Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology: Madrid, Spain 14 - 15 June 2004 9789812565143, 981-256-514-0

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Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology

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Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology Universidad Complutense de Madrid, Spain 14-15 June 2004

S. Cano-Casanova Universidad Pontificia Comillas de Madrid, Spain

J. Lopez-Gomez Universidad complutense de Madrid, Spain

C. Nlora-Corral university of Oxford, UK

editors

K World Scientific N E W JERSEY

*

L O N D O N * SINGAPORE

BElJlNG

S H A N G H A I * HONG KONG

-

TAIPEI * CHENNAI

Published by

World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library

SPECTRAL THEORY AND NONLINEAR ANALYSIS WITH APPLICATIONS TO SPATIAL ECOLOGY Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-256-514-0

Printed in Singapore by Mainland Press

Preface This volume collects the Proceedings of the Complutense International Seminar Spectral Theory and Nonlinear Analysis that we celebrated in Madrid in June 14th and 15th, 2004, at the Department of Applied Mathematics, under the auspices of the Spanish Ministry of Education and Science, under Grant REN2003-00707, and Complutense University through

an International Seminar budget, which allowed us to invite some of the most renowned experts in these fields. Besides the editors, the following experts participated in the International Seminar: F. Cobos (Madrid), E. N. Dancer (Sydney), I. Gohberg (Tel Aviv), D. MacGhee (Glasgow), R. J. Magnus (Reykjavik), J. Mawhin (Louvain La Neuve), A. G. Ramm (Manhattan, Kansas), B. P. Rynne (Edinburgh), and A. SuArez (Sevilla), among many others that attended some of the talks delivered in that International Seminar, whose kind assistance certainly facilitated its success. Such International Seminar was organized to honor the memory of our friend and colleague J. Esquinas, born in March 27th 1960 at Ocaiia (Toledo, Spain), who suddenly died in August 11th 2003 at Covadonga National Park (Asturias, Spain). J . Esquinas was a tremendously gifted mathematician who did some seminal contributions to the theory of generalized algebraic multiplicites in the context of bifurcation theory. His scientific carrier was as short as intense, since he dedicated many of his efforts to the defense of the rights of the workers in Spanish Universities, becoming a renowned very popular personality in both issues. Besides collecting most of the contributions delivered by the participants in this Complutense International Seminar , this volume also includes a number of contributions by well recognized experts in Spectral Theory, Differential Equations and Nonlinear Analysis whose mathematical work is closely related to the one developed by J. Esquinas. The editors are delighted to thank all of them for their contributions to this so special honoring volume. July 2005 The editors

V

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Poster of the Seminar

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Contents

Preface

....................................................

v

........................................

vii

On the Positive Solutions of the Logistic Weighted Elliptic BVP with Sublinear Mixed Boundary Conditions S. Cano-Casanova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Logarithmic Interpolation Spaces F. Cobos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Remarks on Large Solutions J. Garcia-Melicin and J. Sabina de Lis

........................

31

Well Posedness and Asymptotic Behaviour of a Closed Loop Thermosyphon A . Jimknez-Casas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Uniqueness of Large Solutions for a Class of Radially Symmetric Elliptic Equations J. Ldpez-Gdmez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion J. Ldpez-Gdmez and M. Molina-Meyer . . . . . . . . . . . . . . . . . . . . . . . .

111

Local Smith Form and Equivalence for One-parameter Families of F'redholm Operators of Index Zero J. Lbpez-Gdmez and C. Mora-Corral.. ........................

127

Multilump Solutions of the Non-linear Schrodinger Equation A Scaling Approach R. J. Magnus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163

Some Elliptic Problems with Nonlinear Boundary Conditions C. Morales-Rodrigo and A . Sua'rez.. . . . . . . . . ..................

175

Poster of the Seminar

ix

X

Contents

Dynamical Systems Method (DSM) and Nonlinear Problems A . G. Ramm ...................................................

201

Some Recent Results on Periodic, Jumping Nonlinearity Problems B. P. Rynne ....................................................

229

Some Remarks about the Cubic Schrodinger Equation on the Line L. Vega .........................................................

247

Some Remarks on the Invariance of Level Sets in Dynamical Systems J. M. Vegas .................................................... 257

ON THE POSITIVE SOLUTIONS OF THE LOGISTIC WEIGHTED ELLIPTIC BVP WITH SUBLINEAR MIXED BOUNDARY CONDITIONS*

S. CANO-CASANOVA Departamento de Matemdtica Aplicada y Computacidn Escuela Te'cnica Superior de Ingenieria Uniuersidad Pontificia Comillas de Madrid, 28015-Madrid, S P A I N E-mail: [email protected]

This paper is dedicated, with my greatest admiration, to the great mathematician, colleague and f i e n d J. Esquinas Candenas. He, joint with Professor J. Ldpez Gdmez are the main responsible of my scientific formation In this work we prove the uniqueness of positive solution and characterize the existence of positive solutions of a wide class of elliptic BVP of Logistic type with sublinear weighted mixed boundary conditions. The results obtained in this work are an extension of the previous one found in S. Cano-Casanovag. Monotonicity techniques are the main technical tools, used to develop the mathematical analysis.

1. Introduction The main goal of this paper is to prove the uniqueness of positive solution and to characterize the existence of positive solutions of the following nonlinear weighted elliptic boundary value problem of Logistic type, with sublinear weighted mixed boundary conditions given by

{

15= XW(z)u- a ( z ) u T

in R,

T

> 1,

u=O on ro, &u = V ( z ) u- b(z)uq = 0 on rl, q

> 1,

(1)

*This work has been partially supported by the Spanish Ministry of Science and Technology under Grant REN2003-00707. 1

S. Cano-Casanova

2

where by a positive solution of Problem ( 1 )we ~ mean any function u : fi -+ [0,GO) such that u E Wz(R), p > N , u > 0 and satisfies Problem (1) a.e. in R for the value X of the parameter. The results obtained in this work are an extension of the previous one, obtained by S. Cano-Casanovag for Problem (1) in the particular case when W = 1 in R and V = 0 on rl. Throughout this work we make the following assumptions: ( a ) The domain R is a bounded domain of R N ,N 2 1,of class C2, whose boundary dR = ro U rl, where FO and l?l are two disjoint open and closed subset of do. ( b ) X E R and C stands for a linear second order differential operator of the form

which is uniformly strongly elliptic in 52 with ayij = aji E C ' ( f i ) , ai E C(fi), a0 E &(a), 1 5 i , j 5 N . ( c ) The potential a(.) E L,(R) is a nonnegative bounded measurable real weight function in R for which there exists an open subset R: of R with a finite number of components of class C2 with disjoint clousures, satisfying dist(I'1, a@ n R) > 0, such that a = 0 in 52; and it is bounded away from zero on any compact subset of [R \ fi:] U [I'l \ dR:]. ( d ) The potential W ( x ) is a bounded measurable real weight function in $2 with arbitrary sign. (e) As far as the nonlinear mixed boundary conditions, b(x) E C(rl)is a positive function on I'l which is bounded away from zero on I'l n do:, V ( x )E C(F1) with arbitrary sign in rl, u := ( ~ 1 ,. .. , v N ) E

being the conormal field on

rl n a@,

cl(rl;RN)

i.e.,

N

j=1

on I?l

nail:, where n = (nl, .. . ,n N ) is the outward unit normal and finally, a,u := ( V U , u ) .

Throughout this work, for each X E R, we will denote by C(X) the differential operator

C(X) := C - X W ( x ),

Positive Solutions of the Logistic Weighted Elliptic BVP

3

which is uniformly strongly elliptic in R with the same ellipticity constant as the operator L. The following theorem collects the main results of this work.

Theorem 1.1. Problem ( 1 ) possesses ~ a positive solution, if and only if

'0 : [,c(x>,731 stand for the principal eigenvalue in 0 and s2:, respectively, of L(A), subject to the boundary operators B ( - V ( z ) ) and D,respectively, being

where .?[C(X), B ( - V ( ~ ) )and ]

Moreover, in this case, the positive solution of Problem ( 1 ) is ~ unique and if we denote it by ux, then ux is strongly positive in R in the sense that

.A(.)

> 0 f o r each

~t: E

s2 U rl

and @ux(z) < 0

for each E ro and any outward pointing nowhere tangent vector field /3 E C1(l?o; RN). Furthemore, ux

n W;W.

(4)

P> 1

I n particular, ux E C1+a(f=l) f o r all 0 < a < 1 and in addition, ux is a.e. in 52 twice continuously differentiable. 2. Preliminaries, Definitions and Main Notations

In the sequel a function u E Wi(R) is said to be strongly positive in R if u(z)> 0 for each z E R U r l and apu(z) < 0 for each z E I'o with u(z)= 0 and any outward pointing nowhere tangent vector field /3 E Cl(I'0; RN). We now introduce a linear boundary operator, which will play a crucial role throughout this work. Under the assumptions of Sec. 1, given k ( z ) E C ( r l ) we will denote by B ( k ( z ) )the linear boundary operator

B : w;(R) defined by

-

w,2-+ro)x wi-+rl)

S. Cano-Casanova

4

where v := ( ~ 1 , ... , V N ) E C1(I'l;RN) is any outward pointing nowhere tangent vector field to I'l. We want to point out that using the boundary operator B ( k ( z ) )just defined, Problem ( 1 ) ~ can be written in the form

(L(X)+ a(z)uT-l)u o { B(-V(z) + b(~)uP-~)u on dR in R

=

(6)

=0

Now we are going to introduce some basic results concerning the operator B ( k ( z ) ) , that we need to develop our work. Let us consider the eigenvalue boundary value problem

{

Cp=Xp in0 B(k(z))p = 0 on a R

(7)

It is known, thanks to the results found in Theorem 12.1 of H. Amann3, that there exists a least real eigenvalue of Problem (7), denoted in the , called principal eigenvalue of (L, B ( k ( s ) ) ,0). sequel by @[L,B ( k ( z ) ) ] and The principal eigenvalue is simple and associated with it there is a positive eigenfunction, unique up to multiplicative constants and called principal eigenfunction of ( L ,B ( k ( z ) ) ,0). Thanks to Theorem 12.1 of Amann3, the B ( k ( z ) ) R) , belongs to Wi(R) for any p > 1 principal eigenfuntion of (L, and it is strongly positive in a. In fact, cF[L, B ( k ( z ) ) ]is the only eigenvalue of Problem (7) possessing a positive eigenfunction, and it is dominant in the sense that any other eigenvalue (T of Problem (7) satisfies Re(c)> dw,B(k(.))l In the sequel, given any proper subdomain Ro of R of class C2 with dRonR) > 0, we will denote by B ( k ( z ) ,no) the boundary operator dist (rl, build up from B ( k ( z ) )by

and by I$" [C,B ( k ( z ) ,Ro)], the principal eigenvalue of (C, B ( k ( z ) ,Ro), 00). To develop the mathematical analysis of the next sections, are essential the different monotonicity properties of op[C, B ( k ( z ) ) ]joint with the continuous dependence of it with respect to perturbations of the domain and with respect to perturbations of the potential k ( z ) on the boundary, recently proved by S. Cano-Casanova and J. L6pez-G6mez6q5. Suppose p > N . Then, any function u E Wi(R) is said to be a supersoZution (subsolution) of Problem ( l ) ~ if ,it satisfies

+

+

(L(X)u u(z)uT,B(-V(z) b(z)u"-1>u)2 0

(I0)

Positive Solutions of the Logistic Weighted Elliptic BVP

5

and it will be said that it is a strict supersolution (subsolution), if the respective inequality is strict, where 2 stands for the natural product order on Lp(R) x L,(aR). 3. On the Uniqueness and Regularity of Positive Solutions of Problem ( 1 ) ~

In this section we prove the uniqueness of positive solution of Problem ( l ) ~ , if there exists, and we obtain some regularity properties about it.

Theorem 3.1. If ux is a positive solution of Problem ( l ) ~ then , .F[L(X)

+ a(z)u;-l,

B(-V(z) + b(z)u;-')] = 0 ,

(8)

u x is strongly positive in R, (4)is satisfied and in particular, u~E C'+"(fi) f o r all 0 < Q < 1. Moreover, ux is a.e. in R twice continuously differentiable.

Proof. Let u~be a positive solution of Problem ( 1 ) ~Then, . u~E Wz(R) for some p > N, satisfies Eq. (6) and thanks to the embedding

W,2(R) L f C 2 - N*

(Q) -

(9)

for p > N, u~E C2-F(fi). Hence, -XW(X)

+ U(Z)U';-'

E

+ qz)u;-'

L,(R),

Ec(rl),

since a ( z ) ,W ( z )E L,(R) and U X ,b(z),V(z) E C(rl), and therefore, Problem ( 6 ) ~is well defined in the sense of Problem (7) and Eq. (5). Now, since u~is a positive solution of Problem ( 6 ) ~we , have that 0 is an eigenvalue of Problem ( 6 ) ~and u~is a positive eigenfunction of Problem ( 6 ) ~associated with the 0 eigenvalue. Then, by the uniqueness of the principal eigenpair, u~ is the principal eigenfunction of Problem ( 6 ) and ~ Eq. (8) is satisfied. Finally, owing to the structure and regularity of the principal eigenfunction guaranteed by Theorem 12.1 of H. Amann3, we have that ux is strongly positive in R and Eq. (4)is satisfied. The remaining assertions follow again from the embedding Eq. (9) and Theorem VIII.l of E.M. Stein". This completes the proof of the result. Thanks to the previous result, if we set

-

-

and we denote by U+ the cone of non-negative functions of U and by

F :R X U +

v,

0"

:

u+

w,

S. Cano-Casanova

6

the nonlinear operators defined by

+

3 ( A , u):= L(A)u a ( x ) u T , (A, u)E

IW x ZA+ ,

and

Gv(u):= (u, a,u - V ( x ) u+ b(x)uq = B(-V(x) + b(x)u"-1)u),

21 E

u+,

we have that the positive solutions of Problem ( l ) ~ are , the solutions couples (X,ux) with ux > 0 in R satisfying the system F ( A , u x ) = 0 in R = 0 on dR.

{ Gv(ux)

On the other hand, D,Gv(ux) can be written in terms of the boundary opertor B(lc(x))defined in Eq. (5), by

D,Gv(ux)u := B(-V(x) + qb(x)u;-l)u. Theorem 3.2. For each X E R, the positive solution of Problem ( l ) ~i f , there exists, is unique and non-degenerate, in the sense that

.T[a3-(A,ux), ~ u G v ( u x )>l 0 ; i.e., the linearization of Problem ( 1 ) at ~ any positive solution only possesses the null solution. Proof. We argue by contradiction. Suppose ui,i = 1 , 2 are two positive solutions of Problem ( l ) ~ such , that u1 # 212. Then, thanks to Eq. (S),

+

+

O ~ [ L ( A ) a ( z ) u ~ - l~, ( - v ( x ) b(x)uf-l)] = 0,

i = 1,2,

(10)

and the following problem is satisfied

( L ( 4 + a(z)F(x))(.u1- U2)

{ (av

741 - u2

=0

+

=0

in R on I?,,

- V ( x ) b ( z ) G ( z ) ) ( u-~u2) = 0 on r l ,

where F ( x ) E C(R) and G ( x ) E C(l?l) are the functions defined by

and

(11)

Positive Solutions of the Logistic Weighted Elliptic BVP

7

By construction,

F(.) 2

TuY-' > u;-',

G(-) 2 qu;-l > .;-I.

(1'4

Now, using the boundary operator B ( k ( z ) ) defined by Eq. (5), Problem ( 1 1 ) ~can be written in the form

+ 0 { B(-V(z) + b(z)G(z))(ul ua) (L(X) a ( z ) F ( z ) ) ( u 1- ~

2= )

-

in R

= 0 on

dR.

(13)

Thanks to the monotonicity of the principal eigenvalue with respect to the potential (cf. Proposition 3.3 of S. Cano-Casanova and J. L6pez-G6mez5) and with respect to the weight on the boundary (cf. Proposition 3.5 of S. Cano-Casanova and J. L6pez-G6mez5), owing to Eq. (12), and taking into account Eq. ( l o ) , it follows that

.T[.c(W + a ( z ) F ( z )B(-V(z) , + b(z)G(z))l > > CTf[L(X) + a(z)u;-l,B(-V(z) + b(z)u;-l)]= 0. Then, Problem ( 1 3 ) ~is invertible and hence u1 = u2, which gives the contradiction. This completes the proof of the uniqueness of the positive solution of Problem ( l ) ~ if there , exists positive solution. To complete the proof of the theorem, it remains to prove the nondegeneration of each positive solution of Problem ( 1 ) ~ Indeed, . if ux is a positive solution of Problem ( l ) ~ then ,

cr?[DuF(X,UX), DuQv(ux)l =

+

+

= CT?[L(X) Ta(z)u';-l,B(-v(z) qb(z)u;-l)].

Now, owing to Eq. (8), taking into account that T > 1, q > 1, a > 0, b > 0 and that u~is strongly positive in R, it follows from the monotonicity of the principal eigenvalue with respect to the potential and with respect to the weight on the boundary (cf. S. Cano-Casanova and J. L6pez-G6mez5), that

CJ?[DuF(X,4,DuGV('ILX)I > > CT?[C(X) + a(z)ul-l,B(-V(z) + b(z)vq,-l)]= 0 , and therefore, the positive solution ux is non degenerate. This completes the proof.

0

4. On the Existence of Positive Solutions of Problem ( 1 ) ~ In this section we characterize the existence of positive solutions of Problem ( 1 ) ~In . the beginning we give the following necessary condition for the existence of positive solution of Problem ( 1 ) ~ .

S. Cano-Casanova

8

Proposition 4.1. If U A is a positive solution of Problem ( l ) ~ then , Eq. (2) is satisfied, where B(-V(z)) and 2) are the boundary operators defined by (3).

Proof. Let U A be a positive solution of Problem ( 1 ) ~ .Then, thanks to Eq. ( 8 ) , and owing to the monotonicity of the principal eigenvalue with respect to the domain (cf. Proposition 3.2 of S. Cano-Casanova and J. L6pez-G6mez5) and the dominance of the principal eigenvalue under Dirichlet boundary conditions (cf. Proposition 3.1 of S. Cano-Casanova and J. L6pez-G6mez5), the following is satisfied

+

0 = cp[L(X)+ a ( ~ ) u ; - ~ , B ( - V ( z ) b(z)u;-')] < o:'[L(X), B(-V(z) b(z)u;-', a:)] I c:'[L(X), 27.

+

On the other hand, thanks to the monotonicity of the principal eigenvalue with respect to the potential (cf. Proposition 3.3 of S. Cano-Casanova and J. L6pez-G6mez5) and with respect to the weight on the boundary (cf. Proposition 3.5 of S. Cano-Casanova and J. L6pez-G6mez5),it follows from Eq. (8) that

+

0 = @[L(X) a(z>u;-l, B(-V(z)

+ b(z)uI-')]

> o~[L(X),B(-V(+ Z )b ( ~ > ~ : - l > ) ]cF[L(X), B(-V(z))]

since b > 0 on rl and U A is strongly positive in fl. This completes the proof of the result. 0 Now, in order to give a sufficient condition for the existence of positive solutions of Problem ( l ) ~ arguing , as in Theorem 3 of H. Amannl and Theorem 2.1 of H. Amann2, taking into account the results found in Theorem 12.1 of H. Amann3, and using the Characterization of the Strong Maximum Principle given in the work by H. Amann and J. L6pez-G6mez4, the following existence result for the solutions of Problem ( 1 ) is ~ satisfied, which we include without proof in order not to enlarge the exposition.

Theorem 4.1. Let cA be a subsolution and ?,j be a supersolution of Prob, that gA5 i i ~Then, . Problem ( 1 ) ~has at least one solution lem ( l ) ~such in the order interval [cA, F A ] . More precisely, there exists a minimal solution urnin A ' and a maximal solution uyax in the order interval [zA,ii~] such that every solution ux E [cA,iiA]of Problem (1)A satisfies uyin 5 U X I uyax. The next two results give sufficient conditions for the existence of positive strict sub and supersolutions of Problem ( l ) ~respectively. ,

Positive Solutions of the Logistic Weighted Elliptic BVP

9

Proposition 4.2. For each X E R satisfying

.?IW),f?(-V(z))l

a(-v(zc))l , < . F [ W ) , W V ( 4 + PI1

(16)

p+0+

and

for all p > 0. Then, taking into account Eqs. (14), (15) and (16), it is possible to take p > 0 such that

4V(X),

f?(-V(z))l < 4 [ W , f ? ( - V ( 4

+ PI1 < 0

(17)

*

Fix p > 0 satisfying Eq. (17). Now, let us consider the positive function 2 = ~ ( p ) c p ( p )with , ~ ( p> ) 0 small enough, where cp(p) stands for the principal eigenfunction associated to of[L(X),f?(-V(z) p ) ] , normalized so that II~(~)IIL,(~) = 1. By construction we have that in R, g satisfies

+

C(X)u

+ a(z)u'

f?(-V(z> + PI1 + a(+'-l(p)(P'-l(p)) on l?l the following estimate is satisfied =E(P>cp(P)(4v(X),

&u-

V ( ~ ) G +b(z)gq= E(P)V(P)(-P

(18) 7

+~(~)E"-'(P)P"~(P))

(19)

and g = ~ ( p ) c p ( p )= O

on

ro.

(20)

Thus, thanks to Eqs. (17), (18), (19) and (20), 21 satisfies

I

+

in R C(X)u a(z)gT < 0 f?(-V(z) b(z)Zlp-l).u< 0 on 30

+

for ~ ( p >) 0 small enough. Therefore, 3 is a positive strict subsolution of ~ ~ ( p> ) 0 small enough. The remaining assertion follows Problem ( 1 ) for from the fact that the principal eigenfunction p ( p ) is strongly positive in R (cf. H. Amann3). This completes the proof.

S. Cano-Casanova

10

Proposition 4.3. For each X E

R satisfying

[L(X),D]> 0 ,

(21) Problem ( 1 ) possesses ~ a positive strict supersolution, arbitrarily large and bounded away from zero in fi. 0 : :

Proof. In order not to enlarge the exposition, we are going to prove the result in a particular case, sufficiently general, being this proof easily extended to the case when for instance, a(.) belongs to the general class of nonnegative measurable potentials d ( R ) of admissible potentials in R introduced in the works due to S. Cano-Casanova and J. L6pez-G6mez5, S. Cano-Casanova7 and S. Cano-Casanova and J. L6pez-G6mez'). In the following we will suppose that rl = l?: U rq, where r:, i = 1 , 2 are two components of rl, and that R: = 0; U R i E C2, being R6, i = 1 , 2 two components satisfying

Q;nQi=0, fiicn, m;=r;uF, F C R (22) where = aRA n R. Thanks to Eq. (21), it follows from Definition 11.2 of S. Cano-Casanova and J. L6pez-G6mez5 that

r

0 7 q L ( X ) , D ]> 0 , 2 = 1,2. (23) On the other hand, thanks to the results found in Proposition 3.1 of S. Cana-Casanova and J. L6pez-G6mez5, o:A[L(x),a(n,R:)] < 0::[L(X),D]

(24)

: ' [L(x), ~ ( n R;)], = 0 : : [L(x), DI. lim 0

(25)

for all n E N and ntm

Thus, owing to Eqs. (23), (24) and (25), there exists no E N large enough, such that for each n 2 no the following is satisfied

< 07:[L(A),B(n,n;)] < .3L(X),D]. n 2 no. Now, for each 6 > 0 sufficiently small and i 0

Fix consider the 6-neighbourhoods := (a;

+B

~n)R ,

(26) = 1,2, let us

Njt2:= (r: + B ~n)R , @ := (ro+ B6), (27)

where Bg stands for the ball of radius 6 > 0 centered a t the origin. Under the general assumptions, and thanks to Eq. (22), it follows the existence of 60 > 0 small enough such that for each 0 < 6 < 60 and i = 1,2,

fiinfiz = 0 , s2Z,nNj>2 = 0 , fiz c R, Enfi;

= 0 , fljt2n@= 0. (28)

Positive Solutions of the Logistic Weighted Elliptic B V P

11

By construction we have that Rk, i = 1 , 2 is a proper subdomain of R;, and lim

6-0+

a; = at,

in the sense of Definition 6.1 of S. Cano-Casanova and J. L6pez-G6mez5. Thus, it follows from Theorem 4.2 of J. L6pez-G6mez1O, and Theorem 7.1 of S. Cano-Casanova and J. L6pez-G6mez5, that 6-Of lim

cp'

[L(x), ~ ( nR:)], = c? [L(x), ~ ( nR A , ),~

(29)

and &--to+ lim c:: [L(A), D]= 0:; [L(x>, DI.

(30)

Now it follows from Eqs. (21), (23), (26), (29), (30), and Proposition 3.2 and Theorem 10.1 of S. Cano-Casanova and J. L6pez-G6mez5, the existence of 0 < 61 < 60 such that

0
, - S)] + u(z)M'-'(p$)'-'), and taking into account that p; and a(.) are bounded away from zero in NtS2(cf.(c)), there exists M I > 0 large enough such that for each M 2 M: > O

L(X)Ti+ a ( z ) ~ '> O On the other hand, in

Nil2 5

in

\ (fl u R\T U R$I U N l s 2 )we have that B 2

L(X)Ti+ a(s)E' = M ( L @ + a(z)M'-'(@)'),

(39)

Positive Solutions of the Logistic Weighted Elliptic BVP

13

and since by construction, the functions u(x),$ Jare ~ bounded away from zero in f i b \ U R' U 0 2 2 6 U J V ~ "there ~ ) , exists M2 2 M I > 0 such that % 5 5 for each M 2 $2 > 0,

(x

+ U(X)iiT > 0,

L(X)ii in

ST\ (N,"u 2

5

u

2

(40)

UJV:~~). 3

Finally, on the boundary, we have by construction that ; i i = ~ p >; O

on

ro,

(41)

that on I?:,

3,ii - V(z)U+ b(z)$ = Mpi(6 + Mq-lb(x)(pi)q-l)2 M6p; > 0 , (42) and due to the fact that on

+

r: the following estimate is satisfied,

&Ti - V(x)ii b(x)@ = Mp:,,(-(n

+ V ( x ) )+ b ( ~ ) M ~ - ~ ( p ; , , ) ~ - ' ) ,

taking into account that pi,, is bounded away from zero on and b ( x ) is bounded away from zero on rl n aR: = r;, we have that there exists M3 2 M 2 > 0, such that for each M 2 M3 > 0,

a , -~ V ( Z ) E + ~ ( x ) P> o

on

r: .

(43)

Thus, for each X E R satisfying Eq. (21), it follows from Eqs. (38), (39), (40), (41), (42) and (43) that for each M 2 M3 > 0, the funcion Ti = M @ ( z ) satisfies .F(X,ii) > 0 in R { B v(u) - > O onaR and therefore, ii := M G ( x ) with M 2 M3 > 0 is a positive strict superso-

lution bounded away from zero in fi of Problem ( 1 ) ~ . This completes the proof. 0 Now, we are ready to prove Theorem 1.1. Proof of Theorem 1.1. The necessary condition for the existence of positive solution of Problem ( 1 ) ~is Proposition 4.1. We now prove the sufficient condition for the existence of positive solution of Problem ( 1 ) ~ . Let X satisfy Eq. (2). Then, thanks to Proposition 4.2 and Proposition 4.3, Problem ( 1 ) ~possesses a positive strict subsolution ZL, arbitrarily small strongly positive in R and a positive strict supersolution E X , arbitrarily large and bounded away from zero in fi. Then, taking them such that

S. Cano-Casanova

14

we have that < Tix and therefore, the sufficient condition for the existence of positive solution of Problem ( 1 ) follows ~ from Theorem 4.1. Now, Theorem 3.1 and Theorem 3.2 complete the proof of the result. 0 The following result is the adapted version of Theorem 1.1, for the particular case when a = 0 in R, i.e., 52; = R.

Corollary 4.1. Under the general assumptions of Sec. 1, let us consider the BVP

{

cu = XW(z)u

in R, 21=0 on ro, &u = V ( z ) u- b(z)uQ= 0 on rl, q > 1,

(44)

where b ( x ) E C(rl)is a positive function bounded away from zero on F1 and v := ( ~ 1 , .. . ,Y N ) E C1(rl, I R N ) is any outward pointing nowhere tangent vector field to rl. Then, Problem (44) possesses a positive solution, if and only i f

~?[c(~),,13(-v(~c>)l < 0 < .?[c(4,q ,

(45)

ant the remaining assertions of Theorem 1.1 are satisfied. Proof. To build the positive strict supersolution bounded away from zero in 0 of Problem (44)x, for each X E IR satisfying Eq. (45), it suffices to take the positive function ii := M v Y ,

for M > 0, n E N large enough and S > 0 small enough, where @ is the principal eigenfunction associated to the principal eigenvalue DF[I?(X), B ( n , being

o)],

0 :=Ru~;, where I?; is a tubular neighbourhood of ro, and L(X) is an adequate regular extension of the operator L(X)from R to 0. The remainder of the proof is obtained, arguing as in the case of Problem (1)x. This completes the proof.

0

In order to complete the exposition of this work, we include the following result, which is the counterpart of Theorem 1.1 for the case when the potential a(.) is bounded away from zero on any compact subset of 52 U rl. Its proof can be easily obtained, arguing as in the previous sections and doing an easy and straight adaptation of the proof of Proposition 4.3.

Positive Solutions of the Logistic Weighted Elliptic BVP

15

Corollary 4.2. Under the general assumptions of Sec. 1, let us consider the B V P

{

Lu = XW(x)u- a ( x ) u r

in R, r > 1,

u=O on 13,u = V ( x ) u- b(z)uq= 0 o n

ro, rl, q > 1,

(46)

where the potential a ( x ) is bounded away from zero on any compact subset of R U rl, v := (vl,.. . ,V N ) E C1(rl,RN) is any outward pointing nowhere tangent vector field to l?l and b(x) E C(r1) is a nonnegative function o n rl. Then, Problem (46)possesses a positive solution, if and only i f

&w),q - v ( x ) ) l < 0

7

and the remaining assertions of Theorem 1.1 are satisfied. Acknowledgments

I am delighted to thank my colleague and friend J. Lbpez-Gbmez, for inviting to me to contribute with this paper in this memory in tribute to the great mathematician, colleague and friend, J. Esquinas Candenas. Also, I want to thank to J. Esquinas, the advice about the lectures that I had to take to complete my scientific formation in the postgraduate. References 1. H. Amann, Indiana University Mathematics Journal 21, 2, 125-146, (1971). 2. H. Amann, Nonlinear elliptic equations with nonlinear boundary conditions, New Developments in Differential Equations, Ed. W. Eckhaus, 1976. 3. H. Amann, Israel Journal of Mathematics, 45,225-254, (1983). 4. H. Amann and J. L6pez-G6mez, Journal of Differential Equations 146,336374, (1998). 5. S. Cano-Casanova and J. L6pez-G6mez, Journal of Differential Equations 178,123-211, (2002). 6 . S . Cano-Casanova and J. L6pez-G6mez, Nonlinear Analysis T.M.A. 47, 1797-1808, (2001). 7. S. CaneCasanova, Nonlinear Analysis T.M.A., 49,361-430, (2002). 8. S. Cano-Casanova and J. Lbpez-Gbmez, Electronic Journal of Differential Equations, 74,1-41, (2004). 9. S. Cano-Casanova, O n the existence and uniqueness of positive solutions of the Logistic elliptic B VP with nonlinear mixed boundary conditions, Nonlinear Analysis, To appear. 10. J. L6pez-G6mez, Journal of Differential Equations 127,263-294, 1996. 11. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ.,1970.

