Quasilinear Elliptic Equations with Degenerations and Singularities 9783110804775, 9783110154900

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de Gruyter Series in Nonlinear Analysis and Applications 5

Editors A. Bensoussan (Paris) R. Conti (Florence) A. Friedman (Minneapolis) K.-H. Hoffmann (Munich) L. Nirenberg (New York) Managing Editors J. Appell (Würzburg) V. Lakshmikantham (Melbourne, USA)

Pavel Dräbek Alois Kufner Francesco Nicolosi

Quasilinear Elliptic Equations with Degenerations and Singularities

W DE G_ Walter de Gruyter · Berlin · New York 1997

Authors Pavel Drabek Dept. of Mathematics University of West Bohemia Univerzitni 22 306 14 Pilsen Czech Republic

Alois Kufner Mathematical Institute Czech Academy of Sciences Zitnä 25 115 67 Prague 1 Czech Republic

Francesco Nicolosi Dept. of Mathematics University of Catania Viale A. Doria 6 95125 Catania Italy

1991 Mathematics Subject Classification: 35-02; 35J70; 35J65; 35J60; 35D05; 35B45, 35A25 Keywords: Quasilinear elliptic equations, boundary value problems, degeneration, singularity, weak solution © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication

Data

Drabek, P. (Pavel), 1953Quasilinear elliptic equations with degenerations and singularities / by Pavel Drabek, Alois Kufner, Francesco Nicolosi. p. cm. - (De Gruyter series in nonlinear analysis and applications, ISSN 0941-813X ; 5) Includes bibliographical references and index. ISBN 3-11-015490-0 (alk. paper) 1. Differential equations, Elliptic—Numerical solutions. 2. Boundary value problems-Numerical solutions. 3. Bifurcation theory. I. Kufner, Alois. II. Nicolosi, Francesco, 1938- . III. Title. IV. Series. QA377.D76 1997 515'.353-dc21 97-17293 CIP

Die Deutsche Bibliothek — Cataloging-in-Publication

Data

Drabek, Pavel: Quasilinear elliptic equations with degenerations and singularities / b.y Pavel Drabek ; Alois Kufner ; Francesco Nicolosi. — Berlin ; New York : de Gruyter, 1997 (De Gruyter series in nonlinear analysis and applications ; 5) ISBN 3-11-015490-0

ISSN 0941-813 X © Copyright 1997 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typeset using the authors' T g X files: I. Zimmermann, Freiburg. Printing: Gerike GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin. Cover design: Thomas Bonnie, Hamburg.

To our wives Dana, Zlata and Tina for patience, understanding and permanent support

Preface

Boundary value problems for elliptic equations, more precisely, the concept of weak (generalized) solutions, have their background in applications (namely, in the variational approach connected with the critical level of a certain energy functional as well as in numerical methods like FEM etc.). This type of approach is closely related to the concept of Sobolev spaces and is well elaborated for both linear and nonlinear equations. In various applications, we can meet boundary value problems for elliptic equations whose ellipticity is "disturbed" in the sense that some degeneration or singularity appears. This "bad" behaviour can be caused by the coefficients of the corresponding differential operator as well as by the solution itself. The so-called p-Laplacian is a prototype of such an operator and its character can be interpreted as a degeneration or as a singularity of the classical (linear) Laplace operator (with ρ = 2). There are several very concrete problems from practice which lead to such differential equations, e.g. from glaceology, non-Newtonian fluid mechanics, flows through porous media, differential geometry, celestial mechanics, climatology, petroleum extraction, reaction-diffusion problems, etc. In this book, we concentrate mainly on nonlinear (or, more precisely, quasilinear) problems since linear problems have been investigated more frequently, and we will study both types of appearance of "disturbance" — coming from the coefficients and coming from the solution. It turns out that the apparatus of weighted Sobolev spaces is the natural tool for investigation of such problems and that not only differential operators with degeneration, but also operators with singularities can be treated in the same way. As a matter of fact, the combination of properties of weighted Sobolev spaces with abstract methods of nonlinear functional analysis makes it possible to obtain existence results for such boundary value problems and to study basic spectral properties and bifiircations of a certain class of typical nonlinear differential operators. The book consists of two somewhat different parts — Chapter 2 forming the first part, Chapters 3 and 4 the second. But first of all, in Introduction, for the reader's orientation it is shown how the investigation of degenerated and singular boundary value problems via weighted Sobolev spaces appears as a natural extension of the nowadays classical approach to (strongly) elliptic equations via the theory of monotone operators in classical (nonweighted) Sobolev spaces. Chapter 1 is of auxiliary character and we recommend the reader to proceed from the introduction directly to part one or two and to consult Chapter 1 mainly in case

