all t ∈ I and |x| > r, where I = {t ∈ R : |t| < t0 or |t| > t1 }. Therefore, for every u ∈ X, setting 

 Q(u) =

∗

2

RN

V (x)|u| dx +

RN

|u|2s dx

and A = {x ∈ RN : t0 ≤ |u(x)| ≤ t1 }, the following estimate holds true:  Brc

 K(x)f (u)u dx ≤ εCQ(u) + C

K(x) dx. A∩Brc

Due to the boundedness of (un ) ⊂ X, we can find C > 0 such that  RN

V (x)|un |2 dx ≤ C and



∗

RN

|un |2s dx ≤ C ,

for all n ∈ N.

16.3 The Nehari Manifold Argument

531

Therefore,  Brc

K(x)f (un )un dx ≤ εC

+ C

 An ∩Brc

K(x) dx,

where An = {x ∈ RN : t0 ≤ |un (x)| ≤ t1 }. Arguing as in the proof of Proposition 16.2.2 and using (h2 ), we deduce that  An ∩Brc

K(x) dx → 0

as r → +∞

uniformly in n ∈ N and, for ε > 0 small enough,       K(x)f (un )un dx  < (C

+ 1)ε.   Brc  In order to complete the proof, it remains to check that 

 lim

n→∞ B r

K(x)f (un )un dx =

K(x)f (u)u dx Br

 

which easily follows by Lemma 1.4.2.

16.3

The Nehari Manifold Argument

In this section we obtain some preliminary results that serve to overcome the lack of differentiability of the Nehari manifold in which we look for weak solutions to problem (16.1.1). In the following, we search for a nodal or sign-changing weak solution of problem (16.1.1), that is, a function u = u+ + u− ∈ X such that u+ = max{u, 0} = 0, u− = min{u, 0} = 0 in RN and  R2N

(u(x) − u(y))(ϕ(x) − ϕ(y)) dxdy + |x − y|N+2s



 RN

V (x)uϕ dx =

RN

K(x)f (u)ϕ dx,

for every ϕ ∈ X. The energy functional associated with (16.1.1) is given by 1 J (u) = u2 − 2

 RN

K(x)F (u) dx.

532

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

By the assumptions on f , it is clear that J ∈ C 1 (X, R) and that its differential is given by the formula  

V (x)uϕ dx − K(x)f (u)ϕ(x) dx, J (u), ϕ = u, ϕDs,2 (RN ) + RN

RN

for every u, ϕ ∈ X. Then, the critical points of J are the weak solutions of (16.1.1). Let us also observe that one has the decompositions J (u) = J (u+ ) + J (u− ) −

 R2N

u+ (x)u− (y) + u− (x)u+ (y) dxdy, |x − y|N+2s

and

+



+

+



J (u), u  = J (u ), u  −

R2N

u+ (x)u− (y) + u− (x)u+ (y) dxdy. |x − y|N+2s

The Nehari manifold associated with the functional J is given by N = {u ∈ X \ {0} : J (u), u = 0}. Recalling that a nonzero critical point u of J is a least energy weak solution of problem (16.1.1) if J (u) = min J (v) v∈N

and, since our purpose is to prove the existence of a least energy sign-changing weak solution of (16.1.1), we look for u ∈ M such that J (u) = min J (v), v∈M

where M is the subset of N consisting of all the sign-changing weak solutions of problem (16.1.1), that is M = {w ∈ N : w+ = 0, w− = 0, J (w), w+  = J (w), w−  = 0}. If we assume that f is only continuous, the following results are crucial, since they allow us to overcome the non-differentiability of N . Below, we denote by S the unit sphere on X. Lemma 16.3.1 Suppose that (V , K) ∈ K and f fulfills the conditions (f1 )–(f4 ). Then:

16.3 The Nehari Manifold Argument

533

(a) For each u ∈ X \ {0}, let hu : R+ → R be defined by hu (t) = J (tu). Then, there is a unique tu > 0 such that h u (t) > 0

in (0, tu ),

h u (t)

in (tu , ∞).

0, independent on u, such that tu ≥ τ for every u ∈ S. Moreover, for each compact set W ⊂ S, there is CW > 0 such that tu ≤ CW for every u ∈ W. ˆ S is a (c) The map ηˆ : X \ {0} → N given by η(u) ˆ = tu u is continuous and η = η| homeomorphism between S and N . Moreover, η−1 (u) =

u . u

Proof (a) We distinguish two cases. Let us assume that (h3 ) is satisfied. Using assumptions (f1 ) and (f2 ), given ε > 0 there exists a positive constant Cε such that ∗

|F (t)| ≤ ε|t|2 + Cε |t|2s ,

for every t ∈ R,

which in conjunction with the Sobolev embedding result implies that J (tu) =

t2 2 t2 [u] + 2 s 2

t2 ≥ u2 − ε 2 ≥



 

RN

V (x)u2 dx −

RN

K(x)t u dx − Cε

K(x)F (tu) dx 

∗

2 2

RN

RN

∗

K(x)t 2s |u|2s dx

t2 ∗ ∗ u2 − ε K/V L∞ (RN ) t 2 u2 − Cε C KL∞ (RN ) t 2s u2s . 2 (16.3.1)

Taking 0 < ε
0 sufficiently small such that 2 K/V L∞ (RN ) 0 < hu (t) = J (tu), for all t < t0 .

(16.3.2)

534

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

On the other hand, suppose that (h4 ) holds. Then there exists a constant Cm > 0 such that, for each ε ∈ (0, Cm ), one can find an R > 0 such that 

 BRc

K(x)|u|m dx ≤ ε

∗

BRc

(V (x)|u|2 + |u|2s ) dx,

(16.3.3)

for all u ∈ X. Now, using (f˜1 ) and (f2 ), Theorem 1.1.8, (16.3.3) and the Hölder inequality, we have that J (tu) ≥ ≥

t2 u2 − C1 2





RN

t2 u2 − C1 t m ε 2

K(x)t m |u|m dx − C2



≥

t2 u2 − C1 t m ε 2

∗

|u|2s dx ∗

BRc

(V (x)|u|2 + |u|2s ) dx 

− C1 t m K

2∗ s ∗ L 2s −m (BR )



∗

K(x)|u|m dx BR

RN



− C2 t 2s KL∞ (RN )



(V (x)|u|2 + |u|2s ) dx − C1 t m



− C2 t 2s KL∞ (RN )

∗

K(x)t 2s |u|2s dx

∗

BRc

∗

∗

RN

K(x)|u|m dx

 m∗ 2s

BR ∗

RN

|u|2s dx

  t2 2 m 2 2∗s m εu + εCu + CK 2∗s ≥ u − C1 t u ∗ 2 L 2s −m (B ) R

2∗s

2∗s

− C2 Ct KL∞ (RN ) u .

(16.3.4)

This shows that (16.3.2) holds also in this case. Moreover, since F (t) ≥ 0 for all t ∈ R, we have J (tu) ≤

t2 u2 − 2

 K(x)F (tu) dx, A

where A ⊂ supp(u) is a measurable set with finite positive measure. Consequently, J (tu) 1 ≤ − lim inf lim sup 2 t →∞ 2 t →∞ tu





F (tu) K(x) (tu)2 A



u u

2

 dx .

16.3 The Nehari Manifold Argument

535

By (f3 ) and Fatou’s lemma, it follows that lim sup t →∞

J (tu) ≤ −∞. tu2

(16.3.5)

Thus, there exists R > 0 sufficiently large such that hu (R) = J (Ru) < 0.

(16.3.6)

By the continuity of hu and (f4 ), there is tu > 0 which is a global maximum of hu with tu u ∈ N . The next objective is to prove that tu is the unique critical point of hu . Arguing by contradiction, let us assume that there are critical points t1 , t2 of hu with t1 > t2 > 0. Then we have h u (t1 ) = h u (t2 ) = 0, or equivalently,  u − 2

 u2 −

RN

RN

K(x)

f (t1 u)u dx = 0, t1

K(x)

f (t2 u)u dx = 0. t2

Subtracting and taking into account Remark 16.1.1, we obtain 

 0=

RN

K(x)

 f (t1 u) f (t2 u) 2 − u dx > 0 t1 u t2 u

which leads a contradiction. (b) By (a), there exists tu > 0 such that  tu2 u2 =

RN

K(x)f (tu u)tu u dx.

(16.3.7)

Then, estimating the right-hand side of (16.3.7) similarly to (16.3.1) and (16.3.4), we see that there exists τ > 0, independent of u, such that tu ≥ τ . On the other hand, let W ⊂ S be a compact set. Assume, by contradiction, that there exists (un ) ⊂ W such that tn = tun → ∞. Therefore, there exists u ∈ W such that un → u in X. From (16.3.5), we have J (tn un ) → −∞ in R.

(16.3.8)

536

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

Next, by Remark 16.1.1, 1 J (v) = J (v) − J (v), v 2    1 f (v)v − F (v) dx ≥ 0, = K(x) 2 RN

(16.3.9)

for each v ∈ N . Taking into account that (tun un ) ⊂ N , we conclude from (16.3.8) that (16.3.9) is not true, which is a contradiction. (c) Since J ∈ C 1 (X, R), J (0) = 0 and satisfies (a) and (b), the assertion follows by Proposition 2.4.2. The proof is now complete.   ˆ Let us define the maps ψˆ : X → R and ψ : S → R by ψ(u) = J (η(u)) ˆ and ˆ ψ = ψ|S . The next result is a consequence of Lemma 16.3.1, Proposition 2.4.3 and Corollary 2.4.4. Proposition 16.3.2 Suppose that (V , K) ∈ K and f satisfies (f1 )–(f4 ). Then, one has: (a) ψˆ ∈ C 1 (X \ {0}, R) and ψˆ (u), v =

η(u) ˆ J (η(u)), ˆ v , u

for every u ∈ X \ {0} and v ∈ X. (b) ψ ∈ C 1 (S, R) and ψ (u), v = η(u)J (η(u)), v for all v ∈ Tu S. (c) If (un ) is a Palais–Smale sequence for ψ, then (η(un )) is a Palais–Smale sequence for J . Moreover, if (un ) ⊂ N is a bounded Palais–Smale sequence for J , then (η−1 (un )) is a Palais–Smale sequence for the functional ψ. (d) u is a critical point of ψ if and only if η(u) is a nontrivial critical point for J . Moreover, the corresponding critical values coincide and inf ψ(u) = inf J (u). u∈N

u∈S

Remark 16.3.3 We notice that the following equalities hold: d∞ = inf J (u) = u∈N

inf max J (tu) = inf max J (tu).

u∈X\{0} t >0

u∈S t >0

(16.3.10)

In particular, relations (16.3.1), (16.3.5) and (16.3.10) imply that d∞ > 0.

(16.3.11)

16.4 Technical Results

16.4

537

Technical Results

The aim of this section is to prove some technical lemmas related to the existence of a least energy nodal solution. For each u ∈ X with u± ≡ 0, consider the function hu : [0, ∞) × [0, ∞) → R given by hu (t, τ ) = J (tu+ + τ u− ).

(16.4.1)

Its gradient u : [0, ∞) × [0, ∞) → R2 is given by

u (t, τ ) = u1 (t, τ ), u2 (t, τ )   u ∂hu ∂h (t, τ ), (t, τ ) = ∂t ∂s +

= J (tu + τ u− ), u+ , J (tu+ + τ u− ), u−  .

(16.4.2)

Lemma 16.4.1 Suppose that (V , K) ∈ K and f satisfies (f1 )–(f4 ). Then: (i) The pair (t, τ ) is a critical point of hu with t, τ > 0 if and only if tu+ + τ u− ∈ M. (ii) The map hu has a unique critical point (t+ , τ− ), with t+ = t+ (u) > 0 and τ− = τ− (u) > 0, which is the unique global maximum point of hu . (iii) The maps a+ (r) = u1 (r, τ− )r and a− (r) = u2 (t+ , r)r are such that a+ (r) > 0 if r ∈ (0, t+ )

and a+ (r) < 0 if r ∈ (t+ , ∞)

a− (r) > 0 if r ∈ (0, τ− )

and a− (r) < 0 if r ∈ (τ− , ∞).

(16.4.3)

Proof (i) By (16.4.2), we have  u (t, τ ) =

 1 + 1 J (tu + τ u− ), tu+ , J (tu+ + τ u− ), τ u+  , t τ

for all t, τ > 0. Therefore, u (t, τ ) = 0 if and only if J (tu+ + τ u− ), tu+  = 0

and

J (tu+ + τ u− ), su−  = 0,

and this implies that tu+ + τ u− ∈ M. (ii) Firstly we show that hu has a critical point. For each u ∈ X such that u± = 0 and fixed τ0 , we define the function h1 : [0, ∞) → [0, ∞) by h1 (t) = hu (t, τ0 ). Following

538

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

the lines of Lemma 16.3.1-(a), we can infer that h1 has a positive maximum point. Moreover, there exists a unique t0 = t0 (u, τ0 ) > 0 such that h 1 (t) > 0 h 1 (t0 )

if t ∈ (0, t0 ),

= 0,

h 1 (t) < 0

if t ∈ (t0 , ∞).

Thus, the map φ1 : [0, ∞) → [0, ∞) given by φ1 (s) = t (u, s), where t (u, s) satisfies the properties just mentioned with τ in place of τ0 , is well defined. By the definition of h1 , h 1 (φ1 (τ )) = u1 (φ1 (τ ), τ ) = 0

for all τ ≥ 0,

(16.4.4)

that is, 

+ 2

0 = |φ1 (τ )| u  − τ φ1 (τ ) 2

 −

RN

R2N

u+ (x)u− (y) + u− (x)u+ (y) dxdy |x − y|N+2s

K(x)f (φ1 (τ )u+ ) φ1 (τ )u+ dx.

(16.4.5)

Now, we establish several properties of φ1 . (a) The map φ1 is continuous. Let τn → τ0 as n → ∞ in R. We want to prove that (φ1 (τn )) is bounded. Assume, by contradiction, that there is a subsequence, denoted again by (τn ), such that φ1 (τn ) → ∞ as n → ∞. So, φ1 (τn ) ≥ τn for n large. By (16.4.5), we have u+ 2 −

τn φ1 (τn )

 R2N

u+ (x)u− (y) + u− (x)u+ (y) dxdy |x − y|N +2s

 =

RN

K(x)

f (φ1 (τn )u+ ) + 2 (u ) dx. φ1 (τn )u+

(16.4.6)

Since τn → τ0 , φ1 (τn ) → ∞ as n → ∞, assumptions (f3 )–(f4 ) and Fatou’s lemma imply that 

+ 2

u  = lim inf n→∞

RN

K(x)

f (φ1 (τn )u+ ) + 2 (u ) dx ≥ ∞, φ1 (τn )u+

16.4 Technical Results

539

so we reached a contradiction. Hence the sequence (φ1 (τn )) is bounded. Consequently, there exists t0 ≥ 0 such that φ1 (τn ) → t0 . Consider (16.4.5) with τ = τn . Letting n → ∞ we have t02 u+ 2 − τ0 t0

 R2N

 u+ (x)u− (y) + u− (x)u+ (y) dxdy = K(x)f (φ1 (t0 )u+ )φ1 (t0 )u+ dx, |x − y|N+2s RN

that is h 1 (t0 ) = u1 (t0 , τ0 ) = 0. Accordingly, t0 = φ1 (τ0 ), i.e., φ1 is continuous. (b) φ1 (0) > 0. Assume that there exists a sequence (τn ) such that φ1 (τn ) → 0 and τn → 0 as n → ∞. By assumption (f1 ) we get u+ 2 ≤ u+ 2 −  =

RN

τn φ1 (τn )

K(x)

 R2N

u+ (x)u− (y) + u− (x)u+ (y) dxdy |x − y|N+2s

f (φ1 (τn )u+ ) + 2 (u ) dx → 0, φ1 (τn )u+

as n → ∞

so we reached a contradiction. Therefore φ1 (0) > 0. (c) Now we show that φ1 (τ ) ≤ τ for τ large. As a matter of fact, proceeding as in the first part of the proof of a), we can see that there is no sequence (τn ) such that τn → ∞ and φ1 (τn ) ≥ τn for all n ∈ N. This implies that φ1 (τ ) ≤ τ for τ large. Analogously, for every t0 ≥ 0 we define h2 (τ ) = hu (t0 , τ ) and, consequently, we can find a map φ2 such that h 2 (φ2 (t)) = u2 (t, φ2 (t)),

∀t ≥ 0

(16.4.7)

and (a), (b) and (c) hold. By (c), there is a positive constant C1 such that φ1 (τ ) ≤ τ and φ2 (t) ≤ t for every t, τ ≥ C1 . Let   C2 = max max φ1 (τ ), max φ2 (t) τ ∈[0,C1 ]

t ∈[0,C1 ]

and C = max{C1 , C2 }. We define T : [0, C] × [0, C] → R2 by T (t, τ ) = (φ1 (τ ), φ2 (t)). Let us note that T ([0, C] × [0, C]) ⊂ [0, C] × [0, C].

540

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

Indeed, for every t ∈ [0, C], we have that ⎧ ⎨ φ2 (t) ≤ t ≤ C, if t ≥ C1 . φ (t) ≤ max φ (t) ≤ C , 2 2 if t ≤ C1 ⎩ 2 t ∈[0,C1 ]

Similarly, we see that φ1 (τ ) ≤ C for all τ ∈ [0, C]. Moreover, since φi are continuous for i = 1, 2, it is clear that T is a continuous map. Then, by Brouwer fixed point theorem, there exists (t+ , τ− ) ∈ [0, C] × [0, C] such that (φ1 (τ− ), φ2 (t+ )) = (t+ , τ− ). Owing to this fact and recalling that φi > 0, we have t+ > 0 and τ− > 0. By (16.4.4) and (16.4.7), u1 (t+ , τ− ) = u2 (t+ , τ− ) = 0, that is (t+ , τ− ) is a critical point of hu . Next we aim to prove the uniqueness of (t+ , τ− ). Assuming that w ∈ M, we have

w w (1, 1) = w 1 (1, 1), 2 (1, 1)   w ∂hw ∂h (1, 1), (1, 1) = ∂t ∂τ

= J (w+ + w− ), w+ , J (w+ + w− ), w−  = (0, 0), which implies that (1, 1) is a critical point of hw . Now, assume that (t0 , τ0 ) is a critical point of hw , with 0 < t0 ≤ τ0 . This means that J (t0 w+ + τ0 w− ), t0 w+  = 0 and J (t0 w+ + τ0 w− ), τ0 w−  = 0, or equivalently t02 w+ 2

 − τ0 t0

R2N

w+ (x)w− (y) + w− (x)w+ (y) dxdy = |x − y|N+2s

 RN

K(x)f (t0 w+ )t0 w+ dx

(16.4.8) τ02 w− 2 − τ0 t0

 R2N

w+ (x)w− (y) + w− (x)w+ (y) dxdy = |x − y|N+2s

 RN

K(x)f (τ0 w− )τ0 w− dx.