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LOGARITHMIC INTERPOLATION SPACES*

FERNANDO COBOS Departamento de Ancilisis Matemcitico, Facultad de Matemciticas, Universidad Complutense de Madrid, 28040-Madrid1 SPAIN E-mail: cobosOmat.ucm.es

We review several recent results of Fernhdez-Cabrera, Triebel and the author on logarithmic interpolation spaces based on the real method.

1. Introduction The degree of compactness of the embedding from the (fractional) Sobolev space Hp”’”(S2) into the Orlicz space L,(logL)b(S2) has been studied by Triebel [29]. Here R is a bounded domain in Rn with smooth boundary, 1 < p < 00 and b < - 1. Two years later, Edmunds and Triebel 1131, [14], investigated the behaviour of entropy numbers of the embedding (n+w) (0)L,Lp(10gL)b(S22)* An important tool to derive these results is a representation theorem of Zygmund spaces Lp(logL)b(S2) in terms of Lebesgue spaces L,(R). Motivated by this representation, Edmunds and Triebel have studied in [15] abstract spaces based on complex interpolation. More recently Fern6ndezCabrera, Triebel and the present author [lo] have investigated the corresponding spaces using the real interpolation method, giving applications of the results to Lorentz-Zygmund function spaces, Besov spaces of generalized smoothness and Lorentz-Zygmund operator spaces. Next we review these results.

:

‘Work supported in part by grant MTM2004-01888 of the Spanish Ministerio de Educaci6n y Ciencia. 17

18

F. Cobos

2. Function Spaces

Let R be a domain in Rn with finite Lebesgue measure IRI. For 1 < p < 03 and b E R, the Zygmund space Lp(log L)b(R) is defined by all (equivalent classes of) Lebesgue-measurable functions f : R -+C such that

1[lf(.)l n

logb(2 + If(.)I)IPd.

< O3.

Clearly, L,(logL)o(R) = Lp(R). For b < 0, the space L,(logL)b(R) is the set of all measurable functions f : R -+ C for which there exists a constant X > 0 such that

In the literature this space is also denoted by Lexp,-b(R). Zygmund spaces complement the scale of Lebesgue spaces, in the sense that if 0 < E < p then LP(logL)€(R)c L P W and for

-03

c ~p(logL)-€(R),

< b 2 5 bl < 03

LP+€(W c L,(log

L)bl

(0) c -&(log L)bz(Q)c LP-€(R).

Now assume that R has smooth boundary. Then it is known that the (fractional) Sobolev space Hp”’”(R) is continuously embedded in L,(logL)b(R) if, and only if, b 5 - 1. The embedding being compact if, and only if, b < - 1. The study of limiting embeddings of this type goes back to Trudinger [31] and Strichartz [27]. The degree of compactness of these embeddings was studied by Triebel[29],who proved that if b < -1 - $ then the entropy numbers of this embedding satisfy

i

i

c1k-1’p

5 ek 5 c 2 k - l / p

, k E N.

Here c 1 , c 2 are positive constants. Let me recall that if T E L ( E ,F ) is a bounded linear operator between the Banach spaces E and F , then the k-th entropy number of T is defined as

u

2k-1

E

> 0 : T ( U E )&

j=1

{bj

+ EUF}for some b l , . . . ,

b2k-1

EF

Logarithmic Interpolation Spaces

19

where UE,UF are the closed unit balls of the spaces E and F , respectively. It is clear that

llTll 2 el(T) 2 e2(T) 2 . . . 2 0, and T is compact if and only if limk,, e k ( T ) = 0. Hence, the asymptotic decay of the sequence (ek(T))can be considered as a measure of the “degree of compactness” of the operator T . There is a close relation between entropy numbers and eigenvalues. Namely, if E is a complex Banach space, T E C ( E , E ) is compact and ( X k ( T ) ) is the sequence of all non-zero eigenvalues of T , repeated according to algebraic multiplicity and ordered so that IXl(T)I 2

IX2(T)I

L ...L 0

then it turns out that IXk(T)I

I 2 l I 2 e k ( T ) , k E N.

This inequality was established by Carl and Triebel [4] and is the basis for many applications. In particular, Edmunds and Triebel [12] used this inequality and the entropy estimates mentioned before to study the distribution of eigenvalues of some elliptic differential operators. Returning to compact embeddings between function spaces, in 1995 Edmunds and Triebel [13] investigated the behaviour of entropy numbers of the embedding

(a) Lt Lp(logL)b(fi) where 1 < p < 00, s > 0, b < 0 and l/ps = l / p + s / n . In this last paper and in the paper by Triebel [29] a basic tool for the results was a representation theorem of Zygmund spaces in terms of L, spaces. For 1 < p < 00 the result reads as follows:

Theorem 2.1. Let 1 < p < 00 and b E R. Let j o = jo(p) E N such that for all j E W with j L j o , 1 1 1 -1 _ - - + 2 - j < 1 and - - 2-3 > 0. PPj P P”j P If b < 0 , then L,(logL)b(Cl) consists of all measurable functions f : (a) R -+ C such that

F. Cobos

20

and (1) defines a n equivalent norm in Lp(logL)b(R). If b > 0 , then Lp(logL)b(fl) is the set of all measurable functions (22) f : R --+ C which can be represented as 60

f

=

fj

with

fj

E Lpyj(R)

j=jo

such that

Furthermore, the infimum over the expression in (3) with respect to all representations (2), (3), is a n equivalent norm in Lp(1og L)b(R). The proof requires of the domain R merely that it should have finite Lebesgue measure. Theorem 2.1 can also be found in the book I141 where they extended statement (a) covering any 0 < p 5 00, as well as p = 1 in (ii). See also [Ill. As we said, the result is a basic tool for entropy estimates, but it also has intrinsic interest. A consequence of Theorem 2.1 is that assertions which hold for spaces L p ( R ) ,such as properties of integral operators, can be carried over to Zygmund spaces. Edmunds and Triebel have also studied the spaces that come out by replacing in Theorem 2.1 the space Lp(R) by the Sobolev space H,S(R). They called these spaces “logarithmic Sobolev spaces”. Constructions of type (1) to (3) have been considered in the framework of extrapolation theory, especially in the cases p = 1 and p = 00. We refer to the papers by Jawerth and Milman [19] and Milman [22]. Let me give an application of the case (ii) when p = 1 and b = 1. It is taken from the book [14] and refers to the Hardy-Littlewood maximal function. If p = 1, then l/p”J = 1 - 1/23’. So 2j l/(p”j - l), and we get that N

CO

f E L(logL)(R) if and only iff =

1

fj

j=1

CO

1 w i t h 1 p”j - 1 ~

j=1

Consider now the Hardy-Littlewood maximal function

lIfjllLpy,

< 0O.

Logarithmic Interpolation Spaces

21

where the supremum is taken over all cubes Q containing x and with sides parallel to the coordinate axes. It is known that there is a positive constant c such that C

IlfllLp(n, I P 11. P-1 Hence, using the description of L(1og L)(Cl) and sublinearity of M , it follows that IIMfllLpcn, 5

-

IIMfllLI(i2)5 c IlfIIL(logL)(n)l

which is a classical assertion of Hardy and Littlewood from 1930.

3. Abstract Logarithmic Interpolation Spaces

As it is well known, L, spaces can be obtained by complex interpolation. Namely 1

(Lm(R),Li(Q))[e]= L,(Q) if - = 6 . P So, taking Theorem 2.1 as starting point, Edmunds and Triebel [15]studied the corresponding abstract theory based on complex interpolation. In that paper they introduced interpolation spaces which complement the complex interpolation scale and they established the trace theorem for logarithmic Sobolev spaces. But L, spaces can also be obtained by real interpolation. Indeed,

Therefore, it is also natural to investigate the corresponding abstract theory based on the real interpolation method. This was the aim of my joint paper with L.M. Fernhndez-Cabrera and H. Triebel [lo]. Subsequently, we will describe some results of this paper. Let A0 and A1 be Banach spaces with A0 -+ A1 (continuous inclusion). Peetre's K-functional is defined by

K ( t ,a) = K ( t ,a; Ao, Ai) =inf{llaollAo

+ tllalllAl

: a = a0

+ a l , aj E A j } ,

t > 0,

a E Al.

For 0 < 6 < 1 and 1 5 q 5 0;), the real interpolation space A O , = ~ (Ao, Al)e,qconsists of all those a E A1 having a finite norm

F. Cobos

22

Full details on real interpolation can be found in the books [3], [28] or [2]. The theory of real interpolation can be extended by replacing the function te by a more general function parameter e(t) as can be seen in the papers [24], [17], [18]or [25]. For our aim here, the most interesting case is @e,b(t)= te(l

+ I logtl)-b,

t > 0,

where 0 < 8 < 1 and b E R. We put

(with the usual modification if q = 00). As we have already said, interpolating the couple (L,(R), L1(R)) by the real method we get L,(R) if q = p . If q # p we get Lorentz spaces. More precisely, for l/p = 8 and 1 5 q 5 00, we have

Here f* is the non-increasing rearrangement o f f

f * ( t ) = inf{b > O :

I{.

ER :

lf(z)I > 6}1 5 t }

and we have put

f**(t) =

/

t

f*(s)ds.

0

Interpolating with the function parameter @e,bwe get the LorentzZygmund spaces

\f

Namely, for l/p = 8, 1 5 q L

00

and b E R, we have

(Lrn(R), L 1 ( R ) ) Q t J , b ; q

= LP,q(lOgL)b(R>-

Lorentz-Zygmund spaces are considered in the book by Bennett and Sharpley [2] and in the paper by Bennett and Rudnick [l].For p = q we recover the Zygmund spaces Lp,p(logL)b(fl) = Lp(logL)b(fl).

Logarithmic Interpolation Spaces

23

Returning to the real interpolation spaces, we always have = (Ao, Ai)s,qo

Moreover, since A0

~f

~f

if

(Ao, Ai)e,ql =

A1, we have for 1 I p , q 5

(Ao,A I ) ~ , ~(Ao,Ai)e,q ~f

I 00.

00

0 0.

Let 1 I q L: 00. (i) Assume b < 0. Then Ae,q(logA)b is the collection of all a E A,,q which have a finite norm

(with the usual modification if q = 00). (ii) Let b > 0. Then Ae,q(logA)b is the collection of all a E A1 which can be represented as

c 00

a=

aj

(convergence in Al),

with aj E Axj,q

(4)

j=j0

such that

We put

where the infimum is taken over all sequences (iii) If b = 0, then Ae,q(logA)b = Ae,,.

{ a j } satisfying

(4)and (5).

Standard arguments show that Ae,q(logA)b is a Banach space in all cases of b E R. It is independent of j o (equivalence of norms). If we replace the real method by the complex interpolation method, that is to say, if we replace in (2) A,j,q by (AO,AI)[,~]and in (22) we put (Ao,Al)[xjl instead

F. Cobos

24

of A A ~ then , ~ , we obtain the spaces studied by Edmunds and Triebel 1151. They are different from the complex scale and complement it. In our case spaces Ae,q(logA)bare also different from the real interpolation spaces, but they coincide with the spaces generated by the function parameters te(l I logtl)-b :

+

Theorem 3.1. Let 1 I q 5 00, 0 < 6' < 1, and b E R. Let te (1+ I logtl)-b,t > 0. Then we have, with equivalent norms, Aee.b;q

@e,b(t) =

= A6,q(logA)b.

Proof. Sketch of the case b < 0. It is sufficient to prove that the respective norms are equivalent to each other. First note that

Using that A0 -+ Al, one can check that the last expression is equivalent to (cg=,2-majqKq(2m, a ) ) : , moreover constants in equivalences can be chosen independent of aj and 0. Similarly M

For the norm in the other space, we have

I I ~ I A ~ , ~ ( ~ ~ ~C A ) ~ 2jbq C 2-m0q-m2Ti 00

00

9

c

C

m=l

j=jo

(6)

03

00

=

Kq(2m,a )

m=l

j=jo

2-me9K9(2m , a )

2jbq2-m2-jqe

To work with the last sum in (6) let j = k+ [logm],where [.] is the greatest integer function. Then j b - m2-j b [logm] bk - 2-k. We obtain

+

N

c 00

j=jo

C 00

$bq-m2-'q

mbq

2kb92-Wk.

k=jo-[log m]

Now, using that b < 0, the last sum can be estimated from above and from below by positive constants which are independent of m. This yields that

the norms of the two spaces are equivalent.

Logarithmic Interpolation Spaces

25

The proof for the case b > 0 is more involved. Details can be found in [lo]. Let me only mention that it is based on the description of real interpolation spaces in terms of the J-functional. Theorem 3.1 allows us to use the results known on interpolation with function parameter to study spaces Ae,q(logA)b. A first consequence is that A0 is dense in Ae,q(logA)b if q < 00. Concerning duality, if A0 is dense in Al, then we have (Al)’ c_t (Ao)’, so we can consider logarithmic spaces generated by the couple {(Al)’, (Ao)’}. Call them AL,q(logA’)b.

If q < 00, since A0 is dense in Ae,q(logA)b, we get (Ail’

L+

(Ae,q(logA)b)’ ~ - (Ao)’ t

and we can compare the spaces (Ae,q(logA)b)’ with the spaces AL,,(log A’)b. Using Theorem 3.1 and the duality formula for the real method with a parameter function we derive for 1 I q < 00, l / q l/q’ = 1 and 0 < 0 < 1 that

+

(Ae,q(logA)b)’ = Ai-e,,y (log A‘)-b. 4. Concrete Logarithmic Interpolation Spaces

In this final section we apply Theorem 3.1 to concrete situations. We start with the couple (L,(R), L1(R)). We get: Corollary 4.1. Let R be a domain in Rn with finite Lebesgue measure. Let 1 < p < M, 1 5 q 5 03 and let j o = j o ( p ) E N such that for all j E N with j 2 j o , 1 - 1 1 1 _ - - + 2 - j < 1 and - - - - 2-3 > 0. PPJ P P”j P (a) Let b < 0. Then Lp,,(logL)b(R) is the set of all measurable functions f : R -+ CC such that

(equivalent norms). (ii) Let b > 0. Then L,,,(logL)t,(R) is the set of all measurable functions f : R + C which can be represented as

__ f =

fj

with

fj

E

L,uj,,(R)

(7)

F. Cobos

26

such that

Furthermore, the infimum over the expression in (8) with respect to all representations (7), (8), is a n equivalent norm in Lp,,(log L)b(R). For p = q we derive a representation theorem for Zygmund spaces Lp(lOgL)b(R) in terms of Lorentz spaces L,,(R). Note that Theorem 2.1 requires only the simpler Lebesgue spaces. However, the next lemma yields that Theorem 2.1 follows also from our approach. Lemma 4.1. Let Ao, A1 be Banach spaces with A0 -+ Al. Let 0 < 0 < 1, b E R and let j o = j o ( 0 ) E N such that, f o r all j 2 j o , uj = 0

+ 2-j

< 1 and Xj

= 0 - 2-j

> 0.

(i)

Let b < 0. The norm of Ae,l/e(logA)b is equivalent to

(ii)

Let b > 0. Then Ae,l/e(log A)b is formed by all those a E A1 which

c 00

can be represented as a =

aj

(convergence in Al) , with

aj

E Axi,l/xj

is a n equivalent norm in the space Ae,1/e(logA)b. Here the infimum is taken over all representations of the described type. The proof can be found in [9]. The next application refers to Besov spaces of generalized smoothness. Let cp be a C" function in Rn with supp cp Let j E N and cpj(()

cp(t)= 1 if 1c1 I 1. = cp(2-jE) - (p(2-j+'() , ( E Rn. Put cpo = cp. Then

c { E E Rn : IEl 5 2 1 M

k=O

7

Logarithmic Interpolation Spaces

27

is a dyadic resolution of unity. Definition 4.1. Let 1 < p < 00, 15 q 5

00,

s E R, and

Qb(t) = Ilogtlb, 0

< t < 1, b E R.

Then Bk;Qb)(Rn)is the collection of all tempered distributions f on Rn such that

(with the usual modification if q = 00) is finite.

If b = 0 we get the usual Besov spaces B;,q(Rn). Besov spaces of generalized smoothness Bk;*b)(Rn)are Banach spaces and the norms are equivalent to each other for admissible choices of cp. These spaces attracted some attention in connection with fractal analysis and related spectral theory. A short description can be found in Triebel’s book [30]. A detailed study of these spaces is made in the paper by Moura [23]. Let A0 = B;Yq(Rn) and A1 = B;”X(Rn)

with 1 < p < 00,l 5 q 5 00, and Moreover (see [8] and [21])

-00

< s1 < SO < 00. Then Ao

-

AI.

Bk;*b) (W”)= (B;pq(R”)7 qq (Rn)) ee,b;q

where s = (1- @)SO

+ 8.q.

Consequently, using Theorem 3.1 we derive:

Corollary 4.2. Let 0 < B < 1,l 5 q 5 00,b E R, and let 1 < p < 00, --oo < s1 < so < 00 and s = (1 - B)so + Bsl. Let b < 0. Then (i)

F. Cobos

28

where the infimum is taken over all representations 00

f

fj

=

(convergence in S’(Rn)) with fj E Bi:2-’(80-s1) (Rn).

j=j0

The last application refers to operator spaces defined by summability conditions on the singular numbers. Let H be a Hilbert space and given any bounded linear operator T E L ( H ) ,let { s n ( T ) }be the sequence of the singular numbers of T , defined by

sn(T) = inf{llT - RI) : rank R < n } , n E W. For 1 < p < 00,l 5 q 5 00 and b E R, the Lorentz-Zygmund operator space Lp,q,b(H)consists of all T E L ( H ) having a finite norm

(with the usual modification if q = 00). The space L,,,,b(H) is the component over H of the Lorentz-Zygmund operator ideal that has been studied in [5], [6] and [7]. Note that T belongs to Lp,q,b(H)if and only if { s n ( T ) } belongs to the Lorentz-Zygmund sequence space lp,q(lOgl)b. For b = 0 we recover the Lorentz operator space ( L p , q ( H ) , r p , qand ) , for b = 0 and p = q , we get the Schatten p-class ( L p ( H ) , T p ) (see [161, P O I and [261). In order to apply Theorem 3.1 in this context, we put A0 = L l ( H ) and A1 = L ( H ) . Then A0 ~f A1 and, according to [5, Thm. 5.11, we have &w,b(H) = ( L 1 ( H ) L(H))m,,;q* , Theorem 3.1 yields:

Corollary 4.3. Let H be a Halbert space. Let 1 < p < oc),1 5 q 5 03, and let j o = j o ( p ) E N such that for all j E N with j 2 j o ,

(2)

Let b < 0. Then Lp,q,b(H)is the set of all T

(equivalent norms).

E

L ( H ) such that

Logarithmic Interpolation Spaces

29

(ii) Let b > 0. Then C,,,,b(H) consists of all T E C ( H ) which can be represented as T = C,”=,, Tj with Tj E C P p j ,,(H) such that

Furthermore, the infimum over the expression in (9) is an equivalent norm an c p , q , b ( H ) . Remark 4.1. All results we have shown refer to Banach spaces, but logarithmic interpolation spaces can also be considered in the class of quasiBanach spaces. Details can be found in the joint paper by FernhdezCabrera, Manzano, Martinez and the present author [9]. Among other things, we establish there that statement (ii) in Theorem 2.1 holds for 0 < p < 1 as well, and we give applications to operator spaces on Banach spaces. References 1. C. Bennett and K. Rudnick, O n Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980) 1-67.

2. C. Bennett and R. Sharpley, “Interpolation of operators”, Academic Press, Boston, 1988. 3. J. Bergh and J. Lofstrom, “Interpolation spaces. A n introduction”, Springer, Berlin, 1976. 4. B. Carl and H. Triebel, Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spaces, Math. Ann. 251 (1980) 129-133. 5. F. Cobos, O n the Lorentz-Marcinkiewicz operator ideal, Math. Nachr. 126 (1986) 281-300. 6. F. Cobos, Entropy and Lorentz-Marcinkiewicz operator ideals, Arkiv Mat. 25 (1987) 211-219. 7. F. Cobos, Duality and Lorentz-Marcinkiewicz operator spaces, Math. Scand. 63 (1988) 261-267. 8. F. Cobos and D.L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, in Lecture Notes in Mathematics 1302,Springer, Berlin, 1988, pp. 158-170. 9. F. Cobos, L. M. Fernbndez-Cabrera, A. Manzano and A. Martinez Logarithmic interpolation spaces between quasi-Banach spaces, preprint 2004. 10. F. Cobos, L. M. Fernbndez-Cabrera and H. Triebel, Abstract and concrete logarithmic interpolation spaces, J. London Math. SOC.70 (2004) 231-243. 11. D. E. Edmunds and W.D. Evans, “Hardy Operators, Function Spaces and Embeddings”, Springer Monograps in Mathematics, Berlin, 2004.

30

F. Cobos

12. D. E. Edmunds and H. Triebel, Eigenvalue distributions of some degenerate elliptic operators: a n approach via entropy numbers, Math. Ann. 299 (1994) 311-340. 13. D. E. Edmunds and H. Triebel, Logarithmic Sobolev spaces and their applications to spectral theory, Proc. London Math. SOC.71 (1995) 333-371. 14. D. E. Edmunds and H. Triebel, “Function spaces, entropy numbers, differential operators”, Cambridge University Press, Cambridge, 1996. 15. D. E. Edmunds and H. Triebel, Logarithmic spaces and related trace problems, Funct. Approx. Comment. Math. 26 (1998) 189-204. 16. I. C. Gohberg and M. G. Krein, “Introduction t o the theory of linear nonselfadjoint operators”, American Mathematical Society, Providence, R.I., 1969. 17. J. Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (1978) 289-305. 18. S. Janson, Minimal and maximal methods of interpolation, J. Funct. Anal. 44 (1981) 50-73. 19. B. Jawerth and M. Milman, “Extrapolation theory with applications”, Mem. Amer. Math. SOC.440, Providence, 1991. 20. H. Konig, ‘%igenvalue distribution of compact operators”, Birkhauser, Basel, 1986. 21. C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in Lecture Notes in Mathematics 1070, Springer, Berlin, 1984, pp.183-201. 22. M. Milman, “Extrapolation and optimal decompositions”, Lecture Notes in Mathematics 1580,Springer, Berlin, 1994. 23. S. Moura, Function spaces of generalized smoothness, Dissertationes Math. 398 (2001) 1-88. 24. J. Peetre, “A theory of interpolation of n o m e d spaces”, Notes Mat. 39 (1968) 1-86. (Lectures Notes, Brasilia, 1963). 25. L.-E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986) 199-222. 26. A. Pietsch, ‘%igenvalues and s-numbers”, Cambridge University Press, Cambridge, 1987. 27. R. S. Strichartz, A note o n Trudinger’s extension of Sobolev’s inequality, Indiana Univ. Math. J. 21 (1972) 841-842. 28. H. Triebel, “Interpolation theory, function spaces, differential operators”, North-Holland, Amsterdam, 1978; sec. ed. Barth, Leipzig, 1995. 29. H. Triebel, Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces, Proc. London Math. SOC.66 (1993) 589-618. 30. H. Triebel, “The structure of functions”, Birkhauser, Basel, 2001. 31. N. S. Trudinger, O n imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-483.

REMARKS ON LARGE SOLUTIONS

JORGE GARC~A-MELIAN Departamento de Ancilisis Matemcitico Universidad de La Laguna 38271-La Laguna (Tenerife), Spain Email: jjgarmelQul1. es JOSE SABINA DE LIS Departamento de Ancilisis Matemcitico Universidad de La Laguna 38271-La Laguna (Tenerife), Spain Email: josabinaQul1. es This work discusses some aspects of the existence of solutions to the singular Dirichlet problem -Au = X(z)u - a(z)uP, z E 0,ulan = 00 under several assumptions on the null set {a(.) = 0) of the nonnegative coefficient a. In addition, a detailed analysis of the asymptotic profile of the solution to the finite Dirichlet problem -nu.= X(z)u - a(z)uP,z E 0,ulan = uo, as uo -+ 00 is performed.

1. Introduction The main purpose of this work is discussing and also reviewing some features concerning the existence, non existence and other few qualitative properties of positive solutions to the following class of singular boundary value problems:

-Au = A(x)u - ~ ( z ) u P

XER

E

u=0O

aR.

It will be assumed that fl C RN is a bounded domain of class C2@,the coefficients A(z), a(.) lying in P(n)with a(.) 2 0 in R and the exponent p fixed in the range p > 1. All along the work, solutions u will be understood in the classical sense u E C2@(R)the boundary condition meaning that: limu(z) = 00, as dist (x,aR) + O+. 31

32

J . Garcia-Melicin and J. Sabina de Lis

In spite of its possibly artificial look, semilinear elliptic problems subjected as (P) to an “infinite” Dirichlet condition have been studied ever since the beginning of the twentieth century, under the presence of different kinds of nonlinearities instead of the power term in (P). It should be mentioned the pioneering works [2], [30], the former on the theory of automorphic functions. A first more modern and N-dimensional approach to (P) goes back to [21], [22], the inspiring problem being in this occasion a question on electrohydrodynamics, together with [28] where the solution to a problem on the classification of Riemann surfaces is given. The study of the power nonlinearity comes from [29] and [23], but we prefer referring to [7], [14] and [27] for a detailed story and bibliographic quotations on this specific subject. Of course the appeal of (P) goes on and literature on the matter is continuously appearing. It is worthy of mention that [13], aside of being the origin of our interest on the problem, furnished an unexpected new example showing how (P) arises in the most natural population dynamics model. Under the inspiration of that research we are dealing here with different aspects of the subject of existence of solutions when the “self competition” coefficient a(.) vanishes in R in a nontrivial way. Regarding the structure of the coefficient a($) the principal feature is that it can vanish in a whole region 00c R whose boundary 800 could possibly touch somewhere the boundary dR of 0. More precisely, it will be assumed that the null set for a can be expressed as {x E R : a(.) = 0) = where Ro c R is open and locally lies on one side of its boundary dRo which in turn is also locally described -perhaps after a rotation- as the graph in RN of a real C2@function. Equivalently, 520 is a finite union of C2i“ bounded domains Ro,J,. . . ,R O , ~their , closures GO,^ being pair-wise disjoint. This work is preceded by a Section 2 where a brief account of results concerning (P) is presented. Sections 3 and 4 contains the main existence material. Finally, Section 5 develops an analysis of the asymptotic profile to the solutions of the most natural auxiliary problem associated to (P). For future reference in the course of the exposition we are introducing some notation. For a bounded domain Q c RN and a potential q = q(x), say q E Lm(Q), @(-A q ) will stand for the first eigenvalue A to -A$ q$ = A$, x E R, = 0. It is well known that it is the unique eigenvalue with a positive associated eigenfunction $1 E C2i”(n). We finally remark that the results in Sections 3, 4 and a preliminary version of the ones in Section 5 were obtained in the spring of 2000 and later

no

+

+

Remarks on Large Solutions

33

presented in the Spanish Meeting on Differential Equations, celebrated in the University of Salamanca on September 2001 (see [17]). At that time we were still so fortunate as to live and enjoy the warm friendship of our dear colleagues Jestis Esquinas, Chema Fraile and Rene‘ Letelier. This modest work is dedicated to their memory. 2. Preliminaries The purpose of this section is to describe some basic features concerning problem (P). For historical reasons we are focusing the attention in the case where u(x) > 0 in Q but a = 0 on dR. In fact, this was the precise critical framework in which (P) appeared to us, as a short of “degenerate” perturbation limit in a classical population dynamics model (see [13], and [ll]for a preliminary analysis of the associated Dirichlet problem). The main issues to be here reviewed are those of existence, uniqueness and blow-up rates of the solutions at the boundary. It should be stressed that elucidating as better as possible the asymptotic profile of solutions to (P) near dR is a crucial step in order to provide the uniqueness of those solutions (see also Remark 2.1 b) below). At the time where [13] was written the uniqueness of positive solutions to (P) with u positive but vanishing on dR was still an open question (see [14] for a detailed historical account of results when u ( x ) is bounded away from zero in The next result summarizes the main properties of problem (P). We refer to [14] for detailed proofs and more general problems deduced from it by suitable nonlinear perturbations.

n).

Theorem 2.1. Let R c lRN a bounded and C27a domain, A,u E C a ( n ) such that u ( x ) > 0 in R, u ( x ) = 0 for all x E dR. Then the following properties are satisfied.

i ) [Existence] Problem (P) admits a minimal positive solution g ( x ) E C2@(R)and a maximal positive solution n(x) E C2@(R). If in addition X(x) > 0 in then every positive solution u to (P) satisfies the lower estimate,

n,

ii) [Uniqueness] Suppose that u1,u2 E C2(R) are classical positive solutions to ( P ) verifying the asymptotic equivalence near dR, 211 ( X I lim - 1,

4l)-+O

u2(x)

34

J. Garcia-Melidn and J . Sabina de Lis

with d ( z )= dist(z, d o ) . Then u1 and u2 coincide in R. iii) [Asymptotic estimates] If the coeficient a(.) decays to zero according t o

for certain positive constants Co,y, then every classical positive solution u E C 2 ( R )to (P) satisfies,

with ezponent a = (y + 2 ) / ( p - 1) and coeficient A = (a(a+ ~ ) / C O ) ~ / ( ?In ’ - particular, ~). (P) admits a unique positive classical solution u E C2(a). Remark 2.1. a) The uniqueness of positive solutions to (P) was first independently obtained in [lo]and [14]by means of achieving the blow-up rate (2) from the decaying rate ( 1 ) . However, a more accurate asymptotic development of u near the boundary was obtained in [14] on the basis of a decaying profile for a of the form, a(%) = C o f l ( l + C1d

+ o(d))

as d

+ 0,

with CO,C1 either constant or class C2 functions on dR, COpositive on dR (see Remark 1-a), p. 3595 in [14]). Under this assumption, every positive solution u to (P) can be expressed near dR as

+ Bd + o(d)) as d -+0, = ( ( N - l)H - ( a + l>Cl)/(y+ p + 3)), H

u(z)= Ad-a(l

with a , A as in iii), B the mean curvature of dR (see [7] for earlier previous results in this direction). Finally, it must be mentioned that the blow-up rates subject has been recently refined in [27] where (2) is obtained via ( 1 ) by allowing both CO and y to vary continuously on dQ. b) As pointed out in [12],in order to show the uniqueness of positive solutions to (P) (at least when X = 0 ) , what one really needs is an approximate measure of the blow-up rate of the solutions at the boundary, not so precise as (2) in Theorem 2.1. It suffices indeed to know that

Cld(z)-a 5).(it

5 C2d(z)-a

in a neighborhood of dR, for some positive constants C1,C2.

Remarks on Large Solutions

35

c) The existence, uniqueness and blow-up rates of problem (P) with X = 0, a > 0 in R and, more importantly, u(x) -+ 00 as d(z) -+ 0 with an asymptotic rate, U(X)

N

Cod-?

as d -+ 0,

(3)

where d = d(x) and CO is a positive function defined on dR, have been deeply studied in [5] (cf. also [4]for a previous study of the radial case which even covers the complementary range 0 < p 5 1 for the exponent). Among other results, it is shown there that y < 2 is a necessary and sufficient condition for the existence of a positive solution to (P). d) The extension of results as those in Therorem 2.1 to the case of elliptic systems is a very hard task. Of course, the lack of comparison or suitable variational approaches arise as the main sources of difficulty. In fact, only few particular cases have been properly studied at the present moment. To quote some of them, results on existence, uniqueness and asymptotic rates for ,

i

-Au = f ( u , ~ )

X € R

-aw = g(u,w)

X € R

with f = -u*wQ,g= -urws,has been produced in [16] for singular boundary conditions upn = 00, wpn = 00 and even combinations upn = 00, 0180= finite. The analysis has been performed in the so called subcriticalcritical regimes (p-l)(s-1) -qr 2 0. We refer in addition to [15]for recent results concerning the complementary supercritical case f = -wQ, g = -ur and to [18]for cooperative systems. See also [ B ] , [9], [6], [26] where further systems exhibiting related phenomenologies are considered. 3. Contact null set-boundary

We are now discussing the existence of a positive solution to problem (P) when the null set of the coefficient a(.) meets dR in a nontrivial way. More precisely, let {Ro,i}z1 the set of connected pieces of 00. We distinguish between “inner components”, Ri,kl defined as those R o , ~strongly which are those contained in R and “boundary components”, whose boundary has common points with dR. The main assumption in our next result is the existence of interior points to X l n dRo relatively to dR. The conclusion is that under these conditions, problem (P) cannot support positive (and therefore, finite) solutions.