viii

Preface

he needs some information about the tools used in the book, i.e. the properties of the corresponding (weighted) function spaces, namely imbedding theorems and special superposition (Nemytskij) operators on these spaces, and the substantial results about the existence of solutions of abstract operator equations (as the degree theory for nonlinear operators). Chapter 2 is concerned with the first aim of the authors — to deal with disturbed elliptic problems when the disturbance is caused by the coefficients. We provide a survey on the existence of weak solutions, picking out the main ideas and the main methods, comparing various approaches and, last but not least, presenting a number of examples showing the advantages and disadvantages of the methods in question. We distinguish between second order equations, where truncation techniques can be used and where the Leray-Lions theorem is the main functional analytic tool, and higher order boundary value problems where a more general theory of the topological degree of monotone mappings is used. Also, we try to illustrate the mutual interaction of the parameters appearing in the boundary value problems, in particular the growth properties describing the nonlinearity, and the weight functions describing the degeneration or singularity. Here we have collected results obtained by the authors, partially in collaboration with other colleagues, and published in several papers. Chapters 3 and 4 deal with the second aim of the book, namely to collect some results connected with the up to now less investigated field of spectral analysis of certain nonlinear boundary value problems where the disturbance is given mainly by the solution itself. Here we concentrate on the (perturbed as well as nonperturbed) /7-Laplacian because it can serve as a very useful and transparent model case for some more general quasilinear differential operators. In Chapter 3 we deal with problems containing the /7-Laplacian and its more general perturbed version. The case of a bounded domain is considered here and questions connected with the existence and properties of the first eigenvalue, the maximum principle, the existence and the bifurcation of positive solutions are studied. Chapter 4 is devoted to similar problems but on the whole space R N . Many principal troubles arise in this case and so it should be pointed out that these results are not only trivial extensions of the previous ones. Let us emphasize that, in particular, our problems have a non-variational structure and that very fine apriori estimates enable us to deal with very general equations by use of functional analytic approach. Unlike the first part, where a survey is given, the results of this second part — which are mainly due to the research done by the first author — can be used as a starting point for further investigation of such spectral problems. It should be mentioned that the topic dealt with in these two chapters was studied rarely in the past but many papers of numerous authors have appeared recently. Thus, the book addresses readers who wish to become acquainted with modern methods of solving some special types of boundary value problems (graduate students, applied mathematicians and people from related fields of science and engineering etc.) as well as specialists in differential equations who are looking for new problems and approaches in this important topic.

Preface

ix

The authors would like to express their gratitude to Prof. Bronius Kvedaras from Institute of Mathematics and Informatics, Vilnius, Dr. Milan Kucera from Czech Academy of Sciences, Prague and Dr. Jin Bouchala from Technical University, Ostrava whose valuable comments helped to improve the book, to colleagues who helped to produce the manuscript, namely Ms. Iva Sulkovä and Ing. MiloS Brejcha from the University of West Bohemia in Plzen, and also to Dr. Jiff Jarnik from Charles University, Praha who improved their English. It also should be mentioned that the work on the book was made possible by the support of the Grant Agency of the Czech Republic, Grant No. 201/94/0008, as well as by the support of the Italian G. N. A. F. Α., CNR, Grant No. 341/91. This help is gratefully acknowledged.