(16.4.9)

16.4 Technical Results

541

Dividing by τ02 > 0 in (16.4.9), we have w− 2 −



t0 τ0

R2N



w+ (x)w− (y) + w− (x)w+ (y) dxdy = |x − y|N+2s

K(x)

RN

f (s0 w− ) − 2 (w ) dx, τ0 w−

and using the fact that 0 < t0 ≤ τ0 we see that w− 2 −

 R2N

w+ (x)w− (y) + w− (x)w+ (y) dxdy ≥ |x − y|N+2s

 RN

K(x)

f (τ0 w− ) − 2 (w ) dx. τ0 w − (16.4.10)

Since w ∈ M, we also have w− 2 −

 R2N

w+ (x)w− (y) + w− (x)w+ (y) dxdy = |x − y|N+2s

 RN

K(x)

f (w− ) − 2 (w ) dx. w− (16.4.11)

Putting together (16.4.10) and (16.4.11), we get 0≥

 f (τ0 w− ) − 2 f (w− ) − 2 K(x) (w ) − (w ) dx. τ0 w − w− 

 RN

The above relation and assumption (f4 ) ensures that 0 < t0 ≤ τ0 ≤ 1. Let us prove that t0 ≥ 1. Dividing by t02 > 0 in (16.4.8), we have w+ 2 −

τ0 t0

 R2N



w+ (x)w− (y) + w− (x)w+ (y) dxdy = |x − y|N+2s

RN

K(x)

f (t0 w+ ) + 2 (w ) dx t0 w +

and since 0 < t0 ≤ τ0 we deduce that + 2



w  −

R2N

w+ (x)w− (y) + w− (x)w+ (y) dxdy ≤ |x − y|N+2s

 RN

K(x)

f (t0 w+ ) + 2 (w ) dx. t0 w+ (16.4.12)

Further, since w ∈ M, we also have w+ 2 −

 R2N

w+ (x)w− (y) + w− (x)w+ (y) dxdy = |x − y|N+2s

 RN

K(x)

f (w+ ) + 2 (w ) dx. w+ (16.4.13)

542

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

Putting together (16.4.12) and (16.4.13) we get 

 0≥

RN

 f (w+ ) + 2 f (t0 w+ ) + 2 K(x) (w ) − (w ) dx. w+ t0 w+

It follows from (f4 ) that t0 ≥ 1. Consequently, t0 = τ0 = 1, and this proves that (1, 1) is the unique critical point of hw with positive coordinates. Let u± ∈ X be such that u± = 0, and let (t1 , τ1 ), (t2 , τ2 ) be critical points of hu with positive coordinates. By (i) it follows that w1 = t1 u+ + τ1 u− ∈ M

and

w2 = t2 u+ + τ2 u− ∈ M.

We notice that w2 can be written as w2 =

    t2 τ2 t2 τ2 t1 u+ + τ1 u− = w1+ + w1− ∈ M. t1 τ1 t1 τ1

Since w1 ∈ X is such that w1± = 0, we have that (t2 /t1 , τ2 /τ1 ) is a critical point for hw1 with positive coordinates. On the other hand, since w1 ∈ M, we conclude that t2 /t1 = τ2 /τ1 = 1, which gives t1 = t2 and τ1 = τ2 . Finally, we prove that hu has a maximum global point (t¯, τ¯ ) ∈ (0, ∞) × (0, ∞). Let A+ ⊂ supp(u+ ) and A− ⊂ supp(u− ) positive with finite measure. By assumption (f3 ) and the fact that F (t) ≥ 0 for every t ∈ R, hu (t, τ ) ≤

1 tu+ + τ u− 2 − 2

 A+

K(x)F (tu+ ) dx −

 A−

K(x)F (τ u− ) dx

 u+ (x)u− (y) + u+ (y)u− (x) t2 + 2 τ 2 − 2 u  + u  − tτ dxdy 2 2 |x − y|N+2s R2N   + − K(x)F (tu ) dx − K(x)F (τ u− ) dx. ≤

A+

A−

Let us suppose that |t| ≥ |τ | > 0. Then, using the fact that F (t) ≥ 0 for every t ∈ R, we see that  hu (t, τ ) ≤ (t 2 + τ 2 )  − t2

A+

1 + 2 1 − 2 1 u  + u  − 2 2 2

K(x)

F (tu+ ) + 2 (u ) dx. (tu+ )2

 R2N

u+ (x)u− (y) + u+ (y)u− (x) dxdy |x − y|N+2s



16.4 Technical Results

543

By (f3 ), Fatou’s lemma and the fact that 0 < t 2 + τ 2 ≤ 2t 2 , we obtain that 1 hu (t, τ ) ≤ C(u+ , u− ) − lim inf 2 2 2 |t |→∞ |(t,τ )|→∞ t + τ



lim sup

A+

K(x)

F (tu+ ) + 2 (u ) dx = −∞, (tu+ )2

where C(u+ , u− ) > 0 is a constant depending only on u+ and u− . Therefore, lim

|(t,τ )|→∞

hu (t, τ ) = −∞,

(16.4.14)

which upon recalling that hu is a continuous function implies that hu has a maximum global point (t¯, τ¯ ) ∈ (0, ∞) × (0, ∞). The linearity of F and the positivity of K yield 

+

RN

−

K(x)(F (tu ) + F (τ u )) dx =

 RN

K(x)F (tu+ + τ u− ) dx.

(16.4.15)

By (16.4.15), for every u ∈ X such that u± = 0 and for every t, τ ≥ 0, J (tu+ ) + J (τ u− ) ≤ J (tu+ + τ u− ). So, for every u ∈ X such that u± = 0, one has hu (t, 0) + hu (0, τ ) ≤ hu (t, τ ), for every t, τ ≥ 0. Then, max hu (t, 0) < max hu (t, τ ) and max hu (0, τ ) < max hu (t, τ ), t ≥0

t,τ >0

τ ≥0

t,τ >0

and this proves that (t¯, τ¯ ) ∈ (0, ∞) × (0, ∞). (iii) By Lemma 16.3.1-(a) we easily have that ∂hu (r, τ− ) > 0 if r ∈ (0, t+ ), ∂t ∂hu (t+ , τ− ) = 0, u1 (t+ , τ− ) = ∂t ∂hu (r, τ− ) > 0 if r ∈ (t+ , ∞). u1 (r, τ− ) = ∂t

u1 (r, τ− ) =

Therefore, (16.4.3) holds true. The proof of Lemma 16.4.1 is now complete.

 

544

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

Lemma 16.4.2 If (un ) ⊂ M and un  u in X, then u ∈ X and u± = 0. Proof First, note that there is β > 0 such that β ≤ v ±  ∀v ∈ M.

(16.4.16)

Indeed, if v ∈ M, then v ± 2 ≤

 RN

K(x)f (v ± )v ± dx.

Assume that (h3 ) holds true. Then, using (f1 ), (f2 ), and (1.1.1), we see that given ε > 0 there exists a positive constant Cε such that v ± 2 ≤

 RN

K(x)f (v ± )v ± dx 

≤ ε K/V L∞ (RN )

∗

RN

V (x)(v ± )2 dx + Cε C∗ KL∞ (RN ) v ± 2s

(16.4.17)

∗

≤ ε K/V L∞ (RN ) v ± 2 + Cε C∗ KL∞ (RN ) v ± 2s . Choosing 

1 ε ∈ 0, K/V L∞ (RN )

 ,

guarantees that there exists a positive constant β1 such that v ±  > β1 . Analogously, assuming that (h4 ) holds, conditions (f˜1 ), (f2 ), Theorem 1.1.8 and Hölder’s inequality imply that ± 2



v  ≤

RN

K(x)f (v ± )v ± dx ∗

≤ C1 εv ± 2 + C1 C∗ (ε + C2 KL∞ (RN ) )v ± 2s + K

2∗ s ∗ L 2s −m (B

C∗ v ± m . R)

(16.4.18) Since m ∈ (2, 2∗s ), we can choose ε sufficiently small so that it is possible to find a positive constant β2 such that v ±  > β2 . Hence, if we set β = min{β1 , β2 }, then (16.4.16) holds. Then, if (un ) ⊂ M, we have  β2 ≤

RN

± K(x)f (u± n )un dx ,

∀n ∈ N.

(16.4.19)

16.5 Existence and Multiplicity Results

545

Since un  u in X, using Lemma 16.2.2, we can let n → ∞ in (16.4.19). More precisely, using Lemma 16.2.3, it follows that  0 0.

16.5

(16.4.20)

Existence and Multiplicity Results

In this section we prove the existence of least energy nodal weak solutions by using minimization arguments and a variant of deformation lemma. We start by proving the existence of a minimum point of the functional J in M. Let (un ) ⊂ M be such that J (un ) → c∞

in R.

(16.5.1)

We claim that (un ) is bounded in X. Indeed, assume, by contradiction, that that there exists a subsequence, denoted again by (un ), such that un  → ∞ as n → ∞. Set vn =

un , un 

n ∈ N.

Since (vn ) is bounded in X, due to the reflexivity of X, there exists v ∈ X such that vn  v

in X.

(16.5.2)

a.e. in RN .

(16.5.3)

Moreover, by virtue of Lemma 16.2.1, vn (x) → v(x)

546

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

By Lemma 16.4.1-(i) and the fact that (un ) ⊂ M, we have that t+ (vn ) = s− (vn ) = un  and J (un ) = J (un vn ) ≥ J (tvn )  t2 = vn 2 − K(x)F (tvn ) dx 2 RN  t2 − K(x)F (tvn ) dx, = 2 RN

(16.5.4)

for every t > 0 and n ∈ N. Suppose that v = 0. Taking into account (16.5.2) and Lemma 16.2.3, we get  RN

K(x)F (tvn ) dx → 0,

∀t > 0.

(16.5.5)

Passing to the limit in (16.5.4) as n → ∞, and combining (16.5.1) and (16.5.5) we have c∞ ≥

t2 , 2

∀t > 0,

which gives a contradiction. Hence, v = 0. Taking into account definitions of J and (vn ), we have J (un ) 1 = − un 2 2

 RN

K(x)

F (vn un ) (vn )2 dx. (vn un )2

(16.5.6)

Now, since v = 0 and un  → ∞, using (16.5.3) in addition to (f3 ), the Fatou lemma ensures that  F (vn un ) 2 K(x) v dx → ∞ (16.5.7) (vn un )2 n RN which in conjunction with (16.5.1) show that by passing to the limit in (16.5.6) as n → ∞, we reach a contradiction. Therefore, (un ) ⊂ X is a bounded subsequence. Consequently, there exists u ∈ X such that un  u

in X.

(16.5.8)

By Lemma 16.4.2, it follows that u± = 0. Moreover, by Lemma 16.4.1, there are two constants t+ , τ− > 0 such that t+ u+ + τ− u− ∈ M.

(16.5.9)

16.5 Existence and Multiplicity Results

547

Now, our aim is to prove that t+ , τ− ∈ (0, 1]. By (16.5.8) and Lemma 16.2.3,  RN

± K(x)f (u± n )un dx →

 RN

K(x)f (u± )u± dx

(16.5.10)

K(x)F (u± ) dx.

(16.5.11)

and  RN

K(x)F (u± n ) dx

 →

RN

Recalling that (un ) ⊂ M, using (16.5.8) and (16.5.10), and applying the Fatou lemma one obtains that J (u), u± 



= u± 2 −

R2N

u+ (x)u− (y) + u− (x)u+ (y) dxdy − |x − y|N+2s

 RN

K(x)f (u± )u± dx

≤ lim infJ (un ), u± n  = 0.

(16.5.12)

n→∞

Let us assume 0 < t+ < τ− . By (16.5.9), τ−2 u− 2 −t+ τ−

 R2N

u+ (x)u− (y) + u− (x)u+ (y) dxdy = |x − y|N+2s

 RN

K(x)f (τ− u− ) τ− u− dx,

and since t+ < τ− we obtain u− 2 −

 R2N

u− (x)u+ (y) + u− (y)u+ (x) dxdy ≥ |x − y|N+2s

 supp(u− )

K(x)

f (τ− u− ) − 2 (u ) dx. τ− u − (16.5.13)

In view of (16.5.12), we have u− 2 −

 R2N

u− (x)u+ (y) + u− (y)u+ (x) dxdy ≤ |x − y|N+2s

 supp(u− )

K(x)

f (u− ) − 2 (u ) dx. u− (16.5.14)

Putting together (16.5.13) and (16.5.14), we deduce that 

 f (τ− u− ) f (u− ) 0≥ K(x) − (u− )2 dx, τ− u − u− supp(u− ) 

(16.5.15)

which yields τ− ∈ (0, 1] in virtue of (f4 ). Similarly, we can show that t+ ∈ (0, 1]. Now, we prove that J (t+ u+ + τ− u− ) = c∞ .

(16.5.16)

548

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

Using the definition of c∞ , t+ , τ− ∈ (0, 1], (f4 ), and taking into account relations (16.5.9), (16.5.10) and (16.5.11), we get c∞ ≤ J (t+ u+ + τ− u− ) 1 = J (t+ u+ + τ− u− ) − J (t+ u+ + τ− u− ), t+ u+ + τ− u−  2    1 + − + − + − K(x) f (t+ u + τ− u )(t+ u + τ− u ) − F (t+ u + τ− u ) dx = 2 RN    1 = K(x) f (t+ u+ )(t+ u+ ) − F (t+ u+ ) dx 2 RN    1 K(x) f (τ− u− )(τ− u− ) − F (τ− u− ) dx + 2 RN       1 1 + + + − − − ≤ K(x) f (u )(u ) − F (u ) dx + K(x) f (u )(u ) − F (u ) dx 2 2 RN RN    1 K(x) f (u)u − F (u) dx = N 2 R    1 = lim K(x) f (un )un − F (un ) dx n→∞ RN 2   1 = lim J (un ) − J (un ), un  = c∞ . n→∞ 2 Hence, (16.5.16) holds true. Furthermore, the above calculations imply that t+ = τ− = 1. Next we show that u = u+ + u− is a critical point of the functional J , arguing by contradiction. Thus, suppose that J (u) = 0. By continuity, there exist δ, μ > 0 such that μ ≤ |J (v)|, since that v − u ≤ 3δ. Define D = [ 12 , 32 ] × [ 12 , 32 ] and g : D → X± by g(t, τ ) = tu+ + τ u− , where X± = {u ∈ X : u± = 0}. By Lemma 16.4.1, we deduce that J (g(1, 1)) = c∞ , J (g(t, τ )) < c∞

in D \ {(1, 1)}.

(16.5.17)

16.5 Existence and Multiplicity Results

549

Therefore, β = max J (g(t, τ )) < c∞ . (t,τ )∈∂D

(16.5.18)

Now, we apply Lemma 2.2.5 with S = S˜ = {v ∈ X : v − u±  ≤ δ}, 

 c∞ − β μδ , , we deduce that there exists a 4 8 deformation η ∈ C([0, 1] × X, X) such that: and c = c∞ . Choosing ε = min

(a) η(t, v) = v if v ∈ J −1 ([c∞ − 2ε, c∞ + 2ε]); (b) J (η(1, v)) ≤ c∞ − ε for each v ∈ X with v − u±  ≤ δ and J (v) ≤ c∞ + ε; (c) J (η(1, v)) ≤ J (v) for all v ∈ X. It follows from (b) and (c) that max J (η(1, g(t, τ ))) < c∞ .

(t,τ )∈∂D

(16.5.19)

To complete the proof, it suffices to prove that η(1, g(D)) ∩ M = ∅.

(16.5.20)

Indeed, the definition of c∞ and (16.5.20) contradict (16.5.19). Hence, let us define the maps h(t, τ ) = η(1, g(t, τ )),

ψ0 (t, τ ) = J (g(t, 1))tu+ , J (g(1, τ ))τ u− ,   1

+ 1

− ψ1 (t, τ ) = J (h(t, 1))h(t, 1) , J (h(1, τ ))h(1, τ ) . t τ By Lemma 16.4.1-(iii), the C 1 -function γ+ (t) = hu (t, 1) has a unique global maximum point t = 1 (note that tγ+ (t) = J (g(t, 1)), tu+ ). By denseness, given ε > 0 small enough, there is γ+,ε ∈ C ∞ ([ 12 , 32 ]) such that γ+ − γ+,ε C 1 ([ 1 , 3 ]) < ε. Therefore, 2 2





γ+ − γ+,ε C([ 1 , 3 ]) < ε, γ+,ε (1) = 0 and γ+,ε (1) < 0. Analogously, there exists 2 2





γ−,ε ∈ C ∞ ([ 12 , 32 ]) such that γ− − γ−,ε C([ 21 , 23 ]) < ε, γ+,ε (1) = 0 and γ+,ε (1) < 0, where γ− (τ ) = hu (1, τ ).