J . Garcia-Melidn and J. Sabina de Lis

36

Theorem 3.1. Let a E C"(n) be nonnegative and nontrivial so that

where Ro

cR

= 0} =

: a(.)

{x E

no n 0

is smooth in the sense that, =

0'

(c A').,

"

k=l

(E

'$1)

7

being Oc1,C2s"-bounded subdomains of R whose closures Di,k,i$l are pair-wise disjoint and satisfy: -I

'0,k

'7

r l := a@, n 8~ # 0 ,

(4)

each k E (1,.. . ,ml}, 1 E {l,.. . , m 2 ) . Suppose that some 1 5 1 5 exists such that:

for

m2

0

(5)

rl# 0 , 0

the interior l?l of rl being considered with regard to aR. Then, the following properties are satisfied, a) Regardless ( 5 ) , a necessary condition in order that the problem (P) does admit a positive solution u is:

(-A - ~(x)> )o

f o r either R O , ~ = R ; , ~ or

R O , ~ = R:k,

and every 15 Ic 5 ml, 15 15 m2. b) Under the validity of condition (6) the auxiliary problem,

i

-Au = X(x)u - a(x)uP u=m

(6)

XER

x

E

dR,

(7)

admits, for each m E W, a unique positive solution urn E C 2 ~ " ( n ) . c ) I f (6) is satisfied and u, E Czia(n) is the solution to (7), the limit limu,(x)

= 00,

0tl

holds uniformly o n compact sets of every component of Ro fulfilling (5). Therefore (5) implies that problem (P) cannot admit a positive solution u in R.

37

Remarks on Large Sohtions

Figure 1. A configuration for 00having inner components ary connected pieces ."[: One of them fulfills (5).

"A,k

and two kind of bound-

Proof. We begin by proving a) and so let u E C2(R) be a positive solution to (P). If ~ o , isi any inner component, Ro,i c 0, since u is finite in Q,,i it defines a strict positive supersolution to the equation

-Au

- X(Z)U = 0,

(8)

in R o , ~It . is well-known (see [25]) that this implies,

,psi(-A

- A)

> 0.

Let us suppose now that R o , ~is a boundary component and for E > 0 small enough write Q = {x E R : dist(s, R o , ~ < ) E } , Q 2 R o , ~ .Then we have the est irnates , f 0 . i

1

(-A - A) = XYsi(-A - X + q ) > Xp(-A - X + q ) ,

(9)

with q = a u p - l . Take 6 > 0 small and define Q6 = {x E Q : dist(x, as2) > 6). Observe that Q6 c Q together with lim6,o Q6 = &. Similarly, u defines again a strict positive supersolution to the alternative equation:

-Au

+

( ~ ( 5 -)X ( Z ) ) U

+

= 0,

in, QSfor every 6. Thus, X y 6 (-A - X q ) > 0 and o I lim X?~(--A- x q ) = inf xQ6(-A - x q ) = A,Q (-A - x -t 4).

+

6+0+

+

6>0

This and (9) imply the positivity of X " O > a (-A - A) and the necessity of (6) for the existence of positive solutions to (P) is already shown. Next we prove b). As a first remark observe that the uniqueness of positive solutions to (7) follows, for instance, from [3]. To get existence u = 0 can always be taken as a subsolution while to get a comparable

38

J . Garcia-Melidn and J . Sabina de Lis

supersolution ii one only needs to construct a specific supersolution u* E C 2 @ ) which is positive in In fact, it can be checked that ii = Au* also defines a supersolution for every A 2 1. Hence, supersolutions as large as desired can be produced. Following the approach in [24] we are in fact showing that (5) suffices to construct a positive supersolution u*.We are assuming for simplicity that a > 0 on dR \ (see Remark 3.1). To construct u* choose 6 > 0 so small as to have

n.

no

XF6 (-A - X(z)) > 0, where Q is any of the connected pieces R0,i and Q' = {x E RN : dist(x,Q) < 6). For every 1 5 i 5 m, Q = Ro,i, let +i(x) be the first positive eigenfunction associated to XF6 (-A - X(z)) in Q6 which satisfies I+iloo = 1. We define w E C2@) as

where w(x) is a positive and C2 extension of xiq5i to the whole of a, and xi stands for the characteristic function of Ro,i . Direct computation shows

&

that u* = Bw defines a positive supersolution to -Au = Xu - auP in R provided B > 0 is chosen large enough. Thus, the proof of b) is finished. Before proving c) notice that, aside the validity of condition (5), the existence of a positive solution u to (P) easily implies that, u&)

< u(x)

2

E

R,

for every m E W. Indeed such inequality follows from the fact that urn< u near the boundary dR and that the Dirichlet problem for -Au = Xu - aup is uniquely solvable in the class of positive solutions. In particular,

u(x)2 limu,(x) = supu,(x)

x E R:

Hence, once c) is shown the existence of positive solutions to (P) is not possible. Thus, let us finally proceed to show c). Suppose R o , ~= REl is a connected piece satisfying ( 5 ) and choose a connected piece r,

= {x E I' : Notice that r is open in dR. For q > 0 small put dist(z,dr) > Q}, where d r is the boundary of I' relative to dR, and consider a cut off function cp = cp(z) € C:'a(dfl),0 cp 1 such that cp = 1 in r,,,'p = 0 outside l?,,12.

<
>-+ minXTsi(-A - X(z)), as

E

0+, the minimum only extended to those boundary components

R o , ~satisfying (5). Thus, XYL'(-A - X(z)) is positive for small E and the method in the proof of b) can be repeated to get a positive supersolution in RE whose restriction to R is the searched supersolution u*. b) It follows from the analysis in Section 4 that under condition (6) the sequence u, of solutions to (7) converges to a function u E C'~"(R\U ~ o , i ) , the union being extended to all boundary components. The asymptotic behaviour of u, on those boundary components R& not satisfying (5) is still unclear up to complete generality. F'urther features and explicit examples of this limit behaviour will be included in a work in preparation.

J . Garcia-Melidn and J . Sabina de Lis

40

-

c) An interesting problem that one could pose is whether (P) admits positive solutions if a E C"(D) becomes partially negative in R. Suppose, for 0

instance, that R* = {x E R : h ( z ) > 0}, Ro = {a(.) = 0 ) are C2i" domains. It is next remarked that the relative position of the components R i of R- with respect dR is again crucial for this issue. Indeed, if some component R, satisfies (5) no positive solutions for (P) are possible. In fact, if such solution u exists we have in that component,

+

(R,)& = {x E R, : dist(z,dR) > 6 } , since XY"'((-A - A($) sup-') is positive for 6 small. Setting, as in Theorem 3.1, I? a connected piece of the interior of dR, n dR and choosing cp a C2@cut-off function as there, then the problem -Au - Xu = 0 in R, , u = cp on do, can be solved to obtain a positive solution U y E C 2 @ ( F ) Fixed . a positive solution u to (P) we get by comparison that for each m E N,a positive 60 exists so that u > m u y in (a,)& for every 0 < 6 < 60. Thus u > m u y in 0; what is not possible. This proves the remark. -As a conclusion, positive solutions to (P) are only possible when R- c R- c R (of course, together with c R). On the other hand, the existence of positive solutions can be shown, in symmetric cases, provided that R- and R+ keep a suitable balance. Such results, belonging to a work in progress with Prof. C. Flores, will be diffused in near future. 4. Inner null set

We are exploring in this section the existence of positive solutions to (P) when the refuge Ro, the interior in R of the null set {a(.) = 0}, is strongly contained in 0. Suppose that Ro is also a C2i" domain and so it only exhibits finitely many connected pieces R o , ~. ,. . ,Ro,, all of them of class C2i". Then, if (P) admits a positive solution u E C2@(R)we immediately find (see Theorem 3.1),

X?,'(-A

o

- x ( ~ c )>>

i = 1 , . . . ,m.

(10)

Our next result states that (10) is also sufficient to ensure the existence of a positive solution to (P). Theorem 4.1. Let a, X E C"(a),a 2 0 with = {u(x) = 0}, RO c R a C2@domain with connected pieces Ro,i,. . . ,Ro,, satisfying:

-

Ro,icR

i=l,

..., m.

Remarks on Large Solutions

41

Then (10) is a necessary and suficient condition for the existence of a positive solution to (P). Proof. Only the sufficient character of (10) needs to be proved. To construct a positive solution to (P) we first analyze the auxiliary problem (7). In view of (10) and the proof of Theorem 3.1, a positive strict supersolution u* E C2(a) to -Au = X(x)u - a(x)uP can be obtained (see Section 5 , Remarks 5.3 for an alternative construction of u*).Thus, problem (7) admits a unique positive solution u, E C2ta(D)for every m E N. Notice that, thanks to Remark 3.1 a), this assertion holds true even in the case where u vanishes partially or totally on 30. On the other hand, urn is increasing and admits the estimate:

< UD(z)

urn(.) with D = {x E : u(x) > 0}, the singular problem,

UD

z E D,

E C2>a(D) being the minimal solution to

-Au = X(X)U- U(X)U’

(

(11)

XED x E dD.

U=0O

Thus, by using local Loo estimates, W2vQestimates and bootstrapping in a standard way we obtain that um + u in C2>.l(D).It is also easily checked that u + 00 as dist(x,aR) + O+. To complete the existence proof we only need to extend the convergence of Um to an open neighborhood of Ro. Let us proceed separately on each component and so, choose a single component Ro,i and 6 > 0 small so that,

with (Ro,~)’= {X E Rn : dist(x,Ro,i) < 6). Let q$ E C 2 @ ( ( s 2 ~ , ithe ) 6 )associated normalized positive eigenfunction such that, say 14iloo= 1. Since r6/2 =

6

{dist(x, Ro,i) = -} 2

C

D, thanks to (11) we know that:

SUPU,(X) 5 c
0.

EX$'

we

(22)

It follows that problem (21) has a unique bounded classical positive solution v = V ( Z ) which in addition depends only on Z N :

J . Garda-Melidn and J. Sabina de Lis

46

Lemma 5.1. Let a0 > 0 . T h e n problem (21) has a unique bounded positive solution v, which is a function of ZN only, and is explicitly given by:

where A is given by

For convenience, we postpone the proof of Lemma 5.1 until the end of this one. Observe that Lemma 5.1 implies in particular that the whole family,

v(z,cl) -+ ~ ( z=) 2 , a --N in C,,, (R,) as CT 4 00, where A is as in the statement of Lemma 5.1. As a main consequence,

- - auc, uo)

W.,'1LO)

av

3X-N

lx=zo

p+l

- u o 2 (vi(0)

Ix=o

ya?)(.,cl)

=-uo

azN

+ o(1)) = A u y + o (uoq)

-

Jz=O

,

as uo 4 00. This finishes the proof of Theorem 5.1.

0

We proceed now to the proof of Lemma 5.1. Proof of Lemma 5.1. Fix R,h positive and choose DR,h any bounded C23asub domain of R y containing {x : 12'1 < R, 0 < X N < h}, for instance a regularization of the latter domain which contains it. Consider the auxiliary problem,

{

-Av = -a&'

v=l

z E DR,h Z E aDR,h.

(24)

By using a standard weak comparison argument it follows that problem (24) admits at most a unique positive solution. Such solution can be obtained by the method of sub and supersolutions by taking 2 = 0 as subsolution, v = 1 as a supersolution. Thus (24) admits a unique positive solution VR,h E C2ia( B R , h ) which satisfies,

0 < uR,h(Z)

< 1,

z

E DR,h.

Remarks on Large Solutions

47

2 a --N

Now observe that if w = w(z) E C,;,(W+)is any nonnegative bounded solution to (21) (and so satisfying (22)), then wID~,h

defines a subsolution to problem (24) which, as a consequence of weak . comparison, satisfies w(z) 5 w ~ , h ( z )in D R , ~Hence,

0 < u(z) < VR,h(z)

E DR,h,

where strong comparison has been also employed. For n E N set D, := n D ~ , h= {nz : z E D R , ~ while } u,(z) designates the solution to problem (24) in D,. Just for the same argument as the one given above we arrive at: 0 < w(2)

< W , + k ( Z ) < w,(z) < 1

z E

D,,

for every n,k E N. In fact, the restriction of w,+k to D, defines a strict subsolution t o (24) in D, that can be strongly compared with v,. Using the LQand Schauder's estimates as has been already done we get the convergence, z E iiiy,

~ ( z=) limw,(z) = infv,(z) in Ct:(Z:).

In addition,

0 < W ( Z ) Iqz)

z

E

iq,

for any arbitrary bounded positive solution w to (21) satisfying (22). Since 5 ( z ) is also a solution to (21), this means that F ( z ) is in fact the mmimal

positive solution to (21) in such class. Observe now that for every vector 7 E IWN-I,

%(z) =q

z + (7,O ) ) ,

is again as positive solution to (21) bounded by 1. In particular

D(z

+ ( 7 , O ) ) 5 v(z)

2

E

"+N.

This immediately implies that -

v(z'

for all z E Ry and

7

+ 7, Z N ) = q z ' , ZN),

E WN-*. Hence, ?7 depends only on Z N .

J. Garcia-Melicin and J. Sabina de Lis

48

It can be shown in the same way that problem (21) has a minimal solution g in the class of positive solutions bounded by 1, which only depends on ZN. In fact, consider the problem,

{

-Av = --a&'

z E DR,h

(25)

z E dDR,h,

v=c

c

where D R , h is as before while 5 E c 2 > " ( a D ~ , h 0) , 5 5 1, satisfying in addition c ( z ) = 1 for ZN = 0 , lzll < R - 2 ~ E, > 0 small enough, and = 0 outside { < R - E , ZN = 0) in the whole of aDR,h. Problem (25) and the corresponding ones in Dn admit a unique positive classical solution wn(z) such that the relation

c

I'z

0 < 'u)n(z)< ' u ) n + k ( Z ) < v(z), holds for any n, Ic E N,z E D, and an arbitrary nonnegative solution v to (21) satisfying (22). Then, the limit, V(Z) = limw,(z) = supWn(z) 2 a --N exists in C,;, (R+) and defines the minimal solution to (21). Therefore, g only depends on ZN. Finally, observe that the one-dimensional problem,

v"

t>O

= aOvP

v(0)= 1, has a unique positive solution v = v1 ( t )with the property of being defined in the whole interval [0,m). Such solution is explicitely given by

with

v:(O)

= A.

In particular, the maximal and minimal solutions satisfy:

V ( z )= v(z) = V 1 ( z N ) and problem (21) has

Vl(ZN)

.Z

E Rf,

as its unique positive bounded solution.

0

Remark 5.1. a) In many cases it can be shown that (27)

Remarks on Large Solutions

49

for uo large. In fact, notice that since u(z,UO)becomes finite in every J: E !2 as uo -+ 00 (Theorem 4.1), then dist(x,,,R) 3 0 as uo -+ 00 where xu, is any point in where u(., u g ) achieves its maximum. On the other hand X(x) - a(x)u(x,u0)p-I L o at x = xu, if xu, E R. If, for instance, a > 0 on dR then it follows that xu, E dR (and so (27) holds) if uo is large. Otherwise,

a

with d(x) = dist(x, dR) and b some sufficiently small positive number. That is not possible since sup u(., uo) -+ 00 as uo 00. -+

b) It should be stressed that regarding the asymptotic rate (13), a(.) is allowed to freely vary on dR provided it keeps positive there. On the other hand, the limit (23) not only provides asymptotic information on u(., uo) at xo but also in nearby points of the form x = zo n-ly, y fixed, with (T = uo (p-1)’2 and uo -+ 00.

+

We are now studying the asymptotic behavior of the solution u(., uo)to (12) near a point zoE dR where the coefficient a vanishes. To this proposal we are allowing a ( x ) to decay as a power of the distance to the boundary, with a variable rate, possibly depending on the location of the reference point xo E dS2. Since it will be assumed that R is C2@no generality is lost if it is assumed as before that xo = 0 while the outward unit normal to dR at x = 0 is u = - e N . Let us introduce the decaying restriction on the coefficient a to be used in the forthcoming analysis. The notation of the proof of Theorem 5.1 describing dR near x = 0 as Z N = f(x’) is kept. Supposing a(0) = 0 it will be assumed the existence of functions ao, a1 and g defined on a neighborhood of xo = 0 and a positive constant y such that,

4.)

= ao(x’)

+ ai(x’)yL + d x ’ , YN)Y&,

YN = X N - f(x’),

(28)

where a0,al are nonnegative, ao(x’) = o(Jx’17)as )x’1 -+ 0, al(0) # 0 and g ( x ’ ,y ~ -+) 0 uniformly in x’ (Id15 6) as Y N -+ 0. The meaning of the condition is clarified by observing that YN has the status of the “vertical distance” to the boundary dR, a0 gives the restriction alan, i.e., ao(2’) = a(x’,f(x‘)), while a1 measures the rate of deviation of a(.) regarding its value in the boundary as the distance YN tends to 0.

Remark 5.2. Condition (28) is commonly used in the literature under the more restrictive form,

a(.)

= Adr

+ ~(dr),

d

-+

0,

J . Garcfa-Melidin and J. Sabina de Lis

50

d ( z ) = dist(z,dR) with a coefficient A either constant or variable but positive (see [lo], [14], [27], and references quoted there). This corresponds to set ao(z’) = 0 in (28). On the other hand it is remarked that al(z’) is allowed to be variable in (28). We can already state the following improved version of Theorem 5.1.

Theorem 5.2. Let a E C”(n) be nonnegative such that its null set {a(.) = 0 ) = with Ro C C R is a C2ya subdomain ofR fulfilling the conditions of Theorem 5.1. If a(z0) = 0 at zo E dR and a(.) satisfies the decaying condition (28) at zo then the normal derivative of the solution u(.,uo) to problem (12) exhibits the exact asymptotic behavior,

no,

where

K

no

is a universal constant depending only o n p > 1.

Proof. Recall we are taking 20 = 0 with { E N = 0) the tangent hyperplane to dR at z = 0. The course of the proof of Theorem 5.1 can be followed, firstly rectifying dR near z = 0 by means of the change (15) (new coordinates y) and then performing the alternative blow-up scaling, p--l

2

= my,

CT

u ( y ) = uov(2).

= u;+y,

Problem (12) is thus transformed near zero into N-1 b , az2i z N v o-1Z.dz,v = o-2iw - Zivp, -Azv Ci=l

{

with

O

0’’= t71vy

v(0) = 1,

has a unique solution v ( t ) with the property of being defined in the whole interval [O, co). I n addition, v(t) is positive, convex, decreasing and

v ( t ) = At-e with 8 = - A = (8(8 P-1

+ O(t-(’”+e))

+ 1))*

as t -+

00,

and every p satisfying

2 a --N

Thus the whole family v(z,(z) --+ v ( z ~ in ) C,;,(R+) with V ( Z N ) = ~ l ( a : ’ ( ~ + ’ ) z ~ v1 ) , ( t ) being the unique solution to (32) whose existence is ensured by Lemma 5.2. Set, in particular, K

= -V{(O)

> 0.

Then we finally get

as uo

--+ 00.

Thus, the proof of Theorem 5.2 is concluded.

0

Proof of Lemma 5.2. Consider the initial value problem,

t>O

(33)

regarded as a parameter and set v(t,cr) its unique non continuable solution. We claim that a solution to (33) defined for all t > 0 must have v’(t) < 0 in 0 5 t < 00. As a consequence of this fact and arguing by convexity one finds that limv = limv’ = 0 as t -+ co and so 0 < v ( t ) < 1 for t > 0. (T

J. Garcia-Melidn and J. Sabina de La8

52

To show the claim suppose v'(t0) 2 0 at some to. If v(t0) 2 0 (of course, the case v = v' = 0 at t = to is discarded by uniqueness) then v' > 0 together with v " ~ ' 2 tivpv' for t > to. This implies

and v must blow-up at a finite tl > to. If v(to) < 0, v' is positive at the right of to and by convexity v becomes positive with positive derivative at finite time. Thus, blow-up occurs again. Next, let us show the uniqueness of a solution defined in [0, w). It can be checked that v(t,0 1 ) < v(t,1 7 2 ) if a1 < u2 in any interval (0, b) where both solutions are positive. In particular, this must be true if b = 00 and they are two possible different solutions in the conditions of the statement. By integrating in (33) using their behavior at t = oa one gets -a1 =

J,

tYv(t,u1)" < J, t'v(t,

a2)"

= -a2,

which contradicts the assumption on the a i l s . To construct the solution define tmin = tmjn(a), a < 0 , as that t > 0 where inf v(., u ) is achieved. It can be proved that tmin increases when u decreases provided that v(., a) keeps positive. Set

a* = inf{a

< O : inf v(.,a) > 0).

Then -w < a* < 0 while the solution v(t,a*)is defined in all t 2 0 and provides the desired solution. In fact, infv(.,a) > 0 for u < 0 small, by smooth perturbation, and so a* < 0. On the other hand, for 0 < t < t,in(a) and u > a* one gets

O 0, z1 > 0 from (0,O) at T = -00 can only exhibit the following behaviors (see Figure 2):

e(i + e) ( z - z p ) >-

=: h ( z ) for (28 1) 0 < z < 1. Since E ( z , z l ) increases on the orbit this implies that the parameterizing solution Z ( T ) is increasing and blows-up at finite time. ii) The orbit I' reaches z1 = h ( z ) at some 0 < z < 1, enters the region 0 < z < 1, 0 < z1 < h ( z ) remaining there for all future times T . iii) Orbit r behaves as in ii) but enters the region z1 < 0 at some 0 < z < 1. In this case z ( r ) vanishes at some finite 7.

i) The orbit l? keeps in the region

z1

+

Observe now that the solution Z ( T ) obtained from our global solution v(t) = w ( t , u*)satisfies z ( r ) = A-lee7v(e7) A-lee7 as r --+ -00, ~ ' ( 7 )= A-lee7(6v + e7vt(e7)) 6A-lee7 as r -+ -00. Thus, its orbit enters z > 0 , z l > 0 but does not vanish, or blows-up at finite time. Therefore such orbit fulfills ii), which means that it necessarily defines a branch of the stable manifold corresponding to ( 1 , O ) . This implies that N

N

+

~ ( r=)1 O(e-p7)

as

T 3 00,

54

J. Gareta-Meladn and J. Sabina de Lis

Phase space configuration for equation (35).

Figure 2.

for every p 6 (0,- p - ) with p- the negative eigenvalue of the linearization of (34) at the saddle (1,O). This concludes the proof of the lemma. 0

Remark 5.3. a) As an application of the results in this section we are performing an alternative construction of the finite supersolution u*appearing in Sections 3 and 4. Assume that Ro c c R consists of m connected pieces {Ro,i}El fulfilling condition (10). For 6 > 0 small introduce Qi = ( R O , ~and )~ D = R \ UE, Q i , all of them being C2>*domains. The problem

no

-Au = X(x)u - a ( x ) u P u = uo

XED x E dD,

(36)

has a unique positive solution ug(.,U O ) while each problem

{

-Av - X(X)V = 0 v = l

E Qi

(37)

x E dQi,

has also a unique positive solution vi(x) 6 C 2 * a ( a ithe ) , existence being ensured by the conditions A y i (-A - A) > 0, 1 5 i 5 m. Define

u*(x)=

i

u D ( x , UO)

X € D

UOVi (X)

X E Q ~ l, < i < m ,

(38)

Then u*defines a supersolution to -Au = Xu - aup both in D and in each

Qi.Thanks t o Theorem 5.1, the normal derivative dug(x’uo) at x E dQi dU

Remarks on Large Solutions

55

grows faster than -210as uo -+ 00. According to [l]this implies that dU u* defines a positive supersolution for uo large. b) A positive supersolution u*to -Au = Xu - uup in Q having ulaa = 00 can be constructed exactly in the same way by replacing U L ) ( X ,U O ) in (38) by the solution G D ( x , U O ) to,

i

-Au = X ( X ) U 1 ' 1 = 210

u=OO

- U(X)U'

X E D x~dD\dQ x E dQ.

(39)

With an appropriate handling, Theorems 5.1 and 5.2 can be adapted to dGD this new scenario to show that - satisfies the same growth estimates dU regarding U O , as uo -+ 00. The use of this supersolution permits to obtain the existence assertion in Theorem 4.1 under an alternative approach.

Acknowledgments This work has been supported by DGES and FEDER under grant BFM2001-3894.

References 1. H. Berestycki, P. L. Lions, Some applications of the method of super and subsolutions, in Bifurcation and nonlinear eigenvalue problems, Proc., Session, Univ. Paris XIII, Villetaneuse, 1978, pp. 16-41, Lecture Notes in Math., 782, Springer, Berlin, 1980. 2. L. Bieberbach, Au = eu und die Automorphen Funktionen, Math. Ann. 77 (1916), 173-212. 3. H. Brezis, L. Oswald, Remarks o n sublinear elliptic equations, Nonlinear Anal. 10 (1986), no. 1, 5544. 4. M. Chuaqui, C. Cortbar, M. Elgueta, C. Flores, J. Garcia-Meliin, R. Letelier, O n a n elliptic problem with boundary blow-up and a singular weight: the radial case, Proc. Royal SOC.Edinburgh 133 (6) (2003), 1283-1297. 5. M. Chuaqui, C. Corthar, M. Elgueta, J. Garcia-Meliin, Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights, Comm. Pure Appl. Anal. 3 (4) (2004), 653462. 6. N. Dancer, Y . Du, Effects of certain degeneracies in the predator-prey model, SIAM J. Math. Anal. 34 (2002), 292-314. 7. M. Del Pino, R. Letelier, T h e influence of domain geometry in boundary blow-up elliptic problems, Nonlinear Anal. 48 (2002), no. 6, Ser. A: Theory Methods, 897-904.

56

J . Garcb-MelaCin and J. Sabina de Lis

8. Y. Du, Effects of a degeneracy in the competition model. Part I: classical and generalized steady-state solutions, J. Diff. Eqns. 181 (2002), 92-132. 9. Y . Du, Effects of a degeneracy in the competition model. Part 11: perturbation and dynamical behaviour, J. Diff. Eqns. 181 (2002), 133-164. 10. Y. Du, Q. Huang Blow-up solutions f o r a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal. 31 (6) (1999), 1-18. 11. J. Fraile, P. Koch-Medina, J. L6pez-G6mez, S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differential Equations 127 (1996), no. 1, 295-319. 12. J. Garcia-Melitin, A remark o n the existence of positive large solutions via sub and supersolutions, Electronic J. of Differential Equations 2003 (2003), n. 110, 1-4. 13. J. Garcia-Melitin, R. G6mez-Reiiasc0, J. L6pez-G6mez, J. Sabina de Lis, Point-wise growth and uniqueness of positive solutions of sublinear elliptic problems where bifurcation f r o m infinity occurs, Arch. Rational Mech. Anal. 145 (1998), 261-289. 14. J. Garcia-Melitin, R. Letelier-Albornoz, J. Sabina de Lis, Uniqueness and asymptotic behaviour f o r solutions of semilinear problems with boundary blowup, Proc. Amer. Math. SOC.129 (2001), no. 12, 3593-3602. 15. J. Garcia-Melitin, R. Letelier-Albornoz, J. Sabina de Lis, T h e solvability of a n elliptic system under a singular boundary condition, Proc. Royal SOC. Edinburgh, t o appear. 16. J. Garcia-Melitin, J. Rossi, Boundary blow-up solutions to elliptic systems of competitive type, J. Differential Equations 206 (2004), 156-181. 17. J . Garcia-Melitin, J. Sabina de Lis, Elliptic problems under a singular boundary condition, Proceedings of the XVII Conference on Differential Equations and Applications, CD-Rom ISBN 84699-6144-6, Salamanca (Spain), 2001. 18. J. Garcia-Melitin, A. Sutirez, Existence and uniqueness of positive large solutions to some cooperative elliptic systems, Advanced Nonlinear Studies 3 (2003), 193-206. 19. B. Gidas, J. Spruck, A priori bounds f o r positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883-901. 20. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1983. 21. J. B. Keller, Electrohydrodynamics I. T h e equilibrium of a charged gass in a container, J. Rational Mech. Anal. 5 (1956), 715-724. 22. J. B. Keller, O n solution of Au = f ( u ) ,Comm. Pure Appl. Math. 10 (1957), 503-5 10. 23. C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contributions t o Analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 245-272. 24. J. L6pez-G6mez, Permanence under strong competition. Dynamical systems and applications, 473-488, World Sci. Ser. Appl. Anal., 4, World Sci. Publishing, River Edge, NJ, 1995. 25. J. Lbpez-Gbmez, The maximum principle and the existence of principal eigen-

Remarks on Large Solutions

26. 27. 28. 29. 30.

57

values for some linear weighted boundary value problems, J. Differential Equations 127 (1996), no. l, 263-294. J. Lbpez-Gbmez, Existence and metacoexistence states in competing species models, Houston J . Math. 29 (2003), 483-536. J. Lbpez-Gbmez, T h e boundary blow-up rate of large solutions, J . Differential Equations 195 (2003), no. 1, 25-45. R. Osserman, O n the inequality Au 2 f(u),Pacific J. Math. 7 (1957), 16411647. S . I. Pohoiaev, T h e Dirichlet problem f o r Au = u2,Dokl. Akad. Nauk. SSSR 134 (1960), 769-772. H. Rademacher, Einige besondere probleme partieller Differentialgleichungen in Die Differential und Integralgleichungen der Mechanik und Physik (by P. Frank and R. von Mises), vol. I, Rosenberg, New York, 1943, 838-845.

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WELL POSEDNESS AND ASYMPTOTIC BEHAVIOUR OF A CLOSED LOOP THERMOSYPHON'

A. JIMENEZ-CASAS Departamento de Matemcitica Aplicada y Computacidn Grupo de Dincimica No lineal, Uniuersidad Pontificia Comillas de Madrid, 28015-Madrid1 SPAIN E-mail: ajirnenezeec. upco. es

We analyze the motion of a fluid containing a soluble substance in the interior of a closed loop under the effects of natural convection and a given external heat flux. This motion is governing by a coupled differential system. First, we show the well posedness of this system in a framework generalizing the existence results of [ 5 ] . Finally, we prove some result about the asymptotic behaviour for solutions of above system generalizing the results of [12].