Plzen, Praha, Catania Spring 1997

The authors

Contents

List of symbols, theorems, definitions, assumptions, examples List of symbols List of theorems List of definitions List of assumptions List of examples

1 1 3 3 4 4

0

Introduction

5

1

Preliminaries 1.1 The domain Ω 1.2 Function spaces 1.3 Caratheodory functions, Nemytskij (superposition) operators . . . . 1.4 Function spaces (continued) 1.5 Weighted Sobolev spaces 1.6 Leray-Lions theorem 1.7 Degree of mappings of monotone type 1.8 Harnack-type inequality, decay of solution, local regularity and interpolation inequality 1.9 Some technical lemmas

35 37

2

Solvability of nonlinear boundary value problems 2.1 Formulation of the problem 2.2 Second order equations (bounded domains) 2.3 Second order equations (proof of Theorem 2.1) 2.4 Second order equations (unbounded domains) 2.5 Higher order equations (growth conditions) 2.6 Higher order equations (operator representation) 2.7 Higher order equations (degree of the mapping Τ) 2.8 Higher order equations (existence results) 2.9 Examples, remarks, comments

39 39 41 56 65 69 73 81 89 94

3

The degenerated p-Laplacian on a bounded domain 3.1 Basic notation

110 110

17 17 17 19 20 22 31 32

xii

Contents 3.2 3.3 3.4 3.5 3.6

4

Existence of the least eigenvalue of the homogeneous eigenvalue problem Existence of the least eigenvalue of the nonhomogeneous eigenvalue problem Maximum principle for degenerated (singular) equations Positive solutions of degenerated (singular) Β VP Bifurcation from the least eigenvalue

112 122 132 135 143

The p-Laplacian in R N 4.1 Nonlinear eigenvalue problem 4.2 Bifurcation problem for the p-Laplacian in R N

160 160 168

4.3

188

Bifurcation problem for the perturbed p-Laplacian in M^

Bibliography

209

Index

215

List of symbols, theorems, definitions, assumptions, examples List of

symbols

Let us point out, that we will deal here only with real functions of one ore more real variables; c or c, will always denote positive constants. R Ν M.n

the set of real numbers the set of natural numbers the //-dimensional Euclidean space of points * = (X1,X2, ···,XN) Ω a domain in M.N (i.e., an open and connected set) 3Ω the boundary of Ω Ω (μ > 0) = {χ € Ω; u(x) > 0} a = (cti,a2,... an (Ν-dimensional) multiindex (i.e., with components a, which are nonnegative integers) |α| = α j + α 2 Η h α/ν the length of the multiindex a aw Da = (partial) derivative of order \a\ a a2 dx2 ... axNr ρ usually 1 < ρ < oo Ρ . 1 1 = i.e. - + - = 1 P- 1 Ρ Ρ ρ* = — with 1 < ρ < Ν Ν —ρ ρ* — +οο with ρ > Ν m = (Ν + k)\ the number of all Ν-dimensional multiindices of Nikl length k V·7 u = {Dau ; I α I = j} the gradient of order j e Ν of a function u Ap, A ρ the p-Laplacian (p-Laplace operator, ρ > 1), pp. 10,11 X* the adjoint (dual) space to the space X, p. 31 (χ) or w a (λ)

weights or weight functions, i.e., functions measurable and positive a.e.