550

16 Sign-Changing Solutions for a Fractional Schrödinger Equation with. . .

(t), τ γ (τ )) and note that ψ −ψ  Define ψε ∈ C ∞ (D) by ψε (t, τ ) = (tγ+,ε ε 0 C(D) < −,ε

√ 3 2 2 ε,

(0, 0) ∈ ψε (∂D), and, (0, 0) is a regular value of ψε in D. On the other hand, (1, 1) is the unique solution of ψε (t, τ ) = (0, 0) in D. By the definition of Brouwer’s degree [27, 137], we conclude that deg(ψ0 , D, (0, 0)) = deg(ψε , D, (0, 0)) = sgn Jac(ψε )(1, 1), for ε small enough. Since











(1) + γ+,ε (1)] × [γ−,ε (1) + γ−,ε (1)] = γ+,ε (1) × γ−,ε (1) > 0 Jac(ψε )(1, 1) = [γ+,ε

we obtain that



deg(ψ0 , D, (0, 0)) = sgn[γ+,ε (1) × γ−,ε (1)] = 1,

where Jac(ψε ) is the Jacobian determinant of ψε and sgn denotes the sign function. On the other hand, by (16.5.18) we have β + c∞ 2   c∞ − β = c∞ − 2 4

J (g(t, τ )) ≤ β
0 is a parameter, s ∈ (0, 1), N ≥ 3, V ∈ C(RN , R) and A ∈ C 0,α (RN , RN ), α ∈ (0, 1], are the electric and magnetic potentials, respectively, and f : R → R. The fractional magnetic Laplacian is defined by  (−)sA u(x) = C(N, s) lim

u(x)

x+y − eı(x−y)·A( 2 ) u(y)

|x − y|N +2s

r→0 B c (x) r

dy,

C(N, s) = s

4s 



N +2s 2



π N/2 (1 − s)

.

(17.1.2)

This nonlocal operator was defined in [157] as a fractional extension (for an arbitrary s ∈ (0, 1)) of the magnetic pseudorelativistic operator or Weyl pseudodifferential operator defined with mid-point prescription, 1 HA u(x) = (2π)3 =

1 (2π)3

 x + y 2  e  u(y) dydξ ξ − A 2 R6   ı(x−y)· ξ +A x+y 2 e |ξ |2 u(y) dydξ, 

ı(x−y)·ξ

R6

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8_17

553

554

17 Fractional Schrödinger Equations with Magnetic Fields

introduced in [225] by Ichinose and Tamura, through oscillatory integrals. Observe that for smooth functions u, 

 HA u(x) = − lim

ε(0 B c ε



x+ y2





= lim

e−ıy·A

ε(0 B c (x) ε

 u(x + y) − u(x) − 1{|y| V0 = inf V (x) > 0. |x|→∞

x∈RN

(V)

In this context, the presence of the nonlocal operator (17.1.2) makes our analysis more complicated and intriguing, and new techniques are needed to overcome the difficulties that appear. Before to state our results, we introduce the assumptions on the nonlinearity. We suppose that f : R → R satisfies the following conditions: (f 1) (f 2) (f 3) (f 4)

f ∈ C 1 (R, R); f (t) = 0 for t ≤ 0; q−2 there exists q ∈ (2, 2∗s ), where 2∗s = 2N/(N −2s), such that limt →∞ f (t)/t 2 = 0; exists θ > 2 such that 0 < θ2 F (t) ≤ tf (t) for any t > 0, where F (t) = there t 0 f (τ )dτ ;

(f 5) there exists σ ∈ (2, 2∗s ) such that f (t) ≥ Cσ t

σ −4 2

for any t > 0.

A first result we get is the following. Theorem 17.1.1 ([66]) Assume that (V ) and (f 1)–(f 5) hold. Then there exists ε0 > 0 such that problem (17.1.1) admits a ground state solution for any ε ∈ (0, ε0 ). Now, let us introduce the sets M = {x ∈ RN : V (x) = V0 }

and Mδ = {x ∈ RN : dist(x, M) ≤ δ} for δ > 0. (17.1.4)

556

17 Fractional Schrödinger Equations with Magnetic Fields

Secondly, we obtain a multiplicity result for (17.1.1) by using the LusternikSchnirelman category. Theorem 17.1.2 ([66]) Assume that (V ) and (f 1)–(f 5) hold. Then, for any δ > 0 there exists εδ > 0 such that, for any ε ∈ (0, εδ ), problem (17.3.1) has at least catMδ (M) nontrivial solutions. The proof of the above theorems is based on variational methods. In the study of our problem, we will use the diamagnetic inequality established in [157] and the power-type decay at infinity of positive solutions to the limit problem associated with (17.3.1). These facts combined with the Hölder continuity assumption on the magnetic potential, will play an essential role in deriving some useful estimates needed to obtain the existence of solutions and to implement the barycenter machinery.

17.2

A Multiplicity Result Under the Rabinowitz Condition

17.2.1 The Variational Setting Using the change of variable x → ε x, we see that problem (17.1.1) is equivalent to 

(−)sAε u + V (ε x)u = f (|u|2 )u in RN , |u| ∈ H s (RN , R),

where we set Aε (x) = A(ε x). For a function u : RN → C, let us denote (for simplicity, we set  [u]2A =

(17.2.1)

C(N,s) 2

x+y

R2N

|u(x) − eı(x−y)·A( 2 ) u(y)|2 dxdy, |x − y|N+2s

and consider the space  ∗ s,2 N DA (R , C) = u ∈ L2s (RN , C) : [u]2A < ∞ . Then we introduce the Hilbert space   s,2 N (R , C) : Hεs = u ∈ DA ε

 RN

V (ε x)|u|2 dx < ∞

= 1)

17.2 A Multiplicity Result Under the Rabinowitz Condition

557

endowed with the scalar product  u, vε = )

(u(x) − eı(x−y)·Aε( R2N

 +)

RN

x+y 2 )

u(y))(v(x) − eı(x−y)·Aε ( |x − y|N+2s

x+y 2 )

v(y))

dxdy

V (ε x)uv¯ dx

and the noun uε = If u ∈ Hεs , let

√

u, uε .

uˆ j (x) = ϕj (x)u(x)

(17.2.2)

where j ∈ N and ϕj (x) = ϕ(2x/j ) with ϕ ∈ Cc∞ (RN , R), 0 ≤ ϕ ≤ 1, ϕ(x) = 1 if |x| ≤ 1, and ϕ(x) = 0 if |x| ≥ 2. Note that uˆ j ∈ Hεs and uˆ j has compact support. Proceeding as in the proof of Lemma 3.2 in [343], we get the following useful result. Lemma 17.2.1 For any ε > 0, it holds that uˆ j − uε → 0 as j → ∞. The space Hεs enjoys the following fundamental properties. Lemma 17.2.2 Hεs is complete and Cc∞ (RN , C) is dense in Hεs . Proof To prove that Hεs is a complete space, consider a Cauchy sequence (un ) in HAs ε . In √ particular, ( V (ε ·)un ) is a Cauchy sequence in L2 (RN , C), and since V (ε ·) ≥ V0 in RN , √ √ there exists u ∈ L2 (RN , C) such that V (ε ·)un → V (ε ·)u in L2 (RN , C) and a.e. in RN . By Fatou’s lemma, un → u in Hεs . To prove that Cc∞ (RN , C) is dense in Hεs , fix u ∈ Hεs and consider the sequence uˆ j (x) = u(x)ϕ(x/j ) defined as in (17.2.2). In view of Lemma 17.2.1, uˆ j − uε → 0 as j → ∞ and so it is enough to prove the denseness for compactly supported functions in Hεs . Now, we consider u ∈ Hεs with compact support, and assume that supp(u) ⊂ BR . Taking into account that |u(x) − u(y)|2 ≤ 2|u(x) − u(y)eıAε (

x+y 2 )·(x−y)

|2 + 2|u(y)|2|eıAε (

and |eıt − 1|2 ≤ 4 and |eıt − 1|2 ≤ t 2 , we have 

 |u(y)| BR

x+y

2 RN

|eıAε ( 2 )·(x−y) − 1|2 dx |x − y|N+2s

 dy

x+y 2 )·(x−y)

− 1|2

558

17 Fractional Schrödinger Equations with Magnetic Fields



 |u(y)|2

≤C BR



 +

|u(y)|

|x−y|>1

1 dx |x − y|N+2s



max|z|≤ 2R+1 |Aε (z)|2

2

2

|x−y|≤1

BR

dy

|x − y|N+2s−2

 dx

 dy

< ∞, and since V (ε ·) ≥ V0 in RN , we see that u ∈ H s (RN , C). Then, it makes sense to define uε = ρε ∗ u ∈ Cc∞ (RN , C), where ρε is a mollifier with supp(ρε ) ⊂ Bε . Arguing as in the proof of Theorem 3.24 in [211], we have that uε → u in H s (RN , C) as ε → 0. Moreover there exists K > 0 such that supp(uε − u) ⊂ BK for all ε > 0 small enough, and arguing as before, we obtain 

x+y

|eıAε ( 2 )·(x−y) − 1|2 dxdy |x − y|N+2s R2N    1 2 2 ≤ 2[uε − u]s + C |(uε − u)(y)| dx dy N+2s BK |x−y|>1 |x − y|  $   (max|z|≤ 2K+1 |Aε (z)|)2 2 2 |(uε − u)(y)| dx dy + |x − y|N+2s−2 BK |x−y|≤1  ≤ 2[uε − u]2s + C |(uε − u)(y)|2 dy → 0 as ε → 0.

[uε − u]2Aε ≤ 2[uε − u]2s + 2

|(uε − u)(y)|2

BK

  Now we recall the following fractional diamagnetic inequality: s,2 N Lemma 17.2.3 ([157]) For any u ∈ DA (R , C), we have that |u| ∈ Ds,2 (RN , R) and

[|u|]s ≤ [u]A .

(17.2.3)

We also have the following pointwise diamagnetic inequality ||u(x)| − |u(y)|| ≤ |u(x) − u(y)eıA(

x+y 2 )·(x−y)

|

a.e. x, y ∈ RN .

Proof The proof is very simple. Indeed, since |eıt | = 1 for all t ∈ R and )z ≤ |z| for all z ∈ C, we have for a.e. x, y ∈ RN |u(x) − u(y)eıA(

x+y 2 )·(x−y)

|2 = |u(x)|2 + |u(y)|2 − 2)(u(x)u(y)e−ıA(

x+y 2 )·(x−y)

)

≥ |u(x)|2 + |u(y)|2 − 2|u(x)||u(y)| = ||u(x)| − |u(y)||2

17.2 A Multiplicity Result Under the Rabinowitz Condition

559

 

which immediately gives desired inequality. Then, using Theorem 1.1.8, Lemmas 17.2.3 and 6.2.3, we get

Lemma 17.2.4 The space Hεs is continuously embedded in Lr (RN , C) for r ∈ [2, 2∗s ], and compactly embedded in Lrloc (RN , C) for r ∈ [1, 2∗s ). Moreover, if V∞ = ∞, then, for any bounded sequence (un ) in Hεs , we have that, up to a subsequence, (|un |) is strongly convergent in Lr (RN , R) for r ∈ [2, 2∗s ). For compact supported functions in H s (RN , R), we prove the following result. Lemma 17.2.5 If u ∈ H s (RN , R) and u has compact support, then w = eıA(0)·x u ∈ Hεs . Proof Assume that supp(u) ⊂ BR . Since V is continuous, it is clear that 

 V (ε x)|w| dx =

V (ε x)|u|2 dx ≤ Cu2L2 (RN ) < ∞.

2

RN

BR

Therefore, it is enough to show that [w]Aε < ∞. Recalling that A is continuous, |eıt − 1|2 ≤ 4 and |eıt − 1|2 ≤ t 2 for all t ∈ R, we have  [w]2Aε =

R2N

|eıA(0)·x u(x) − eıA(0)·y eıAε ( |x − y|N+2s 

x+y 2

)·(x−y)

u(y)|2

dxdy

x+y

u2 (y)|eı[Aε ( 2 )−A(0)]·(x−y) − 1|2 dxdy 2N |x − y|N+2s R # $   2 |Aε ( x+y 4 2 ) − A(0)| ≤ 2[u]2s + 2 u2 (y) dx + dx dy N+2s |x − y|N+2s−2 BR |x−y|≥1 |x − y| |x−y| 0 such that  lim sup j →∞

Bj \Br

|unj |τ dx ≤ σ

(17.2.4)

560

17 Fractional Schrödinger Equations with Magnetic Fields

for all r ≥ rσ . We conclude this section with some properties on the nonlinearity that will be useful in the proofs of our results. Lemma 17.2.7 The nonlinearity f satisfies the following properties: (i) for every ξ > 0 there exists Cξ > 0 such that θ F (t 2 ) ≤ f (t 2 )t 2 ≤ ξ t 2 + Cξ |t|q 2

for all t ∈ R;

(ii) there exist C1 , C2 > 0 such that, F (t 2 ) ≥ C1 |t|θ − C2 (iii) if unj  u in Hεs and uˆ j is defined as in (17.2.2), then

for all t ∈ R;

 RN

F (|unj |2 ) − F (|unj − uˆ j |2 ) − F (|uˆ j |2 ) dx = oj (1)

as j → ∞;

(iv) if (un ) ⊂ Hεs is bounded, (unj ) a subsequence as in Lemma 17.2.6 such that unj  u in Hεs and uˆ j is defined as in (17.2.2), then  RN

[f (|unj |2 )unj − f (|unj − uˆ j |2 )(unj − uˆ j ) − f (|uˆ j |2 )uˆ j ]φdx → 0

as j → ∞

uniformly with respect to φ ∈ Hεs with φε ≤ 1. Proof Properties (i) and (ii) are easy consequences of (f 1)–(f 4). Let us prove (iii). Recalling that uˆ j = ϕj u with ϕj ∈ [0, 1], it follows from (i) and the Young inequality that  |F (|unj | ) − F (|unj − uˆ j | )| ≤ 2 2

2

0

1

|f (|unj − t uˆ j |2 )||uj − t uˆ j ||uˆ j | dt

% & ≤ C (|unj | + |u|)|u| + (|unj | + |u|)q−1 |u| ≤ ξ(|unj |2 + |unj |q ) + C(|u|2 + |u|q ) for any ξ > 0. Then |F (|unj |2 ) − F (|unj − uˆ j |2 ) − F (|uˆ j |2 )| ≤ ξ(|unj |2 + |unj |q ) + C(|u|2 + |u|q ).

17.2 A Multiplicity Result Under the Rabinowitz Condition

561

Now, let  ξ Gj = max |F (|unj |2 ) − F (|unj − uˆ j |2 ) − F (|uˆ j |2 )| − ξ(|unj |2 + |unj |q ), 0 . ξ

ξ

Note that Gj → 0 as j → ∞ a.e. in RN and 0 ≤ Gj ≤ C(|u|2 + |u|q ) ∈ L1 (RN , R). Then, applying the dominated convergence theorem, we deduce that  RN

ξ

Gj dx → 0

as j → ∞.

ξ

On the other hand, by the definition of Gj , ∗

ξ

|F (|unj |2 ) − F (|unj − uˆ j |2 ) − F (|uˆ j |2 )| ≤ ξ(|uj |2 + |uj |2s ) + Gj . Hence, since (unj ) is bounded in Hεs , we have  lim sup

RN

j →∞

|F (|unj |2 ) − F (|unj − uˆ j |2 ) − F (|uˆ j |2 )| dx ≤ Cξ

and, from the arbitrariness of ξ , we get the thesis. To prove (iv), let us consider φ ∈ Hεs such that φε ≤ 1 and σ > 0. Note that, for any r ≥ max{rσ,2 , rσ,q }, where rσ,τ has been introduced in Lemma 17.2.4,    

[f (|unj | )unj − f (|unj − uˆ j | )(unj 2

RN

 ≤  +

Br

Brc

2

  − uˆ j ) − f (|uˆ j | )uˆ j ]φ dx  2

|f (|unj |2 )unj − f (|vj |2 )vj − f (|uˆ j |2 )uˆ j ||φ| dx |f (|unj |2 )unj − f (|vj |2 )vj − f (|uˆ j |2 )uˆ j ||φ| dx

= Dj + Ej . Taking into account Lemmas 17.2.4 and 17.2.1, we can apply the dominated convergence theorem to obtain that Dj → 0 uniformly in φ ∈ Hεs with φε ≤ 1. On the other hand, by (i) in Lemma 17.2.7 and uˆ j = 0 in Bjc for any j ≥ 1, we deduce that, for j large enough,  Ej =

Bj \Br

|f (|unj |2 )unj − f (|unj − uˆ j |2 )(unj − uˆ j ) − f (|uˆ j |2 )uˆ j ||φ| dx



≤C

Bj \Br

(|unj | + |uˆ j | + |unj |q−1 + |uˆ j |q−1 )|φ| dx.

562

17 Fractional Schrödinger Equations with Magnetic Fields

Since φε ≤ 1, Hölder’s inequality and Lemma 17.2.4 imply that  Bj \Br

⎡  q−1 ⎣ (|unj | + |unj | )|φ| dx ≤ C

1 Bj \Br

|unj |2 dx

 q−1 ⎤



2

+

q

Bj \Br

|unj |q dx

⎦

which combined with Lemma 17.2.4 yields  lim sup j →∞

1

Bj \Br

(|unj | + |unj |q−1 )|φ| dx ≤ C(σ 2 + σ

q−1 q

).

Moreover, Lemmas 17.2.4 and 17.2.1 ensure that uˆ j → u in L2 (RN , C) ∩ Lq (RN , C) as j → ∞. This and the Hölder inequality give 

 lim sup j →∞

Bj \Br

(|uˆ j | + |uˆ j |q−1 )|φ| dx =

1

Brc

(|u| + |u|q−1 )|φ| dx ≤ C(σ 2 + σ

q−1 q

)

for r large enough. The arbitrariness of σ > 0 implies that Ej → 0 as j → ∞ uniformly   with respect to φ, φε ≤ 1. The proof of Lemma 17.2.7 is thus complete.

17.2.2 A First Existence Result The goal of this section is to prove Theorem 17.1.1. We want to find solutions of (17.2.1) in the sense of the following definition. Definition 17.2.8 We say that u ∈ Hεs is a weak solution to (17.2.1) if for any v ∈ Hεs  u, vε = )

RN

 f (|u|2 )uv¯ dx .

Such solutions can be found as critical points of the functional Jε : Hεs → R defined as Jε (u) =

1 1 u2ε − 2 2

 RN

F (|u|2 ) dx.