1. Introduction

In the engineering literature a thermosyphon is a device composed of a closed loop containing a fluid where some soluble substance has been dissolved. The motion of the fluid is driven by several actions such as gravity and natural convection. In particular, we will consider the convective movements caused by inner solute fluctuations generated by a temperature gradient; this fact is known as the Soret effect, and it has been studied experimentally by Hart [l]between others. We study the evolution of the velocity of the fluid v, of the temperature T of the fluid and of the solute concentration S. We assume that the section of the loop is constant and small compared with the dimensions of the physical device, so that the arc length coordinate along the loop (x)gives the position in the circuit. The velocity v of the fluid is assumed to be independent of the position in the circuit, i.e., it is *Work supported by projects BFM2003-03810 and BFM2003-07749-C05-05/FISICICYT, Spain. 59

A . Jamknez- Casas

60

assumed to be a scalar quantity depending only on time, v = v(t). The other relevant quantities, namely temperature, T ( t ,x ) , and concentration of the solute, S ( t , x ) , are assumed to depend on time and position along the loop. We assume that the average circulation is generated by the net buoyancy torque exerted by both solute concentration and temperature, and is retarded by viscous drag at the wall, i.e., by friction forces. In addition we assume that the variations of temperature are independent on the solute concentration. We consider the distribution equation of solute into the loop as in [l]and [8],where has been used the conservation of mass for the solute and has been assumed that the fluid also transports the solute, and be generated by Soret diffusion and reduced by molecular diffusion. The evolution of the above quantities is given by the following ODE/PDE system (cf. [4],[5],[6],[14] for further details)

dv

f

+ G(+)+= (T - S ) f, ~ ( 0=) vo dT dT - + V= h(v)(Ta- T ) , T(0,X ) = TO(.) at ax as as = C- a2s - b-a2T S(0,X)= SO(X) -++at ax ax2 6x2' *

(1)

It is important to note that all functions considered are 1-periodic respect to the spatial variable. The function f describes the geometry of the loop and the distribution of gravitational forces, so note that f f = 0 where $ = dx denotes integration along the closed path of the circuit. We consider the general geometries as in [14]. In the sequel we assume that G and h, are given continuous functions, such that G(v) 2 Go > 0, h(v) 2 ho > 0 and T, is the (given) ambient temperature. The functions G(v), which specifies the friction law at the inner wall of the loop ([8],[9],[11],[14]), and h which prescribes the heat flux at the wall of the loop (cf. [9]),are given by different forms. The diffusion coefficients b, c are positive constants and we note that b is proportional to the Soret coefficient, therefore if we assume it to be zero, i.e., if we neglect the Soret effect, and we start with an homogeneous initial concentration of solute, then S remains constant in time and space in Eq. (1)and, since $ f = 0, then Eq. (1) reduces exactly to the model in [11],[14] and [12]. In the present work two main results are presented:

s,'

- First, in Section 2, we prove that Theorem 3.1 in [5] is also true when we consider initial data in other more general space.

Well Posedness and Asymptotic Behaviour of a Closed Loop Thermosyphon

61

- Finally, in Section 3, under suitable conditions we prove that any solution either converges to the rest state or the oscillations of velocity around w = 0 must be large enough. This result generalizes the one proposed in [12] for a thermosyphon model including a twocomponent fluid. 2. Well posedness, existence and uniqueness of solutions

To prove the existence and uniqueness of solutions we will consider the suitable phase space. 2.1. Well posedness. Discussion about the suitable

framework Using the notation in [5], we define:

Definition 2.1. We denote by admissible space any Banach space X of 1-periodic functions, verifying the following properties

+

i) 11g(. k)llx = 11g11x, for every Ic E R and g E X. ii) Given g E X, then the function R + X, defined by k is continuous.

H

g(.+k) E X

Remark 2.1. Examples of admissible spaces are W$;*(O, l),C:eT(O,1) or C:;F(O, 1) among others. In this section, we will show the reasons to consider the phase space y , used in the next theorem, as suitable framework. Indeed, we assume that To,T, E X, where X is an admissible space in the sense of Definition 2.1. In order to prove the existence of solution of system Eq. (1) with initial data (WO,TO, SO)E Y , we develop a more general framework for SOand f . Let consider the third equation of system Eq. (1) St

d2 + c(-- 8x2 + 1)s= -b-T3x2 a2

- W-

dS

dX

+ CS

It is known that it is possible solve this working in the fractional power spaces, Z", associated to the sectorial operator (-& I) (cf. [2]). We note also Za+' is an admissible space in the sense of Definition 2.1 for T. We know that the map t H T ( t )is continuous in X but we also need that the map t H &T(t) is well defined in 2" that is t H T ( t )E Z"+l. To get this we consider a suitable relationship between X and Z", that is,

+

x

Lf Z"+1.

A . Jamdnez-Casas

62

&

On the other hand we need that the operator : Za+i -+ 2" will be well defined, for this S ( t ) E Z a + i , and hence it suffices to consider

so E .a+;. Therefore, we can consider the initial data (210, TO,So) E Y = R x X x ~f Zff+l,and choosing cr such that the map F ( t , v , S ) = -b&T cS, has good properties that allow us to use a fixed point argument. Arguing as in [5],first to prove the existence of solutions of system Eq. (1) considering an initial data in Y , we note that the condition iii) of Lemma 3.1 in [5],is now given by this

Za++ with X

vg +

Now, taking into account that

we have that Eq. (2) is true if

is satisfied. Ta have Lipschitz Next, we know that Eq. (3) is true if the functions TO, translations in Zff+'. Also it is suffices to suppose that To,Ta E Za+$to get Eq. (3) (cf. [4],[5]and [lo]). Thus, from the above arguing we get the following result: Lemma 2.1. Let T > 0 and v E C[O,T]and we consider To,T, E X being X an admissible space, then the jhnction T" defined by

is the integral solution of

dT" aT" V= h(v)(Ta- T") with T"(O,Z)= To(.) (5) dX dt and verifies the following properties i) For every t E [0,T ] and f o r every X admissible space, we have that

+

-

IIT"(t)IlxI m=4IITollx, IITallx).

(6)

ia) Suppose that the translations of Ta and To are Lipschitz in Zff+', i.e., there exists a positive constant Cd > 0 with d = a and d = 0 such that

Well Posedness and Asymptotic Behauaour of a Closed Loop Thermosyphon

63

+

llTd(. k ) - Td(.)llZa+i I cdllcl for every k E W. Then, the function T", defined by Eq. (4) is Lipschitz in t with values in Za+'. iii) W e consider the particular case X = H;er = W&; and we denote by H&: = (HjeT)' the dual space of Hiel. = Wig:. Thus, if T0,Ta E H;er, then Eq. (4) satisfies Eq. ( 5 ) pointwise, and T" E C([O,T],H;~,)n C'((0, T),H j e T ) .Moreover, if h E C1then T" E C2((0,T ) , LieT)and given IJ~E C[O,T]with i E {1,2}, we have that

for some M > 0 , where JIv1- ~

Im(t)

2 1 = 1 S~ U ~ ~ ~ [ ~ , ~v2(t)l. I

Finally, we have to choose the suitable space for the function f such that $(T - S)f will be well defined. Taking into account that T ( t ) E X, and S ( t ) E Za+i, it is suffices to take f E X' n (Z*+i)'. In this way, if X ~f Za+%+ Za+;then we can consider f E (Za+i)'. az Using the sectorial operator theory from [2], we have that --m I is sectorial in L$, = Z , 1 < p < cm with domain W;;; = Z1 and Za C W i Z p ,a! 2 0 , is the domain of (-& I)O. In the same way we have that z 1 / 2 = WLP and Z - 1 / 2 is the dual space of W;;;, with = 1. Under the above assumptions, taking different values of a we get the following particular cases with Y = R x X x Za+i. i) If cy = -;, then the phase space is Y = R x X x LZeT with X an admissible space such that X W:$, Ta E W;;; and f E L&T. It should be pointed out that this choice in the particular case when p = 2, corresponds to the hypothesis of Theorem 3.1 in [5]. ii) If we consider now a = -1 instead of a = -;, then the phase space is Y = R x X x 2-i with 2-i= (Wig:)' and X an admissible space such that X 24 = Wig;, with Ta E Wi;; and f E W$. In this situation, if we consider p = 2, we get T,, f E and (VO,TO, SO)E R x Hi,, x H i : .

+

+

4+

-

-

HieT

2.2. Existence and uniqueness of solutions

In this section we will show that under the above conditions we have the existence and uniqueness of solutions of system Eq. (1).

Theorem 2.1. Suppose that H ( r ) = rG(r) is locally Lipschitz and T, E X where X is an admissible space such that one of the following assertions is satisfied:

A . Jam&ez-Casas

64

+

a) X c W;;: and f E L&,, where $ f = 1,1< p < 00. In this case we consider Y = R x X x Lger. ii) X c W&$, and f E W:;:. In this case we consider y = R x X x

Cw;;:!)'.

Then, in both situations, given (wo,TO,SO)E Y , there exists a unique solutions (v,T , S ) E C([0,GO), y ) . Moreover the map:

S*(t)(vo,To,So) = ( v ( t ) T , ( t ) S(t)), , t20 defines a Co semigroup an Y . Proof. Taking into account the above section together Lemma 2.1, and arguing as in [5], we conclude the result. To get this we cover several steps. First, we prove there exists a unique local solution (v, T , S ) E C([O,T ] ,Y ) of Eq. (1). To prove this we use a fixed point argument. We consider the space from i) (analogous working in ii)), W = {(v,S) E C([O,T];R x L~e,),v(O)=vo,S(O)=So such that Iv(t)-vol I ~ I , / I S ( ~ ) - S O ( I LI ~ 72 ~, for every t E [ O , T ] } , with ~ , y i , ZE {1,2}, some fked positive conis a Banach space with the norm ll(wlS)IIw = stants. Then (W,11 . SUPt,[O,7] II(V(t)? s(t))llWxLP,,, = ll~lloo+ llSllL=(LP,,,)* Let U = (v,S) E W and J ( U ( t ) )= (w(t), R(t,x ) ) be the solution of the system

dw -

= -G(v)zJ

+

f

(T" - S ) . f , ~ ( 0=) 210, d2T" (7) - - C+ C R= CS - W- b-, R(0, X ) = SO(X), at ax2 ax ax2 dR

dt d2R

dS

i.e.,

where A is the operator defined by A =

F is given by F ( t , U ) =

(:4-g++ 0

, the nonlinearity

1)) $(T"(t)- S)f

-WE+ b&T"(t) + CS ) and -G(v)v

T" is defined in Eq. (4). Then we have J ( U ( t ) )=

(:i:ii)

with

Well Posedness and Asymptotic Behaviour of a Closed Loop Thennosyphon

R(t) = J2(U(t))= ,-C(-&)tS

+

I'

65

0

as

a2

e ' - c ( - ~ + ' ) ( t - T ) [ c S (-r )v(r)- - b-T"(r)]dr. ax ax2

(10)

Since T,, TOE X c WP$, from Lemma 2.1, T" verifies T" E C([O,71, X ) and &Tu E C([O,T],L~,,).Thus, from the continuity of G, we have Moreover e-c(-s+')tSo E C([O,.r];L;,,)and that J l ( U ( t ) )E C([O,T];R). cS(t)- v ( t ) E -b&T"(t) E L"(O,T, so applying the regularity results from [lo],we obtain

and hence J ( U ( t ) )E C([O,71;R x LFe,.). Next, we prove that there exist T > 0, y1 and 7 2 small enough such that J is a contraction in (W,(1 . llm). In second step, we prove the local solution is defined for every t 2 0. For this, we suppose the solution has been extended to a maximal interval of time [0,T ) , with (w,T ,S ) E C([O,T ) ; y ) and we prove the norm of the solution in y remains bounded in finite time intervals. 0 If we consider the case p

= 2,

we get the next result.

Corollary 2.1. Suppose that H ( r ) = rG(r) is locally Lipschitz and the functions T, and f E Hie,.. Then given (VO,TO,SO)E Y = R x Hic, x H;:, there exists a unique solution (w,T,S) E C([O,m),Y). Moreover we have that

w E C'(O,m),T E C(O,oo;H~,,)nCl(o,m;L;,,),

s E c ( ( 0 , H~;)A; st E C((0, m ) ;HA;) for every

E

E (0,l).

Proof. We note that the nonlinearity F z ( t , S ) = b&T - v asz is locally Lipschitz in S and Lipschitz in t , when we consider the map F2 : R x H;: + H;:, where Hp7e"T is the dual space of H:er. Now applying the sectorial theory from [2], we get S € C((O,T);L&,)and St E c((0,~); H;:), for every E E (0,l). Next, we note that S verifies the equation - c B = Fz(t,S ) by this way taking into account the above regularity of S and T we obtain

g,

A. Jamdnez-Casas

66

that FZ(t, S) E C((0,T),H$), and by elliptic regularity we get that S E c ( ( 0 ,Hier). ~); 0 From above section, we have also the following result:

Corollary 2.2. W e suppose that H ( r ) = rG(r) is locally Lipschitz, h E C1, T a E Hier and f E Lger. Then, given (210, To,SO)E Y = R x H$.r x L;,,, there exists a unique solution of system Eq. (1) globally defined, i.e., (v,T ,S) E C([O,m),Y ) . Moreover, the function S*(t)(vo,TO,SO)= (v(t),T ( t ) ,S ( t ) )t, 2 0 defines a Co semigroup in y . We will see a result about the existence and behavior of constant solutions with respect to the spatial variable, i.e., depending only on time.

Proposition 2.1. Under the hypothesis of Theorem 2.1 i f we suppose that T a is a constant finction, i e . , T, E R, then the unique solutions of system Eq. (1) depending only on time, i.e., the solutions such that T = T ( t ) and S = S ( t ) , are given by the solute concentration constant in ( z , t ) ,i.e., S ( t , x ) = SO E R. Moreover these solutions ( v ( t ) , T ( t ) , S o )converge to (0,T,,SO)exponentially when the time goes to infinity. Proof. If $$ = = 0 then taking into account that $ f = 0 we have that $(T - S)f = 0. Therefore from the system for the constant solutions in x , we get that S ( t )= So E R, Y = v0e-J; G(v)dr and T ( t )= T a (To T,)e-JJh(v)dr,and hence v ( t ) 3 0 and T ( t ) + T, exponentially, when

+

t+m.

0

3. Asymptotic behaviour for solutions under orthogonality

condition In previous works, like [4]and 151, the asymptotic behaviour of the system Eq. (1) for large enough time is studied. In this sense the existence of a inertial manifold associated to the functions f (loop-geometry) and T, (ambient temperature) have been proved. The abstract operators theory ([2],[13] and [lo]) has been used for this purpose. In this section we prove in Proposition 3.1 the results which rise an important consequence: for large time the velocity reaches the equilibrium -null velocity-, or takes a value to make its integral diverge, which means that either it remains with a constant value without changing its sign or

Well Posedness and Asymptotic Behaviour of a Closed Loop Thennosyphon

67

it will alternate an infinite number of times so the oscillations around zero become large enough to make the integral diverge.

3.1. Previous results and notations First, we note that in this section we consider the case in which all periodic functions in Eq. (1) have zero average, i.e., we work in y = R x x Lie,, where = {T E Hier with $ T = 0) and = {S E Lie,. with $ s = 0). In effect, we note that integrating the third equation of Eq. (1) with = -v $ 8.5 respect to x , since T and S are periodic functions we have $

Hier

Hier

tier

+

b$ @ c$ = 0. Therefore, $ [$ Sdx] = 0 and $ S is constant respect to t , i.e., $ S = $So = mo. Moreover, integrating with respect to x the second equation of Eq. (1) and taking into account again the periodicity of T , we have that $($ T ) = h(v)($T,-$T). there fore if we consider now^ = T - $ T a n d a = S-$So, then from the second and third equation of system Eq. (l), we obtain that T and a verify the equations

dr dt + v-da0X = h(v)(r,

- r ) , r ( 0 ) = ro = TO-

f

To

(11)

where I-, = T, - $Ta. Finally, since $ f = 0 , we have that $(T - S)f = $(r - a)f,and the equation for v reads

dv

dt + G(v)v =

f

(T - U ) . f ,

~ ( 0= ) VO.

(13)

Thus, from Eqs. ( l l ) , (12) and (13) we have (v, T,a) verifies system Eq. (1) with r,, TO, a0 replacing Ta,TO,SOrespectively and now $ T = $ u = $ T, = $a0 = $70 = 0. Note that if T a is constant, then T, = 0. Observe moreover that from the equation above for the average of T and taking into account that h 2 ho > 0 , we have that $ T converges to $ T, exponentially as t --f 00. Also note that to obtain the original dynamics we put v, T = T+$ T ,S = a mo, which shows that the dynamics is essentially independent of mo. Thus, using again variables v,T and S instead of v, T and a we consider the system Eq. (1) with $TO= 0 , $SO = 0 , $ T, = 0 and $ T ( t )= $ S ( t ) = 0 for every t 1 0. We then consider the semigroup S*(t)generated by Eq.

+

A . JamdnetCasas

68

(1) and given by Corollary 2.2, which is defined by S*(t)(vo,To,So) = (v(t), T ( t ,.I, s(t,.)I. In this section we assume also that G*(r)= T G ( T )is locally Lipschitz, h E C1, T, E and f E L:eT are given by following Fourier expansions

HieT

c

Ta(z)=

f (z) =

bke2akix;

kEZ*

c

cke2*kix;where Z* = Z \ (0)

,

kEZ*

(14) while TOE H:eT and SOE LEeT are given by

To(.) =

c

akoe2skix,So(.)

kEZ*

kEZ'

Finally assume that T ( t , x )E fize,. and S ( t ,x) E LBeTare given by dk(t)e2"kiZ Z' = Z \ (0).

ak(t)e2"kix and S ( t , x ) =

T ( t , z )=

kEZ*

kEZ*

(15) We note that Cik = - a k (& = -dk) since all functions consider are real and also a0 = do = 0 since they have zero average. Now we observe the dynamics of each Fourier mode and from Eq. (l), we get the following system for the new unknowns, v and the coefficients ak(t) and dk(t): dv

{

-k G(v)v = xkEZ*(ak(t) - dk(t))C-k &(t) [2TkiV(t) h('U(t))]ak(t) = h(v(t))bk &(t) -k [2TkZW(t) 4CT2k2]dk(t) = -4bT2k2ak(t).

+

+

+

(16)

We consider the functions T, and f given by following Fourier expansions

Ta(S) =

c

bke2"kix;

kEK

f (x)=

c

Cke2"kix;

(17)

kEJ

where J={kEZ*/Ck # O } , K = { k E Z * / b k #O}withZ*=Z\{O}. First, from the equations Eq. (16) we can observe the velocity of the fluid is independent of the coefficientsfor temperature ak(t) and the salinity dk(t) for every k E Z* \ ( K f l J ) . That is, the relevant coefficients for the evolution of the velocity are only ak(t) and dk(t) with k belonging to the set K n J. This important result about the asymptotic behaviour has been proved in [4] and [5] using the inertial manifold theory (cf. [2],[13],[10]).

Well Posedness and Asymptotic Behaviour of a Closed Loop Thennosyphon

69

The aim is to prove the Proposition 3.1 which generalize the result of thermosyphon model without solute of [12]. To do so we examine which are these steady-state solutions, also called equilibrium points. We have to make the difference between equilibrium points (constants respect to the time) null velocity, called rest equilibrium, and equilibrium points with non-vanishing constant velocity. Equilibrium conditions. i) The system Eq. (16) presents the rest equilibrium v = 0 , ak = bk and d k = k b k V k E K f l J under the assumption of the following orthogonality condition:

This equilibrium corresponds to v = 0, T = T, and S = SO= tT,, with the orthogonality condition $ T,f = 0. ii) Any other equilibrium position will have a non-vanishing velocity and the equilibrium is given by:

G(v)v= CkEKnJ

h(v

ho(v)+irkiv bkc-k h(v ak = h(v)+2!rkivbk - 4 6 ~ 'k 2 h ( v )

-

CkEKnJ d k C - k (19)

dk = ( h ( v ) + 2 r k i v ) ( 2 r k i v + 4 ~ r ~ k ~ ~ ~ ~ .

3.2. Asymptotic behaviour

Lemma 3.1. If we assume that a solution of Eq. (16) satisfies Iv(s)lds < 00, then for every q > 0 there exists to such that

S F

l t h ( r ) e - J : h(e-.f:2riku - 1)dr I Q

with t 2 t o

(20)

where h ( r ) = h(v(r)).Moreover lim sup I t-cc

lo t

(uk (r)e- 1 : 2rkiv

-

bk )

-4Clr2

k2( t-r)

I limsup luk(t) - bkl t+cc

S F

d

TI

I

+ vlbkl

with t 2 to.

(21)

Proof. If Iv(s)lds < 00, then for all 6 there exists to > 0 such that for t every t o 5 r I t we have I v1 I 6. Then, for any v > 0 we can take t o large enough such that

s,

Ie-Jp2rikv - 1 I q for all t o 5 r 5 t.

(22)

A . Jim4nez-Casas

70

Therefore writing h(r) = h(v(r))we get

lo t

5 ~ ( -e--f."h) 1 5 v with t 2 to.

h ( r ) e-.f:h(e-J,2Znikv -I)&

Taking into account that 7 -+ 0 for t from zero, we get Eq. (20). To prove Eq. (21), we write

lo

-+ 0;)

and h is strictly bounded away

t

(ak(r)e- -f:

t

4

27rkiv - bk)e-4cn2k2(t-r)dT =

t

T+Lo

(ak(r)-bk)e- J: 2nikve-4cn2k2(t-r)d

bk(e- J:

2i~ik~-l)~--4~~n~k~(t-r)d

and taking modulus in this expression the first term in the right member remains ( a k ( r )- bk)e- J:

2~nikv -4c~n~k~(t-r)d

rI 5

e

Now, for the second term in the right, considering the previous result together with Eq. ( 2 2 ) ,we have

and taking into account that

5 1, we get Eq. (21).

(1-e-4e"2k2(t-t0)) 4cZn2k2

Proposition 3.1. i ) W e assume that I = C k E K n j bkc-k = 0, with K n J finite set, and that a solution of Eq. (16) satisfies &O0 Iv(s)lds < 0;). Then the system reaches the rest stationary solution, that:

{

v ( t ) -+ 0, as t -+ 0O a&)

--$

b k , as

t -+ 00

d k ( t ) -+ kbk, as t

-+ 0;)

ii) Conversely, i f I = XkEKnJbkc-k # 0 then f o r every solution Iv(s)lds = 00, and v ( t ) does not converge to zero.

Proof. i) First, we study the behaviour for large time. The distance between the coefficients that represents the solution of the system, U k ( t ) and

Well Posedness and Asymptotic Behaviour of a Closed Loop Thennosyphon

71

to the values of those coefficients in the equilibrium, bk and pbk are computed. For to enough large, we known that for every t > t o and noting h(w(r))= h(r)

dk(t)

-o,f-

=ak(t0)e

ak(t)

+ bk

Pirikv+h

l:

h(r)e-j: 2aikv+hdr

and then

(23)

lo t

ak(t)-(l-eJi

2irkiv )bk

= ak(t0)e-j:

2aikv+h +bk

h(r)e-S:h(e-S~2irik~-l)dr.

Taking limits when t --+ 00, we get ( a k ( t ) - ( 1 - e - - f i 2 * i k v ) b k ) --+ 0, since ak(to)e--fi2*ikv+h 0 and from Eq. (20) we have that bkSttoh(r)e--f:h(e--:2akiv - 1 ) --+ 0. Now taking into account that ( 1 - e- -fi2*ikv)bk converges to bk for large time we conclude that --+

{

ak(t)

Integrating the equation for dk(t)

--+

I1 ( t )= X k E K n J a k ( t 1 c - k dk(t)

+

X k E K n J bkc-k = I.

(24)

we have that t

- J,k (2aikv+4cir2k 2 )

=dk(t0)e

bk

---$

lo

(- 4 b n 2 k 2 ) a k (r)e- .f:

(2Xikv+4C~Zk2)dr

(25)

from Eq. ( 2 5 ) we have that dk(t)

+

b - ( 1 - e- 4cir2k2(t-to) );bk = d&)e

st", (-4Cn2k2)e-4cR2k2(t-r)

- J,b (2irikv+4ca2 k 2 )

bc(ak(r)e--f:2"ikv

-bk)dr

(26)

working as in the a k case, we prove that Eq. ( 2 6 ) tends to zero and then we obtain the result about d k . In fact, taking limits when t + 00, we obtain d k ( t ) - ( 1 - J,o (2irkvi+4cn2k 2 ) --+ 0 and taking into e-4~ir2k2(t-to));bk b --+ 0, since d k ( t 0 ) e account Eq. (21) from Lemma 3.1 together Eq. (24),we have that

So that we get

A . Jimdnez-Casas

72

To conclude, we study now when the velocity v(t) goes to zero. Reading the equation for v , the first equation of system Eq. (16), as dv dt we have that

-

b + G(v)v = ( I l ( t )- I ) + (I2(t)- -bI ) + (1 + -)I C

C

where

11( t )= C k E K r l J 4 ) c - k I z ( t ) = C k E & q J dk(t)C-k. Now from Eqs.(24) and (27) for every 6 > 0 there exists t o such that IIl(s) - I1 5 6 and lI~(s)- $11 5 6 for every t o 5 s 5 t < 00. Let F ( t ) = J ,: e- J; Gdr, with

and then using L'Hopital's Lemma from [12] for the function F , we have lim sup F ( t ) 5 t-co

eJto Gdr

1 G(W)

< limsup -

GeJ;oGdT-

t+co

Hence, we find that

i.e., since 6 is arbitrary, we get

v(t) - (1

+ -Cb) I F ( t )

4

0.

(32)

We note that all above result, are valid for every I always we have the conditions Iv(s)lds < 00. Now, if I = 0 we get from Eq. (32) that v(t) 4 0. ii) If I = bkc-k # 0 and we assume that Iv(s)lds < 00. Then using again L'Hopital's Lemma from [12], for F, with Eq. (29) we have that

sow

zkEKnJ

lim inf F ( t ) 2 lim inf t-m

t+m

eJtoGdT

2 liminf

G ~ J ~ ~ G t+m ~ T

1 G(v)

-> 0,

Well Posedness and Asymptotic Behaviour of a Closed Loop Thermosyphon

73

and therefore from this together with Eq. (32) we conclude liminf Iw(t)l t-02

> 0,

which implies that Joo0Iv(s)lds = 03. This result is in contradiction with Iv(s))ds < 03, what implies that it is not a valid the initial condition hypothesis. 0

3.3. Concluding remarks Recalling that functions associated to circuit geometry, f, and to ambient temperature, T,, are given by f(x) = xkEJcke2.rrkix and T,(z) = bkeaTkiX,respectively. In [4],using the operator abstract theory, it is proved that if K n J = 0 , then the global attractor for system Eq. (1)is reduced to a point { (0, T,, :T,)} . In this sense the Proposition 3.1 offers the possibility to obtain the same asymptotic behaviour for the dynamics, i.e., the attractor is also reduced to a point taking functions f and T, without this condition, that is with K n J # 0 , it s enough that the set ( K n J ) # 0 , but xkc(Knj)bkc-k = 0. We note, the result about the inertial manifold ([4],[5])reduces the asymptotic behaviour of the initial system Eq. (1) to the dynamics of the reduced explicit system Eq. (16) with k 6 K n J. We observe also that from the analysis above, it is possible to design the geometry of circuit, f, and/or ambient temperature, T,, so that the resulting system has an arbitrary number of equations of the form N = 4n+ 1. Note that it may be the case that K and J are infinite sets, but their intersection is finite. Also, for a circular circuit we have f (z) a sin(z) bcos(z), i.e., J = {fl}and then K n J is either {fl}or the empty set.

xkEK

N

+

Acknowledgments

I would like to thank Francisco Javier Alcazar for his numerical experiments that confirm the result about the asymptotic behaviour for solutions under orthogonality conditions. Also, I would like to thank Professors Santiago Cano-Casanova and JuliBn Lbpez-Gbmez for give me the opportunity to participate in this special volume. References 1. J.E. Hart, J . of Heat Transfer 107,840-849,(1985).

74

A . Jamdnez-Casas

2. D. Henry, Lectures Notes in Mathematics 840, Springer- Verlag, (1981). 3. M.A. Herrero, J.J. L6pez-Velazquez, European J. Appl. Math. 1, 1-24, (1990). 4. A. JimBnez-Casas, “Dindmica en dimensidn infinita: modelos de campos de fase y un termosifdn cerrado”, PhD Thesis, U. Complutense de Madrid, (1996). 5. A. JimBnez-Casas, A. Rodriguez-Bernal, Math. Meth. in the Appl. Sci. 22, 117-137 (1999). 6. A. JimBnez-Casas, A. M. Lozano-Ovejero, Appl. Math. and Comp.124, 289318 (2001). 7. A. JimBnez-Casas, Nonlinear Analysis 47, 687-692 (2001). 8. J.B. Keller, J. Fluid Mech. 26, 3,599-606, (1966). 9. A. Liiian, in Fluid Physics, Lecture Notes of Summer Schools, (M.G. Velarde, C.I. Christov, Eds.) pp. 507-523, World Scientific, Singapore (1994). 10. A. Rodriguez-Bernal, Appl. Anal. 37,95-141, (1990). 11. A. Rodriguez-Bernal, E.S. Van Vleck, SIAM J. Appl Math. 58, 4, 1072-1093, (1998). 12. A. Rodriguez-Bernal, E. S. Van Vleck, Int. J. Bif. Chaos 8 , 1,41-56 (1998). 13. R. Temam, Appl. Math. Sci. 68, Springer-Verlag, New York (1988). 14. J.J. L6pez-Vel&quez, SIAM J.AppZ. Math. 54, 6, 1561-1593 (1994). 15. P. Welander, J. Fluid Mech. 29, 1, 17-30 (1967).

UNIQUENESS OF LARGE SOLUTIONS FOR A CLASS OF RADIALLY SYMMETRIC ELLIPTIC EQUATIONS*

JULIAN LOPEZ-GOMEZ Departamento de Matema'tica Aplicada Universidad Cornplutense de Madrid 28040-Madrid, Spain Email: LopetGomezOmat.ucm.es

This paper is dedicated to the memory of Jeszis Esquinas; exceptional mathematician and extraordinary f i e n d . . . In this paper we show the uniqueness and ascertain the boundary blow-up rates of the large solutions in a general class of radially symmetric semilinear elliptic boundary value problems. Our results are optimal in many circumstances and provide us with substantial generalizations of extremely sharp existing results. The main uniqueness theorem is a direct consequence of the maximum principle, rather than of the explicit knowledge of the boundary blow-up of the large solutions, as in most of the available references.

1. Introduction

Throughout this paper, given we shall denote

ZO E

B N , N 2 1, and R > 0, Ra > R1 > 0,

BR(ZO) := { z E BN :

IZ

- 201 < R } ,

and consider 0E{BR(~O)~ARI,R~(~O))

Also, for each x E R, we set

d(s):= dist (x,aR) . *Work supported by the Ministry of Education and Science of Spain under grant REN2003-00707. 75

J. L6pet-Gdmez

76

Using these notations, our main existence theorem can be stated as follows.

Theorem 1.1. Suppose X 2 0 , p > 1, and f E C[O, 00) satisfies

f ( t )2 f(s) > 0 if t 2 s > 0 .

(1)

Then, the singular boundary value problem

{

-Au

= XU - f ( d ( 2 ) ) ~ '

u=OO

in a , on d Q ,

(2)

possesses a unique (positive) solution. By a solution of (2) it is meant a positive strong solution L (as discussed by D. Gilbarg & N. S. Trudingers) satisfying

These solutions are usually referred to as large (or explosive) solutions of

-Au

= Xu - f(d(z)>up

in 0 .