Wk'2(Q)

Sobolev (Hilbert) space with norm || · \\ki2, p. 5 w)

weighted Sobolev (Hilbert) space with norm || · \\k,2,w and weight w = {ιυα ; |α| < fc}, pp. 6,7

Wk'p(£i)

Sobolev (Banach) space with norm || · \\ ktP , pp. 8, 20

c\ > 0

for a.e. * € Ω

i = 0, 1 , . . . , Ν.

and

(0.39)

Then, similarly as in the linear case, all conditions (0.24)-(0.26) remain satisfied and we can look for a weak solution in the same space as for the operator (0.36), namely in the Sobolev space However, again as in the linear case, the situation changes dramatically if some of the coefficients a, (x) violate condition (0.38) and/or condition (0.39). Since the form (0.27) can be written as Ν a(u,

υ) = Σ Ζ ί

I

du

at (x)

p~2

dxi

J a

du 3 dv -~^-dx+ dx,i dxi

C / ao(x)\u\p~2uvdx Ja

(notice that in our particular example, the "coefficients" aa(x', ciiix- ξ ο , ξ ι , . . . , ξ Ν ) = a i ( x M \ P ~ H i ,

ξ)

i = 0,1

(0.40)

have the form Ν),

(0.41)

we can immediately see that N

a(u,u)

= Y ]

Γ I ai(x) Ja

du — dxi

p

dx +

I

f

Ja

a0(x)\u\p

is nothing else than the p-th power of the norm in the weighted

dx

Sobolev

space

12

0 Introduction

with the family of weights u> = {fli(x); i = 0 ,

(0.42)

and the norm \u\pao(x)dx

lulli.,.« =

+ J 2 f

du dxi

n

di (X) dx

)

(0.43)

Since the coefficients α, (JE) of the operator A are weight functions, we have a(u,u) = \\u\\plpw

(0.44)

and we can derive the estimate \a(u,v)\

> X, and if this operator is, moreover, compact, we will speak about a compact imbedding which will be denoted by Y

X.

The most famous result concerning Sobolev spaces is the following imbedding theorem: Theorem 1.2 (i) The Sobolev space nach space X, i.e., the estimate

is continuously imbedded into the Ba-

M i x < c||«||jklP holds for every function u € Wk'p(tt)

with a constant c > 0 independent of u, where

(a) X = Ι*(Ω) with q provided kp < Ν \

(1.7)

ρ

Ν

22

1 Preliminaries

(b) X = Ζ/(Ω) with arbitrary r > 1 ifkp = N; (c) X = Cm'k(Q) provided kp > Ν, where the nonnegative integer m and the number λ are chosen in such a way that either (Jc — m — l)p < Ν < (k - m)p

and

0 < λ < ^ ~ m)P ~ Ρ

N

^

or {k — m — \)p = Ν

and

λ € (0,1)

is arbitrary.

(1.10)

(ii) Moreover, the imbedding (b) is compact, and the imbeddings (a) and (c) are compact if the inequalities in (1.8) and (1.9) are sharp. Example 1.1 (Sobolev inequality) Let us take k = 1 and Ω = Then the spaces hp N 1,;> w W (R ) and W0 (R ) coincide (see, e.g., [1] or [38]) and the expression

pjx

(L ™ f is also a norm in Wl,p(RN).

If we denote by p* the value Ρ* —

(P 1) can be investigated using repeatedly the results for order one. The easiest way how to derive imbeddings for weighted Sobolev spaces is to reduce them to unweighted spaces. This can be done, e.g., by the following procedure: Example 1.3 (a) Let us consider, for simplicity, the weighted Sobolev space Wl'p(Q., w) with a special choice of the family w: w0(x) = 1,

wi(x) = w2(x) = •'• = wN(x) = v(x).

In this case, we will use for the space W 1,p (Q,iu) the special notation W x,p (v, Ω). This space is normed by \\u\\i,p,v = [Jju(x)\pdx

+JjVu(x)\pv(x)dxy

.