Using Lemmas 17.2.4 and 17.2.7, we see that Jε is well defined and that Jε ∈ C 1 (Hεs , R). Let us show that for any ε > 0 the functional Jε satisfies the geometrical assumptions of the mountain pass theorem [29]. Lemma 17.2.9 The functional Jε satisfies the following conditions: (i) there exist α, ρ > 0 such that Jε (u) ≥ α with uε = ρ;

17.2 A Multiplicity Result Under the Rabinowitz Condition

563

(ii) there exists e ∈ Hεs such that eε > ρ and Jε (e) < 0. Proof In view of item (i) in Lemma 17.2.7, Lemma 17.2.4, and the condition (V ), we have that for ξ ∈ (0, V0 ) Jε (u) ≥

   ξ 1 2 1 Cξ [u]Aε + 1− V (ε x)|u|2 dx− |u|q dx ≥ C1 u2ε −C2 uqε , 2 2 V0 2 RN RN

from which we deduce the first statement. To prove (ii), we observe that by item (ii) in Lemma 17.2.7 and taking ϕ ∈ Cc∞ (RN , C) such that ϕ ≡ 0 we have Jε (tϕ) ≤

t2 ϕ2ε − t θ C1 ϕθLθ (RN ) + C2 |supp(ϕ)| → −∞ 2

as t → ∞  

since θ > 2.

Using a variant of mountain pass theorem without the Palais–Smale condition (see Remark 2.2.10), we deduce that there exists a sequence (un ) ⊂ Hεs such that Jε (un ) → cε

and

Jε (un ) → 0,

(17.2.5)

where cε is the minimax level of the mountain pass theorem, namely cε = inf max Jε (γ (t)) γ ∈ t ∈[0,1]

with  = {γ ∈ H ([0, 1], Hεs ) : γ (0) = 0, Jε (γ (1)) < 0}. The sequence (un ) is bounded in Hεs . Indeed, by (17.2.5) and (f 4), 1 C(1 + un ε ) ≥ Jε (un ) − Jε (un ), un  θ      1 1 1 1 2 2 2 2 − f (|un | )|un | − F (|un | ) dx = un ε + 2 θ 2 RN θ   1 1 − un 2ε . ≥ 2 θ Moreover, it is standard to verify the characterization cε =

inf

sup Jε (tu) = inf Jε (u),

u∈Hεs \{0} t ≥0

u∈Nε

564

17 Fractional Schrödinger Equations with Magnetic Fields

where Nε = {u ∈ Hεs \ {0} : Jε (u), u = 0} is the usual Nehari manifold associated with Jε . Lemma 17.2.10 We have: (i) there exists K > 0 such that, for all u ∈ Nε , uε ≥ K; (ii) for every u ∈ Hεs \ {0} there exists a unique t0 = t0 (u) such that Jε (t0 u) = maxt ≥0 Jε (tu) and then t0 u ∈ Nε . Proof Property (i) follows easily from item (i) in Lemmas 17.2.7 and 17.2.4, since, if u ∈ Nε , then, for all ξ > 0  u2ε =

RN

f (|u|2 )|u|2 dx ≤ ξ u2ε + Cuqε .

To prove (ii), fix u ∈ Hεs \ {0} and consider the smooth function h(t) = Jε (tu) for t ≥ 0. Arguing as in Lemma 17.2.9, we get that Jε (tu) ≥ C1 t 2 u2ε − C2 t q uqε and t2 Jε (tu) ≤ u2ε − t θ C1 2

 |u|θ dx + C2 | | → −∞

as t → ∞,



where is a compact subset of supp(u) with | | > 0. Then there exists a maximum point of h. To prove the uniqueness, let 0 < t1 < t2 be two maximum points of h. Then, since h (t1 ) = h (t2 ) = 0,  u2ε

=

 f (|t1 u| )|u| dx = 2

RN

2

RN

f (|t2 u|2 )|u|2 dx,

which contradicts the strict monotonicity of f assumed in (f 5).

 

To prove the compactness of the Palais–Smale sequences at the level d, for suitable d ∈ R, we will use the following preliminary result. Lemma 17.2.11 Let d ∈ R and (un ) ⊂ Hεs be a Palais–Smale sequence for Jε at the level d and such that un  0 in Hεs . Then, one of the following alternatives occurs:

17.2 A Multiplicity Result Under the Rabinowitz Condition

565

(a) un → 0 in Hεs ; (b) there are a sequence (yn ) ⊂ RN and constants R, β > 0 such that  |un |2 dx ≥ β > 0.

lim inf n→∞

BR (yn )

Proof Assume that (b) does not hold. Then, for every R > 0,  |un |2 dx = 0.

lim sup

n→∞

y∈RN

BR (y)

Since (un ) is bounded in Hεs , it follows from (17.2.3) that (|un |) is bounded in H s (RN , R), and applying Lemma 1.4.4 we deduce that un Lq (RN ) → 0. Then, using the fact that (un ) is a Palais–Smale sequence for Jε at the level d, and (i) in Lemma 17.2.7, we see that, for every ξ > 0,  0≤

un 2ε

=

RN

f (|un |2 )|un |2 dx + on (1) q

≤ ξ un 2L2 (RN ) + Cξ un Lq (RN ) + on (1) ≤

ξ q un 2ε + Cξ un Lq (RN ) + on (1). V0  

Taking ξ small enough, we get (a).

To continue our argument, we need to consider the following family of scalar limit problems associated with 

(−)s u + μu = f (u2 )u in RN , u ∈ H s (RN , R),

with μ > 0, whose corresponding C 1 functional Iμ : H s (RN , R) → R is given by 1 1 Iμ (u) = u2μ − 2 2

 RN

F (u2 ) dx,

where

1 2 uμ = [u]2s + V0 u2L2 (RN ) . Even in this case we can introduce the Nehari manifold Mμ = {u ∈ H s (RN , R) : Iμ (u), u = 0},

(Pμ )

566

17 Fractional Schrödinger Equations with Magnetic Fields

and then mμ = inf max Iμ (γ (t)) = γ ∈'μ t ∈[0,1]

inf

sup Iμ (tu) = inf Iμ (u)

u∈H s (RN ,R)\{0} t ≥0

u∈Mμ

with 'μ = {γ ∈ C([0, 1], H s (RN , R)) : γ (0) = 0, Iμ (γ (1)) < 0}. We will call ground state for (Pμ ) each minimum of Iμ in Mμ , wich is also a solution of (Pμ ). Remark 17.2.12 Arguing as in Lemma 17.2.10, we can prove that for every fixed μ > 0 there exists K > 0 such that, for all u ∈ Mμ , uε ≥ K and that for any u ∈ H s (RN , R)\ {0} there exists a unique t0 = t0 (u) such that Iμ (t0 u) = maxt ≥0 Iμ (tu), and then t0 u ∈ Mμ . Following the same arguments as in Theorem 6.3.11, we obtain the following result: Lemma 17.2.13 Let (wn ) ⊂ Mμ be a sequence satisfying Iμ (wn ) → mμ . Then (wn ) is bounded in H s (RN , R) and, up to a subsequence, wn  w in H s (RN , R). If w = 0, then wn → w ∈ Mμ in H s (RN , R) and w is a ground state for (Pμ ). If w = 0, then there exist (y˜n ) ⊂ RN and w˜ ∈ H s (RN , R) \ {0} such that, up to a subsequence, wn (· + y˜n ) → w˜ ∈ Mμ in H s (RN , R) and w˜ is a ground state for (Pμ ). Remark 17.2.14 In view of Lemma 17.2.13 and arguing as in the proof of Theorem 3.2.11, we can see that (Pμ ) admits a positive ground state w ∈ H s (RN , R) that is Hölder continuous and has a power-type decay at infinity, more precisely 0 < w(x) ≤

C |x|N+2s

if |x| > 1.

Now we prove a fundamental property on the Palais–Smale sequences for Jε in the noncoercive case (V∞ < ∞). Lemma 17.2.15 Let d ∈ R. Assume that V∞ < ∞ and let (vn ) be a Palais–Smale sequence for Jε at the level d with vn  0 in Hεs . If vn → 0 in Hεs , then d ≥ mV∞ . Proof Let (tn ) ⊂ (0, ∞) be such that (tn |vn |) ⊂ MV∞ . First we prove that lim supn tn ≤ 1. Assume, by contradiction, that there exist δ > 0 and a subsequence, still denoted by (tn ), such that tn ≥ 1 + δ

∀n ∈ N.

(17.2.6)

17.2 A Multiplicity Result Under the Rabinowitz Condition

567

Since (vn ) is a Palais–Smale sequence for Jε at the level d,  [vn ]2Aε +

RN

 V (ε x)|vn |2 dx =

RN

f (|vn |2 )|vn |2 dx + on (1).

(17.2.7)

On the other hand, tn |vn | ∈ MV∞ . Thus we get  [|vn |]2s + V∞ vn 2L2 (RN ) =

RN

f (tn2 |vn |2 )|vn |2 dx.

(17.2.8)

Putting together (17.2.7), (17.2.8) and using (17.2.3), we conclude that 

% RN

 & f (tn2 |vn |2 ) − f (|vn |2 ) |vn |2 dx ≤

RN

(V∞ − V (ε x)) |vn |2 dx + on (1). (17.2.9)

Now, by condition (V ), for every ζ > 0 there exists R = R(ζ ) > 0 such that V∞ − V (ε x) ≤ ζ

for any |x| ≥ R.

(17.2.10)

Combining (17.2.10) with the fact that, by Lemma 17.2.4, vn → 0 in L2 (BR , C), so that |vn | → 0 in L2 (BR ), and with the boundedness of (vn ) in Hεs , we get 

 (V∞ − V (ε x)) |vn | dx =



2

RN

(V∞ − V (ε x)) |vn | dx + 2

BR

≤ V∞



 |vn | dx + ζ 2

BR

≤ on (1) +

BRc

BRc

(V∞ − V (ε x)) |vn |2 dx

|vn |2 dx

ζ vn 2ε ≤ on (1) + ζ C. V0

Then, in view of (17.2.9), we deduce that 

% RN

& f (tn2 |vn |2 ) − f (|vn |2 ) |vn |2 dx ≤ ζ C + on (1).

(17.2.11)

Since vn → 0, we can apply Lemma 17.2.11 to deduce the existence of a sequence (yn ) ⊂ ¯ β, such that RN , and the existence of two positive numbers R,  |vn |2 dx ≥ β > 0. BR¯ (yn )

(17.2.12)

568

17 Fractional Schrödinger Equations with Magnetic Fields

Now, consider the functions wn = |vn |(· + yn ). Thanks to condition (V ), (17.2.3), and the boundedness of (vn ) in Hεs , we have that wn 2V0 = |vn |2V0 ≤ vn 2ε ≤ C. Therefore, wn  w in H s (RN , R) and wn → w in Lrloc (RN , R) for all r ∈ [2, 2∗s ). By (17.2.12) 

 w dx = lim 2

BR¯

n→∞ B R¯

wn2 dx ≥ β,

and so there exists a set ⊂ RN of positive measure such that w = 0 in . It follows from (17.2.6) and (17.2.11) that 

f ((1 + δ)2 wn2 ) − f (wn2 ) wn2 dx ≤ ζ C + on (1).

Applying Fatou’s lemma and using (f 5) we obtain 

0< f ((1 + δ)2 w2 ) − f (w2 ) w2 dx ≤ ζ C,

so by the arbitrariness of ζ > 0 we reached a contradiction. Now, two cases can occur. Case 1

lim supn→∞ tn = 1.

In this case there exists a subsequence, still denoted by (tn ), such that tn → 1. Taking into account that (vn ) is a Palais–Smale sequence for Jε at the level d, mV∞ is the minimax level of IV∞ , and (17.2.3), we have d + on (1) = Jε (vn ) ≥ Jε (vn ) − IV∞ (tn |vn |) + mV∞ 

1 − tn2 1 2 V (ε x) − tn2 V∞ |vn |2 dx ≥ [|vn |] + 2 2 RN  % & 1 F (tn2 |vn |2 ) − F (|vn |2 ) dx + mV∞ . + 2 RN

(17.2.13)

Since (|vn |) is bounded in H s (RN , R) and tn → 1, we see that (1 − tn2 ) [|vn |]2s = on (1). 2

(17.2.14)

17.2 A Multiplicity Result Under the Rabinowitz Condition

569

Now, using (V ), we deduce that for every ζ > 0 there exists R = R(ζ ) > 0 such that, for every |x| > R, V (ε x) − tn2 V∞ = (V (ε x) − V∞ ) + (1 − tn2 )V∞ ≥ −ζ + (1 − tn2 )V∞ . Thus, since (vn ) is bounded in Hεs , |vn | → 0 in Lp (BR ), and tn → 1, we get  



V (ε x) − tn2 V∞ |vn |2 dx = V (ε x) − tn2 V∞ |vn |2 dx RN

BR

+

 BRc



V (ε x) − tn2 V∞ |vn |2 dx 

≥ (V0 − tn2 V∞ )

 |vn |2 dx − ζ BR



+ V∞ (1 − tn2 ) ≥ on (1) −

BRc

BRc

|vn |2 dx

|vn |2 dx

C ζ. V0 (17.2.15)

Finally, using the mean value theorem, item (i) in Lemma 17.2.7, the fact that tn → 1, and the boundedness of (|vn |), we get  % &    2 2 2  F (tn |vn | ) − F (|vn | ) dx  ≤ |f (θn |vn |2 )||tn2 − 1||vn |2 dx  RN

RN

q

≤ (C1 vn 2L2 (RN ) + C2 vn Lq (RN ) )|tn2 − 1| = on (1). (17.2.16) Combining (17.2.13), (17.2.14), (17.2.15) and (17.2.16) we can infer that d + on (1) ≥ on (1) − ζ C + mV∞ , and taking the limit as n → ∞ we get d ≥ mV∞ . Case 2

lim supn→∞ tn = t0 < 1.

In this case there exists a subsequence, still denoted by (tn ), such that tn → t0 and tn < 1 for all n ∈ N. Since (vn ) is a bounded Palais–Smale sequence for Jε at the level d, we have  1 1 [f (|vn |2 )|vn |2 − F (|vn |2 )]dx. d + on (1) = Jε (vn ) − Jε (vn ), vn  = 2 2 RN (17.2.17)

570

17 Fractional Schrödinger Equations with Magnetic Fields

Observe that, by (f 5), the function t → f (t)t − F (t) is increasing for t > 0. Hence, since tn |vn | ∈ MV∞ and tn < 1, from (17.2.17), we obtain mV∞ ≤ IV∞ (tn |vn |) 1 = IV∞ (tn |vn |) − tn IV ∞ (tn |vn |), |vn | 2 

1 = f (tn2 |vn |2 )tn2 |vn |2 − F (tn2 |vn |2 ) dx 2 RN 

1 f (|vn |2 )|vn |2 − F (|vn |2 ) dx ≤ 2 RN = d + on (1). Passing to the limit as n → ∞, we get d ≥ mV∞ .

 

Thus we are ready to give conditions on the levels c which ensure that Jε satisfies the (PS)c condition. Proposition 17.2.16 The functional Jε satisfies the (PS)c condition at any level c < mV∞ if V∞ < ∞, and at any level c ∈ R if V∞ = ∞. Proof Let (un ) be a Palais–Smale sequence for Jε at the level c. Then (un ) is bounded q in Hεs and, up to a subsequence, un  u in Hεs and un → u in Lloc (RN , C) for any ∗

q ∈ [1, 2s ). Using (f 1)–(f 3), it is easy to deduce that Jε (u) = 0, which combined with (f 4) gives 1 1 Jε (u) = Jε (u) − Jε (u), u = 2 2

 RN

[f (|u|2 )|u|2 − F (|u|2 )] dx ≥ 0.

(17.2.18)

In view of Lemma 17.2.6, we can find a subsequence (unj ) ⊂ Hεs satisfying (17.2.4). Now, let vj = unj − uˆ j , where uˆ j is defined as in (17.2.2). We claim that Jε (vj ) = c − Jε (u) + oj (1)

(17.2.19)

Jε (vj ) = oj (1).

(17.2.20)

and

17.2 A Multiplicity Result Under the Rabinowitz Condition

571

To prove (17.2.19), note that  Jε (vj ) − Jε (unj ) + Jε (uˆ j ) = [uˆ j 2ε − unj , uˆ j ε ] +

RN

[F (|unj |2 ) − F (|vj |2 ) − F (|uˆ j |2 )] dx

= Aj + Bj .

In view of the weak convergence of (unj ) to u in Hεs and Lemma 17.2.1, we can see that Aj → 0 as j → ∞. Moreover, by (iii) in Lemma 17.2.7, we have that Bj → 0 as j → ∞. To show (17.2.20), we observe that      

Jε (vj ) − Jε (unj ) + Jε (uˆ j ), φ = )  ≤

RN

RN

  [f (|unj |2 )unj − f (|vj |2 )vj − f (|uˆ j |2 )uˆ j ]φ¯ dx 

|f (|unj |2 )unj − f (|vj |2 )vj − f (|uˆ j |2 )uˆ j ||φ| dx

and applying (iv) in Lemma 17.2.7 we get that Jε (vj ) − Jε (unj ) + Jε (uˆ j ), φ → 0 for any φ ∈ Hεs such that φε ≤ 1. Then, since Jε (unj ) → 0 and Jε (uˆ j ) → Jε (u) = 0, we can infer that (17.2.20) is satisfied. Let us assume that V∞ < ∞ and c < mV∞ . By (17.2.19) and (17.2.18), we have that c − Jε (u) ≤ c < mV∞ . Since (vj ) is a Palais–Smale sequence for Jε at the level c − Jε (u) and vj  0 in Hεs , it follows from Lemma 17.2.15 that vj → 0 in Hεs . Hence, Lemma 17.2.1 implies that unj → u in Hεs as j → ∞. If V∞ = ∞, then, by Lemma 17.2.4, vj → 0 in Lr (RN , C) for any r ∈ [2, 2∗s ) and by (17.2.20) and item (i) in Lemma 17.2.7, we deduce that  vj 2ε =

RN

f (|vj |2 )|vj |2 dx + oj (1) = oj (1).

Hence, as before, unj → u in Hεs as j → ∞ which completes the proof.