Besides their own intrinsic interest, these uniqueness results provide us with the dynamics of the positive solutions in a large number of sublinear and superlinear indefinite parabolic problems in the absence of steady state solutions, when the dynamics is governed by the metasolutzons -appropriate extensions by infinity of the large solutions- (cf. R. G6mez-Reiiasco & J. L6pez-G6mezg, J. J. L6pez-G6mez & P. Quittner21, J. L6pez-G6mez & M. Molina-Meyer2', and the references there in). The interest in these problems goes back to the pioneering works of J. B. Kellerl' and R. Osserman22,whose a priori bounds still conform the basis to obtain existence results for (2). Actually, thanks to them, (2) admits a solution. But yet the problem of the uniqueness remains widely open. As the usual strategy adopted in the literature to obtain uniqueness results consists in characterizing the boundary blow-up rates of the large solutions, only partial uniqueness results are available. Basically, because in order to capture the boundary blow-up rates one needs to impose some additional (severe) restrictions on f ( t ) . In the special case when f(0) > 0, Theorem 1.1 follows from well known results of C. Loewner & L. Nirenberg13, V. A. Kondratiev & V. A. Nikishin", C. Bandle & M. M a r c ~ s ' ~L.~ V~ ~B, ~ Oand ~ ~M.~ Marcus , & L. VQron12. Later, Y. Du & Q. Huang6 and J. Garcia-Melibn et al.7 established it for the more general case when f(0) = 0 and

f ( t ) = Pt' [I + o(l)]

as t 1 0

(3)

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

for some positive constants 0> 0 and y

77

> 0. Theorem 1.1follows from the

fact that in case (3) any solution L of (2) satisfies

In this result, the uniqueness of L ( x ) is a direct consequence of the fact that any solution of (2) must satisfy (4). All these results where substantially generalized by F. C. Cirstea and V. R a d u l e ~ c u ~who * ~ ,proved the following general result. Theorem 1.2. Suppose f E C1[O,

00)

satisfies f(0) = 0 ,

Then, (2) has a unique positive solution, L , and

and h ( t ) is the unique solution to the integral equation

By differentiating (6) with respect to t and integrating reveals that, for each t > 0,

but yet no general uniqueness result within the spirit of Theorem 1.1 seems to be available, even in the presence of radial symmetry. Our proof of Theorem 1.1 relies upon (1) and the strong maximum principle, through a re-scaling argument showing that the minimal and the maximal large solutions are equal. Therefore, Theorem 1.1substantially differs from Theorem 1.2 because, instead of being based upon the boundary blow-up rates of the large solutions, relies upon the monotonicity of f. Nevertheless, in a further stage, the knowledge of the boundary blowup rates of the large solutions is imperative to use the localization method of J. L6pez-G6mezl6 in obtaining general uniqueness results for general problems, either in the absence of radial symmetry, or in the presence of radial symmetry without imposing (1) (cf. J. L6pez-G6mezlg for further

J. Ldpez-Gbmez

78

details). Consequently, this paper also analyzes the boundary blow-up of the solutions of (2). Our main result concerning this issue can be stated as follows. Theorem 1.3. Suppose X L 0 , p (11, and

lim tl0

> 1, and f

E C[O, m)

satisfies f(0) = 0,

F (t)F” (t ) = l o E (0,m) [F’(t)]2

where

Then, lim --

where L is the unique solution of (2). This result provides us with the boundary blow-up rate of the large solution of (2), in terms o f f , in a rather explicit way. A sufficient condition for (8) is given through the following lemma of technical nature. Lemma 1.1. Suppose p > 1, f E C[O, m) satisfies f(0) = 0 and (l),and there exists E > 0 such that:

Cl. f E C2(0,€] and, for each t E ( O , E ] , f’(t) > 0 and (Logf)”(t) < 0. C2. The function

is non-oscillating in (0, E ] , in the sense that either Btpl(t) > 0 for each t E ( O , E ] , or Bbl(t) < 0 for each t E ( O , E ] , or Blpl = Bo in ( O , E ] f o r some constant Bo. Then, the limit ( 8 ) exists and, actually, 10 E [l,m). The local logarithmic concavity of f in condition C1 is a rather natural and very weak non-oscillation condition, because, as a consequence of f ( 0 ) = 0, we have that limLogf(t) = -m. tl0

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

79

Subsequently, we will consider the function

a(t) := f(t) > o ,

O 0 if and only if Thus, in such case, := lima(t)

a0

tl0

20

is well defined. Moreover, by L’Hbpital’s rule,

Consequently, the local logarithmic concavity of f entails the existence of the first limit of (5). Necessarily, a0 = 0 if f(0) = 0. The mathematical analysis carried out in this work goes beyond revealing that such kind of restrictions on f are only necessary to characterize the boundary blow-up rates of the solutions of (2) through the solution of the associated onedimensional problem

{

u”=

fU*,

u(0) = 0 0 ,

t>O, u(0O) = 0

By Proposition 3.3 (see Section 3), if f E C 3 ( O , & ] and a and a‘ are nonoscillating in (0,E ] , then the functions Bbl defined in (11) are non-oscillating for every p 2 1. Moreover, for any p 2 1, P+l limB t t l 0 kJl( = a1(p+ 1) 1

+

if a1 := lima’(t) E [O, 00) . tl0

Consequently, as for any t E (0,E ] the following is satisfied

it turns out that second restriction of condition (5) can be equivalently expressed in the form

J. Ldpez-Gdmez

80

provided a' is non-oscillating and a1 < 00. Therefore, whereas Theorem 1.2 is basically imposing a non-oscillation condition on B[1],our Theorem 1.3 only requires the non-oscillation of Bbl, where p equals the growth rate of the non-linearity at infinity, which seems far more natural. More crucially, by Holder inequality, we have that, for each t > 0 and p > 1,

and, consequently, formula (9) is substantially sharper than (7), because it gives the boundary blow-up rate even when f E L* , while for the validity of (7) one must impose f E La, which implies f E L A for all p > 1. Therefore, in the radially symmetric case, Theorem 1.3 is substantially sharper than Theorem 1.2. At a further stage, combining the previous results with the localization method of J. L6pez-G6mezl6, where the solutions of (2) were estimated by the large solutions in sufficiently small interior balls and sufficiently large exterior annuli -tangent to the boundary-, gives rise to much more general uniqueness results, where the decay rates of the function coefficient in front of the non-linearity can vary along dR, though we send to the interested reader to J. L6pez-G6mezlg for further details in this direction. This paper is distributed as follows. In Section 2 we prove Theorem 1.1. In Section 3 we study the one-dimensional problem (12). Finally, Section 4 contains the proof of Theorem 1.3. The proof of Lemma 1.1 will be given in Section 3. 2. Proof of Theorem 1.1

Subsequently, for each sufficiently small Q 2 0, we consider

R,:=(ZER

: dist(z,dR)>v}

and, for each M > 0, we denote by O [ M , , ~ the unique positive solution of the truncated problems

-Au { ?L=M

= Xu - f(d(x))uP

in R, , on do,.

Then, according to J. L6pez-G6mez1*,the limits

(13)

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

81

provide us with the minimal and the maximal solutions of (2), because limMtc4OlM,,I equals the minimal solution of

{ -Au

= Xu - f ( d ( z ) ) u p

U=0O

in R q , on dR,,

for sufficiently small q 2 0. By construction, Lmin I L a

x

*

Moreover, as the unique solution of (13) -already denoted by O[M,,~- is radially symmetric, &in and Lmax also are radially symmetric. This ends the proof of the existence. To prove the uniqueness we have to distinguish two cases. First, suppose

R = BR(z0).

(14)

Then, for each z E 0, d(z) = R - r ,

r

:= 1z - 2 0 ) .

Subsequently, for each E E (0, R), we set

Z(z) := Lmin(z0 + ps(z - 2 0 ) ) ps := R--E’

0 5 1z-zoI

IR--E.

This function satisfies

E

= 00

on dBR-,(zo).

Moreover, for each z E B R - ~ ( z owe ) , have that

-AE(z)

5

+

-P:ALmin(zo P ~ (Z ZO)) - pZf(R - pEJz- ZO))L.”(X)

= p:XZ(z)

2 XE(Z) - p:f(R - )II: - SOJ)Z.”(Z) = XE(z) - p:f(d(z))L.P(z), since pE > 1, X 2 0, and

f ( R - Psla: - 501) 5 f ( R - 12 - zol)

7

because f is non-decreasing. Thus, the function __

E := p~p’l L.

in B R - & ( ~ O ,)

provides us with a supersolution of -Au = Xu - f ( d ( z ) ) u p in B R - & ( X O ) , on ~ ’ B R - & ( ~ o )

U=0O

J. Ldpez- Gdmez

82

Therefore, by the construction of the maximal solution, as a consequence of the strong maximum principle, we find that for each E E (0,R ) and 2

E BR--E(~O), 2 -

PEP-'L m i n ( z 0

+

- 20)) = i ( z ) 2 L m a x ( z )

PE(Z

I 0 shows that, for each 2 E B ~ ( z o ) ,

Consequently, passing to the limit as E Lmin(2)

*

2 Lmax(z)

7

which concludes the proof in case (14). Now, suppose =A

R ,~R( ~5 0 ).

(15)

Then, setting

R, := Ri

+ R2 2

'

we have that

Moreover, since Lmin and L,,, Lmin(z) Lmax(z)

= $min(r) = $max(r)

if R , < r < R 2 , if R 1 < r < R m .

R2-r r-R1

d(z) := dist (z,dR) =

are radially symmetric,

r:=(z-z01,

7

where $ J ~ ~ , , ( Tand ) $max(r) are the reflections around mal and the maximal solutions of -$'I

-

Y

$

I

XER,

1

A$ - f(R2 - T ) $ P ,

T = R,

of the mini-

R, < T < R2,

(16)

Now, we will show that any positive solution $ of (16) satisfies

$'(r) 2 0

for each

T

E

[&, R2).

(17)

This is clear if X = 0, and, more generally, if X < 0, because multiplying the differential equation by rN-' and integrating in (Rm,r ) shows that, for each T E (Rm,R2),

The argument in the case when X > 0 will proceed by contradiction. Suppose X > 0 and (16) possesses a solution 1c, for which there exists F E (Rm,R2) such that

$(?)

< 0.

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

83

Then, since $J'(&) = 0 and limrtR2+ ( T ) = 00, there exist

R,,,Iro_

$min(T)

$rnax(r)

,

E

[Rni R 2 ) .

This shows the uniqueness in case (15) and concludes the proof. 3. A pivotal one-dimensional problem

In this section we study the problem

{ where p

> 1, and f

u“ = f U P , t > 0 , u(0) = 0 3 , u(03)= 0 ,

E C[O, 03) satisfies

(1).

3.1. E x i s t e n c e a n d uniqueness

The main result of this section is the following.

Theorem 3.1. Suppose p > 1 and f E C[O,m) satisfies (1). Then, (21) possesses a unique solution.

Proof. Subsequently, for each M > 0 and b > 0, we consider the auxiliary problem U”

=

UP , t E [0,b] , , u(b) = 0 .

~ ( 0= ) M

Clearly, (I, 21) = (0, M ) provides us with an ordered sub-supersolution pair of (22). Hence, the problem possesses a solution u E [0,MI. Necessarily, u ( t ) > 0 for each t E [0,b ) , since u(t) > 0 for sufficiently small t > 0, because u(0) = M , and, for each t o E (0, b ) , the pair ( u , v ) = (0,O) is the unique solution of uI = 2, ,

v /= fUP ,

U(t0)

= v(t0) = 0 .

Thus, u(t) > 0 for each t E [O,b), since u 2 0. Note that u is decreasing, since u”(t) = f(t)uP(t)> 0 for each t E (O,b]. Moreover, the solution is unique. Indeed, if u1 # u 2 are two solutions of (22) with u ~ ( t o> ) u2(t0) for some t o E (0, b ) , then there exist tl E [0,t o ) and t 2 E ( t o , b] such that Ul(tj)

=U2(tj)9

j E {L2)

Pick t , E ( t l ,t 2 ) such that

,

Ul(t)

> u 2 ( t >, t E ( t l , t 2 ) .

J. L6pe.z-G6met

86

Then, 02

(u1 - U 2 ) 1 1 ( t , ) =

f (tm)[uY(tm)- U ; ( t m ) ] > 0 ,

which is impossible. Therefore, the solution of (22) is unique. Subsequently, we will denote it by 'LL[M,b].Note that '1L[M,b]5 M . Now, we claim that

imply Ml

2 U[Ml,bl]2 U[Mz,bz]

Indeed, in the interval [0,b 2 ] , the function persolution of

{

U"

= fQ

[o, b2] -

in

u [ M 1 , b l ] provides

(23)

us with a su-

, t E [0,bz] ,

~ ( 0=) M 2 ,

~ ( b 2= ) 0

,

while zero is a subsolution. Thus, this problem has a solution in between zero and ? + , f l , b l ] .As the solution is unique, (23) holds true. Thanks to (23), for each M > 0, the point-wise limit

is well defined in [0,00). Moreover, U M 5 M . In particular, the set of solutions '11[M,b],b > 0 , is bounded in C[0,00). Now, by a rather standard compactness argument involving the theorem of Ascoli-Arzela, it is apparent that U M provides us with a solution of

u" = f u P , t > 0 , u(O)=M, ~(00)=0. By construction, u M ( t ) > 0 and u)M(t)5 0 for each t 2 0. Actually, uh(t)< 0 for each t > 0 , since uL(t)= f (t)uR(t)> 0. Hence,

L := lim u ~ ( t2)0 . tta Necessarily L = 0 , since f is bounded away from zero at infinity. Consequently, U M is a decreasing function approximating zero as t t 00. We now show that U M is the unique solution of (25). The argument proceeds by contradiction. Suppose u1 # u2 are two (positive) solutions of (25) with ul(t0) > uz(to) for some to > 0. Then, there exist tl E [O,to)

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

87

and t 2 E (to,001 such that u l ( t j ) = u2(tj),j E {1,2}, and ul(t)> uz(t) for each t 6 (tl, t 2 ) . Pick t, E (tl, t 2 ) such that ‘1Ll(trn)- U2(trn) =

t€[ti,tal

{‘1Ll(t)- U Z ( t ) } *

Then,

0 2 (‘1L1- uz)”(tm) = f ( t r n ) [ q t m )- ‘LL3tm)l> 0 , which is impossible. Therefore, the function U M defined through (24) provides us with the unique solution of (25). Now, due to (23), for each 0 < M < and b > 0 we have that u [ M , b ] 5 U G . , ~ and, ] hence, passing to the limit as b t 00 gives U M 5 UG. Actually, u ~ ( t 0 for ) ufi(to),necessarily uh(t0) = uX(t0) and, hence, by the which U M ( ~ O= uniqueness of the Cauchy problem

u’= 2 1 , one has that

= fUP,

U(t0) =UM(t0)

U M ( ~= ) u f i ( t )for

, v(t0) = ‘Uh(tO),

each t 2 0, which is impossible, because

M = U M ( 0 ) < A2 = U f i ( 0 ) . Subsequently, for each b

> 0, we denote by Cb the minimal solution of

{

d‘= fG’, 0 < t < b , ~ ( 0= ) 0 0 , u(b) = 0 0 ,

whose existence was established in J. L6pez-G6mez14 through the a priori bounds of J. B. Kellerll and R. Osserman22. Fix M > 0 and b > 0. Then, there exists a sufficiently small E > 0 such that [b

2 M 2 U M in [0,E ] U [b - E , b]

and, hence, C b provides us with a supersolution of

{

u”= f U P , U(E)

E

0 and b > 0, UM

5 Cb

in [O, b ] .

Consequently, by the monotonicity in M , the point-wise limit

C : = lim u M Woo

(26)

J. Lbpez-Gdmez

88

is well defined in [0, m). By a rather standard compactness argument based on (26), it is apparent that C solves (21). It should be noted that P(t) < 0 and C l l ( t ) > 0 for each t > 0, and that limttm C(t) = 0. Actually, C provides us with the minimal solution of (21), since any other solution L must satisfy L 2 C, because L 2 U M for each M > 0. Consequently, we will subsequently denote it by Cmin. To establish the existence of a maximal solution one can proceed as follows. For each E > 0, let CLin denote the minimal solution of

{

u”= f U P , U ( & )= C Q ,

t > E ,

u ( m )= 0 ,

whose existence follows easily from the previous results. Now, for each M > 0 and E > 0, let u& denote the unique solution of

{

UII

= fUP

U ( & )= M

, t >E , ,

u(CQ)

=0 ,

and pick 0 < E < 2. As in the interval [E,m),uk provides us with a subsolution of (28), we find that U L5 u& there in, and, hence, passing to the limit as M t 00 shows that Ckin I Ckin in [Z, CQ). Consequently, the point-wise limit

Cmax

:= limCLin El0

is well defined and it provides us with a solution of (21). From the construction itself, it is apparent that Cmax is the maximal solution of (1). So far, we have established the existence of a minimal and a maximal solution, Cmin and Cmax, for (21), in the sense that any other solution C must satisfy emin

I C I emax

*

To complete the proof of the theorem it remains to show that emin = Cmax. Note that in all previous cases, the limiting solutions are non-negative and, hence, they must be positive by the uniqueness of the associated Cauchy problem. Thus, they cannot admit an interior local maximum in [0,m) as they are convex. Let l be any solution of (21) and consider, for each E > 0, the shifted function f(t) := C(t - E ) , t > E . By (l),for each t > E we have that

T ( t )= Y ( t - E ) and, hence,

= f (t - E ) P ( t - E )

I f (t)P(t- E ) = f (t)P(t)

provides us with a supersolution of (27). Note that

89

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

where !&in is the minimal solution of (27) and u& is the unique solution of (28). Now, we shall show that for each M > 0 and E > 0, (30)

.?>Uh.

Indeed, since Z(0) = 00, (30) holds true for sufficiently small t - E > 0. Thus, if (30) fails to be true, there exist tl > E and t z E (tl,001 such that j ( t j )= u&(tj),j E {1,2}, and

u&(t)> l ( t )= q t - E ) ,

t

E

(tl,t z ) .

Pick tm E (tl,t z ) such that

Then,

o L ( u b - .?)"(tm) L f ( t m ) [ ( U h ) P ( t m ) - .?p(tm)I > 0

7

which is a contradiction. Thus, (30) holds true and, passing to the limit as M t 00, we find from (29) that for each t > E ,

l&,,(t) 1.F(t) = l ( t - E ) . Consequently, passing to the limit as E

t,,,(t)

10, we find that, for each t > 0,

= limlki,(t)

5 l ( t ).

El0

Therefore, l = lmax. This concludes the proof.

0

Subsequently, we shall denote by l ( t ) ,t > 0, the unique solution of (21).

Remark 3.1. In the special case when, for some constants p > 0 and Y > 0,

f ( t )= Pt7,

t L0,

it is easy to see that

which is the most common case analyzed in the literature. The following result collects some useful properties of l ( t ) . Lemma 3.1. Suppose p > 1 and f E C[O, 00) satisfies (1). Then, liml'(t) = -00 tl0

,

lim ['(t)= 0 ,

ttm

lim [ f ( t ) l P f ' ( t= ) ]0 .

ttm

(32)

J. Ldpez-Gdrnez

90

Proof. Since C’‘(t)

> 0, t > 0, P(t) increases. Thus, L := limC’(t) E

[-00,

tl0

01

is well defined. Suppose, L E (-00, 01. Then, for each 0 < s

< t,

t

C(t)- C(s) =

C’

1 L(t - s)

and, hence, C(s)

5 C(t) - L(t - s)

which is impossible, because lim,lo C(s) = 00. This shows the validity of the first limit of (32). Now, for each integer n 12, there exists t, E [n - 1,n] such that C(n) = C(n - 1)

+

Ll n

C‘ = C(n - 1)

+ C’(t,)

and, hence, there exists a sequence t,, n 2 1, such that t, t 00 and C’(t,) -+ 0 as n 4 00, since C(n) -+ 0. Therefore, by the monotonicity of C’, we obtain that C’(t) 1 0 as t 00, which concludes the proof of the second limit. Finally, note that

0 < fP+l= C”C = (Pa)’- ( . e l ) ?

(33)

We already know that limttoo(C‘!) = 0. Moreover, !‘C is increasing, because (.ell)’

= C”C

+ (C’)2

> 0.

Thus, limtroo(&’l)’= 0 and the third limit of (32) falls down from (33). 0 3.2. Global lower estimates for t ( t )

The main result of this section reads as follows.

Proposition 3.1. Suppose p > 1 and f E C[0,00) n C1(O,0o) f ( t ) > 0 and f’(t) 2 0 for each t > 0. Then, for each t > 0,

where C is the unique solution of (21) and

satisfies

Uniqueness of Large Solutions of Radially S y m m e t r i c Elliptic Equations

Therefore, for each t > 0, where

Proof. Multiplying

= f P by

el’

P

and rearranging terms gives

Moreover , according to Lemma 3.1,

Thus, integrating (37) in [t,00) and using f’ 2 0 gives

>-

z

p + 1f ( t ) [ P + l ( t )

and, hence, for each t Going back to the differential equation, (39) implies

and, hence, for each t > 0,

Note that the change of variable

-t’(t) = [ q ( t ) ] - S transforms (40) into

> 0.

91

J. Ldpez-Gbmez

92

Moreover, by Lemma 3.1, ~ ( 0 = ) 0, because limtlot’ = -co. Hence, integrating (42) in [0, t] shows that, for each t > 0,

The estimate (34) follows readily by substituting (43) into (41). Finally, the estimate (35) follows from (34) integrating it in [t,m). Note that, due to Lemma 3.1, limttm !’(t) = 0. 0 In the special case when f ( t ) = P t Y , t 2 0, for some positive constants and y, the function F ( t ) introduced in (36) is given by

,

P

t>O.

Therefore, it has the same behaviour as the solution of (21), given by (31). Consequently, the general global lower estimate (35) seems really sharp. Actually, if f is assumed to be constant for sufficiently large t , then (38) becomes 2 [W2 =P -f(t)PeP+1(t) +l for sufficiently large t , and, consequently, it is indeed modulated by F ( t ) . 3.3. Global properties of the function F ( t )

In this section we shall study some general properties satisfied by the function F ( t ) introduced in (36). It will be useful to express it in the form

F(t)=

lm a (Jd

p+l

t

where A ( t ) :=

,

f”)

t >0

The following result shows that it is well defined if f E C[O, co) satisfies (l), and it establishes some of its main properties.

Lemma 3.2. Suppose p > 1 and f E C[O,oo) satisfies (1). Then, F E C2(0, co) and, for each t > 0, F(t)> 0 ,

F’(t) < 0 ,

F”(t) > 0 .

Moreover, limF(t) = ca tl0

limF’(t) = -m, tl0

lim F ( t ) = lim F’(t) = 0

ttm

ttw

(44)

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

93

and

Proof. As f is non-decreasing, for each t 2 E > 0, we have that A(t) 2

f*(E)(t

p+t

-E)P-~

(46)

and, hence,

Therefore, F ( t ) is well defined for each t

> 0 and

lim F ( t ) = O

ttw

Moreover, F E C2(0,00), since A E C’(0,oo) satisfies A(t) (46) implies that

> 0, t > 0, and

-1 lim F’(t) = lim - = O . ttw A(t)

ttw

Also, A(0) = 0 and

Thus, for each t

> 0,

The second limit of (44) follows from the second identity of (48), because A(0) = 0. Now, since f is non-decreasing, we find that p+t

A(t)=

(Ltfh)’-’ ~ f . - ’ ( t ) t p p+l -l,

t>o,

and, hence, for each t E (0,1], we have

which ends the proof of (44). Similarly, (47) implies

A’(t) 2 ’*A(t)t-’ P-1

,

t > 0,

(49)

J . L6pe.z-Gbmez

94

and, hence, we find from (48) and (49) that

Now, we shall show that

Indeed, for each E

Thus, for each E

> 0 and t

E ( O , E ] , we have that

> 0,

because A(0) = 0. Moreover, A ( s ) 2 A ( t ) for each s E [ t , ~since ] , A’ and, hence,

> 0,

Therefore,

As this inequality is satisfied for all E > 0 and we are taking the limit of positive quantities, (51) holds. Consequently, we find from (48) and (51) that

-F’(t) lim -= lim tlo F ( t ) tlo Note that (50) and (52) also imply that lim-

F”(t)

F(t)

=lim-

F”(t)

. lim- -F’(t)

-F’(t) t l 0 F ( t ) This completes the proof of (45) and concludes the proof. tl0

tl0

0

Our next result establishes an extremely sharp property of F ( t ) , whose significance will be apparent in the following section. To state it is appropriate to introduce the following concepts.

Definition 3.1. A function h E C ’ ( ~ , E ]E, > 0, is said to be non-oscillating in (0, E ] when some of the following alternatives occurs:

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

95

1. h’(t) > 0 for each t E ( O , E ] . 2. h’(t) < 0 for each t E (O,E]. 3. h = ho in ( O , E ] for some constant ho. The function h is said to be non-oscillating at t = 0 if it is non-oscillating in (0,6] for some 0 < 6 5 E .

Definition 3.2. Given p > 1 and f E C[O, m) with f ( 0 ) = 0 , it is said that E Cp if there exists E > 0 such that:

f

C1. f E C2(0,&jand, for each t E ( O , E ] , f’(t) > 0 and (Logf)”(t) < 0. C2. The function

is non-oscillating in (0,E ] .

A detailed discussion concerning the meaning of conditions C1, C2 in the definition of the class C p will be carried out once concluded the proof of the next result.

Proposition 3.2. Suppose p > 1 and f following limit is well defined (see (48))

E

C p is non-decreasing. Then, the

Proof. Subsequently, we consider the function cp(t)defined through

By Lemma 3.2, cp(0) = 0. Moreover, for any t > 0 , cp(t) > 0 and, differentiating and rearranging terms, we find that

1

p’(t) = A’@)

O01 --1 t A

and

Also, after some straightforward calculations, it is apparent that

(55)

J. L6pez-Gdmez

96

where B ( t ) is the function introduced in (53). Note that setting

a ( t ) := f ( t ) (> O),

tE

f '(4

the function B ( t ) satisfies

a(t)B'(t)= -B(t) [d(t)+

-1

1

P+l

(O,E]

+ 1,

,

t

(58)

E (O,E] .

(59)

Moreover, by condition C1 of Definition 3.2, for each t E (0,el,

+ 1)n (0,E ] , we find from (59) that a(t")B'(t")= - ( p + 1) [d(t") + 1 + 1 = - ( p + l)al(t") < 0 .

Thus, for each t" E B - l ( p

Consequently, B'(i) < 0, and, hence, B(t) > p + 1 for each t E (0,t"), if such a t" exists. Suppose this is the case. Then, (59) implies

B (uB)' = 1 - p + l < 0 in

(0~5)

and, hence,

--

dt a B

in(0,ij.

>O

Thus, since

we find that ;d2 iii(Logl

t f")

>O

foreach t ~ ( O , t " ) ,

which is impossible, because

and, so, the second derivative should be somewhere negative in ( 0 , t ) for each t > 0. Therefore,

B(t) < p

+1

for each t E (0,E ] .

(61)

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

97

Subsequently we shall restrict ourselves to consider t E ( O , E ] . We claim that, for any sufficiently small r] E (0, E ] , the following holds

cp’(t)2 0

1

v

t E (O,V].

(62)

To prove (62) we argue by contradiction. Since cp(0) = 0 and cp(t) > 0 if t > 0, must be somewhere positive in any interval of the form (0,6], 0 0 (62) fails to be true, then cp’(t) must change sign in (0,6] for any 6 > 0. In such case, there exist two sequences t, 1 0 and s, L O , as n t 00, such that

cp’(tn)> 0 ,

cp”(tn)= ‘ p l l (S , ) = 0 ,

cp‘(s,)

1, [A’(t,)12 cp’(t,) + 1 [A’(%# cp!(sn)+ 1 since A ( t ) > 0 and A’(t) > 0 for each t > 0. Therefore, the function A ( t )A” (t) -1, t>O, [A‘(t)l changes sign infinitely many times as t 1 0. This is impossible, since (57)

implies

A(t )A” (t) - 1 = p - l ( ” ” - 1) [A’(t)I2 p+l p+l and, therefore, B ( t ) - ( p + 1) must change of sign infinitely many times as t L O , which contradicts (61). Consequently, c > 0 can be shortened, if necessary, to get (62) with 17 = E. Actually, by (62), if there exists 8 E (0, E ] such that cp’(8) = 0, then cp”(8) = 0 and (56) implies

+

A (8)A“(8) =I, [A‘(8)1

and, hence, B(8) = p 1. This contradicts (61) and, therefore, we can assume that E > 0 has been chosen to satisfy

cp’(t)> 0 , By (55), this shows that

v t E (O,E].

lm a,

1 < A’(t)

t

E

(O,E] .

Now, thanks to (61), we find from (57) that

A(t)A”(t) < 1, ps-l< [A‘(41 2

t E (O,&].

(63)

J. L 6pez- G6mez

98

Moreover, E can be reduced, if necessary, so that some of the following three alternatives occurs: (a) cp” 2 0 in ( O , E ] . (b) cp” 5 0 in ( O , E ] . (c) cp” changes of sign infinitely many times in any interval of the form (0,6],6 < E .

Suppose Alternative (a) occurs. Then, cp’(t)is non-decreasing and, hence, the following limit exists L:=limcp‘(t)>O.

(64)

tL0

Thus, by (55),

which concludes the proof of (54). Suppose Alternative (b) occurs. Then, it follows from (55), (56) and (63) that, for each t E (0,E ] ,

Therefore, the limit (64) is well defined, since cp’(t) is non-increasing, and, consequently, (54) follows. To conclude the proof of the proposition it remains to show that cp(t) cannot satisfy Alternative (c) if B ( t ) is non-oscillating in ( O , E ] . Indeed, by differentiating (56) and rearranging terms it is easy to see that

Thus, for any t^ E ( O , E ] such that

since

> 0.

cp”(0 = 0, we find from (57) that

Consequently, cp” cannot change sign in ( O , E ] if either

B’(t) > 0 for each t E ( O , E ] , or else B’(t) < 0 for each t E ( O , E ] . Suppose B(t)= BO for each t E ( O , E ] . Then, (59) and (61) imply 1 1 O < B o < p + l and d ( t ) = - - - = Bo P + 1

p+l-Bo

Bo(P+~)

99

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

for each t E ( O , E ] , and, hence, setting

y := Bo(P + 1)

p+l-Bo’ we find that there exists a constant C 0 such that, for each t E ( O , E ] ,

a(t)= 7-9

+ c.

Thus, it follows from (58) that there exists p

f(t) = P(t + YCIY7

> 0 such that

tE

(0,EI

*

Necessarily C = 0, since we are assuming that f(0) = 0. Thus, f ( t )= P t Y , for each t E ( O , E ] , and, hence,

Therefore,

which concludes the proof.

0

Remark 3.2. The limit (65) ranges in [l,(p+ 1)/2) as y fore, in general, the value of (54)depends upon f and p.

E [O,oo).

There-

The local logarithmic concavity (Log f)” < 0 in the definition of the class Cp is rather natural, since f(0) = 0 and, hence, lim Log f ( t )= --ca. tl0

-

-

Indeed, it is satisfied if f ( t ) P t Y for t 0 and some constants P > 0, y > 0, or f ( t ) -tLogt for t 0, or f ( t ) e-llt for t 0. Actually, by (60),(Log f ) ” ( t ) < 0 if, and only if, a’(t) > 0, and, therefore, condition C1 of Definition 3.2 holds if and only if the function a = f / f’ is non-oscillating. The following result provides us with a very simple criterion in terms of f ensuring that B(t) is non-oscillating for all p L 1. N

N

N

N

Proposition 3.3. Suppose E > 0, f E C [ O , E ] n C3(0,&], f(o) = 0, f’(t) > 0 for each t E ( O , E ] , and a := flf‘ and a’ are non-oscillating in ( O , E ] . Then, the function B ( t ) defined in (53) is non-oscillating for each p 2 1. Moreover, limB(t) = tl0

P+l

(P + 1)al + 1

i f a1 = lim a’(t) E [O, ca). tl0

J. L6pez-Gdmez

100

Proof. By the proof of Proposition 3.2, we already know that a’@) > 0 for each t E (0, E ] . Moreover, differentiating (59) shows that, for each p 2 1,

+

a”(t)B(t)+ 2a’(t)B’(t) a(t)B”(t)= -B’(t) p+

1

t E (014.

Suppose there is a p 2 1 for which B ( t ) is not non-oscillating. Then, there exist two sequences, tn and Snl n 2 1, such that t, 10 and s, 0 as n t 00 and, for each n 2 1,

I0,

B’(t,)B’(s,) Thus, for each n 2 1 and (2a’(Tn)

Tn

+

E

B”(t,) = B”(S,) = 0 .

{tn,s,},

we find that

5)

B’(7-n)= -a”(Tn)B(Tn).