(1.19)

1.5 Weighted Sobolev spaces

25

Let us suppose that the weight function υ satisfies — besides (1.13), (1.14) — also the condition €L!(n) (1.20) with a certain s > 0 which will be specified later. (Notice that condition (1.20) is stronger than (1.13) since here we claim the integrability over the whole domain Ω while in (1.13) we only need the local integrability.) Introducing the parameter ps by ps Ps = ——r < Ρ s + 1

and using the Holder inequality with the parameter r =

s + 1

ρ

= — > 1, we obtain

s ί |υ(χ)|p'dx= f Jn Jo.

Ps

\v(x)\Psvf(x)v~f(x)dx ps Ä , ' "

< ^\v(x)\pv(x)dx^

v

1 7+T

(1.21)

"

du

Taking here ν = — , i = 1 , 2 , . . . , N, and considering, moreover, a bounded domain, dxi

we see that a function u e Wi,p(v,

Ω) belongs to the nonweighted space Μ Ι ι , λ < c|l"lli,p,v

i.e. Whp(v,

Ω) ^

Wl'Ps(a),

(1.22)

of course, with a parameter ps which is less than p. Now, we can use Theorem 1.2 and obtain that WUp(v,

Ω) ^

Ζ/(Ω)

(1.23)

where l N(s + 1) (cf. Theorem 1.2 (a), (b)). Moreover, we have the compact imbedding Whp(v,a)

provided 1 < r < p*.

^ ^

Ζ/(Ω)

(1.24)

26

1 Preliminaries Ν In particular, we have ρ* > ρ if s > —, and consequently, Ρ Whp

(υ, Ω)

ί/(Ω)

for

s>

(1.25) Ρ

We will use this compact imbedding later. Let us point out that (1.25) holds if υ" 5 e ί Λ Ω )

with

s e

Ο·26)

oo^ η

since, to be in accordance with (1.13), we have to suppose that also s >

—-—. Ρ~ 1 w) with the special

(b) Let us consider a higher order weighted Sobolev space Wk,p(£l, choice of w, namely wa(x)

= 1

for

| a | < k,

wa(x)

This special space will be denoted by Wk'p(v,

Mk.p,v=\

Τ f \Dau(x)\pdx+T Vl«!^-!7"

= v(x)

for

| a | = k.

Ω) and normed by

[ \Dau(x)\pv(x)dx\ W**

.

Ja

(1.27)

)

If we use the results of part (a) for the (k — l)-st derivatives of u, we immediately obtain from (1.25) that Wk'p(v,

Ω)

νΡ*-1,ρ(Ω)

provided (1.26) holds. (c) Let us consider the space W l ' p ( v , t t ) WQ'p(V,

from part (a) and its subspace

Ω) with Ω a bounded domain. The imbeddings derived for W{ p(y, Ω)

hold also for W 0 1,p (v, Ω). For ρ < ρ* we have, in virtue of the imbedding WQPi(&)

^jju(x)\

p

d

x

y q ,

may occur. In the latter case, the corresponding imbedding is not only continuous but, moreover, compact, while in the former case, some additional assumptions are needed to guarantee compactness. Example 1.4 (the one-dimensional case) For Ν = 1 and Ω = (a, b) with —oo < a < b < oo, the conditions which guarantee the validity of (1.30) have been described completely (see, e.g., [55]). Here, let us consider the case of functions u satisfying the additional condition u(a) = 0; (1.33) if we denote by side in the Hardy

W^'p((a,

b), wi) the set of all functions u for which the right-hand

inequality

^

\u{t)\^w(t)dty

(1.34)

i)

norm

(1.35)

b), ω).

LH(a,

in the

Now, inequality (1.34) holds — and the imbedding (1.35) is continuous — for \ < ρ (t)dty

1

QT

w \ -

p

\ t ) d t y w\~p\x)d^j


o(*) = uiiO) = · · · = u>n(x) = [dist(jc, 3Ω)] λ and ω(χ) = [distU, 9Ω)]*, i i c K " bounded, λ, κ eR, λ < ρ - 1. Then the inequality (1.31) holds for every « €