 

Now we show that Nε is a natural constraint, namely that the constrained critical points of the functional Jε on Nε are critical points of Jε in Hεs . Proposition 17.2.17 The functional Jε restricted to Nε satisfies the (PS)c condition at any level c < mV∞ if V∞ < ∞, and at any level c ∈ R if V∞ = ∞. Proof Let (un ) ⊂ Nε be a Palais–Smale sequence at the level c for Jε restricted to Nε . Then, by Proposition 2.3.10, Jε (un ) → c as n → ∞ and there exists (λn ) ⊂ R such that Jε (un ) = λn Tε (un ) + on (1),

(17.2.21)

572

17 Fractional Schrödinger Equations with Magnetic Fields

where Tε : Hεs → R is defined by  Tε (u) =

u2ε

−

RN

f (|u|2 )|u|2 dx.

Thanks to (f 5), Tε (un ), un  = 2un 2ε − 2  = −2

RN





RN

f (|un |2 )|un |2 dx − 2

RN

f (|un |2 )|un |4 dx

f (|un |2 )|un |4 dx ≤ −2Cσ un σLσ (RN ) < 0.

Up to a subsequence, we may assume that Tε (un ), un  → % ≤ 0. If % = 0, then on (1) = |Tε (un ), un | ≥ Cun σLσ (RN ) , so we obtain that un → 0 in Lσ (RN , C). Observe that, since (un ) ⊂ Nε and Jε (un ) → c as n → ∞, the sequence (un ) is bounded in Hεs . By interpolation, we also have un → 0 in Lq (RN , C). Hence, by (i) in Lemma 17.2.7, we get  un 2ε =

RN

f (|un |2 )|un |2 dx ≤

ξ ξ q un 2ε + Cξ un Lq (RN ) = un 2ε + on (1), V0 V0

which implies that un → 0 in Hεs . This is impossible in view of item (i) of Lemma 17.2.10. Therefore, % < 0 and by (17.2.21) we deduce that λn = on (1). Moreover, by the assumptions on f , |Tε (un ), φ| ≤ 2un ε φε + 2

 RN

 |f (|un |2 )||un ||φ| dx + 2

RN

|f (|un |2 )||un |3 |φ| dx

s ≤ Cun ε (1 + un q−2 ε )φε for every φ ∈ Hε .

Then, the boundedness of (un ) implies the boundedness of Tε (un ) and so, by (17.2.21), we infer that Jε (un ) = on (1), that is, (un ) is a Palais–Smale sequence for Jε at the level c. Hence, it remains to apply Proposition 17.2.16.   Consequently, we have the following result. Corollary 17.2.18 The constrained critical points of the functional Jε on Nε are critical points of Jε in Hεs . Now we are ready to give the proof of the main result of this section.

17.2 A Multiplicity Result Under the Rabinowitz Condition

573

Proof of Theorem 17.1.1 By Lemma 17.2.9, we know that Jε has a mountain pass geometry. Then, by a variant of mountain pass theorem without the Palais–Smale condition (see Remark 2.2.10), there exists a Palais–Smale sequence (un ) ⊂ Hεs for Jε at the level cε . If V∞ = ∞, then by Lemma 17.2.4 and Proposition 17.2.16, we deduce that Jε (u) = cε and Jε (u) = 0, where u ∈ Hεs is the weak limit of (un ). Now, suppose that V∞ < ∞. In view of Proposition 17.2.16, it is enough to show that cε < mV∞ . Without loss of generality, we suppose that V (0) = V0 = inf V (x). x∈RN

Let μ ∈ (V0 , V∞ ). Clearly, mV0 < mμ < mV∞ . Let w ∈ H s (RN , R) be a positive ground state to the autonomous problem (Pμ ) and η ∈ Cc∞ (RN , R) be a cut-off function such that η = 1 in B1 and η = 0 in B2c . Define wr (x) = ηr (x)w(x)eıA(0)·x , with ηr (x) = η(x/r) for r > 0, and observe that |wr | = ηr w and wr ∈ Hεs in view of Lemma 17.2.5. Take tr > 0 such that Iμ (tr |wr |) = max Iμ (t|wr |) t ≥0

Let us prove that there exists a sufficiently large r such that Iμ (tr |wr |) < mV∞ . If, by contradiction, Iμ (tr |wr |) ≥ mV∞ for any r > 0, then by using the fact that |wr | → w in H s (RN , R) as r → ∞ (see Lemma 1.4.8), we have tr → 1 and mV∞ ≤ lim inf Iμ (tr |wr |) = Iμ (w) = mμ , r→∞

which contradicts the inequality mV∞ > mμ . Hence, there exists r > 0 such that Iμ (tr |wr |) = max Iμ (τ (tr |wr |)) and Iμ (tr |wr |) < mV∞ . τ ≥0

(17.2.22)

Now, we show that lim [wr ]2Aε = [ηr w]2s .

ε→0

First, note that 

[wr ]2Aε

x+y

|eıA(0)·x ηr (x)w(x) − eıAε ( 2 )·(x−y) eıA(0)·y ηr (y)w(y)|2 = dxdy |x − y|N +2s R2N x+y  ηr2 (y)w2 (y)|eı[Aε ( 2 )−A(0)]·(x−y) − 1|2 2 dxdy = [ηr w]s + |x − y|N +2s R2N

(17.2.23)

574

17 Fractional Schrödinger Equations with Magnetic Fields

 + 2)

(ηr (x)w(x) − ηr (y)w(y))ηr (y)w(y)(1 − e−ı[Aε ( |x − y|N +2s

R2N

x+y 2 )−A(0)]·(x−y)

)

dxdy

= [ηr w]2s + Xε + 2Yε .

√ Since |Yε | ≤ [ηr w]s Xε , it suffices to show that Xε → 0 as ε → 0 to deduce that (17.2.23) holds. Observe that, for 0 < β < α/(1 + α − s), 

 Xε ≤

RN

w2 (y) 

 +

|eı[Aε (

RN

w (y)

− 1|2

|x − y|N+2s

|x−y|≥ε−β

|eı[Aε (

2

x+y 2 )−A(0)]·(x−y)

x+y 2 )−A(0)]·(x−y)

|x

|x−y| 0 be the unique positive number such that Jε (tε ε,y ) = max Jε (tε ε,y ) t ≥0

576

17 Fractional Schrödinger Equations with Magnetic Fields

and let us introduce the map ε : M → Nε by setting ε (y) = tε ε,y . By construction, ε (y) has compact support for any y ∈ M. We begin by proving the following result. Lemma 17.2.19 ε,y 2ε → ω2V0 as ε → 0, uniformly with respect to y ∈ M. Proof Applying the dominated convergence theorem we easily see that 

 V (εx)|ε,y (x)| dx → V0 2

RN

RN

ω2 (x) dx.

Thus, we only need to prove that, as ε → 0 

|ε,y (x1 ) − ε,y (x2 )eı(x1 −x2 )·Aε (

x1 +x2 2 ) |2

|x1 − x2 |N+2s

R2N

 dx1 dx2 →

R2N

|ω(x1 ) − ω(x2 )|2 dx1 dx2 . |x1 − x2 |N+2s

Using the change of variable ε xi − y = ε zi (i = 1, 2), we can write  R2N

|ε,y (x1 ) − ε,y (x2 )eı(x1 −x2 )·Aε ( |x1 − x2 |N+2s

 = 

R2N

x1 +x2 2

) 2 |

dx1 dx2

|ψ(| ε z1 |)ω(z1 )eıτy (z1 ) − ψ(| ε z2 |)ω(z2 )eıτy (z2 ) eı(z1 −z2 )·A(ε |z1 − z2 |N+2s

z1 +z2 2

+y) 2 |

dz1 dz2

|ψ(| ε z1 |)ω(z1 ) − ψ(| ε z2 |)ω(z2 )|2 dz1 dz2 |z1 − z2 |N+2s ! "  2 ψ 2 (| ε z2 |)ω2 (z2 ) 1 − cos (z1 − z2 ) · [A(ε( z1 +z 2 ) + y) − A(y)] +2 dz1 dz2 |z1 − z2 |N+2s R2N % & z1 +z2  [ψ(| ε z1 |)ω(z1 ) − ψ(| ε z2 |)ω(z2 )]ψ(| ε z2 |)ω(z2 ) 1 − eı(z2 −z1 )·[A(ε( 2 )+y)−A(y)] + 2) dz1 dz2 |z1 − z2 |N+2s R2N =

R2N

= Xε + Yε + 2Zε .

Since ψ(|x|) = 1 for x ∈ Bδ/2 , we can use Lemma 1.4.8 to get  Xε =

R2N

|ψ(| ε z1 |)ω(z1 ) − ψ(| ε z2 |)ω(z2 )|2 dz1 dz2 → |z1 − z2 |N +2s

as ε → 0. On the other hand, by the Hölder inequality, |Zε | ≤



 Xε Yε .

Therefore, it is enough to show that Yε → 0 as ε → 0.

 R2N

|ω(z1 ) − ω(z2 )|2 dz1 dz2 |z1 − z2 |N +2s

17.2 A Multiplicity Result Under the Rabinowitz Condition

577

Since ψ = 0 in Bδc , we have 

 Yε = 2 Bδ/ε (0)

 +

! " 2 1 − cos (z1 − z2 ) · [A(ε( z1 +z 2 ) + y) − A(y)] dz1 |z1 − z2 |N+2s |z1 −z2 | 0. Moreover, proceeding as in the proof of Lemma 17.2.23 we have: Lemma 17.4.22 For every δ > 0 we have lim sup dist(βε (u), Mδ ) = 0.

ε→0

ε u∈N

We end this section with the proof of a multiplicity result for (17.4.13). Theorem 17.4.23 For every δ > 0 such that Mδ ⊂ , there exists ε˜ δ > 0 such that, for any ε ∈ (0, ε˜ δ ), problem (17.4.13) has at least catMδ (M) nontrivial solutions. Proof Given δ > 0 such that Mδ ⊂ , we can use Lemmas 17.4.19, 17.4.20, 17.4.22 and argue as in [146] to deduce the existence of ε˜ δ > 0 such that, for any ε ∈ (0, εδ ), the diagram ε

βε

ε −→ Mδ M −→ N is well defined and βε ◦ ε is homotopically equivalent to the embedding ι : M → ε ) ≥ catMδ (M). It follows from Proposition 17.4.16 and standard Mδ . Thus, catN ε (N ε ) critical points on Nε . Lusternik-Schnirelman theory that Jε possesses at least catN ε (N Applying Corollary 17.4.17, we can deduce that (17.4.13) has at least catMδ (M) nontrivial solutions.  

17.4.4 Proof of Theorem 17.4.2 In this last subsection we prove that the solutions obtained in Theorem 17.4.23 verify √ |uε (x)| ≤ ta for any x ∈ R \ ε and ε small. We begin with the following lemma which will play a fundamental role in the study of the behavior of the maximum points of solutions.

638

17 Fractional Schrödinger Equations with Magnetic Fields

εn be a solution to (17.4.13) such that Lemma 17.4.24 Let εn → 0 and un ∈ N m = lim sup un 2εn < 1. n→∞

Then vn = |un |(· + y˜n ) belongs to L∞ (R, R) and there exists C > 0 such that vn L∞ (R) ≤ C

for all n ∈ N,

where (y˜n ) is given by Lemma 17.4.21. Moreover lim vn (x) = 0,

|x|→∞

uniformly in n ∈ N.

Proof Recalling that Jεn (un ) ≤ mV0 + h(εn ) with h(εn ) → 0 as n → ∞, we can proceed as in the first part of the proof of Lemma 17.4.21 to deduce that Jεn (un ) → mV0 . Invoking Lemma 17.4.21, we find a sequence (y˜n ) ⊂ R such that εn y˜n → y0 ∈ M and vn = 1 |un |(· + y˜n ) has a convergent subsequence in H 2 (R, R). Now we use a Moser iteration argument. For each n ∈ N and L > 0, we set uL,n = 2(β−1) min{|un |, L} and vL,n = uL,n un , where β > 1 will be chosen later. Taking vL,n as test function in (17.4.13) we can see that ⎞ ⎛ x+y  1 ⎝ (un (x) − un (y)eıAε n ( 2 )(x−y) ) 2(β−1) 2(β−1) ıAε n ( x+y )(x−y) 2 (un uL,n (x) − un uL,n (y)e ) dxdy ⎠ ) 2π |x − y|2 R2  =

2(β−1)

R

g(εn x, |un |2 )|un |2 uL,n

 dx −

R

2(β−1)

V (εn x)|un |2 uL,n

dx.

(17.4.46)

By assumptions (g1 ) and (g2 ), for every ξ > 0 there exists Cξ > 0 such that g(εn x, |un |2 )|un |2 ≤ ξ |un |2 + Cξ |un |2 B(un )

for all n ∈ N,

(17.4.47)

where B(un ) = (eωτ |un | − 1) ∈ Lq (R) for some q > 1 close to 1, τ > 1 such that τ qm < 1, and 2

B(un )Lq (R) ≤ C

for any n ∈ N.

(17.4.48)

On the other hand, the same calculations performed in Remark 17.3.10 show that ⎛  )⎝

(un (x) − un (y)eıAε n ( |x − y|2 R2

≥ [(|un |)]21 . 2

x+y 2 )(x−y) )

⎞ x+y 2(β−1) 2(β−1) (un (x)uL,n (x) − un (y)uL,n (y)eıAε n ( 2 )(x−y) ) dxdy ⎠

(17.4.49)

17.4 A Multiplicity Result for a Fractional Magnetic Schrödinger. . .

639

Since 1 β−1 |un |uL,n , β

(|un |) ≥ 1

and recalling that H 2 (R, R) ⊂ Lr (R, R) for all r ∈ [2, ∞), we can see that 1 β−1 C|un |uL,n 2Lγ (R) , β2

[(|un |)]21 ≥ C(|un |)2Lγ (R) ≥ 2

(17.4.50)

where γ > 2q . Combining (17.4.46), (17.4.49) and (17.4.50) we can infer that  2   1 β−1 2(β−1) 2(β−1) C|un |uL,n 2Lγ (R) + V (εn x)|un |2 uL,n dx ≤ g(εn x, |un |2 )|un |2 uL,n dx. β R R

(17.4.51) Taking ξ ∈ (0, V0 ) and using (17.4.47) and (17.4.51) we have  wL,n 2Lγ (R)

≤ Cβ

2 R



2(β−1) B(un )|un |2 uL,n dx

= Cβ

2 R

2 B(un )wL,n dx,

(17.4.52)

β−1

where wL,n = |un |uL,n . We derive from (17.4.48) and Hölder’s inequality that  1 

 wL,n 2Lγ (R)

≤ Cβ

2

q

R

q

(B(un )) dx

R

2q

wL,n dx

 1

q

≤ Cβ 2 wL,n 2L2q (R) ,

and letting L → ∞ we obtain 1

1

un Lβγ (R) ≤ C 2β β β un L2q β (R) . Since γ > 2q , we can use an iteration argument to deduce that, for all m ∈ N, un Lτ k(m+1) (R) ≤ C where k = that

γ 2q

m

1 i=1 2k i

m

k

i i=1 k i

un Lγ (R) 1

and τ = 2q , which together with the boundedness of (un ) in Hε2 implies

un L∞ (R) ≤ K

for all n ∈ N.

(17.4.53)

640

17 Fractional Schrödinger Equations with Magnetic Fields

Arguing as in the proof of Lemma 17.3.8 and using (V1 ), we see that vn = |un |(· + y˜n ) solves 1

(−) 2 vn + V0 vn ≤ hn

in R,

(17.4.54)

where hn (x) = g(ε n x + εn y˜n , vn2 )vn . Since (17.4.53) yields vn L∞ (R) ≤ C for all n ∈ N, by interpolation we know that vn → v converges strongly in Lr (R, R) for all r ∈ [2, ∞), for some v ∈ Lr (R, R). Furthermore, in view of the growth assumptions on f , we also have that hn → f (v 2 )v 1 in Lr (R, R) and that hn L∞ (R) ≤ C for all n ∈ N. Let zn ∈ H 2 (R, R) be the unique solution of 1

(−) 2 zn + V0 zn = hn

in R.

(17.4.55)

Then, zn = K ∗ hn , where K(x) = F −1 ((|k| + V0 )−1 ) satisfies the following properties (see [105, 198, 199] and Remark 3.2.18): (1) K is positive and even in R \ {0}; (2) there exists a positive constant C > 0 such that K(x) ≤ (3) K ∈

Lq (RN )

for any q ∈ [1, ∞].

K1 |x|2

for any x ∈ R \ {0};

Arguing as in Remark 7.2.10, we obtain that |zn (x)| → 0 as |x| → ∞ uniformly in n ∈ N. Since vn satisfies (17.4.54) and zn solves (17.4.55), a comparison argument shows that 0 ≤ vn ≤ zn in R and for all n ∈ N. Furthermore, we can infer that vn (x) → 0 as |x| → ∞, uniformly in n ∈ N.   Now, we are ready to present the proof of the main result of this section. Proof of Theorem 17.4.2 Let δ > 0 be such that Mδ ⊂ , and we show that there exists ε of (17.4.13), εˆ δ > 0 such that, for every ε ∈ (0, εˆ δ ) and every solution u ∈ N uL∞ (R\ε )
0 such that, for some subsequence still denoted (y˜n ), we have that (y˜n − r, y˜n + r) ⊂  for all n ∈ N. Hence, (y˜n − εrn , y˜n + εrn ) ⊂ εn for all n ∈ N, which implies that R \ εn

  r r ⊂ R \ y˜n − , y˜n + εn εn

for all n ∈ N.

Invoking Lemma 17.4.24, there exists R > 0 such that vn (x)
R it holds that   r r ⊂ R \ (y˜n − R, y˜n + R). R \ εn ⊂ R \ y˜n − , y˜n + εn εn √ Then |un (x)| < ta for any x ∈ R \ εn and n ≥ ν, and this contradicts (17.4.57). Let ε˜ δ > 0 be given by Theorem 17.4.23 and set εδ = min{˜εδ , εˆ δ }. Theorem 17.4.23 1

provides at least catMδ (M) nontrivial solutions to (17.4.13). If u ∈ Hε2 is one of these ε , and, in view of (17.4.56) and the definition of g, we can infer that solutions, then u ∈ N u is also a solution to (17.4.13). Since uˆ ε (x) = uε (x/ ε) is a solution to (17.4.1), we can deduce that (17.4.1) has at least catMδ (M) nontrivial solutions. Finally, we study the behavior of the maximum points of |uˆ n |. Take εn → 0 and (un ) a sequence of solutions to (17.4.13) as above. We first note that (g1 ) implies that there exists √ γ ∈ (0, ta ) small such that g(ε x, t 2 )t 2 ≤

V0 2 t 2

for all x ∈ R, |t| ≤ γ .