Since a’ is non-oscillating in ( O , E ] , either a”(t) > 0 for each t E ( O , E ] , or a”(t) < 0 for each t E (0, E ] , or a’ = a1 for some constant a1 > 0. Suppose a” > 0, or a” < 0, Then, for each 7, E {tn,s,}, n 2 1, we have that sign B’(T,)

=

-sign a”(~,)

and, hence,

B’(tn)B’(sn)> 0 , which is a contradiction. Consequently, a’ = a1 in (0, E ] for some constant a1 > 0, and, hence, by the argument given in the last part of the proof of Proposition 3.2, a ( t ) = y-lt and f ( t )= P t Y for each t E ( O , E ] , where y = l / a l and P > 0. Therefore, B must be constant in (O,E]. This shows that B ( t ) is non-oscillating in (0, E ] for every p 2 1. Finally, by (61), Bo E [O,p 11. Thus, in case a1 E [0,co),it follows from L’H6pital’s rule that

+

and , therefore, P+l

Bo = - [ ( l P+2

which concludes the proof.

- a l p o + 11,

Uniqueness of Large Solutions of Radial& Symmetric Elliptic Equations

101

3.4. Global estimates of e ( t ) in terms of F ( t )

The main result of this section is the following.

Theorem 3.2. Suppose p > 1, and f E C[0,00) is a bounded function satisfying f(0) = 0 , (l),and

where F ( t ) is the function defined in (36). Then, there exist 0 < EO < CO such that, for each E E (O,EO] and C 2 CO,( E F , C F )provides us with a sub-super-solution ordered pair of (21). Thus, E F ( ~5) [ ( t )L C F ( t ),

t >0,

(67)

where l ( t ) is the unique solution of (21). Therefore, for each t > 0 , the following global estimates hold

Consequently, limC(t) = 0 0 ,

limC‘(t) = - 0 0 ,

tl0

-eyt) e(t)

eyt)

lim -= lim -=limtl0

tl0

(70)

tl0

-P(t)

tl0

eyt) =00. e(t)

Proof. Clearly, for sufficiently small E > 0 and sufficiently large C > 0, the functions EF and C F will provide us with a subsolution and a supersolution of (21) if, and only if, the auxiliary function

is separated away from zero and globally bounded in (0,00). Indeed, by Lemma 3.2, G ( t ) > 0 for each t > 0. Moreover, (47) and (48) imply

F y t ) = -pP +1 l 4t)

(I’

f&)

-1

(’f

*+I

t)> o ,

t

>o.

J. L6petGdmez

102

By (66), G(t) is separated away from zero and bounded on any compact subset of (0, m). Moreover, since f is non-decreasing and bounded above, the following limit exists

L := lim f ( t ) ttw

and, hence, there exists T

> 0 such that

L - 1 5 f ( t ) 5 L for each t 2 T . Thus, for each t 2 T ,

and, hence,

IT

( L - 1)’- (t - T ) s

+ts

5 A ( t ) 5 L‘-

Consequently, there exist two constants 0 < CI< C2 such that

Also, there exist 0 < C3 < C4 such that,

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

103

Therefore,

and, hence, G ( t )is separated away from zero and bounded in [0,00). This shows the existence of 0 < EO < Co such that EF is a subsolution of (21) and CF is a supersolution of (21) for each E E (0, EO] and C E [Co,00). To obtain (67) we consider E 5 eo < CO 5 C and, for each natural number n 2 2, the boundary value problem

{

u” = f u P , t E [n-’,n), u(n-1) = q F ( n - l ) , u(n)= q F ( n ) .

This problem has a positive solution, because (EF,C F ) is an ordered pair of sub and supersolutions. Moreover, by the sublinear structure of the problem, it is unique. Let denote it by un,n 2 2. By construction, for each n 2 2 we have that

EF 5 un 5 Cu,

in [n-l,n].

By a rather standard compactness argument, we can extract a subsequence of u,, n 2 2, converging to a solution of (21) uniformly in compact intervals of (0,oo); necessarily l , by uniqueness. This concludes the proof of (67). Now, by (67), we have that

~ ~ f ( t ) F 5~ l”(t) ( t ) = f ( t ) l P ( t )5 C P f ( t ) F p ( t )

for each t

> 0 . (74)

t >0,

(75)

Moreover,

&P-’f(t)Fp(t) 5 F”(t) 5 CP-lf(t)Fp(t),

because EF is a subsolution and C F a supersolution. The estimate (68) is an easy consequence of (74) and (75). Now, integrating in [t,00)the estimate (68) and using the fact that F ( t ) 4 0, F’(t) -+ 0 and l ( t )-+ 0 as t T 00, gives (69). The fact that l(t)T 00 as t 1 0 is satisfied by definition of l(t). Thus, (70) is an easy consequence of (44) and (69). Similarly, (71) follows from (45) and the global estimates (67)-(69). This concludes the proof. 0 3.5. The blow-up rate of l ( t ) at t = 0 The following result ascertains the blow-up rate of l ( t ) at t = 0.

J . Ldpez-Gbmez

104

Theorem 3.3. Suppose p > 1 and f E C[O,m) is a bounded function satisfying f(0) = 0, ( l ) ,and (66). Then,

Proof. By (72) and (73), it follows from (66) that

Now, consider the auxiliary function

By the global estimate (67), q ( t ) satisfies E 5 q ( t ) 5 C , t O

< E 5 qL

:= iiminfq(t) ti0

5 qM

:= limsupq(t)

> 0. Thus, 5 C.

tl0

To show the existence of limtloq(t) we argue by contradiction. Suppose q L < q M . Then, there exist two sequences t,, s,, n 2 1 such that

and, hence, by (78), we obtain that

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

105

because, due to Lemma 3.2, F ( t ) > 0 and F”(t) > 0 for any t > 0. Therefore, by (45) and (76), passing to the limit as n + 00 in these inequalities gives

and, consequently,

which contradicts the assumption exists

qL

< q M . Therefore, the following limit

w

qo := lim -

tio F ( t ) .

Finally, by L’H8pital rule, we find from (77) that

which concludes the proof.

0

4. Boundary blow-up rate of the large solution

The main result of this section is the following.

> 1, and f

Theorem 4.1. Suppose X 1 0 , p ( l ) ,and (66). Then

E C[0,00) satisfies f(0) = 0,

where L is the unique solution of ( 2 ) . Proof. The existence and uniqueness of L have been already shown by Theorem 1.1. It remains to prove (79). First, suppose

a = BR(20). Then,

L b ) = +(TI

7

7- = 12 - 201

,

where $J is the unique solution of

-+” $’(O)

-

v+’

= 0,

= A+ - f

$(R) = 0 3 .

(- ~ T)+P

in (0, R) ,

(80)

J. Ldpez-Gdmez

106

Now, we shall show that, for each sufficiently small E > 0, there exists a constant A, > 0 such that, for every A > A,, the function

& ( r ) := A + (1+ E ) (f )

2

C( R - r ) ,

0 Ir 5 R ,

is a positive supersolution of (go), where C stands for the unique solution of (21). Indeed,

&(O) = 0

and

lim$,(r) = 0 0 . rtR

4,is a supersolution of (80) if, and only if, 1+E r(1 + E ) -2NL(R - T ) + ( N + 3)L’(R-r) - ( 1 + ~ ) R2 R2

Thus,

(s)

A L X C ( R - r )[ [ ( R - T ) + ( l + E ) ( f ) - f ( R - r ) Cp(R- r )

[L(R - r ) A

Dividing this inequality by C”(R-r), it is apparent that of (80) if, and only if, 1 + C~( R - r ) ~ ( 1 E+) C’(R- r ) -2N( N 3)R2 P’(R - T ) R2 C”(R - T )

2

+(l+E)(f)2]p.

4,is a supersolution

C(R-r) A + ( I + & (;)2] ) ’ P ( R - r ) L(R - r )

where we have used C‘‘ = f L P (81) reduces to

C”(R-r)

]

+ +

[

2

2

(81)

> 0. By Theorem 3.2, at r = R the inequality

-(1+

+

E)

L -(1+

E)P,

which is satisfied because 1 E > 1 and p > 1. By continuity, there exists 6 = a(&) > 0, such that (81) holds for each r E ( R - 6, R ] . By choosing a sufficiently large A , (81) is satisfied in [0,R ] , since p > 1 and C is positive and bounded away from zero in compact intervals of (0,m). This shows that & is a supersolution of (80) for sufficiently large A. Now, we will show that, for each sufficiently small E > 0, there exists C < 0 such that

107

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

is a non-negative subsolution of (80). Indeed, by reversing the inequality (81), it is obvious that 14, is a subsolution of (80) if in the region where 2

c + (1 - E ) (f)L(R -

20

T)

the following inequality is satisfied

-2N-

T ( l - &) P(R - T ) + ( N + 3 ) - R' P ( R- T ) R2 P ( R - T ) e(R-r) A % ( R - T ) l ( R- T )

2

1-& l(R-T)

[

-t-(l--~)(i)~] (82)

As the mapping

.-(i) 2

[(R-T)

in increasing, for each C < 0 there exists

I = z(C) E

(0,R)such that

while

Moreover, C H z(C) is decreasing and lim z ( C ) = R ,

CI-00

Thus, proceeding as above, at

T

lim z(C) = 0 .

(83)

cto

= R inequality (82) reduces to

-(1 - E ) 5 - ( 1 -

€)*,

which is satisfied because 1 - E < 1 and p > 1. By continuity, there exists 6 = 6(a) > 0 such that (82) holds in [R- 6, R). Moreover, by (83), there exists C < 0 such that z ( C ) = R - 6. For this choice of C,& is a subsolution of (80). By construction, lim '&(')

T t R l ( R- T )

=1

+E

and

lim

+

(7.1 7qR I ( R - r )

Moreover, choosing a sufficiently large A gives

=I-&.

(84)

J. Ldpez-Gdmet

108

Thus, by the uniqueness of the solution L ( z ) ,we have that, for each E

11,

(I.

4

I L(.) I4,(I.

- .ol)

- .ol)

,

.I

- 201

> 0,

0, we obtain that

Clearly, (79) follows readily combining (86) with Theorem 3.3, which concludes the proof if R = BR(xo).Now, suppose

a = A R ~ ,(ZO) R ~. Then, setting d(r) := min(R2 - r, T - R1) , we have that

L ( z ) := 11,(r),

r =I z- 201,

where 11, is the unique solution of

-11,” -11,’ X11, { 11,(Ri) 11,(R2)

= f(d(r))+P = m.

-

=

in (R1, R2) ,

(87)

Now, consider the function

where R, := (R1 + R2)/2. Arguing as above, it is easy to check that, for each sufficiently small E > 0, there exists a constant A, > 0 such that, for any A > A,, the function

&(r) := A

+ (1+ &)B(r),

r E ( R I Rz) , ,

is a positive supersolution of (87). Similarly, there exists C < 0 for which the function

11, ( r ) := max(0, C + (1- &)e(r)}

4

is a non-negative subsolution of (87). Thus, thanks to Theorem 3.2 of J. Lopez-G6mezl6, (87) possesses a positive solution, say $, in between-J$

Uniqueness of Large Solutions of Radially Symmetric Elliptic Equations

109

and & for sufficiently large A. Consequently, by the uniqueness of $ ( T ) , as a consequence of Theorem 1.1, we find that & 5 $ 5 & and, hence,

As (88) holds true for any sufficiently small E > 0, (86) holds. Theorem 3.3 ends the proof in this case as well. This concludes the proof. El References 1. C, Bandle and M. Marcus, Sur les solutions maximales de problbmes elliptiques nonlinbaires: bornes isopbrimetriques et comportement asymptotique, C, R. Acad. Sci. Paris 311 (1990), 91-93. 2. C, Bandle and M. Marcus, Large solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior, J . D ’Analyse Math. 58 (1991), 9-24. 3. C. Bandle and M. Marcus, On second order effects in the boundary behavior of large solutions of semilinear elliptic problems, Di8. Int. Eqns. 11 (1998), 23-34. 4. F.C. Cirstea and V. Radulescu, Uniqueness of the blow-up boundary solution of logistic equations with adsorbtion, C.R. Acad. Sci. Paris, Ser. I, 335 (2002), 447-452. 5. F. C. Cirstea and V. Radulescu, Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris, Ser. I, 336 (2003), 231-236. 6. Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic problems, SIAM J . Math. Anal. 31 (1999), 1-18. 7. J. Garcia-MeliBn, R. Letelier-Albornoz and J. C. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. SOC.129 (2001), 3593-3602. 8. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin 1977. 9. R. Gbmez-Refiasco and J. Lbpez-Gbmez, On the existence and numerical computation of classical and Non-classical solutions for a family of elliptic boundary value problems, Nonl. Anal. T.M.A. 48 (2002), 567-605. 10 V. A. Kondratiev and V. A. Nikishin, Asymptotics, near the boundary, of a solution of a singular boundary value problem for a semilinear elliptic equation, Diff. Eqns. 26 (1990), 345-348. 11 J. B. Keller, On solutions of Au = f(u),Comm. Pure Appl. Math. X (1957), 503-5 10. 12. M. Marcus and L. Vbron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. Henri Poincare‘ 14 (1997). 237-274. . 13. C. Loewner and L. Nirenberg, 1974, Partial differential equations invariant

110

14.

15. 16. 17. 18.

19. 20. 21. 22. 23.

J. Ldpez-Gdmez

under conformal or projective transformations, in Contributions to Analysis, p. 245-272, Academic Press, New York, 1974. J. Lbpez-Gbmez, Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, El. J. Diff. Eqns. Conf. 05 (2000), 13.5171. 3. Lbpez-Gbmez, Dynamics of parabolic equations. From classical solutions to metasolutions, Diff. Int. Equations. 16 (2003), 813-828. J. Lbpez-Gbmez, The boundary blow-up rate of large solutions, J. Diff. Eqns. 195 (2003), 25-45. J. Lbpez-Gbmez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sc. Math. Jap. 61 (2005), 493-517. J. Lbpez-Gbmez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra, in Handbook of Differential Equations: Stationary Partial Differential Equations (M. Chipot and P. Quittner eds.), Elsevier. In press. J. Lbpez-Gbmez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Eqns. Submitted. J . Lbpez-Gbmez and M. Molina-Meyer, Superlinear indefinite systems: Beyond Lotka-Volterra models, J . Difl. Eqns. In press. J. Lbpez-Gbmez and P. Quittner, Complete and energy blow-up in indefinite superlinear parabolic problems, Disc. Cont. Dyn. Syst. In press. R. Osserman, On the inequality Au 2 f(u),Pacific J. Maths. 7 (1957), 1641-1647. L. VBron, 1992, Semilinear elliptic equations with uniform blow up on the boundary, J. D'Analyse Math. 59 (1992), 231-250.

COOPERATION AND COMPETITION, STRATEGIC ALLIANCES, AND THE CAMBRIAN EXPLOSION*

JULIAN LOPEZ-GOMEZ Departamento de Matemcitica Aplicada, Universidad Complutense de Madrid, 2804O-Madridt Spain E-mail: Lopez-GomezOmat.ucm. es MARCELA MOLINA-MEYER Departamento de Matemciticas, Universidad Carlos 111 de Madrid, 2891 I-Leganis, Madrid, Spain, E-mail: [email protected]. es

Dedicated to the memory of J. Esquinas Ecology, Economy and Management require a huge interdisciplinary effort to ascertain the hidden mechanisms driving the evolution of communities and firm networks. This article shows that strategic alliances in competitive environments provoke an explosive increment of productivity and stability through a feedback mechanism promoted by cooperation, while competition causes segregation within cooperative profiles. Some further speciation and radiation mechanisms enhancing innovation, facilitated by environmental heterogeneities and specific market regulations, might explain the biodiversity of life and the high complexity of industrial and financial markets. Extinctions occur by the lack of adaptation of strongest competitors to sudden environmental stress.

1. Introduction

This article uses some very recent advances in the theory of nonlinear Partial Differential Equations (PDEs) to show that combining cooperative and competitive interactions in spatially heterogeneous environments promotes *Work partially supported by grant REN2003-00707 of the Spanish Ministry of Science and Education.

111

112

J. L6pe.z-GGo’mezand M. M o h a - M e y e r

persistence and a substantial increment of productivity, diversity and stability in Ecology and E ~ o ~ o ~ The ~ mathematical ~ ~ ~ ~analysis ~ ~ of these issues is necessary to maintain existing communities and design optimal production strategies. To predict and evaluate these phenomena, we use a model combining hybrid interactions with spatial dispersion, based on the competition paradigm of and V ~ l t e r r a whose ~ ~ , analysis is imperative to ascertain the balance of cooperation and competition. Very simple prototype models of this type were actually used by Black & Scholes6 and Merton31 to evaluate stock options; in such models, dispersion and distribution rates are inter-exchanged by volatility rates. The most striking prediction from the mathematical analysis carried out here is that strategic alliances, even at a very localized level, can provoke a huge added value (overyielding) after some period of time if the strength of the alliance is sufficiently high. As the benefits of strategic alliances between competing firms tend to remain hidden, by fiscal reasons, and the periods of time necessary for revealing the real impact of localized cooperative interactions in ecosystems tend to be extraordinarily large -as suggested by the time span necessary to recover diversity in tropical ecosystems after mass using mathematical models is imperative for evaluating the effects of very localized, but possibly strong, alliances in competitive environments. This article shows that these alliances can indeed provoke an explosive increment of productivity, but eventually at the price of segregation of the “weaker competitor” within cooperation lines (patches). The underlying segregation mechanisms might help to explain the extraordinary diversity of life and industrial and financial markets, through some further adaptation mechanisms to specific market regulations or spatial heterogeneities. Basically, competition provokes segregation, whereas cooperation promotes huge added values through a feedback mechanism that will be discussed later. The mathematical theory necessary to conduct this analysis is very r e ~ e n t ~ ~ ~ ~ ~ .

2. The Cambrian explosion

544 millions years ago there were three animal phyla with their corresponding variety of external forms, though 538 millions years ago there were thirty-eight; the same number that exists today, except for one or two extinction^^^. Consequently, the vast existing diversity of body architectures appeared during a five-million-year interval beginning 543 millions year ago. This phenomenon is referred to as the “Cambrian explosion”,

~

~

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion 113

since the first fossils of that period of time where found in the Cambrian Hills in Wales by Sedgwick who name it as the “Cambrian”. The prior time span is called the Precambrian. Cambrian explosion supports that the history of life consists of long time intervals of “micro-evolution” combined with short “macro-evolution”periods, which have been the most prolific generators of biodiversity; in strong contrast with the classical theory of Darwin and Wallace, according to which evolution occurred gradually. Parker34 defends the theory that Cambrian explosion was originated by the sudden evolution into eyes -in less than one million years- of many of former skin energy receptors. The architecture of eyes alone can provide a lot of information on how animals lived. For instance, the position of the eyes on the head alone can reveal the position of the animal in the chain food. Eyes positioned at the sides of the head, facing sideways, can scan wide angles and detect any movement from nearly all direction, tend to belong to preys, while eyes positioned together at the from of the head, facing forward, generally belong to predators. Eyes of predators are extremely better for pinpointing targets and estimating the distance between them, even though see less part of the surroundings. As a consequence of vision, the complexity of the interactions between species grew drastically, promoting all kind of mutual competition, cooperation and predating strategies, that originated the extraordinary biodiversity of the Earth’s biosphere in the blink of an eye. Therefore, the Cambrian explosion supports the theory that combining cooperative and competitive interactions in communities provokes explosive increments of productivity and diversity.

3. Cooperation and competition in Ecology

Although ecologists realize that cooperative interactions arise very often in nature, they are not paid the deserved attention in ecological studies yet. However cooperation was recognized as a crucial driving force in population dynamics by the most pioneering ecologist^^!^, during the last decades many ecologists focused their efforts towards the understanding of competition in communities, as, within the paradigm of Darwin and Wallace, yet competition is thought to be the most important process governing communities evolution and b i o d i v e r ~ i t y Although ~ ~ ~ ~ ~ cooperative ~~~~~~ interactions are well documented between organisms from different kingdoms, as they can make significant contributions to each other’s needs

114

J. Ldpez-Gbmez and M. Molina-Meyer

without sharing the same resources, documenting cooperative interactions between similar organisms in the abundance seems to be a hard task in empirical studies, because they do not arise in isolation but in combination with competition, which validates the “abiotic stress hypothesis” of Bertness and Callaway4. According to it, the importance of cooperation in plant communities increases with abiotic stress or consumer pressure8i5. Alternatively, the importance of competition increases when abiotic stress is low. Besides the abiotic stress hypothesis seemingly contradicts the huge number of cooperative mechanisms observed in tropical e c o ~ y s t e m s ~ it ~ , seems rather controversial to think of cooperation as the main driving force governing evolution during mass extinctions1, when abiotic stress suddenly grew in a geological time scale -by whatever cause, unimportant here. Actually, the huge number of “Lazaro” species (those disappearing from fossil registers whose persistence is confirmed after some period of time) documented by zoo-paleontologists after mass extinctions rather supports the theory that most of the species tend to segregate into small refuge areas (deep see waters, caves, islands. . . ) where they safely stay until environmental stress decreases. Therefore, competition seemingly is the main driving force governing evolution after mass extinctions, instead of cooperation, even though local cooperation does play a crucial role for avoiding massive extinctions, as it becomes apparent from the mathematical analysis carried out here. As a matter of fact, extinction rules tend to reverse during mass extinction periods, up to provoke extinction of the most powerful competitors during background extinctions -less adapted to afford sudden environmental stress, e.g., Nautilus and mammals, apparently weaker than ammonites and dinosaurs, could successfully pass through several mass extinctions, while amonites and dinosaurs did not’*20.Consequently, the main interaction patterns governing evolution in the abundance tend to be rather different from those in harsh environmental conditions. Elton’s hypothesis12 that greater diversity causes greater productivity (the rate of biomass production) and stability (low susceptibility to invasion), further strengthened by Pimente135 and MargaleP9, remains among the most controversial principles in plant communities of the last decades. Although most of empirical studies have confirmed it18341,yet pure competition models have not been able to corroborate it30342- rather naturally. Although the available competition models in heterogeneous environments predict that coexistence leads to overyielding (phenomenon explaining why several competing species have greater productivity than any individual

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion 115

species in monoculture) and that overyielding has a strong stabilizing effect on total community biomass, the mathematical problem of validating Elton’s hypothesis remains utterly open42. Quite strikingly, yet local cooperative interactions have not been incorporated into the mathematical discussion of the diversity-stability debate. As a consequence of the mathematical analysis carried out here, it is apparent that overyielding caused by local cooperative interactions is substantially higher than overyielding promoted by sheer competition, and that stability indeed increases, which might definitively validate the Elton hypothesis mathematically. Local cooperative interactions have remained outside the diversity-stability debate during the last decades because of the tremendous number of mathematical difficulties in searching for the combined effects of spatial heterogeneities, dispersion, cooperation and competition.

4. Cooperation and competition in Management

Most of industrial, commercial and service sector firms make strategic alliances of rather competitive nature. These alliances have typically two effects. Besides they allow sustaining the “weaker competitor”, that otherwise would become extinct, they do increment productivity and benefits in a rather substantial way. Such hybrid interactions are utterly apparent by considering a number of recent alliances in the automobile sector (among others, BMW-Chrysler, PSA-Toyota, Honda-Isuzu, and OpelRenault-Suzuki). Most of these agreements focus on cooperation in R e search & Development and manufacturing of one or several components or product lines, while distribution typically remains competitive. These alliances are widely known under the label of “allied in costs, rival in markets”. The importance of strategic alliances in competitive markets has been so much recognized in the late nineties that the concept of “coopetition” has emerged in management literature to name the hybrid behavior cooperation-competition7~22~16. Undoubtedly, strategic alliances combined with competition have shown to be an extraordinary mechanism for value creation, even though not paid the deserved attention until very recently, still being an under-researched theme. Whether or not co-opetition strategies may be maintained for either short of long time intervals, experience dictates that only if extended during a sufficiently large period of time proves to be really helpful for the creation of knowledge and economic value. Actually, the added value of co-opetition during short periods of time may be only a small fraction more than the

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one provided by sheer competitionlo. Extrapolating these ideas to an ecological context readily there emerge some of the main difficulties in isolating hidden local cooperative interactions between competitive species. Indeed, after all documented mass extinctions most of oceanic and terrestrial systems recovered in around one million years, except for the notable exception of tropical ecosystems, which needed many million years to attain their former biodiversity levels’. Thus, it is necessary to give the appropriate time to a co-opetition system up to attain its maximal overyielding, or added value. Such time might depend upon the strength of strategic alliances in a rather hidden way. Therefore, searching for general principles governing the dynamics of co-opetitive systems from empirical studies might be almost impossible, since a long time span might be necessary to attain them. Consequently, using mathematical models in co-opetition studies seems certainly imperative. Ecology and Economy are closely related fields, as they are nothing more than a miraculous combination of matter and energy, even though humans still think of themselves in a Platonic way they are at the center of the Universe! Adopting this perspective, strategic alliances among firms might provide an extraordinary empirical model to analyze the balance between cooperation and competition in life communities. Firms are well acquainted that benefits (ecological overyielding) can blow-up as a result of an appropriate alliance, even though real figures are not usually published.

5. The importance of mathematical models

The importance of mathematical models relies upon the fact that they provide an idealized behavior against which reality can be judged and measured, promoting further interdisciplinary studies. The main paradigm of this being Newton synthesis in Physics. Newton introduced the concept of derivative and, hence, the differential equations to obtain the planetary laws of Keppler. Consequently, the Newtonian potential was a combined consequence of the intensive empirical studies of Brache and his predecessors, the exquisite mathematical taste of Keppler, and the tremendous audacious and genius of Newton. Bringing together the three methodologies promoted one of the most important scientific revolution of Physics. Actually, in many circumstances mathematical models have shown to be imperative for making predictions that on the empirical side are “unpredictable”. Among them, the discovery of Hertztian waves from Maxwell equations, the prediction of the curvature of light by Einstein, or the pre-

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion 117

diction of the existence of black holes by Hawking and Penrose. 6. Modeling cooperation in competitive environments

Subsequently, we adopt the language of population dynamics, though most of our findings should have a counterpart in Economics and Management Science. To analyze the effects of local alliances in the dynamics of competing species we think of two populations U and V randomly dispersing in the inhabiting area s2 c RN,N 2 1, according to the Fourier-Fick-Fisher's law. Precisely, we assume the evolution to be governed by the system of PDEs dtU - D1V2U = XU (1 - KY1U - K,'a12V) (1) &V - D2V2V = pV (1- KT'V - KT1a21U)

{

where D1 > 0, D2 > 0 measure the dispersion rates of U , V, dt stands for time derivation, and V2 is the spatial dispersion operator the Laplacian19i17t33732.Typically, for each location z E 0 and time t > 0, U ( z ,t ) and V(z, t ) measure the density of the populations, X and p are the intrinsic growth rates of U and V ,and K1 > 0, K2 > 0 are their carrying capacities. The functions alz(z) and ( ~ 2 l ( zmeasure ) the nature and strength of the interactions. In competition models, Q I ~ ( Z )> 0 and a 2 1 ( 2 ) > 0 for each z E R. The non-spatial counterpart of system (1) (Dl = D2 = 0) goes back to and V ~ l t e r r a ~The ~ . spatial models where introduced by Fisher", to describe the spread of a novel alele, by Skellam37,to study invasion problems, and by Black & Scholes6 and Merton31, to evaluate stock options. Random dispersion entails the organisms to move from densely populated areas to less populated regions3', thought such tendency can change as a result of competition and cooperation. Setting b = (XK2/pK1) ~

1 2 ,

c = (pK1/XK2) ~

2 1 ,

u = (Kl/X)U, v = ( K 2 / p ) u , system (1) can be written as

{

- D1V2u = XU - u2 - buu 6'tU - D2V2u = p~ - u2 - cuu &U

(2)

where we will focus our attention. This system must be completed with initial conditions U(2,O) = u o ( z ) ,

4 z , o ) = vo(z),

(3)

for each 2 E R (UO and 210 are the initial population distributions), and some appropriate boundary conditions along the habitat edges, called dR.

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J. Lbpez-Gdmez and M . Molina-Meyer

In this work we assume the region outside R is immediately lethal and, hence, we impose u(5,t ) = v(5, t ) = 0

(4)

for every x E 8R and t > 0, i.e., R is surrounded by an absorbing boundary. Terrestrial-aquatic edges are absorbing boundaries for seeds of plant species incapable of surviving in both habitat^'^, as is the legislative boundary of Yellowstone National Park for bison dispersing into Montana, where they are shot to control the spread of brucellosis''. In model (2)-(4), the

Figure 1. An admissible spatial configuration of interactions. The species compete in 0; n 02 and cooperate in 0, n OF, whereas u predates on v in 0, n 0 $ ,'u predates on u in 0; n OF, and u (resp. v) is free from any action of v (resp. u ) in (resp. a:).

coefficient functions b = b(x) and c = ~ ( zprovide ) the nature and strength of the interactions at every location x E R, which is of hybrid type, since b(z)and c(z) are arbitrary functions. Precisely, we suppose that R consists

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion 119

of fll,the region where b(s)> 0 (u receives aggressions from w), the region R,, where b(z) < 0 (u is facilitated by w), and R t , where b ( z ) = 0 (u is free from w). In particular, 0: is a refuge area (Figure 1). Similarly, the sign of c(z) divides R into R,: where w receives aggressions from u, R;, where v is facilitated by u, and RE, where v is free from u (Figure 1). These hybrid models, within the spirit of V ~ l t e r r a *and ~ to Lbpez-Gbmez & M ~ l i n a - M e y e r ~ ~ ? ~ ~ .

go back

7. The balance of cooperation and competition

As occurs in classical non-spatial models, in many circumstances the dynamics of (2)-(4) is regulated by its steady states, i.e., by the non-negative solution pairs (u,v) of

{

-D1V2u = XU - u2 - ~ ( Z ) U W in R -D2V2w = pv - w2 - C ( Z ) U V u=w=o on Xl

(5)

Besides the “trivial state’’ ( O , O ) , problem (5) admits three types of nonnegative solutions. Namely, the solutions having one component positive and the other vanishing, (u,O) or (O,w), refereed to as the “semi-trivial positive solutions”, and the solutions having both components positive, the so called LLcoexistence states”. By well known results in nonlinear PDEs, (5) has a semi-trivial positive solution of the form (u,0) if and only if X > a[-D1V2;R] .

(6)

In such case, (ux,0 ) is the unique among such states, where U A > 0 stands for the unique positive solution of

-D1V2u = X u - u 2 {u=O

in R on

(see Lbpez-Gbme~~~). Subsequently, given a dispersion rate D > 0 , a patch P C R, and a “potential” V in P , we denote by a[-DV2 + V ;P] the “principal eigenvalue” of -DV2 + V in P subject to zero boundary conditions on the edges dP3’. Condition (6) provides the critical A, or D1, for R to maintain the species u in the absence of v. Similarly, (5) has a semi-trivial positive solution of the form (0,w) if and only if p > u[-D2V2;R] and, in such case, (0,.up) is the unique among these solutions; v p being the unique positive solution of -D2V2v=pv-v2

{w=o

in R on

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J. L6pez-Gdmez and M. Molina-Meyer

The states ux and v, equal the asymptotic profiles of u and v in isolation astfm. Subsequently, we regard to X and p as the main parameters of the model. Analyzing the linearized stability of the states (O,v,) and ( u x , O ) reveals that X = f ( p ) := c[-D1V2 bv,; R] (7) p = g(X) := 8[--02V2 cux; R]

+ +

are their curves of change of stability in the (A, p)-plane. Setting

u := a[-V2; R], these curves satisfy f ( D 2 8 )= 010,

g(01c) = 0 2 c .

Moreover, their tangents at infinity are the straight lines X = bL x p and p = CL x A, where bL < 0 and CL < 0 are the minimum values of b and c in R25. Actually, the region where X > f(p) and p > g(X) provides us with the set of values of (A, p ) where any semi-trivial solution is unstable. Both species are expected to persist there25. By using standard techniques in local and global bifurcation theory23, it is easy to see that a two-dimensional variety of coexistence states bifurcates along these curves from the semi-trivial solution^^^*^^. To compute these curves in the following one-dimensional examples, we coupled centered finite differences, a pure spectral method with collocation, and a path-following solver together with the inverse power method. This gives high accuracy with low computational work13~27~24~25~26. We have computed these curves in case

N = 1, R

= (0, l), 0 1 = 0 2 =

1, b = C ,

for each of the following choices of b:

and 5

b=

Pix[(i-i)h,ih]

sin [5n(- ih

i=l

+ h)]

(9)

where

PI = P 3 = P5 = -0.4, ,& = 20,

p4 = 40,

h = 0.2,

and, given an interval J, we have denoted by x J the function defined by x J = 1 in J and J = 0 in its complement. According to Figure 2, in case

x

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion 121

(8) the theory predicts permanence in the region enclosed by the curves, while extinction occurs outside, since one semi-trivial states is stable there. Computing gave

27.02 = maxg(X) = m y f(p) . X>r2

In case (9), the region enclosed by the curves consists of two components. In the bounded component, (ux,O) and ( 0 , ~ are ~ ) stable and the model

exhibits a founder control competition. Therefore, it behaves much like the classical non-spatial model under severe aggressions. In the unbounded component, any semi-trivial state is unstable and the species persist. 250

B 200 150

Permanence

3.

100

50 0'

0

I0

Figure 2. Computed graphs of X = f(p) and p = g(X) for the choices (8), (A), and (9),(B). In both cases, u = x 2 and f(u)= g ( u ) = n2. Their tangents at infinity are X = -0.4 x p and p = -0.4 x A, since bL = -0.4.

To compute the coexistence states, we fixed X and used p as the main continuation parameter, by coupling a pure spectral method with collocation and a path continuation (Figure 3). In Figure 3, we have plotted the parameter p versus IuI

+ lvl := maxu + maxv n n

in (A,B,C), and versus the &-norm in (D). By simply looking at Figures 2 and 3, it becomes apparent that, whatever the value of X be, these models possess a coexistence state for sufficiently large p, in strong contrast with the behaviour exhibited by the pure competition model, where, according to the Competitive Exclusion Principle (CEP), v should drive u to extinction. Rather strikingly, the CEP fails

122

J. L6pez-Gdmez and M. M o h a - M e y e r L . .

400

5.300.

+ 3

.

-200 100

0‘ 0

-0

50

mu

100

150

mu 180 160 N > +J 3 -

D

140 20 100. 80

“0

50

100

mu

150

60 60

80

100

120 mu

140

160

180

Figure 3. Bifurcation diagrams for the choice (8) (at X = -10 (A), X = 27.018 < 27.02 (B), and X = 48.98 > 27.02 (C)), and (9) (at X = 140 (D)). Each point along any curve provides a state. Stable states are indicated by solid lines, unstable by dashed lines; (0,O) is unstable for each p > x 2 . The other horizontal line in (B,C,D) represents ( U X , 0); it does not exist in case (A), because X = -10 < x 2 . It is stable if p < g(X), and unstable if p > g(X). In all cases, p = g(X) is a bifurcation value to coexistence states from (.A, 0). For the choice (8), all coexistence states are stable. When they are unique (A,B,C), they should be global attractors. The inclined curve represents (0, v ~ )p, > x 2 ; it bifurcates from v = 0 at p = x 2 . In (A), ( 0 , ~ is ~ )a global attractor if p < f - l ( - l O ) , while the coexistence state is a global attractor if p > f - l ( - l O ) . In (B), ( 0 , ~ changes ~) character twice, because f - l (A) has two points. As a consequence, the model exhibits two curves of coexistence states: one bounded and another unbounded. As X approximates 27.02, the two values of f - l ( X ) approach, until collision at X = 27.02 and disappear for all further X > 27.02 (see (C)). (D) shows a situation where the high level of the aggressions and the spatial distribution of the interaction patches causes a complex dynamics through multiple coexistence states. Although there is a unique curve of coexistence states, there is a range of p’s where the model possesses 9 coexistence states, four stable and five unstable.

for our prototype model at every A, and, in particular, for each A < r2 (Figure 3A), where SZ cannot support u in isolation. Consequently, permanence for this range of is utterly attributable to cooperation with w. Rather naturally, the intrinsic productivity of o, measured by p, must be

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion 123

sufficiently high so that u can be maintained by w. If not, extinction might occur.

F-7

11' 1' 1'1

I t _t

0

0

w 0.5 1

0

0.5

1

0

0.5

1

2ooo

t

1000

500 0

0.5

1

I t1' T I

200

0

0.5

1

0 0

0.5

1

Figure 4. Four series of super-imposed coexistence states along some of the curves plotted in Figure 3. Each series shows the components u (left) and v (right) versus 5 € (0,l). The sense of the arrows, T or 1, indicates if they grow or decay as p increases. Eventually, they might first increase and then decrease for larger p's, which has been indicated by T l . (A) shows a series of states along the coexistence curve of Figure 3A. As p increases, w grows everywhere, while u segregates into the cooperation patch increasing there. (B) shows a series of states along the coexistence curve of Figure 3B. This behaviour is reminiscent of competition models, but these states are global attractors, even though the associated non-spatial model ( D l = D2 = 0) exhibits extinction, since maxn 6 = 2 > 1; the classical theory predicts founder control competition. Therefore, permanence here is attributable to cooperation. The plots along the unbounded coexistence curve of Figure 3B are similar to those shown in (A). (C) shows a series of states along the coexistence curve of Figure 3C; sort of superposition of the previous patterns. Although for small p's the model exhibits a competitive behavior, it reveals a rather cooperative behavior for large p's, even though u segregates into the cooperation patch. (D) shows a series of states after the last turning point of the coexistence curve of Figure 3D. As p increases, z1 grows everywhere, while u segregates into cooperation patches, where it increases.

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J . L6pez-G6mez and M. Molina-Meyer

8. Searching for general principles in co-opetition theory

Most of previous co-opetition patterns hold true for the general model (2)(4),not exclusively for the previous one-dimensional prototype^^^^^^^^^. In general, ( 5 ) possesses a coexistence state for every X E JR and sufficiently large p26 if cooperative interactions are weak ( b >~ -1, CL > -1). Naturally, the more localized and weaker the alliance is, the larger must be p so that w can maintain u. Quite strikingly, and, hence, extremely interestingly, u and w blow-up in finite time within cooperation patches if the alliance is reinforced up to attain bL < -1 and CL < -1, because (5) loses the coexistence states for large p’s and, hence, the dynamics of the model is governed by its metaConsequently, the model predicts that strong alliances, even of a very partial nature (producing one single component, sharing a flight line service. . . ) provide a dramatic increment of added value; the necessary strength of the alliance depending upon the intensity of competition, of course, even though the weaker competitor might be forced to specialize (segregate) into cooperation fields by the strongest competitor. Actually, the mathematical theory predicts productivity to blow-up as time passes by as soon as some of the competitors be sufficiently good and alliances strong. The alliance helps to support the weaker firm. The productivity of the weaker firm within cooperation lines improves the productivity of the stronger one. The associated added value favors the productivity of the weaker, and so on.. . This feedback mechanism explains why the balance between cooperation and competition is so strikingly fruitful. When productivity is very high, the weaker competitor must segregate into the cooperation lines, even though some sudden market change might reverse the sense of the former productivity flow. Co-petition certainly stabilizes the system, as invasion by other firms is less facilitated by the extraordinary increment of added value. At this is point of the discussion, ecologists should note that, in the abundance, some minor facilitation effects might be extremely difficult to be detected and isolated in short periods of time, though they can dramatically change the dynamics of the community in the large. Also, note that Figure 4D suggests a very simple mechanism to generate biodiversity (complexity) by combining competition with cooperation through segregation into cooperation areas and further differentiation promoted by their adaptation to spatial and/or temporal heterogeneities. In such scheme,

Cooperation and Competition, Strategic Alliances, and the Cambrian Explosion 125

competition is imperative for spatial segregation, cooperation is crucial to avoid extinction and increment productivity within facilitation patches, and spatial heterogeneities promote speciation. Sudden abiotic stress in environmental conditions might alter dramatically the interaction patterns of the species, eventually reversing the nature of the former interactions, and promoting massive extinctions. Further radiation of some of the weaker sub-species -better adapted to sudden changes- might explain the extraordinary biodiversity of Gaia and that this paper has not been written by two dinosaurs.

References 1. J. Agusti, The Logic of Extinctions (Tusquets, Barcelona, 1996). 2. W. C. Allee, A. E. Emerson, 0. Park, K. P. Schmidt, Principles of Animal Ecology (Saunders, New York, 1949). 3. G. E. Belovsky, C. Mellison, C. Larson, P. A. Zandt, Science 286, 1175 (1999). 4. M. D. Bertness, R. M. Callaway, fiends in Ecol. and Evol. 9,191 (1994). 5. M. D. Bertness, G. H. Leonard, Ecology 78 1976 (1997). 6. F. Black and M. Scholes, J . Polit. Econ. 81,637 (1973). 7. A. M. Branderburger, B. J. Nalebuff, Co-opetition (Doubleday, New York, 1996). 8. R. M. Callaway, L. R. Walker, Ecology 78,1958 (1997). 9. F. E. Clements, J. Weaver, H. Hanson, Plant Competition (Carnegie, Washington, 1926). 10. G. B. Dagnino, G. Padula, paper presented at the 2nd European Academy of Management Conference, Stockholm, Sweden, May 2002. 11. A. Dobson, M. Meagher, Ecology 177,1026 (1996). 12. C. S. Elton, The Ecology of Invasion by Animals and Plants (Methuen, London, 1958). 13. J. C. Eilbeck, SIAM J . Sci. Stat. Comp. 7,599 (1989). 14. W. F. Fagan, R. S. Cantrell, C. Cosner. The Amer. Natur. 153,165 (1999). 15. R. A. Fisher, Ann. Eug. 7,353 (1937). 16. D. R. Gnywali, R. Madhavan, Acad. Manag. Rev. 26,431 (2001). 17. A. Hastings, Ecology 77,1675 (1996). 18. A. Hector et al., Science 286,1123 (1999). 19. E. E. Holmes, M. A. Lewis, J. E. Banks, R. R. Veit, Ecology 75,17 (1994). 20. D. Jablonski, Science 253,754 (1991). 21. M. J. Keeling, H. B. Wilson, S. W. Pacala, Science, 290,1758 (2000). 22. A. A. Lado, N. Boyd, S. C. Hanlon, Acad. Manag. Rev. 22, 110 (1997). 23. J. Lbpez-Gbmez, Spectral Theory and Nonlinear Functional Analysis (RNM 426, CRC Press, Boca Raton Florida, 2001). 24. J. Lbpez-Gbmez, J. C. Eilbeck, M. Molina-Meyer, K. Duncan, IMA J . Numer. Anal. 12 405 (1992).

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25. J. Lbpez-Gbmez, M. Molina-Meyer, Res. Inst. Math. Sci. KYOTO 12, 118 (2003). 26. J. Lbpez-Gbmez, M. Molina-Meyer, Adv. Math. Sci. Appns. 14, 87 (2004). 27. J. L6pez-G6mez, M. Molina-Meyer, M. Villarreal, S I A M J. Numer. Anal. 29, 1075 (1992). 28. A. J. Lotka, 1932. J. Washington Acad. Sci. 22, 461 (1932). 29. R. Margalef, Brookhaven Symp. Biol. 22, 25 (1969). 30. R. M. May, Nature 238, 413 (1972). 31. R. C. Merton, Bell J. of Econ. and Manag. Science 4, 141 (1973). 32. J. D. Murray, Mathematical Biology (Springer, Berlin, 1989). 33. A. Okubo, Diffusion and Ecological Problems (Springer, New York, 1980). 34. A. Parker, In the Blink of a n Eye (Free Press, London, 2003). 35. D. Pimentel, Ann. Ent. SOC.Amer. 54, 76 (1961). 36. E. Schrodinger, What is Life? (Cambridge Univ. Press, Cambridge, 1944). 37. J. G. Skellam, Biometrika 38, 196 (1951). 38. D. Tilman, Resource Competition and Community Structure (Princeton Univ. Press, Princeton, 1982). 39. D. Tilman, Ecology 75, 2 (1994). 40. D. Tilman, P. Kareiva, Spatial Ecology (Princeton Univ. Press, Princeton, 1997). 41. D. Tilman, Science 285, 1099 (1999). 42. D. Tilman, Ecology 80, 1455 (1999). 43. V. Volterra, LeGons sur la thiorie rnathgmatique de la lutte pour la vie (Gauthier-Villars, Paris, 1931). 44. S. A. West, I. Pen, A. S. Griffin, Science 296, 72 (2002). 45. E. 0. Wilson, The Diversity of Life (Harvard Univ. Press, Cambridge, Massachusetts, 1992).

LOCAL SMITH FORM AND EQUIVALENCE FOR ONE-PARAMETER FAMILIES OF FREDHOLM OPERATORS OF INDEX ZERO*

JULIAN L ~ P E Z - G O M E Z Departamento de Matemcitica Aplicada Universidad Complutense de Madrid 28040-Madrid, Spain Email: LopetGomezQmat.ucm.es CARLOS MORA-CORRAL Mathematical Institute University of Oxford 24-29 St Giles’, Oxford OX1 3LB, United Kingdom Email: mora-coramaths. ox. ac.uk

Dedicated to the memory of Jestis Esquinas Let C be a smooth family from a neighborhood of the (real or complex) number Xo t o the set of Fredholm operators of index zero between two Banach spaces. This paper shows that the following assertions are equivalent: (a) C admits a local Smith form at Xo; (b) Xo is an algebraic eigenvalue of C; (c) the lengths of the Jordan chains of C at Xo are uniformly bounded from above; (d) the determinant of any finitedimensional reduction of C has a zero of finite order at Xo. To prove this, we use two theories of algebraic multiplicity: the one given by Jordan chains and the one given by transversalization. It is also shown that two families are locally equivalent at XO if and only if their multiplicity invariants at Xo coincide.

1. Introduction In this paper we will consider IK E {W,C},two Banach spaces U and V over K) an open subset R of IK,a point A0 in 0, and a family

c : R + L(U, V ) *Work supported by the Ministry of Education anb Science of Spain under grant REN2003-00707. 127

128

J . L6pez-Gdmez and C. Mora-Corral

of class C' for some r E M U { m } such that C(X0) E @o(U,V). Here C(U,V) denotes the Banach space of linear bounded operators from U to V, and @o(U,V) denotes the subset of L(U,V) formed by the Fredholm of index zero operators, that is, by those operators T E C(U,V) for which the (finite) dimension of its kernel coincides with the codimension of its range. This paper studies local properties of the family C at an eigenvalue. The point XO E R is said to be an eigenvalue of the family C if C(X0) is not an isomorphism; as C(X0) is Fredholm of index zero, the condition that C(X0) is not an isomorphism is equivalent to the condition that C(X0) has a non-zero kernel. This notion of eigenvalue of a family does not coincide with the concept of eigenvalue of a n operator, unless

U-V

and C ( X ) = X j - T

for some T E C(U,V), where j stands for the canonical inclusion. Let XO be an eigenvalue of C. It is said that Xo is an algebraic eigenvalue of C if 6, C > 0 and rn 2 1 exist such that for each 0 < J X - XoJ < 6, the operator C(X) is an isomorphism whose inverse satisfies the estimate I~C(X)-1115 CIA - Xol-".

This is the class of eigenvalues for which a multiplicity of C at XO will be defined. We will give two definitions that will turn out to be equivalent. The first one uses Jordan chains, while the second one is based upon the concept of transversal eigenvalue. The concept of Jordan chain of a family at an eigenvalue is an extension of the concept of chain of generalized eigenvectors of an operator. Jordan chains were introduced in the mid forties to develop a spectral theory within the context of matrix polynomial pencils (see I. Gohberg et a1.6 and the references therein). Since then, they have shown to be a powerful technical tool in several areas; among them, Jordan chains have shown to be fruitful in the study of the structure of the solution set of classes of linear systems of differential equations (see J. T. Wloka et al.14), in the analysis of nonlinear eigenvalue problems in the context of bifurcation theory (see P. J. Rabierl2) or in the study of the completeness of eigenvectors of nonlinear families (see Friedman and Shinbrot4). Although most of the available mathematics about Jordan chains focused into complex polynomial or analytic operator pencils, in many applications (among them in most of non-linear mathematical real world models) the underlying operator family is far from being holomorphic, but rather real analytic or of class CT regularity (see J. L6pez-G6mezg and the references therein).

Local Smith Form and Equivalence for Families of Fredholm Operators

129

Closely related to Jordan chains is the so-called Smith form. Polynomial families admit a global Smith form (see, for example, I. Gohberg et a1.6). On the other hand, P. J. Rabier12 showed that Coo families admit a local Smith form provided no Jordan chain can be continued indefinitely; this is what occurs, for example, when the family is of polynomial type and it is invertible at some point. The local Smith form of C will be, roughly speaking, the simplest form that can be reached by multiplying C by families of isomorphisms. In this paper we characterize the existence of the local Smith form in terms of the invariants of the multiplicity, thus refining the results of Rabier12. This will be done with the help of the theory of multiplicity due to Esquinas and L15pez-G6mez~~~*~~~. In fact, relating the invariants of both multiplicities will allow us to prove one of the main results of this paper, namely, that the following conditions are equivalent: C admits a local Smith form at XO; XO is an algebraic eigenvalue of C; 0

the lengths of the Jordan chains of C at XO are uniformly bounded from above; the determinant of any finite-dimensional reduction of the family has a zero of finite order at XO.

The local Smith form provides the natural background to study the local equivalence of families. Rmghly speaking, two families are equivalent when one can be obtained from the other by multiplication by families of isomorphisms. The local Smith form of a family is the simplest family equivalent to it. We will see in this section that two families are equivalent at a point if and only if their multiplicity invariants coincide. The plan of this paper is as follows. In Section 2 we define the concepts of Jordan chain, canonical sets of Jordan chains, partial multiplicities, and multiplicity based on Jordan chains. In Section 3 we show a preliminary result about the construction of the local Smith form. In Section 4,transversal eigenvalues are defined. Then, Section 5 shows that every family C satisfying our general requirements and for which XO is an algebraic eigenvalue can be tranversalized by means of a polynomial family Qi : K --+ L(V,V ) such that @(A,) is an isomorphism; this means that the family CQ,is transversal at Xo. The multiplicity of C at XO is then defined with the help of any of these transversal families CQ, at XO. In Section 6 we explain the relationships between the concepts of multiplicity given by Jordan chains and by transversalization; in particular, we show that both multiplicities coincide.

J . L6pez-Gbmez and C. Mora-Corral

130

The key point here is that the structure of the canonical sets of Jordan chains at transversal eigenvalues is very simple. All this information will be brought together in Section 7 to prove the characterization of the existence of the local Smith form. Finally, in Section 8 we explain the notion of local equivalence of families and use the results about the existence of local Smith form to give some necessary and sufficient conditions for two families to be locally equivalent at a point. 2. Jordan chains and canonical sets

In this section we recall the classical concept of Jordan chain and some related concepts, as discussed by P. J. Rabier12. Throughout this paper, we will use the following notation for the Taylor coefficients of C at XO

Definition 2.1. Take T E M U (00) and assume that C E C‘(R, L(U,V)) satisfies C(X0) E @o(U,V). Let 0 5 s 5 T be an integer. Every ordered set of s 1 vectors (uo,. . . , u s )E US+l with uo # 0, and

+

i=O

is said to be a Jordan chain of length s + 1 of the family C at XO starting at uo. When j = 0, equation (2) is reduced to Couo = 0. Therefore, a necessary and sufficient condition for the existence of Jordan chains is that C(X0) is not an isomorphism. In such case, the vector uo is said to be an eigenvector of C associated to the eigenvalue XO. By definition, the eigenvectors of C at XO are the elements of N[Co]\ (0). Consider the following special case: U is continuously contained in V (U L) V ) ,the map j E L(U,V )stands for the inclusion, and C(X) = X j -T for all X E K and a fixed T E L(U, V). Then, (2) becomes

Tuo = XOUO,

+ U O , .. .

T u= ~X o ~ l

,

TuS= XouS

+~ ~ - 1 ,

and, therefore, (uo, . . . ,u s ) is a chain of generalized eigenvectors of the operator T at the eigenvalue XO, in the classical sense.

Definition 2.2. Take uo E U ,an integer Ic 2 0, an T E M U {oo}, and a family C E Cr(R, L(U,V)) satisfying C(X0) E @o(U,V). Then, it is said

Local Smith Form and Equivalence for Families of Bedholm Operators

131

that the rank of uo is k when k 5

T and k is the maximum length of all Jordan chains of C at XO starting at uo. It is said that the rank of uo is infinity if T = w and for each each integer k there exist Jordan chains of C at XO starting at uo with length k. The rank of uo is denoted rankuo.

The rank of uo is not defined if C is of class C r , but not of class Cr+' in any neighborhood of XO, and there are Jordan chains of C at Xo starting at uo of length T 1. Suppose 2 is of class C". A related concept of infinity rank is the following: it is said that the Jordan chain (210, . . . ,u,) of C at XO can be continued indefinitely if there exists a sequence { U ~ } ~ ? , + I in U such that for all n 2 0 the ordered family (uo, . . . ,urn)is a Jordan chain of C at XO. Obviously, if the Jordan chain formed by the element uo can be continued indefinitely then rankuo = 00. Under our general assumption 20 E @o(U,V),the reciprocal is also true, as we will see in Theorem 7.2. The following example provides us with an eigenvector of infinite rank: if U = V = K = R, XO = 0, and C is the family defined by

+

C(X) :=

if X E R \ { O } if X = O

(3)

then Cj = 0 for each j 2 0, and, hence, rankuo = 00 for all uo E R \ { 0 } , since for each n 2 1, the ordered set (uo, . . .,U O ) E Rn constitutes a Jordan chain of length n of C at 0. Now we describe the construction of a canonical set of Jordan chains. To that aim, we assume that for some T E N U {m} the family C E Cr(n,C(U,V ) )satisfies that C(X0) is a non-invertible Fredholm operator of index zero, and that the length of all Jordan chains of C at XO is uniformly bounded from above by some integer less than or equal to T . Let kl be the optimal bound, so that k l is an integer with 1 5 k l 0 such that for each 0 < IX - Xol < 6, the operator C(X) is an isomorphism and

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J. L6pe.z-Gdmez and C. Mom-Corral

The least natural number u 2 1for which (14) holds true for some C, 6 > 0 and all 0 < IX - XOI < 6 is called the order of XO. Given an integer k L 0 and a XO E R, it is said that Xo is a k-algebraic eigenvalue if either k = 0 and C(X0) is an isomorphism, or k 2 1 and XO is an algebraic eigenvalue of order k of C. When the family C is (real or complex) analytic, the order of XO as an algebraic eigenvalue of C is nothing but the order of A0 as a pole of 2-l. If C is not analytic there is no concept of pole, but in the authorslo paper it is shown that the right counterpart of the concept of pole of C-l is that of the order as an algebraic eigenvalue of C. Actually, such concept of pole shares many of the local properties characteristic of the analytic case, among them the existence of (finite) Laurent expansions. We will not explain that theory here, but only present a corollary that will be useful later. The following result was proved by L6pez-G6mezg by using the theory of multiplicity explained in this section and the one introduced by Magnusll . Lemma 5.1. Take r E N U {m}. Suppose C E C'(R,L(U,V)) satisfies C(X0) E @o(V,V),and (A - XO)~C(X)-' exists and is bounded for X in a perforated neighborhood of XO, and for some integer k 5 r . Then, the function X H (A - XO)~C(X)-' is of class Cr-k in a neighborhood of XO.

The main result of this section establishes that if XO is a v-algebraic eigenvalue of C for some integer 1 5 v 5 r , then there exists a polynomial CP : IK --+ L ( U ) such that CP(X0) is an isomorphism, and Xo is a v-transversal eigenvalue of the product family'2 := 2@. Moreover, the dimensions of C:(N[C$] n . . n N[C7-,]) for 0 5 j < r 1 do not depend on the family of isomorphisms CP. Of course, a polynomial @ : IK + L ( V ) is a family of the form

+

n

j=O

for some integer n 2 0 and some operators To,. . . ,Tn E C(U). Also, the product of two families is defined in the natural way. Namely, if W is another Banach space, 52 and 52' are two neighborhoods of XO, and

C :R

+ L(U, V ) ,

332 : R'

--f

L(W,V )

are two maps, then the product map 2332 : R n R'

cm(X) := C(X>m(x),

--+

L(W,V) is defined as

x E R n R'.

Local Smith Form and Equivalence f o r Families of Fredholm Operators

143

The next theorem is attributable to Esquinasl and L6pez-G6mezg. Theorem 5.1. Take r E NU (00) and let C E C'(R, C(U,V ) )be such that C(X0) is a non-invertible Fredholm operator of index zero.

(a) If A0 is an algebraic eigenvalue of C of order 1 5 u 5 T , then there exists a polynomial CP : IK -+ C(V) such that CP(X0) = Iu and Xo is a u-transversal eigenvalue of C@. (b) If @, S E C'(R, C ( U ) ) satisfg 0

CP(X0) and S(X0) are isomorphisms, XO is a kl-transversal eigenvalue of C* := CCP for some 1 5 kl 5 r , Xo is a kz-transversal eigenvalue of CQ := CS f o r some 1 I k2 I T .

Then kl = k p , the point XO is an algebraic eigenvalue o f 2 of order kl and, f o r each 1 5 j 5 k l ,

nN[C?])

j-1

i=O

i=O

j-1

dimC;(

= dimCT(

nN[Cy]).

Theorem 5.1 makes the following concept of multiplicity consistent (see Esquinasl and L6pez-G6mezg).

Definition 5.2. Taker E NU{oo} and let C E C'(R,C(U, V)) be such that C(X0) is a Fredholm operator of index zero for some XO E R. If XO is an algebraic eigenvalue of C of order 1 5 v 5 r , then the algebraic multiplicity of C at XO, denoted by x[C; XO], is defined by k

Cj

j-1

x [ ~ ; ~= o ] . d i m ~ ; ( n N [ @ ] ) , j=O

i=O

where CP E C'(R,C(U)) is such that @(XO) is an isomorphism, and XO is a k-transversal eigenvalue of the family

P := CCP . If C(X0) is an isomorphism we define

x[C;Xo] = 0. If C is of class C" and XO is a non-algebraic eigenvalue of C we define

x[C;Xo] = 00. The multiplicity of C at XO is said to be defined when some of the following options holds:

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J. L6pe.z-Gdmez and C. Mora-Corral

(a) C is of class C" in a neighborhood of XO; (b) C is of class Cr in a neighborhood of XO, for some integer r 2 0, and is a k-algebraic eigenvalue of C, for some 0 5 k 5 r.

A0

The multiplicity of C at XO is said to be finite when option (b) above holds. The multiplicity x[C;Xo] is left undefined when XO is a non-algebraic eigenvalue of C and C is not of class C" in any neighborhood of XO. Also, it is left undefined when XO is a k-algebraic eigenvalue of 2,but C is not of class C k in any neighborhood of XO. By the results of Section 6 , x[C;XO] is equivalent to the concept of multiplicity reviewed in Section 2. As an example, consider the family C E C"(R) defined by (3). That family is real analytic in R \ { 0 } , but zero is not an algebraic eigenvalue of C. Moreover, C j = 0 for all j 1 0, and, therefore, 0 is not a k-transversal eigenvalue of CCP for any integer k 2 0 and any @ E C"(R), as Theorem 5.1 predicts. 6. Canonical sets at transversal eigenvalues

In this section we prove that the theories of multiplicity according to Jordan chains (reviewed in Section 2) and through transversalization (reviewed in Section 5) are equivalent.

Lemma 6.1. Let k 2 0 be an integer, consider operators 20,. . . ,Ck E L(U, V ) , and suppose the following spaces conform a direct sum k- 1

R[CO],~l("CO]),

' " 7

C k ( n "Cj]).

(15)

j=O

Then, for every k vectors

UO,.

. . ,'ilk E u,

i=Q

i f and only i f

Proof. Obviously, (17) implies (16). Suppose (16). We will prove by

induction on k that (17) holds. It follows from (16) that COUO= 0, and,

Local Smith Form and Equivalence for Families of Redholm Operators

145

hence, (17) holds true for Ic = 0. Assume that, for every (yo,. . . ,yk-1) E U k such that j

0 5 j 5 k - 1,

= 0,

p y j - i i=O

one has

n

k-1-j

yjE

N[C~I, O 0 and for all X in a perforated neighborhood of Xo. Therefore, Xo is an algebraic eigenvalue of C. C3 implies Cd: Suppose XO is a k-algebraic eigenvalue of C. Thanks to Proposition 6.3, the length of every Jordan chain of C at XO is bounded from above by k. Cd implies C5 is obvious, C5 implies C1 is Proposition 3.1, and C3 is equivalent to C6 is Definition 5.2. This completes the proof of the theorem when T E (00,w). The statement of the theorem is obvious if T = 0. Now suppose that T 2 1 is an integer. D1 implies D2: If equations (6)and (7) hold for some neighborhood R' of Xo, and some continuous families C and $ defined in R' with values in the set of invertible m x m matrices, then !D(X)-'

. . ,(A - X O ) - ~ = ,1 , . . . , l},

= diag {(A - X O ) - ~ ~. ,

X E 0' \ {Xo},

and it is easily seen that lp(A)-lll 5 CIA - Xol-k for all X in a perforated neighborhood of XO, for some constant C k := max(k1,. . . ,kn}. D2 implies D1: Let us define the family k

> 0 , and

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J. L6pez-Gdmez

and C. Mom-Corral

By Proposition 7.1, XO is a k-algebraic eigenvalue of the analytic function h, there exists a function 5 of class C'-k in a neighborhood of XO with values in L ( V ) such that SC = h and ~ ( X O = ) Iv. By Proposition 6.3, there exist two analytic families & $ defined on a neighborhood of XO whose values are m x m matrices such that d(&) and $(XO) are isomorphisms, h(X)= $(X)D(X)$(X) for X N XO, and (7). Then, C = 3-1&D$. D3 implies D2: We define the family h through (25). Then, k is analytic and k is an upper bound for the length of its Jordan chains at XO. Let kl 5 k be the optimal bound. By the equivalence of Cl-C6 of the first part of the theorem and by Proposition 6.3, XO is a kl-algebraic eigenvalue of k. By Proposition 7.1, XO is a kl-algebraic eigenvalue of 2. D2 implies D3: We define the family k through (25). By Proposition 7.1, XO is a &algebraic eigenvalue of k,and, by Proposition 6.3, the length of the Jordan chains of k at XO are bounded from above by k . Then, by Propositions 2.3 and 7.1, C and k have the same Jordan chains at XO. Therefore, the lengths of the Jordan chains of C at Xo are bounded from above by k. D2 is equivalent to 04:This is Definition 5.2. 0 The following result allows us to reduce an infinite-dimensional problem with the Fredholm condition to a finite-dimensionalsetting; it is taken from Gohberg et al.5 (see also Gohberg and Siga17).

Lemma 7.1. Suppose r E M U (00,w) and C E C'(R,L(U,V)) satisfies C(X0) E @o(U,V). Then, there exist a n open neighborhood R' of XO contained in R, a decomposition U = Uo @ U1 with dimUo < 00 and U1 a closed subspace of U ,and three families E E C'(R', L(U,V ) ) , 5 E C'(R', L ( U ) ) and ?.?X E C'(R', L(U0))

such that E(X0) and ~ ( X O are ) isomorphisms and C(X) = @(A)

[rn(X)

@ IUJ

S(X),

XE

a'.