(17.4.58)

Arguing as above, we can take R > 0 such that un L∞ (R\(y˜n−R,y˜n +R)) < γ .

(17.4.59)

642

17 Fractional Schrödinger Equations with Magnetic Fields

Up to a subsequence, we may also assume that un L∞ (y˜n −R,y˜n +R) ≥ γ .

(17.4.60)

Indeed, if (17.4.60) is not true, by (17.4.59) we get un L∞ (R) < γ , and then from the fact that Jε n (un ) = 0, (17.4.58) and Lemma 17.2.3 we obtain  [|un |]2 +

 R

V0 |un |2 dx ≤ un 2εn =

R

g(εn x, |un |2 )|un |2 dx ≤

V0 2

 R

|un |2 dx,

whence |un |V0 = 0, which is a contradiction. Hence (17.4.60) holds. Taking into account (17.4.59) and (17.4.60), we can infer that if pn is a global maximum point of |un |, then pn ∈ (y˜n − R, y˜n + R), that is pn = y˜n + qn for some qn ∈ (−R, R). Recalling that the associated solution of (17.4.1) is of the form uˆ n (x) = un (x/ εn ), we can see that ηn = εn y˜n + εn qn is a global maximum point of |uˆ n |. Since qn ∈ (−R, R), εn y˜n → y0 and V (y0 ) = V0 , the continuity of V implies that lim V (ηn ) = V0 .

n→∞

Next we study the decay properties of |uˆ n |. By Lemma 3.2.17 and Remark 3.2.18, we can find a function w such that 0 < w(x) ≤

C 1 + |x|2

for all x ∈ R,

(17.4.61)

and 1

(−) 2 w +

V0 w≥0 2

in R \ [−R1 , R1 ],

(17.4.62)

for some suitable R1 > 0. Using Lemma 17.4.24, we know that vn (x) → 0 as |x| → ∞ uniformly in n ∈ N, so there exists R2 > 0 such that hn (x) = g(εn x + εn y˜n , vn2 )vn ≤

V0 vn 2

in R \ [−R2 , R2 ].

Let wn be the unique solution to 1

(−) 2 wn + V0 wn = hn

in R.

(17.4.63)

17.4 A Multiplicity Result for a Fractional Magnetic Schrödinger. . .

643

Then wn (x) → 0 as |x| → ∞ uniformly in n ∈ N, and, by comparison, 0 ≤ vn ≤ wn in R. Moreover, by (17.4.63), (−)1/2wn +

V0 V0 wn = hn − wn ≤ 0 2 2

in R \ [−R2 , R2 ].

(17.4.64)

Choose R3 = max{R1 , R2 } and set d=

min w > 0

w˜ n = (b + 1)w − d wn .

and

[−R3 ,R3 ]

(17.4.65)

where b = supn∈N wn L∞ (R) < ∞. Clearly, by (17.4.64) and (17.4.65), we get w˜ n ≥ bd + w − b d > 0 1

(−) 2 w˜ n +

in [−R3 , R3 ],

V0 w˜ n ≥ 0 2

in R \ [−R3 , R3 ].

Applying Lemma 1.3.8 we deduce that w˜ n ≥ 0

in R,

which combined with (17.4.61) and vn ≤ wn shows that 0 ≤ vn (x) ≤ wn (x) ≤

C˜ (b + 1) w(x) ≤ d 1 + |x|2

for all n ∈ N, x ∈ R,

for some constant C˜ > 0. Recalling the definition of vn , we have 

x |uˆ n |(x) = |un | εn ≤



 = vn

x − y˜n εn



C˜ − y˜n |2

1 + | εxn

=

C˜ ε2n ε 2n +|x − εn y˜n |2

≤

C˜ ε2n ε2n +|x − ηn |2

The proof of Theorem 17.4.2 is now complete.

for all x ∈ R.  

Bibliography

1. N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part. Math. Z. 248(2), 423–443 (2004) 2. R.A. Adams, Sobolev Spaces. Pure and Applied Mathematics, vol. 65 (Academic Press, New York-London, 1975), xviii+268 pp 3. F.J. Almgren, E.H. Lieb, Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc. 2(4), 683–773 (1989) 4. C.O. Alves, Local mountain pass for a class of elliptic system. J. Math. Anal. Appl. 335(1), 135–150 (2007) 5. C.O. Alves, Existence of a positive solution for a nonlinear elliptic equation with saddle–like potential and nonlinearity with exponential critical growth in R2 . Milan J. Math. 84(1), 1–22 (2016) 6. C.O. Alves, V. Ambrosio, A multiplicity result for a nonlinear fractional Schrödinger equation in RN without the Ambrosetti-Rabinowitz condition. J. Math. Anal. Appl. 466(1), 498–522 (2018) 7. C. O. Alves, V. Ambrosio, T. Isernia, Existence, multiplicity and concentration for a class of fractional p&q Laplacian problems in RN . Commun. Pure Appl. Anal. 18(4), 2009–2045 (2019) 8. C.O. Alves, F.J.S.A. Corrêa, G.M. Figueiredo, On a class of nonlocal elliptic problems with critical growth. Differ. Equ. Appl. 2(3), 409–417 (2010) 9. C.O. Alves, F.J.S.A. Corrêa, T.F. Ma, Positive solutions for a quasi-linear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49(1), 85–93 (2005) 10. C.O. Alves, J.M. do Ó, O.H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in R2 involving critical growth. Nonlinear Anal. 56(5), 781–791 (2004) 11. C.O. Alves, J.M. do Ó, O.H. Miyagaki, Concentration phenomena for fractional elliptic equations involving exponential critical growth. Adv. Nonlinear Stud. 16(4), 843–861 (2016) 12. C.O. Alves, J.M. do Ó, M.A.S. Souto, Local mountain-pass for a class of elliptic problems in RN involving critical growth. Nonlinear Anal. Ser. A: Theory Methods 46(4), 495–510 (2001) 13. C.O. Alves, G.M. Figueiredo, Multiplicity of positive solutions for a quasilinear problem in RN via penalization method. Adv. Nonlinear Stud. 5(4), 551–572 (2005) 14. C.O. Alves, G.M. Figueiredo, Existence and multiplicity of positive solutions to a p-Laplacian equation in RN . Differ. Integr. Equ. 19(2), 143–162 (2006) 15. C.O. Alves, G.M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN . J. Differ. Equ. 246(3), 1288– 1311 (2009)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8

645

646

Bibliography

16. C.O. Alves, G.M. Figueiredo, M.F. Furtado, Multiplicity of solutions for elliptic systems via local mountain pass method. Commun. Pure Appl. Anal. 8(6), 1745–1758 (2009) 17. C.O. Alves, G.M. Figueiredo, M.F. Furtado, Multiple solutions for critical elliptic systems via penalization method. Differ. Integr. Equ. 23(7–8), 703–723 (2010) 18. C.O. Alves, G.M. Figueiredo, M.F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ. 36(9), 1565–1586 (2011) 19. C.O. Alves, O.H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in RN via penalization method. Calc. Var. Partial Differ. Equ. 55(3), Art. 47, 19 pp (2016) 20. C.O. Alves, S.H.M. Soares, Existence and concentration of positive solutions for a class of gradient systems. NoDEA Nonlinear Differ. Equ. Appl. 12(4), 437–457 (2005) 21. C.O. Alves, M.A.S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity. J. Differ. Equ. 254(4), 1977–1991 (2013) 22. C.O. Alves, M.A.S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains. Z. Angew. Math. Phys. 65(6), 1153–1166 (2014) 23. C.O. Alves, M.A.S. Souto, M. Montenegro, Existence of a ground state solution for a nonlinear scalar field equation with critical growth. Calc. Var. Partial Differ. Equ. 43(3–4), 537–554 (2012) 24. C.O. Alves, M.A.S. Souto, M. Montenegro, Existence of solution for two classes of elliptic problems in RN with zero mass. J. Differ. Equ. 252(10), 5735–5750 (2012) 25. C.A. Alves, M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasilinear Choquard equation via penalization method. Proc. R. Soc. Edinburgh Sect. A 146(1), 23–58 (2016) 26. A. Ambrosetti, V. Felli, A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. (JEMS) 7(1), 117–144 (2005) 27. A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies in Advanced Mathematics, vol. 104 (Cambridge University Press, Cambridge, 2007), xii+316 pp 28. A. Ambrosetti, A. Malchiodi, S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159(3), 253–271 (2001) 29. A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) 30. A. Ambrosetti, Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials. Differ. Integr. Equ. 18(12), 1321–1332 (2005) 31. V. Ambrosio, Periodic solutions for a pseudo-relativistic Schrödinger equation. Nonlinear Anal. 120, 262–284 (2015) 32. V. Ambrosio, Ground states for superlinear fractional Schrödinger equations in RN . Ann. Acad. Sci. Fenn. Math. 41(2), 745–756 (2016) 33. V. Ambrosio, Multiple solutions for a fractional p-Laplacian equation with sign-changing potential. Electron. J. Differ. Equ. 151, 12 (2016) 34. V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator. J. Math. Phys. 57(5), 051502, 18 pp (2016) 35. V. Ambrosio, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method. Ann. Mat. Pura Appl. (4) 196(6), 2043–2062 (2017) 36. V. Ambrosio, Ground states for a fractional scalar field problem with critical growth. Differ. Integr. Equ. 30(1–2), 115–132 (2017) 37. V. Ambrosio, Periodic solutions for a superlinear fractional problem without the AmbrosettiRabinowitz condition. Discret. Contin. Dyn. Syst. 37(5), 2265–2284 (2017)

Bibliography

647

38. V. Ambrosio, Zero mass case for a fractional Berestycki-Lions type problem. Adv. Nonlinear Anal. 7(3), 365–374 (2018) 39. V. Ambrosio, Mountain pass solutions for the fractional Berestycki-Lions problem. Adv. Differ. Equ. 23(5–6), 455–488 (2018) 40. V. Ambrosio, Concentration phenomena for critical fractional Schrödinger systems. Commun. Pure Appl. Anal. 17(5), 2085–2123 (2018) 41. V. Ambrosio, Periodic solutions for critical fractional problems. Calc. Var. Partial Differ. Equ. 57(2), Art. 45, 31 pp (2018) 42. V. Ambrosio, Multiple solutions for superlinear fractional problems via theorems of mixed type. Adv. Nonlinear Stud. 18(4), 799–817 (2018) 43. V. Ambrosio, Boundedness and decay of solutions for some fractional magnetic Schrödinger equations in RN . Milan J. Math. 86(2), 125–136 (2018) 44. V. Ambrosio, Existence and concentration results for some fractional Schrödinger equations in RN with magnetic fields. Commun. Partial Differ. Equ. 44(8), 637–680 (2019) 45. V. Ambrosio, Concentration phenomena for a fractional Choquard equation with magnetic field. Dyn. Partial Differ. Equ. 16(2), 125–149 (2019) 46. V. Ambrosio, Multiplicity and concentration results for a fractional Choquard equation via penalization method. Potential Anal. 50(1), 55–82 (2019) 47. V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in RN . Rev. Mat. Iberoam. 35(5), 1367–1414 (2019) 48. V. Ambrosio, Multiplicity and concentration results for a class of critical fractional Schrödinger-Poisson systems via penalization method. Commun. Contemp. Math. 22(1), 1850078, 45 pp (2020) 49. V. Ambrosio, On a fractional magnetic Schrödinger equation in R with exponential critical growth. Nonlinear Anal. 183, 117–148 (2019) 50. V. Ambrosio, Concentrating solutions for a fractional Kirchhoff equation with critical growth. Asymptot. Anal. 116(3-4), 249–278 (2020) 51. V. Ambrosio, Multiplicity and concentration results for a fractional Schrödinger-Poisson type equation with magnetic field. Proc. R. Soc. Edinburgh Sect. A 150(2), 655–694 (2020) 52. V. Ambrosio, Multiplicity of solutions for fractional Schrödinger systems in RN . Complex Var. Elliptic Equ. 65(5), 856–885 (2020) 53. V. Ambrosio, Multiplicity and concentration of solutions for a fractional Kirchhoff equation with magnetic field and critical growth. Ann. Henri Poincaré 20(8), 2717–2766 (2019) 54. V. Ambrosio, Multiplicity and concentration of solutions for fractional Schrödinger systems via penalization method. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30(3), 543–581 (2019) 55. V. Ambrosio, Infinitely many periodic solutions for a class of fractional Kirchhoff problems. Monatsh. Math. 190(4), 615–639 (2019) 56. V. Ambrosio, On the multiplicity and concentration of positive solutions for a p-fractional Choquard equation in RN . Comput. Math. Appl. 78(8), 2593–2617 (2019) 57. V. Ambrosio, A local mountain pass approach for a class of fractional NLS equations with magnetic fields. Nonlinear Anal. 190, 111622, 14 pp (2020) 58. V. Ambrosio, Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields. Discret. Contin. Dyn. Syst. 40(2), 781–815 (2020) 59. V. Ambrosio, Multiplicity and concentration results for fractional Schrödinger-Poisson equations with magnetic fields and critical growth. Potential Anal. 52(4), 565–600 (2020) 60. V. Ambrosio, Existence and concentration of nontrivial solutions for a fractional magnetic Schrödinger-Poisson type equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XXI, 1043–1081 (2020)

648

Bibliography

61. V. Ambrosio, Fractional p&q Laplacian problems in RN with critical growth. Z. Anal. Anwend. 39(3), 289–314 (2020) 62. V. Ambrosio, Concentration phenomena for a class of fractional Kirchhoff equations in RN with general nonlinearities. Nonlinear Anal. 195, 111761, 39 pp (2020) 63. V. Ambrosio, The nonlinear fractional relativistic Schrödinger equation: existence, multiplicity, decay and concentration results. submitted 64. V. Ambrosio, On the fractional relativistic Schrödinger operator. submitted 65. V. Ambrosio, Concentration phenomena for fractional magnetic NLS. submitted 66. V. Ambrosio, P. d’Avenia, Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity. J. Differ. Equ. 264(5), 3336–3368 (2018) 67. V. Ambrosio, G.M. Figueiredo, Ground state solutions for a fractional Schrödinger equation with critical growth. Asymptot. Anal. 105(3–4), 159–191 (2017) 68. V. Ambrosio, G.M. Figueiredo, T. Isernia, Existence and concentration of positive solutions for p-fractional Schrödinger equations. Ann. Mat. Pura Appl. (4) 199(1), 317–344 (2020) 69. V. Ambrosio, G.M. Figueiredo, T. Isernia, G. Molica Bisci, Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations. Adv. Nonlinear Stud. 19(1), 113–132 (2019) 70. V. Ambrosio, H. Hajaiej, Multiple solutions for a class of nonhomogeneous fractional Schrödinger equations in RN . J. Dynam. Differ. Equ. 30(3), 1119–1143 (2018) 71. V. Ambrosio, T. Isernia, Sign-changing solutions for a class of fractional Schrödinger equations with vanishing potentials. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29(1), 127–152 (2018) 72. V. Ambrosio, T. Isernia, A multiplicity result for a fractional Kirchhoff equation in RN with a general nonlinearity. Commun. Contemp. Math. 20(5), 1750054, 17 pp (2018) 73. V. Ambrosio, T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem. Math. Methods Appl. Sci. 41(2), 615–645 (2018) 74. V. Ambrosio, T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discret. Contin. Dyn. Syst. 38(11), 5835–5881 (2018) 75. V. Ambrosio, T. Isernia, On the multiplicity and concentration for p-fractional Schrödinger equations. Appl. Math. Lett. 95, 13–22 (2019) 76. D. Applebaum, Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, vol. 93. (Cambridge University Press, Cambridge, 2004), xxiv+384 pp 77. G. Arioli, A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Ration. Mech. Anal. 170(4), 277–295 (2003) 78. N. Aronszajn, K.T. Smith, Theory of Bessel potentials. I. Ann. Inst. Fourier (Grenoble) 11, 385–475 (1961) 79. A. Arosio, S. Panizzi, On the well-posedness of the Kirchhoff string. Trans. Am. Math. Soc. 348(1), 305–330 (1996) 80. A.I. Ávila, J. Yang, Multiple solutions of nonlinear elliptic systems. NoDEA Nonlinear Differ. Equ. Appl. 12(4), 459–479 (2005) 81. J. Avron, I. Herbst, B. Simon, Schrödinger operators with magnetic fields. I. General interactions. Duke Math. J. 45(4), 847–883 (1978) 82. A. Azzollini, P. d’Avenia, A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2), 779–791 (2010) 83. A. Azzollini, P. d’Avenia, A. Pomponio, Multiple critical points for a class of nonlinear functionals. Ann. Mat. Pura Appl. 190(3), 507–523 (2011) 84. A. Azzollini, A. Pomponio, On a “zero mass” nonlinear Schrödinger equation. Adv. Nonlinear Stud. 7(4), 599–627 (2007)

Bibliography

649

85. A. Bahri, H. Berestycki, A perturbation method in critical point theory and applications. Trans. Am. Math. Soc. 267(1), 1–32 (1981) 86. S. Barile, G.M. Figueiredo, Existence of least energy positive, negative and nodal solutions for a class of p&q-problems with potentials vanishing at infinity. J. Math. Anal. Appl. 427(2), 1205–1233 (2015) 87. B. Barrios, E. Colorado, A. de Pablo, U. Sánchez, On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252(11), 6133–6162 (2012) 88. T. Bartsch, Z. Liu, T. Weth, Sign changing solutions of superlinear Schrödinger equations. Commun. Partial Differ. Equ. 29(1–2), 25–42 (2004) 89. T. Bartsch, A. Pankov, Z.-Q. Wang, Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 4(4), 549–569 (2001) 90. T. Bartsch, Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN . Commun. Partial Differ. Equ. 20(9–10), 1725–1741 (1995) 91. T. Bartsch, T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology. Ann. Inst. H. Poincaré Anal. Non Linéaire 22(3), 259–281 (2005) 92. T. Bartsch, T. Weth, M. Willem, Partial symmetry of least energy nodal solutions to some variational problems. J. Anal. Math. 96, 1–18 (2005) 93. P. Belchior, H. Bueno, O.H. Miyagaki, G.A. Pereira, Remarks about a fractional Choquard equation: ground state, regularity and polynomial decay. Nonlinear Anal. 164, 38–53 (2017) 94. A.K. Ben-Naoum, C. Troestler, M. Willem, Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. 26(4), 823–833 (1996) 95. V. Benci, G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc. Var. Partial Differ. Equ. 2(1), 29–48 (1994) 96. V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11(2), 283–293 (1998) 97. V. Benci, C.R. Grisanti, A.M. Micheletti, Existence of solutions for the nonlinear Schrödinger equations with V (∞) = 0, in Contributions to Non-linear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol. 66 (Birkhäuser, Basel, 2006), pp. 53–65 98. V. Benci, A.M. Micheletti, Solutions in exterior domains of null mass nonlinear field equations. Adv. Nonlinear Stud. 6(2), 171–198 (2006) 99. H. Berestycki, T. Gallouët, O. Kavian, Equations de Champs scalaires euclidiens non linéaires dans le plan. C. R. Acad. Sci. Paris Sér. I Math. 297(5), 307–310 (1983) 100. H. Berestycki, P.L. Lions, Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 82(4), 313–345 (1983) 101. H. Berestycki, P.L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Ration. Mech. Anal. 82(4), 347–375 (1983) 102. H. Berestycki, P.L. Lions, Existence d’états multiples dans des équations de champs scalaires non linéaires dans le cas de masse nulle. C. R. Acad. Sci. Paris Sér. I Math. 297(4), 267–270 (1983) 103. S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4, 17–26 (1940) 104. G. Bianchi, J. Chabrowski, A. Szulkin, On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear Anal. 25(1), 41–59 (1995) 105. R.M. Blumenthal, R.K. Getoor, Some theorems on stable processes. Trans. Am. Math. Soc. 95, 263–273 (1960) 106. K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, Z. Vondra˘cek, Potential Analysis of Stable Processes and its Extensions, ed. by P. Graczyk, A. Stos. Lecture Notes in Mathematics, vol. 1980 (Springer, Berlin, 2009), x+187 pp

650

Bibliography

107. D. Bonheure, J. Van Schaftingen, Ground states for the nonlinear Schrödinger equation with potential vanishing at infinity. Ann. Mat. Pura Appl. (4) 189(2), 273–301 (2010) 108. J. Bourgain, H. Brezis, P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations (IOS, Amsterdam, 2001) 109. C. Brändle, E. Colorado, A. de Pablo, U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinburgh Sect. A 143(1), 39–71 (2013) 110. L. Brasco, S. Mosconi, M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality. Calc. Var. Partial Differ. Equ. 55(2), Art. 23, 32 pp (2016) 111. H Brezis, J.-M. Coron, L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz. Commun. Pure Appl. Math. 33(5), 667–684 (1980) 112. H. Brezis, T. Kato, Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. (9) 58(2), 137–151 (1979) 113. H. Brezis, E.H. Lieb, A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983) 114. H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437–477 (1983) 115. C. Bucur, E. Valdinoci, Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20 (Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016), xii+155 pp 116. H. Bueno, A.H.S. Medeiros, G.A. Pereira, Pohozaev-type identities for a pseudo-relativistic Schrödinger operator and applications. preprint arXiv:1810.07597 117. H. Bueno, O.H. Miyagaki, G.A. Pereira, Remarks about a generalized pseudo-relativistic Hartree equation. J. Differ. Equ. 266(1), 876–909 (2019) 118. J. Busca, B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 163(1), 41–56 (2000) 119. T. Byczkowski, J. Malecki, M. Ryznar, Bessel potentials, hitting distributions and Green functions. Trans. Am. Math. Soc. 361(9), 4871–4900 (2009) 120. J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity. Trans. Am. Math. Soc. 362(4), 1981–2001 (2010) 121. J. Byeon, L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch. Ration. Mech. Anal. 185(2), 185–200 (2007) 122. J. Byeon, O. Kwon, J. Seok, Nonlinear scalar field equations involving the fractional Laplacian. Nonlinearity 30(4), 1659–1681 (2017) 123. J. Byeon, Z.-Q Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II. Calc. Var. Partial Differ. Equ. 18(2), 207–219 (2003) 124. X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31(1), 23–53 (2014) 125. X. Cabré, J. Solá-Morales, Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58(12), 1678–1732 (2005) 126. X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010) 127. L.A. Caffarelli, L.Silvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007) 128. A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, in 1961 Proceedings of Symposia in Pure Mathematics, vol. IV (American Mathematical Society, Providence, 1961), pp. 33–49 129. D.M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2 . Commun. Partial Differ. Equ. 17(3–4), 407–435 (1992)

Bibliography

651

130. D.-M.M. Cao, H.-S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in RN . Proc. R. Soc. Edinburgh Sect. A 126(2), 443–463 (1996) 131. A. Capella, J. Davila, L. Dupaigne, Y. Sire, Regularity of radial extremals solutions for some non-local semilinear equation. Commun. Partial Differ. Equ. 36(8), 1353–1384 (2011) 132. R. Carmona, W.C. Masters, B. Simon, Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Func. Anal 91(1), 117–142 (1990) 133. A. Castro, J. Cossio, J. Neuberger, A sign-changing solution for a superlinear Dirichlet problem. Rocky Mountain J. Math. 27(4), 1041–1053 (1997) 134. G. Cerami, An existence criterion for the critical points on unbounded manifolds. Istit. Lombardo Accad. Sci. Lett. Rend. A 112(2), 332–336 (1978, 1979) 135. J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc. Var. Partial Differ. Equ. 3(4), 493–512 (1995) 136. J. Chabrowski, J. Yang, Existence theorems for elliptic equations involving supercritical Sobolev exponent. Adv. Differ. Equ. 2(2), 231–256 (1997) 137. K.-C. Chang, Methods in Nonlinear Analysis. Springer Monographs in Mathematics (Springer, Berlin, 2005), x+439 pp 138. W. Chen, C. Li, B. Ou, Classification of solutions for an integral equation. Commun. Pure Appl. Math. 59(3), 330–343 (2006) 139. X.J. Chang, Z.Q. Wang, Ground state of scalar field equations involving fractional Laplacian with general nonlinearity. Nonlinearity 26(2), 479–494 (2013) 140. C. Chen, Y. Kuo, T. Wu, The Nehari manifold for a Kirchhoff type problem involving signchanging weight functions. J. Differ. Equ. 250(4), 1876–1908 (2011) 141. G. Chen, Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations. Commun. Pure Appl. Anal. 13(6), 2359–2376 (2014) 142. M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal. 30(7), 4619–4627 (1997) 143. W. Choi, J. Seok, Nonrelativistic limit of standing waves for pseudo-relativistic nonlinear Schrödinger equations. J. Math. Phys. 57(2), 021510, 15 pp (2016) 144. S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equations with an external magnetic field. J. Differ. Equ. 188(1), 52–79 (2003) 145. S. Cingolani, M. Clapp, S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. 63(2), 233–248 (2012) 146. S. Cingolani, M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations. Topol. Methods Nonlinear Anal. 10(1), 1–13 (1997) 147. S. Cingolani, M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ. 160, 118–138 (2000) 148. S. Cingolani, S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths. J. Math. Phys. 46(5), 053503, 19 pp (2005) 149. C.V. Coffman, A minimum-maximum principle for a class of non-linear integral equations. J. Analyse Math. 22, 391–419 (1969) 150. E. Colorado, A. de Pablo, U. Sánchez, Perturbations of a critical fractional equation. Pac. J. Math. 271(1), 65–85 (2014) 151. D.G. Costa, C.A. Magalhaes, Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23(11), 1401–1412 (1994) 152. V. Coti Zelati, M. Nolasco, Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22(1), 51–72 (2011) 153. V. Coti Zelati, M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type. Rev. Mat. Iberoam. 29(4), 1421–1436 (2013)

652

Bibliography

154. V. Coti Zelati, P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on Rn . Commun. Pure Appl. Math. 45(10), 1217–1269 (1992) 155. A. Cotsiolis, N.K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295(1), 225–236 (2004) 156. P. d’Avenia, G. Siciliano, M. Squassina, On fractional Choquard equations. Math. Models Methods Appl. Sci. 25(8), 1447–1476 (2015) 157. P. d’Avenia, M. Squassina, Ground states for fractional magnetic operators. ESAIM Control Optim. Calc. Var. 24(1), 1–24 (2018) 158. J. Dávila, M. del Pino, S. Dipierro, E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal. PDE 8(5), 1165–1235 (2015) 159. J. Dávila, M. del Pino, J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256(2), 858–892 (2014) 160. D.G. de Figueiredo, O.H. Miyagaki, B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range. Calc. Var. Partial Differ. Equ. 3(2), 139–153 (1995) 161. D.C. de Morais Filho, M.A.S. Souto, Systems of p-Laplacian equations involving homogeneous nonlinearities with critical Sobolev exponent degrees. Commun. Partial Differ. Equ. 24(7–8), 1537–1553 (1999) 162. M. de Souza, Y.L. Araújo, On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth. Math. Nachr. 289(5–6), 610–625 (2016) 163. L.M. Del Pezzo, A. Quaas, A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian. J. Differ. Equ. 263(1), 765–778 (2017) 164. L.M. Del Pezzo, A. Quaas, Spectrum of the fractional p-Laplacian in RN and decay estimate for positive solutions of a Schrödinger equation. Nonlinear Anal. 193, 111479 (2020) 165. M. del Pino, P.L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4(2), 121–137 (1996) 166. F. Demengel, G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations. Universitext (Springer, London; EDP Sciences, Les Ulis, 2012), xviii+465 pp 167. A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional p-minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire, 33(5), 1279–1299 (2016) 168. E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. math. 136(5), 521–573 (2012) 169. Y. Ding, Variational Methods for Strongly Indefinite Problems. Interdisciplinary Mathematical Sciences, vol. 7 (World Scientific Publishing Co. Pte. Ltd., Hackensack, 2007), viii+168 pp 170. S. Dipierro, M. Medina, E. Valdinoci, Fractional elliptic problems with critical growth in the whole of Rn . Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 15 (Edizioni della Normale, Pisa, 2017), viii+152 pp 171. S. Dipierro, L. Montoro, I. Peral, B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential. Calc. Var. Partial Differ. Equ. 55(4), Paper No. 99, 29 pp (2016) 172. S. Dipierro, G. Palatucci, E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche (Catania) 68(1), 201–216 (2013) 173. J.M. do Ó, O.H. Miyagaki, M. Squassina, Nonautonomous fractional problems with exponential growth. NoDEA Nonlinear Differ. Equ. Appl. 22(5), 1395–1410 (2015) 174. J.M. do Ó, O.H. Miyagaki, M. Squassina, Critical and subcritical fractional problems with vanishing potentials. Commun. Contemp. Math. 18(6), 1550063, 20 pp (2016) 175. J.M. do Ó, M.A.S. Souto, On a class of nonlinear Schrödinger equations in R2 involving critical growth. J. Differ. Equ. 174(2), 289–311 (2001) 176. I. Ekeland, On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

Bibliography

653

177. I. Ekeland, Convexity Methods in Hamiltonian Mechanics. (Springer-Verlag, Berlin, 1990), x+247 pp 178. A. Erdelyi, Higher Trascendental Functions (McGraw-Hill, New York, 1953) 179. M. Esteban, P.L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, in Partial Differential Equations and the Calculus of Variations. Progress in Nonlinear Differential Equations, 1, vol. I (Birkhäuser Boston, Boston, 1989), pp. 401–449 180. E.B. Fabes, C.E. Kenig, R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982) 181. M. Fall, V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discret. Contin. Dyn. Syst. 35(12), 5827–5867 (2015) 182. M.M. Fall, F. Mahmoudi, E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28(6), 1937–1961 (2015) 183. P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinburgh Sect. A 142(6), 1237–1262 (2012) 184. P. Felmer, I. Vergara, Scalar field equation with non-local diffusion. NoDEA Nonlinear Differ. Equ. Appl. 22(5), 1411–1428 (2015) 185. P. Felmer, Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian. Commun. Contemp. Math. 16(1), 1350023, 24 pp (2014) 186. G.M. Figueiredo, Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems with critical growth. Commun. Appl. Nonlinear Anal. 13(4), 79–99 (2006) 187. G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J. Math. Anal. Appl. 401(2), 706–713 (2013) 188. G.M. Figueiredo, M. Furtado, Positive solutions for some quasilinear equations with critical and supercritical growth. Nonlinear Anal. 66(7), 1600–1616 (2007) 189. G.M. Figueiredo, M.F. Furtado, Multiple positive solutions for a quasilinear system of Schrödinger equations. NoDEA Nonlinear Differ. Equ. Appl. 15(3), 309–333 (2008) 190. G.M. Figueiredo, M. Furtado, Positive solutions for a quasilinear Schrödinger equation with critical growth. J. Dyn. Differ. Equ. 24(1), 13–28 (2012) 191. G.M. Figueiredo, J.R. Santos Júnior, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method. ESAIM Control Optim. Calc. Var. 20(2), 389–415 (2014) 192. G.M. Figueiredo, J.R. Santos Júnior, Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity. J. Math. Phys. 56, 051506 18 pp (2015) 193. G.M. Figueiredo, G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in RN . NoDEA Nonlinear Differ. Equ. Appl. 23(2), Art. 12, 22 pp (2016) 194. A. Fiscella, A. Pinamonti, E. Vecchi, Multiplicity results for magnetic fractional problems. J. Differ. Equ. 263(8), 4617–4633 (2017) 195. A. Fiscella, P. Pucci, Kirchhoff-Hardy fractional problems with lack of compactness. Adv. Nonlinear Stud. 17(3), 429–456 (2017) 196. A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014) 197. A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986) 198. R.L. Frank, E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in R. Acta Math. 210(2), 261–318 (2013)

654

Bibliography

199. R.L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math. 69(9), 1671–1726 (2016) 200. R.L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430 (2008) 201. G. Franzina, G. Palatucci, Fractional p-eigenvalues. Riv. Math. Univ. Parma (N.S.) 5(2), 373– 386 (2014) 202. J. Fröhlich, B. Jonsson, G. Lars, E. Lenzmann, Boson stars as solitary waves. Commun. Math. Phys. 274(1), 1–30 (2007) 203. N. Garofalo, Fractional thoughts, in New Developments in the Analysis of Nonlocal Operators. Contemporary Mathematics, vol. 723 (American Mathematical Society, Providence, 2019), pp. 1–135 204. A.R. Giammetta, Fractional Schrödinger-Poisson-Slater system in one dimension. preprint arXiv:1405.2796 205. B. Gidas, Bifurcation phenomena in mathematical physics and related topics, in Proceedings of the NATO Advanced Study Institute held at Cargèse, June 24–July 7, 1979, ed. by C. Bardos, D. Bessis, NATO Advanced Study Institute Series. Ser. C Mathematical and Physical Sciences 54, 1980 206. B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979) 207. B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn , in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a (Academic Press, New York/London, 1981), pp. 369–402 208. D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics (Springer, Berlin, 2001), xiv+517 pp 209. L. Grafakos, Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250, 3rd edn. (Springer, New York, 2014), xvi+624 pp 210. Z. Guo, S. Luo, W. Zou, On critical systems involving fractional Laplacian. J. Math. Anal. Appl. 446(1), 681–706 (2017) 211. D.D. Haroske, H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics (European Mathematical Society (EMS), Zürich, 2008) 212. X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations. Z. Angew. Math. Phys. 62(5), 869–889 (2011) 213. Y. He, G. Li, Standing waves for a class of Schrödinger-Poisson equations in R3 involving critical Sobolev exponents. Ann. Acad. Sci. Fenn. Math. 40(2), 729–766 (2015) 214. Y. He, G. Li, S. Peng, Concentrating bound states for Kirchhoff type problems in R3 involving critical Sobolev exponents. Adv. Nonlinear Stud. 14(2), 483–510 (2014) 215. X. He, W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R3 . J. Differ. Equ. 252(2), 1813–1834 (2012) 216. X. He, W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc. Var. Partial Differ. Equ. 55(4), Art. 91, 39 pp (2016) 217. I.W. Herbst, Spectral theory of the operator (p2 + m2 )1/2 − Ze2 /r. Commun. Math. Phys. 53(3), 285–294 (1977) 218. J. Hirata, N. Ikoma, K. Tanaka, Nonlinear scalar field equations in RN : mountain pass and symmetric mountain pass approaches. Topol. Methods Nonlinear Anal. 35(2), 253–276 (2010) 219. F. Hiroshima, T. Ichinose, J. L˝orinczi, Kato’s inequality for Magnetic Relativistic Schrödinger Operators. Publ. Res. Inst. Math. Sci. 53(1), 79–117 (2017) 220. L. Hörmander, The analysis of linear partial differential operators. III. Pseudodifferential operators, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274 (Springer, Berlin, 1985), viii+525 pp

Bibliography

655

221. L. Hörmander, The analysis of linear partial differential operators. IV. Fourier integral operators, in Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275. (Springer, Berlin, 1985), vii+352 pp 222. A. Iannizzotto, S. Mosconi, M. Squassina, Global Hölder regularity for the fractional pLaplacian. Rev. Mat. Iberoam. 32(4), 1353–1392 (2016) 223. A. Iannizzotto, M. Squassina, 1/2-Laplacian problems with exponential nonlinearity. J. Math. Anal. Appl. 414(1), 372–385 (2014) 224. T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, in Mathematical Physics, Spectral Theory and Stochastic Analysis. Operator Theory: Advances and Applications, vol. 232 (Birkhäuser/Springer Basel AG, Basel, 2013), pp. 247–297 225. T. Ichinose, H. Tamura, Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field. Commun. Math. Phys. 105(2), 239–257 (1986) 226. N. Ikoma, Existence of solutions of scalar field equations with fractional operator. J. Fixed Point Theory Appl. 19(1), 649–690 (2017) 227. T. Isernia, Positive solution for nonhomogeneous sublinear fractional equations in RN . Complex Var. Elliptic Equ. 63(5), 689–714 (2018) 228. S. Jarohs, T. Weth, On the strong maximum principle for nonlocal operators. Math. Z. 293(1–2), 81–111 (2019) 229. L. Jeanjean, Two positive solutions for a class of nonhomogeneous elliptic equations. Differ. Integr. Equ. 10(4), 609–624 (1997) 230. L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28(10), 1633–1659 (1997) 231. J. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on RN . Proc. R. Soc. Edinburgh Sect. A 129(4), 787–809 (1999) 232. L. Jeanjean, S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations. Adv. Differ. Equ. 11(7), 813–840 (2006) 233. L. Jeanjean, K. Tanaka, A positive solution for an asymptotically linear elliptic problem on RN autonomous at infinity. ESAIM Control Optim. Calc. Var. 7, 597–614 (2002) 234. L. Jeanjean, K. Tanaka, A remark on least energy solutions in RN . Proc. Am. Math. Soc. 131(8), 2399–2408 (2003) 235. L. Jeanjean, K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. 21(3), 287–318 (2004) 236. L. Jeanjean, K. Tanaka, A positive solution for a nonlinear Schrödinger equation on RN . Indiana Univ. Math. J. 54(2), 443–464 (2005) 237. T. Jin, Y. Li, J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. (JEMS) 16(6), 1111–1171 (2014) 238. T. Kato, Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972, 1973) 239. H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation. Adv. Nonlinear Stud. 7(3), 403–437 (2007) 240. G. Kirchhoff, Mechanik (Teubner, Leipzig, 1883) 241. S.G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math. 1(3), 193–260 (1983) 242. M.A. Krasnoselski, Topological Methods in the Theory of Nonlinear Integral Equations, Translated by A.H. Armstrong; translation ed. by J. Burlak (A Pergamon Press Book The Macmillan Co., New York, 1964), xi + 395 pp

656

Bibliography

243. K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. Ser. A: Theory Methods 41(5–6), 763–778 (2000) 244. N.S. Landkof, Foundations of Modern Potential Theory, Translated from the Russian by A.P. Doohovskoy. Die Grundlehren der mathematischen Wissenschaften, Band 180 (Springer, New York/Heidelberg, 1972), x+424 pp 245. N. Laskin, Fractional quantum mechanics and Lèvy path integrals. Phys. Lett. A, 268(4–6), 298–305 (2000) 246. N. Laskin, Fractional Schrödinger equation. Phys. Rev. E (3) 66(5), 056108, 7 pp (2002) 247. N. Laskin, Fractional Quantum Mechanics (World Scientific Publishing Co. Pte. Ltd., Hackensack, 2018), xv+341 pp 248. E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Stud. Appl. Math. 57(2), 93–105 (1976/1977) 249. E.H. Lieb, M. Loss, Analysis. Graduate Studies in Mathematics, vol. 14 (American Mathematical Society, Providence, 1997), xviii+278 pp 250. E.H. Lieb, H.T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Commun. Math. Phys. 112(1), 147–174 (1987) 251. E.H. Lieb, H.T. Yau, The stability and instability of relativistic matter. Commun. Math. Phys. 118(2), 177–213 (1988) 252. E. Lindgren, P. Lindqvist, Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49(1–2), 795– 826 (2014) 253. J.L. Lions, On some questions in boundary value problems of mathematical physics, in Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), (NorthHolland Math. Stud., 30, North-Holland, Amsterdam-New York, 1978), pp. 284–346 254. P.L. Lions, The Choquard equation and related questions. Nonlinear Anal. 4(6), 1063–1072 (1980) 255. P.L. Lions, Symetrié et compacité dans les espaces de Sobolev. J. Funct. Anal. 49(3), 315–334 (1982) 256. P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(4), 223–283 (1984) 257. P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. Part I . Rev. Mat. Iberoamericana 1(1), 145–201 (1985) 258. J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, vol. I. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181 (Springer, New York/Heidelberg, 1972), xvi+357 pp 259. S.B. Liu, On ground states of superlinear p-Laplacian equations in RN . J. Math. Anal. Appl. 361(1), 48–58 (2010) 260. Z.S. Liu, S.J. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl. 412(1), 435–448 (2014) 261. B. Liu, L. Ma, Radial symmetry results for fractional Laplacian systems. Nonlinear Anal. 146, 120–135 (2016) 262. Z. Liu, M. Squassina, J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. NoDEA Nonlinear Differ. Equ. Appl. 24(4), Art. 50, 32 pp (2017) 263. C. Liu, Z. Wang, H.S. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity. J. Differ. Equ. 245(1), 201–222 (2008)

Bibliography

657

264. Z. Liu, J. Zhang, Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth. ESAIM Control Optim. Calc. Var. 23(4), 1515–1542 (2017) 265. L. Ljusternik, L. Schnirelmann, Méthodes topologique dans les problèmes variationnels (Hermann and Cie, Paris, 1934) 266. L. Ma, L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195(2), 455–467 (2010) 267. J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, (Springer-Verlag, New York, 1989), pp. xiv+277 268. V.G. Maz’ja, Sobolev Spaces, Translated from the Russian by T.O. Shaposhnikova. Springer Series in Soviet Mathematics (Springer, Berlin, 1985), xix+486 pp 269. C. Mercuri, M. Willem, A global compactness result for the p-Laplacian involving critical nonlinearities. Discret. Contin. Dyn. Syst. 28(2), 469–493 (2010) 270. C. Miranda, Un’osservazione sul teorema di Brouwer. Boll. Unione Mat. Ital. 3(2), 5–7 (1940) 271. O.H. Miyagaki, M.A.S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition. J. Differ. Equ. 245(12), 3628–3638 (2008) 272. X. Mingqi, P. Pucci, M. Squassina, B. Zhang, Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discret. Contin. Dyn. Syst. A 37(3), 503–521 (2017) 273. G. Molica Bisci, V.D. R˘adulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems, vol. 162 (Cambridge University Press, Cambridge, 2016) 274. V. Moroz, J. Van Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265(2), 153–84 (2013) 275. V. Moroz, J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations. Trans. Am. Math. Soc. 367(9), 6557–6579 (2015) 276. V. Moroz, J. Van Schaftingen, A guide to the Choquard equation. J. Fixed Point Theory Appl. 19(1), 773–813 (2017) 277. S. Mosconi, K. Perera, M. Squassina, Y. Yang, The Brezis-Nirenberg problem for the fractional p-Laplacian. Calc. Var. Partial Differ. Equ. 55(4), Art. 105, 25 pp (2016) 278. J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Commun. Pure Appl. Math. 13, 457–468 (1960) 279. J. Moser, A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970/1971) 280. D. Motreanu, V.V. Motreanu, N. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems (Springer, New York, 2014), xii+459 pp 281. E. Murcia, G. Siciliano, Positive semiclassical states for a fractional Schrödinger-Poisson system. Differ. Integr. Equ. 30(3–4), 231–258 (2017) 282. R. Musina, A.I. Nazarov, Strong maximum principles for fractional Laplacians. Proc. R. Soc. Edinburgh Sect. A 149(5), 1223–1240 (2019) 283. N. Nyamoradi, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type. Math. Commun. 18(2), 489–502 (2013) 284. Y.G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V )a . Commun. Partial Differ. Equ. 13(12), 1499–1519 (1988) 285. T. Ozawa, On critical cases of Sobolev’s inequalities. J. Funct. Anal. 127(2), 259–269 (1995) 286. R.S. Palais, Lusternik-Schnirelman theory on Banach manifolds. Topology 5, 115–132 (1966) 287. R.S. Palais, S. Smale, A generalized Morse theory. Bull. Am. Math. Soc. 70, 165–172 (1964) 288. G. Palatucci, A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50(3– 4), 799–829 (2014) 289. S. Pekar, Untersuchung uber die Elektronentheorie der Kristalle (Akademie, Berlin, 1954).

658

Bibliography

290. R. Penrose, Quantum computation, entanglement and state reduction. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356(1743), 1927–1939 (1998) 291. K. Perera, Z.T. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221(1), 246–255 (2006) 292. A. Pinamonti, M. Squassina, E. Vecchi, Magnetic BV functions and the Bourgain-BrezisMironescu formula. Adv. Calc. Var. 12(3), 225–252 (2019) 293. S.I. Pohožaev, On the eigenfunctions of the equation u+λf (u) = 0. Dokl. Akad. Nauk SSSR 165, 36–39 (1965) 294. S.I. Pohožaev, A certain class of quasilinear hyperbolic equations. Mat. Sb. 96(138), 152–166, 168 (1975) 295. P. Pucci, S. Saldi, Critical stationary Kirchhoff equations in RN involving nonlocal operators. Rev. Mat. Iberoam. 32(1), 1–22 (2016) 296. P. Pucci, M. Xiang, B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in RN . Calc. Var. Partial Differ. Equ. 54(3), 2785–2806 (2015) 297. P.H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems. Indiana Univ. Math. J. 23, 729–754 (1973/1974) 298. P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, in CBMS Regional Conference Series in Mathematics, vol. 65, 1986 299. P.H. Rabinowitz, On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43(2), 270–291 (1992) 300. M. Ramos, Z.-Q. Wang, M. Willem, Positive solutions for elliptic equations with critical growth in unbounded domains, in Calculus of Variations and Differential Equations (Haifa, 1998) Chapman & Hall/CRC Res. Notes Math., vol. 410 (Chapman & Hall/CRC, Boca Raton, 2000), pp. 192–199 301. M. Reed, B. Simon, Methods of Modern Mathematical Physics, I, Functional Analysis (Academic Press, New York/London, 1972), xvii+325 pp 302. X. Ros-Oton, J. Serra, The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213(2), 587–628 (2014) 303. D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237(2), 655–674 (2006) 304. M. Ryznar, Estimate of Green function for relativistic α-stable processes. Potential Anal. 17(1), 1–23 (2002) 305. M. Schechter, W. Zou, Superlinear problems. Pac. J. Math. 214(1), 145–160 (2004) 306. S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in RN . J. Math. Phys. 54(3), 031501, 17 pp (2013) 307. S. Secchi, On fractional Schrödinger equations in RN without the Ambrosetti-Rabinowitz condition. Topol. Methods Nonlinear Anal. 47(1), 19–41 (2016) 308. S. Secchi, On some nonlinear fractional equations involving the Bessel potential. J. Dyn. Differ. Equ. 29(3), 1173–1193 (2017) 309. R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, in Recent Trends in Nonlinear Partial Differential Equations. II. Stationary Problems, Contemporary Mathematics, vol. 595 (American Mathematical Society, Providence, 2013), pp. 317–340 310. R. Servadei, E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367(1), 67–102 (2015) 311. X. Shang, J. Zhang, Y. Yang, On fractional Schrödinger equations with critical growth. J. Math. Phys. 54(12), 121502, 20 pp (2013)

Bibliography

659

312. Z. Shen, F. Gao, M. Yang, Ground states for nonlinear fractional Choquard equations with general nonlinearities. Math. Methods Appl. Sci. 39(14), 4082–4098 (2016) 313. L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007) 314. M. Squassina, B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operators. C. R. Math. Acad. Sci. Paris 354(8), 825–831 (2016) 315. E. Stein, Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30 (Princeton University Press, Princeton, 1970), xiv+290 pp 316. P. R. Stinga, User’s guide to the fractional Laplacian and the method of semigroups. Handbook of fractional calculus with applications, vol. 2, De Gruyter, Berlin, 235–265 (2019) 317. P.R. Stinga, J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35(11), 2092–2122 (2010) 318. W.A. Strauss, Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55(2), 149–162 (1977) 319. M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34, 4th edn. (Springer, Berlin, 2008), xx+302 pp 320. C.A. Stuart, H.S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on RN . Commun. Partial Differ. Equ. 24(9–10), 1731–1758 (1999) 321. A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems. J. Funct. Anal. 257(12), 3802–3822 (2009) 322. A. Szulkin, T. Weth, The Method of Nehari Manifold. Handbook of Nonconvex Analysis and Applications (Int. Press, Somerville, MA, 2010), pp. 597–632 323. M.H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space. I. Principal properties. J. Math. Mech. 13, 407–479 (1964) 324. J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 42(1–2), 21–41 (2011) 325. K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent. J. Differ. Equ. 261(6), 3061–3106 (2016) 326. H. Triebel, Theory of Function Spaces. Monographs in Mathematics, vol. 78 (Birkhäuser Verlag, Basel, 1983), 284 pp 327. N.S. Trudinger, On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967) 328. B.O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics, vol. 1736 (Springer, Berlin, 2000), xiv+173 pp 329. J.L. Vázquez, The Dirichlet problem for the fractional p-Laplacian evolution equation. J. Differ. Equ. 260(7), 6038–6056 (2016) 330. X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 53(2), 229–244 (1993) 331. J. Wang, L. Tian, J. Xu, F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253(7), 2314–2351 (2012) 332. K. Wang, J. Wei, On the uniqueness of solutions of a nonlocal elliptic system. Math. Ann. 365(1–2), 105–153 (2016) 333. Z. Wang, H.S. Zhou, Positive solutions for a nonhomogeneous elliptic equation on RN without (AR) condition. J. Math. Anal. Appl. 353(1), 470–479 (2009) 334. M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42(2), 499–547 (2015) 335. M. Warma, The fractional Neumann and Robin type boundary conditions for the regional fractional p-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 23(1), Art. 1, 46 pp (2016)

660

Bibliography

336. G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944), vi+804 pp 337. R.A. Weder, Spectral properties of one-body relativistic spin-zero Hamiltonians. Ann. Inst. H. Poincaré Sect. A (N.S.) 20, 211–220 (1974) 338. R.A. Weder, Spectral analysis of pseudodifferential operators. J. Funct. Anal. 20(4), 319–337 (1975) 339. J. Wei, M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation. J. Math. Phys. 50(1), 012905, 22 pp (2009) 340. M. Willem, Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24 (Birkhäuser Boston, Inc., Boston, 1996), x+162 pp 341. J. Zhang, Z. Chen, W. Zou, Standing waves for nonlinear Schrödinger equations involving critical growth. J. Lond. Math. Soc. (2) 90(3), 827–844 (2014) 342. J. Zhang, M. do Ó, M. Squassina, Fractional Schrödinger-Poisson Systems with a General Subcritical or Critical Nonlinearity. Adv. Nonlinear Stud. 16(1), 15–30 (2016) 343. B. Zhang, M. Squassina, X. Zhang, Fractional NLS equations with magnetic field, critical frequency and critical growth. Manuscripta Math. 155(1–2), 115–140 (2018) 344. X. Zhang, B. Zhang, D. Repovš, Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials. Nonlinear Anal. 142, 48–68 (2016) 345. J. Zhang, W. Zou, A Berestycki-Lions theorem revisited. Commun. Contemp. Math. 14(5), 1250033, 14 pp (2012) 346. J. Zhang, W. Zou, The critical case for a Berestycki-Lions theorem. Sci. China Math. 57(3), 541–554 (2014) 347. J. Zhang, W. Zou, Solutions concentrating around the saddle points of the potential for critical Schrödinger equations. Calc. Var. Partial Differ. Equ. 54(4), 4119–4142 (2015) 348. L. Zhao, F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent. Nonlinear Anal. 70(6), 2150–2164 (2009) 349. X.P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation. J. Differ. Equ. 92(2), 163–178 (1991) 350. X.P. Zhu, H.S. Zhu, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains. Proc. R. Soc. Edinburgh Sect. A 115(3–4), 301–318 (1990) 351. W. Zou, M. Schechter, Critical Point Theory and its Applications (Springer, New York, 2006), xii+318 pp

Index

Symbols ε-dependent concentration-compactness lemma, 320

A Ambrosetti–Rabinowitz condition, 58, 146, 195, 251, 256, 295, 336, 418 B Barycenter map, 229, 250, 356, 410, 475, 579 Bessel kernel, 9 Bessel potential, 9 Bessel potential spaces, 10 C Caffarelli-Silvestre extension, 11, 234, 312, 360, 412, 478, 498 Cerami condition, 42 Cerami sequence, 150, 509 Compactness lemma, 177 Concentration-compactness lemma, 28, 281, 463 Critical point, 36

F Fractional Choquard equation, 335 Fractional critical Sobolev exponent, 3 Fractional diamagnetic inequality, 558 Fractional Kato’s inequality, 593 Fractional Kirchhoff equation, 363, 379, 483 Fractional Kirchhoff–Schrödinger–Poisson system, 483 Fractional Laplacian operator, 6 Fractional magnetic Laplacian, 553 Fractional p-Laplacian operator, 254 Fractional Schrödinger-Poisson system, 443 Fractional Schrödinger system, 291 Fractional Sobolev spaces, 1 Fréchet derivative, 36

G Gateaux derivative, 35 Ground state, 53, 98, 108, 117, 147, 148, 162, 187, 216, 224, 264, 352, 383, 429, 465, 555, 566

H Hölder regularity for (−)s , 14 Harnack inequality, 19

D Deformation lemma, 39 Del Pino-Felmer condition, 255, 296, 335, 379, 583

K Krasnoselski genus, 43

E Ekeland’s variational principle, 38

L Lusternik-Schnirelman category, 44

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN , Frontiers in Mathematics, https://doi.org/10.1007/978-3-030-60220-8

661

662 M Maximum principles, 15 Method of moving planes, 80, 133 Moser iteration, 76, 232, 270, 288 Mountain pass theorem, 41

N Nehari manifold, 46, 118, 207, 214, 264, 343, 383, 398, 430, 448, 466, 532, 564, 565

P Palais–Smale condition, 39 Penalization method, 258, 276, 298, 339, 381, 419, 447, 585 Pohozaev identity for (− + m2 )s , 136 Pohozaev identity for (−)s , 52, 98, 102, 176, 308, 493, 517 Pseudogradient, 39 Pseudo-relativistic operator, 107

Index R Rabinowitz condition, 195, 555 Riesz kernel, 8 Riesz potential, 8

S Sign-changing solution, 521 Splitting lemmas, 162 Struwe–Jeanjean monotonicity trick, 161, 484, 500

V Vanishing potentials, 522

W Weighted Lebesgue space, 17