Proof. Since C(X0) is Fredholm of index zero, there exists a finite-rank operator F E L(U,V ) for which the family E defined by E(X) := C(X)

+ F,

X E 52,

takes invertible values for X in a neighborhood of XO. As F has finite rank, N [ F ] has a topological finite-dimensional complement UOin U ,so

U = Uo @ " F ] .

Local Smith Form and Equivalence for Families of Fredholm Operators

153

Let P E ,C(U) be the projection onto Uo with kernel N [ F ] . Then P has finite rank and F P = F , because Iu - P is a projection onto N [ F ] .Thus,

Iu for X

E

- E(X)-lF =

[Iu - PcE(X)-~FP] [Iu - ( I u - P ) E(X)-'FP]

XO. Set, for each X sufficiently close to XO,

S(X)

:= Iu - (Iu - P)e(X)-lFP.

This family 5 is of class C' and invertible in a neighborhood of Xo. In fact,

T(X)-'

= Iu

+ (Iu - P)E(X)-lFP,

X

Xo.

Thus, C(X) = E(X)[Iu- E(X)-'F] = E(X) [Iu - PE(X)-lFP]s(X), X

N

Xo.

Define

@(A) := Iu - PE(X)-lFP for X e Xo. With respect to the decomposition U family 6 can be expressed in the form @(A) = diag{@(~)luo,INIF]},

=

N

R [ P ]@ "PI, the

Xo,

since @ ( X ) ( V o ) c UOand @(X)IN[F]= h [ F ]7

N

Xo.

This concludes the proof.

0

Lemma 7.1 allows us to generalize Theorem 7.1 to the infinitedimensional case. The following characterization is thus obtained. Again, C1 and D1 below give us the local Smith form. Theorem 7.2. Take r E NU{m,w}, XO E K, a neighborhood R of XO, and a family C E C'(R, C(U, V ) )with C(X0) E @o(U,V ) .

If r E

(00,

w}, then the following conditions are equivalent:

Cl. There exist an open neighborhood R' of XO contained in R, two families E E C'(R',C(U,V)) and 5 E C"(R',L(U)) such that E(X0) and ~ ( X O ) are isomorphisms, and a decomposition U = UO@ UIwith dim UO< 00 and U1 a closed subspace of U such that C(X> = W ) [ W@)Iu,IS(X),

E Q',

(26)

J . Ldpez-Gdmez and C. Mora-Corral

154

and 9 ( X ) = diag {(A - X O ) ~ ' ,. . ., (A - X O ) ~ " } , X E R',

(27)

for some integers kl, . . . ,k, 2 1, and n = dim"&]. C2. For every neighborhood R' of XO, any Banach spaces U and any topological decompositions U = U1 @ U 2 and P = fi @ v 2 such that dim U1 = dim < 00, any families

v,

c? E c

y n ,C(P, V ) ) ,

cfl E C 6 " ( a 2 / , C ( U 2 , ?2)), such that the three operators

Wo),

5 E c-(R', C(U,U ) ) , M E CW(R', q U 1 , fi))

S(XO),

cfl(A0)

are isomorphisms, and every decomposition of the f o r m C(X) = W) [M(4@ n(x)l S(X),

E

a',

(28)

one has that the order of det M at XO is finite. C3. XO is a n algebraic eigenvalue of 2. Cd. The lengths of all Jordan chains of C at XO are uniformly bounded f r o m above. C5. No Jordan chain of C at XO can be continued indefinitely. C6. The multiplicity x [ C ;XO] is finite. If r 2 0 then the following conditions are equivalent:

D1. There exist a n open neighborhood R' of XO contained in 0, two families (E E C(R', C(U,V ) )and 5 E C(W,C ( U ) ) such that ~ ( X O and ) ~ ( X O are ) isomorphisms, and a decomposition U = UO@ U1 with dimUo < 00 and U1 a closed subspace of U such that (26) and (27) hold for some integers 1 I k l , . . . ,k, I r with n = dim N[Co]. D2. XO is a n algebraic eigenvalue of C of order k I r. D3. The lengths of all Jordan chains of C at Xo are bounded above by k 5 r. 04.The multiplicity x [ C ;XO] exists and is finite. Moreover, in this case the families C and is of class C'-k and 5 is analytic.

5 of D1 can be chosen so that E

Proof. Suppose r E (00,w). We prove the equivalence between Cl-C6. Cl implies C3: Owing to (26), for each X in a perforated neighborhood of Xo, C(X)-' = 5(X)-1[9(X)-1 @ IU,]e(x)-',

Local Smith Form and Equivalence for Families of h d h o l m Operators

155

whence, thanks to (27), for each X belonging to some perforated neighborhood of Xo, and some constant C > 0 one has

where k is the maximum of k l , . . . ,k,. C5 implies C1: We use Lemma 7.1 and, following its notation,

c = e(m@ Iv,)5.

(29)

By Proposition 2.2, no Jordan chain of 9 T. @ I c ~can , be continued indefinitely. It is easy to verify that the Jordan chains of M @ Iul at XO are of the form

(:) ,...,(;)), where (210,. . . ,21), is a Jordan chain of M at XO. Therefore, no Jordan chain of the finite-dimensional family M can be continued indefinitely. Applying the implication C5 implies Cl of Theorem 7.1, one concludes the result easily. C3 implies C2: Suppose the expression (28) as described in C2. Then, for X in a perforated neighborhood of XO,

m(x)-' a3 %(A)-'

= s(X)c(x)-'e(x),

whence

IlM(X)-'ll

I cIIc(X)-'II

for some constant C > 0. Then, XO is an algebraic eigenvalue of M, and by Theorem 7.1, det M has a zero of finite order at XO. C2 implies C3: We use Lemma 7.1 and, following its notation, (29). Then, d e t M has a zero of finite order at XO. By Theorem 7.1, XO is an algebraic eigenvalue of M and, hence, of 2. C3 implies C4 follows from Proposition 6.3, C4 implies C5 is obvious, and C3 is equivalent to C6 is Definition 5.2. This completes the proof in the case T E (00,w). The statement of the theorem is obvious if T = 0. Now suppose T 2 1 is an integer. Dl implies D2 is similar to the proof of C1 implies C3. The implication D2 implies 0 3 follows from Proposition 6.3. The proof of 0 3 implies Dl is similar to the one of C5 implies Cl. The equivalence D2 is equivalent t o 0 4 is Definition 5.2. This concludes the proof. 0

J. L6pez-Gdmez and C.Mora-Corral

156

8. Equivalence of families

The local Smith form of a family is the simplest form that can be obtained from that family by multiplication by families of isomorphisms. This fact is fully understood within the theory of equivalence of families. Definition 8.1. Let U1, U2, V I ,VZ be four Banach spaces. Take XO E K and r, s E N U (03, w } . Let SIo(U1,V1) be the set of functions C defined on a neighborhood of A0 such that C(X0) E 'Po(U1,Vi). We say that two families C E Sio(U1, Vi) and LJJl E SIo(Uz, V2)are C" equivalent at XO when there exist a neighborhood 52 of XO in K,and two families E E Si0(V2,Vl) and 3 E Sio(U1, U Z )such that C(X) = E(X)m(X)3(X),

X E 52.

Moreover, SIo(U1) will denote SIo(U1,U1). We also define SIoas the set of families C such that there exist two Banach spaces U , V for which C E SI0(U,V ) . Take r, s E N U (03, w } . It is immediate to verify that the C" equivalence is a relation of equivalence in SIo. Many statements of the previous sections can be rephrased in a shorter way using the language of equivalence introduced in Definition 8.1. For example, Proposition 7.1 may be expressed as follows. Proposition 8.1. Take r E N U ( o o , ~ } . Let C,m E SI,(U,V). Suppose that XO is a k-algebraic eigenvalue of C for some integer 0 5 k 5 r , and that m, = 2, for all 0 5 n 5 k. Then, there exists 5 E S;ok(V) such that s(X0) = I" and C = 3!Bl. The families C and ?JJlare C'-k equivalent at XO, and XO is a k-algebraic eigenvalue of M.

Suppose that A0 is a k-algebraic eigenvalue of Take r E N U (03). C E SIo,for some integer 0 5 k 5 r. Then, as shown in Theorem 7.2, C has a local Smith form at XO. Now we are going to see that this local Smith form is unique, up to the order of the exponents.

Proposition 8.2. Let U, V be two Banach spaces such that

where rn

X and Y are closed subspaces of U ; W and Z are closed subspaces of V ;

157

Local Smith Form and Equivalence for Families of Fredholm Operators

m = dimX = dim W for some integer m 2 0 .

Take T E N u {co} and let 6 E Si,(Y, 2 ) be such that ~ ( X O is ) a n isomorphism. Let k l , . . . ,Ic, E N n [ O , T ] , and define M ( X ) := diag{(X - X O ) ~. ~. .,,(A - Xo)",6(X)}

E L(U,V),

X

21 Xo,

(31) with respect to certain bases of X , W , and to the decompositions (30). Consider a family C E SI, that is C equivalent to M at XO. Call n := dim N[Co]. Then n 5 m, and there exists a permutation T of (1,.. . ,m} such that 'T(n+l)

= ' ' ' = 'T(,)

(32)

= '7

and the integers IcT(1)

2

' ' '

2 'T(T2)

(33)

are the partial multiplicities of C at XO. Proof. By Proposition 2.3, the families C and M have the same partial multiplicities at XO. Define the family D as D(X) = diag{(X - X O ) ~.~. .,, (A - A,)"}

E

L ( X ,W ) ,

X

IIT

XO,

with respect to the bases of X and W for which (31) is satisfied. Fix an integer s 2 0. As ~ ( X O is ) an isomorphism, the Jordan chains of M at XO of length s 1 are precisely of the form

+

(:)

,...,

(;)

,

where ( 2 1 0 , . . . , u s ) is a Jordan chain of D at XO. In particular, M and D have the same partial multiplicities at XO. Let T be a permutation of (1,.. . ,m} such that ',(I)

2

' ' '

1

kT(T71) *

As C and M are C equivalent at XO, and 60is an isomorphism, then the dimension of the kernels of CO and 90coincide. This implies that n 5 m and (32). Finally, from the diagonal structure of 9, it is easy to see that the integers (33) are the partial multiplicities of D at XO. This concludes 0 the proof. Now, we present'the main result of this section, where several necessary and sufficient conditions for the equivalence of families are presented.

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J. L6pez-Gdmes and C. Mom-Corral

Theorem 8.1. Take r E NU(oo,w} and four Banach spaces U1, U2, VI, V2. Let C E Sg, ( V l ,V1) and 9l E Sg0(U2, V2) be such that XO is an algebraic eigenvalue of both C and M of (possibly different) order less than or equal to k , for some integer 0 5 k 5 r. Then, the following assertions are equivalent:

(a) (b) (c) (d)

The families C and 9.Jl are CT-k equivalent at XO. The families C and !Dl are C equivalent at XO. The families C and !Dl have the same partial multiplicities at XO. For all integers 0 5 s 5 r, dim N[trng(Co, . . . ,C,}] = dim N[trng(Mo, . . .,!TI,}].

(34)

(e) For all 0 5 s 5 k - 1, equality (34) holds. (f)There exist Eo,.. . ,E k - 1 E L(U1, U2) and Fo,. . . ,F k - 1 E L(V2, K ) such that EOand FO are isomorphisms, and trng(C0, * * * 7 Ck-I} = trng(E0,. . . , E ~ - I } .trng{mo,. . . , m k - l } .

trng(F0,. . .,Fk--l}.

(9) There exist Go,. . . ,G k - 1 E L(%,V2) such that Go is an isomorphism, and

-

“trng(C0, * . ,&-I}] = trng(G0,. . . ,Gk-1}N[tITlg(mo,. . . ,mk-l}].

(h) There exist polynomials Q : IK 3 C(U1) and 9 : IK L(U2) such that Qo = Iul and 90= Iu2, the families C@:= CQ and mq := !Dl9 are transversal at Xo and, for all 0 5 j 5 k - 1, --f

j-1

dims;(

nN [ c ~ )

= dimMF(

i=O

j-1

n~ [ m 3 ) .

(35)

i=O

(i) For all Q , 9 E SxTo such that QO and 90are isomorphisms, the families C@:= CQ and Mq := 9XP are transversal at XO, and, for all integers 0 5 j 5 r , equality (35) holds.

Proof. The following implications are obvious: (a) implies (b); (d) implies (e); (f) implies (g). The following implications have already been proved: the implication (b) implies (c) follows from Theorem 7.2 and Proposition 2.3; the equivalence between (e) and (h) follows from Theorem 5.1 and Proposition 6.4; the implication (h) implies (c) follows from Proposition 6.3; the equivalence between (h) and (i) follows from Lemma 4.1 and Theorem 5.1.

Local Smith Form and Equivalence for Families of FredhoIm Operators

159

The implications below require a proof. (e) implies (d): Proposition 6.4and Lemma 4.1 imply that dim N[trng{Co,

. . .,Ck-l}]= dim N[trng{Co, . . . ,C,}]

and dim N[trng{!Vlo,. . . ,9&-1}]

= dim N[trng{!Vlo, . . . , m,}].

for all k - 1I sI T. (c) implies (a): As C and M have the same partial multiplicities at Xo, in particular dimN[Co] = dimN[Mo]. By Theorem 7.2, the family C is Cr-k equivalent at XO to a 9 E S,",(Ul) that expresses in the following form D(X) = diag{(X - X O ) ~. .~.,,(A - X~)~",lx}, X

N

Xo,

for n = dim N[&] , some integers 1 5 kl,. . . ,k, 5 T and X a closed subspace of U1.Without loss of generality, we suppose that kl 2 . . 2 kn. By Proposition 8.2,the integers kl,. . . , k, are the partial multiplicities of C at Xo. Similarly, is Crek equivalent at XO to a fJ E S,"o(U2) that expresses as

-

%(A) = diag{(X - X O ) ~ ~.,. ,. (A - X O ) ~ " ly}, ,

X N XO,

where Y is a closed subspace of UZ.As U1 and U2 are isomorphic, and the codimension of X in U1 coincides with the codimension of Y in UZ, then X and Y are isomorphic and the families 9 and fJ are C" equivalent. Therefore, C and !Bl are Cr-k equivalent. (g) implies (e): Consider 0 5 s 5 k - 1 and II: E N[trng{Co, . . . ,C,}]. Then the column vector col[O,z] in U,k belongs to N[trng{Co, . . . ,&-I}]. Therefore, there exists a z E N[trng{!7&,

. . . ,?lRk-1}]

such that col[O,y] = trng{Go,. . .,Gk-1)~.

It is easy to see that necessarily z is of the form col[O,z] with to N[trng{Tmo,. . .,!Vl,}]. Therefore, y = trng{Go,.

II:

belonging

. . ,G,}z.

As 60 is an isomorphism, the previous reasoning can be reversed and, consequently, N[trng{Co,. . . ,C,}]= trng(G0,. . . ,G,}N[trng{Mo,

. . . ,M,}],

J. Ldpez-Gdmez and C. Mora-Corral

160

which implies (e). (b) implies (f): Define the polynomials

as k

C(X)

k

:= C C j ( X - Xo)k, j=O

%(A)

:=

CMj(X - XI#,

X E K.

j=O

By Proposition 8.1, C and k are C equivalent at XO, and so are the families M and ?k. Moreover, by assumption, the families C and M are C equivalent at XO. Therefore, the analytic families k and 9%are C equivalent at XO. Since we have proved the equivalence between conditions (a) and (b), we obtain that k and ?% are C" equivalent at Xo, and, hence, there exist C,3 E Syo such that C(X0) and ~ ( X O are ) isomorphisms and

k=c?%3. Thus, trng(C0,. Ck-1) = trng{ Eo, . . . , Ck-1)

. trng{%Xo, . . . ,M k - 1 ) . trng{&, . . . ,S k - 1 ) .

This concludes the proof of this implication and of the theorem.

0

References 1. J. Esquinas, Optimal multiplicity in local bifurcation theory, 11: General case. J . Differential Equations 75 (1988) 206-215. 2. J. Esquinas and J. Lbpez-Gbrnez, Resultados 6ptimos en teoria de bifurcaci6n y aplicaciones, Actas del IX Congreso de Ecuaciones Diferenciales y Aplicaciones, Universidad de Valladolid, 1986, 159-162. 3. J. Esquinas and J. L6pez-G6mez, Optimal multiplicity in local bifurcation theory, I: Generalized generic eigenvalues. J. Diferential Equations 71 (1988) 72-92. 4. A. F'riedman and M. Shinbrot, Nonlinear eigenvalue problems. Acta Math. 121 (1968) 77-125. 5. I. Gohberg, S. Goldberg, and M. A. Kaashoek: Classes of Linear Operators, vol. 1, Operator Theory: Advances and Applications 49, Birkhauser, Bassel 1990. 6. I. Gohberg, P. Lancaster and L. Rodman, Matrix polynomials, Computer Science and Applied Mathematics, Academic Press, New York 1982. 7. I. C. Gohberg and E. I. Sigal, An Operator Generalization of the Logarithmic Residue Theorem and the Theorem of RouchB. Math. SbornQ 84( 126) (1971), 607-629. English Trans.: Math. USSR Sbornik 13 (1971) 603-625.

Local Smith Form and Equavalence f or Families of Fredholm Operators

161

8. T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin 1995. 9. J. Lbpez-Gbmez, Spectral Theory and Nonlinear Functional Analysis, CRC Press, Chapman and Hall RNM 426, Boca Raton 2001. 10. J. Lbpez-Gbmez and C. Mora-Corral, Finite Laurent Developments and the Logarithmic Residue Theorem in the Real Non-analytic Case. Integr. Equ. Oper. Theory 51 (2005) 519-552. 11. R. J. Magnus, A Generalization of Multiplicity and the Problem of Bifurcation. Proc. London Math. SOC.(3) 32 (1976) 251-278. 12. P. J. Rabier, Generalized Jordan Chains and Two Bifurcation Theorems of Krasnoselskii. Nonlinear Analysis, Theory, Methods t'd Applications 13 (1989), 903-934. 13. P. Sarreither, Zur algebraischen Vielfachheit eines Produktes von Operatorscharen. Math. Scand. 41 (1977) 185-192. 14. J. T. Wloka, B. Rowley and B. Lawruk, Boundary Value Problems for Elliptic Systems, Cambridge University Press, New York 1995.

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MULTILUMP SOLUTIONS OF THE NON-LINEAR SCHRODINGER EQUATION - A SCALING APPROACH

ROBERT J. MAGNUS Science Institute University of Iceland Dunhaga 3 107 Reykjavik, Iceland E-mail: [email protected]

+

The problem -AIL F ( V ( c z ) , u )= 0 is considered in Wn for small E . A special case, -Au+ V(EZ)U - U P = 0, is equivalent to the problem of finding semi-classical stationary states of the non-linear Schrodinger equation. For small E > 0 solutions are obtained that approach, as E + 0, a linear combination of translates, through a distance O(l/e), of solutions of -Au F ( a k , u ) = 0 in W2~2(Wn),where a k is a finite sequence of non-degenerate critical values of V . These are the secalled multi-bump solutions. The method involves a rescaling of the variables and the use of a modified implicit function theorem. The usual implicit function theorem is inapplicable owing t o lack of convergence of the derivative of the non-linear Hilbert space operator, obtained after an appropriate rescaling, in the operatornorm topology. An asymptotic formula for the solution for small E is obtained.

+

1. Statement of the problem Consider the problem

-Au

+ F(u,u) = 0

(1)

where F is a smooth function, typically a polynomial in u,and a a real parameter in some interval I. Suppose that for each a we know a solution of (l),&(x), with the properties that 0 0

&(x) decays exponentially at infinity; &(x) has minimal degeneracy.

By minimal degeneracy we mean the following: 0

All solutions ~ ( xdecaying ) as 11x11 -+

163

00

of

R. J . Magnus

164

are spanned by the n partial derivatives Dj&(z), which are assumed to be linearly independent. The equation

is solvable if and only if

for j = 1,.. . , n . We are being a little vague here because we don’t wish at this stage to get involved in specifying function spaces, but a precise description of minimal degeneracy would say that with respect to certain function spaces the linearization of (1) at the solution &(x), that is, the F’rechet derivative at this point, is a Fredholm operator of index 0. Moreover its kernel is n-dimensional and contains no more than it is forced to, as revealed by differentiating (1) with respect to the coordinates xi. We now pose:

((XI(

+ CCI when a is replaced by a Problem Find solutions decaying as function of x with range in I that is “nearly constant.”

There are of course many ways to understand the expression “nearly constant.” We shall study the case

+ F ( V ( E Z U) ,) = 0

-Au where E in I .

(2)

> 0 is a small parameter and V a given smooth function with range

Example Take F ( a ,u)= au-up with I = 10,m[. This gives the problem

+

-AIL V ( E Z )Uup = 0.

(3)

Putting y = E X leads to the problem of semiclassical states of the non-linear Schrodinger equation with potential V ,

-e2A,u

+ V(y)u - up = 0.

The function &(x) is the ground state solution of

-Au defined as the minimizer of

+ uu

-

up = 0 ,

(4)

Multilump Solutions of the Non-linear Schrodinger Equation

165

subject to the constraints u # 0 and

1

( l l V ~ 1 1+~au2 -

dx = 0.

(The existence of &a for 2 5 p if n 5 2 and for p < if n 2 3 was established in a series of papers starting from that of W. Strauss [5]. Minimal non-degeneracy of the ground state was established in a series of papers of which we may cite those of M. Weinstein [7] and Kwong [2].) Although we shall concentrate entirely on the problem (2) it is worth mentioning some other examples.

Example TakeF(a,u) = u--up. Replace a by the function a+h(X-ix), X > 0 small. The problem obtained is equivalent to the non-linear eigenvalue problem

A,w where x = f i y , w = X*u.

+ ( a + h(y))wP = XW Note that the problem

-Au

+ u - UU’

=0

has the minimally non-degenerate solution u = a - A & ( x ) where &(x) is the ground state of -Au u - up = 0.

+

Example Another case is F ( a ,u ) = u - aup. Replace a by the function a ~ h ( x )We . have the perturbation problem

+

-Au

+ u - (a + E~(Z))U’

= 0.

2. Background

Floer and A. Weinstein [l](1986) showed the existence of solutions of (4) that accumulate near a non-degenerate critical point of V for small E . Then Y. G. Oh [4] (1990) showed the existence of solutions that accumulate near several critical points (multi-hump solutions). The author proposed a conceptually simple method to obtain multi-bump solutions in 1998 [3]. These solutions are illustrated below both for equation (4) (with spatial variable y; the solutions accumulate at critical points bl and b2 of V) and for (3) (spatial variable x; the “humps” keep a nearly constant profile and have separations of order O ( ~ / E ) ) .

R. J. Magnus

166

3. Method of scaling

By a scaling, or perhaps more properly a rescaling, we mean an €-dependent change of coordinates in the state space that becomes singular at E = 0. We illustrate this first with a simple finite-dimensional example in which all the main steps occur that are encountered in infinite-dimensional ones.

Example (First year calculus) Let f : R x R -+R. Suppose that f(0,x) = 0 for all x. Problem: to find c such that solutions of f ( E , x) = 0 exist for which E # 0 is small and x is near to c. Assume that: -(o,c) af

a€

= 0,

-(o,c) a2f

aEax

# 0.

Introduce the scaling

x = c + €W. Substitute the scaling and expand by Taylor’s Theorem: a2f a2f (0,C)E2W + 0 ( € 3 ) . + Ew)= -21 (0,c)E2 + d€2 d d X

f(€,

Divide by c2. This gives the rescaled problem:

+ =a2f (O,C)W

Take the limit

E

Id”f(0,C) 2 dE2 -+ 0

limE-2f(E,c+ew) = --(o,c) 1d2f 2 dE2 This gives the limit equation: €+O

q ( 0 , c ) 2

aE

+ -(o,c)w a2f a€aX

+ O(E) = 0. + -a2f (o,c)w. = 0.

Multilump Solutions of the Non-linear Schrodinger Equation

167

Solve the limit equation; it has the solution wo where

We conclude by the implicit function theorem that the problem f ( ~z), = 0 has solution such that 2 = c EW(E) and W(E) -, wo.

+

We propose to solve

+ F ( V ( E X )U), = 0

-Au

(5)

by scaling. First we must pose our problem in some function spaces. We shall simply assume that the map u

-Au

+ F ( V ( E Z )U),

is differentiable from the Hilbert space W2,2(Rn) (the state space of our problem and hereafter denoted W2>2) to L2(Rn)(hereafter L 2 ) for each E > 0 and that its Frechet derivatives can be calculated in the obvious way. Let b be a non-degenerate critical point of V ;let a = V(b).Let 4(z) be the solution &(z) of

-Au

+ F ( a ,U) = 0.

In order to describe the scaling we define a new state variable (s,w) where s E Rn and w E W where

*s

W = {W E W2*2 . wDj4 = 0, j = 1 , . . . , n } . The scaling is defined by

u ( 5 ) = 4 ( z - ; +bs

)

+ E2

w (z - - e+ bs ,

or for short

u(z) = 4(z


h > 0. T h e n there exists K > 0 independent o f 6 such that

Multilump Solutions of the Non-linear Schrodinger Equation

173

References 1. Floer, A. and Weinstein, A. Nonspreading wave packets for the cubic Schrodinger equation with a bounded potential. J. Funct. Anal. 69, 397-408 (1986). 2. Kwong, M. K., Uniqueness of positive solutions of Au - u u p = 0 in Rn. Arch. Rational Mech. Anal. 105, 243-266 (1989). 3. Magnus, R. J., On multi-lump solutions to the non-linear Schrodinger equation. Elec. J. Diff. Eq. 1998, no. 29, 24pp. 4. Oh, Y. G., On positive multi-lump bound states of nonlinear Schrodinger equations under multiple well potential. Comm. Math. Phys. 131, 223-253 (1990). 5. Strauss, W. A,, Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977) no. 2, 149-162. 6. Wang, X., On concentration of positive bound states of nonlinear Schrodinger equations. Comm. Math. Phys. 153, 22S244 (1993). 7. Weinstein, M., Modulational stability of the ground states of nonlinear Schrodinger equations. SIAM J. Math. Anal. 16, 567-576 (1985).

+

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SOME ELLIPTIC PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS

c. MORALES-RODRIGO AND A. SUAREZ* Dpto. Ecuaciones Diferenciales y Andisis Numkrico, C/ Tarfia s/n, 41080, Univ. of Sevilla, Seville, Spain E-mails: cristianmatematicasOyahoo.comand suarezOus. es

In memory of Prof. J. Esquinas This paper concerns with some elliptic equations with non-linear boundary conditions. Sub-supersolution and bifurcation methods are used in order to obtain existence, uniqueness or multiplicity of positive solutions.

1. Introduction

In this paper we study positive solutions of some nonlinear elliptic problems with mixed nonlinear boundary conditions. Throughout it, we consider the following assumptions: (1) R c RN, N 2 1, is a bounded domain with boundary dR of class C2. Moreover

3~ := rou rl, where ro and rl denote two disjoint open and closed sets in the relative topology of dR. (2) L is a uniformly elliptic differential operator in R of the form

with coefficients aij = aji E C27a(n)l bi E C1ta(n) and c E C"(G), a E (0,l). *Supported by the Spanish Ministry of Science and Technology under grant BFM200306446.

175

176

C. Morales-Rod?%go and A . Szldrez

(3) We define the mixed boundary operator, B, by

Bu :=

u

on ro,

Bu on rl,

+

where the operator B := 3, b with Y E C1(I'l,RN) an outward pointing nowhere tangent vector-field and b E C1@(I'l). In this paper we study the following problems where a is a positive or negative regular function on rl and 0 < q < 1 < p , r . We first study an elliptic equation with a logistic term on the boundary

{

Lu=O u=O

Bu = pu +

in R, on ro, a(z)uron r1,

(3)

where p E R will be regarded as bifurcation parameter. We do not know previous works in which (3) was analyzed. We characterize the existence, uniqueness and stability of positive solution in terms of the parameter p (see Theorem 5.2). Second, we study of the sublinear-superlinear equation

-Au = Xu - u p in 0, on 30,

(4)

where n is the outward normal vector-field of R. (The case -uT instead ur has been studied in Ref. [S].) Equation (4) has attracted a lot of attention in the last years with X = 0, see Refs. (61, [lo], (181, [21], [22] and [26], among others, where basically the equation and its corresponding parabolic problem were analyzed in the particular case X = 0, and in Refs. [28], [29] where the local bifurcation was studied. We complete this study giving existence, non-existence and stability results in function of X (see Theorem 5.3). Finally, we study the concave-convex equation

Lu = Xm(z)u* in 52, - a(z)ur in

dR.

(5)

where m E C(a)is nonnegative and non-trivial. Equation (5) was studied previously in Ref. [14] when Lu = -Au u, and m = a z 1 by variational methods. When a < 0 we prove that there exists a positive solution of (5) if and only if X > 0. If a > 0 we complete and improve the results of Ref. [14] (see Theorems 5.4 and 5.5).

+

Some Elliptic Problems with Nonlinear Boundary Conditions

177

In order to study these equations we employ mainly sub-supersolution and bifurcation methods. We present in Sec. 2 results related with principal eigenvalues associated to these problems. In Sec. 3 we prove a general result of bifurcation from the trivial solution when the bifurcation parameter appears in both equation and boundary. As consequence, we can use it for equations (3) and ( 4 ) . For the study of (5) we need a different result of bifurcation, where the parameter is in front of a non-linear term. In Sec. 4 we present results concerning to uniqueness, stability and a-priori bounds of positive solutions for general equations with nonlinear boundary conditions. Finally, in Sec. 5 we apply the results to the cited equations. 2. Some Preliminary Results: Eigenvalues Problems

Along this paper, we use the positive cone

P := {u E C1@) : u 2 0, u # 0 in R U rl, Bu = 0 on d o } , and we say that u is positive if u E P and that u is strongly positive if u E int(P) := {uE P : u > 0 in R U rl, d u / d n < 0 on ro},where n is the outward normal vector-field of 52. On the other hand, the mixed operator B m, m E C(rl), means a similar operator to (2) with b m instead of b in B. Finally, given two functions u,v we write (u, w) > 0 if u,w 2 0 and some of the inequalities non-trivial. Consider the eigenvalue problem

+

+

R, ondR.

= X p in

p { LBp=O

H. Amann [2]proved the existence of a unique simple eigenvalue, the principal eigenvalue, whose associated eigenfunction can be chosen strongly positive in R. We denote this eigenvalue by a l [ L ,B]. a l [ L , D ]and al[L,Nl stand for the principal eigenvalues under Dirichlet and Neumann homogeneous boundary conditions, respectively. Some properties of al[L,B] have been studied in details by S. CanoCasanova and J. Lbpez-Gbmez [9] (see also Ref. [4]),we state some of them. Proposition 2.1.

[L,B] > 0 if and only if there exists a positive supersolution of ( L ,B , R), i.e., a positive function E such that LE 2 0 in R and BE 2 0 o n dR with some inequality strict. (2) The map q E Lm(R) H 01 [ L q, B] is increasing and continuous. (1)

01

+

C. Morales-Rodrigo and A . Sudrez

178

+

(3) The map m E C(r1) H crl[L, B m] is increasing and continuous. (4) Suppose I?l # 0 and consider a sequence b, E C(r1) such that limn++w minr, b, = +w. Then,

lim ol[L, B

n++W

(5) Suppose I'l #

+ bn] = ai[L,D].

0, then 61[L,B] < 61[L,D].

Consider now the eigenvalue problem

{

Lcp = Xm(z)cpin R, cp=o on Fo, f3p = Xr(z)cp on rl.

(6)

We suppose the following condition mE

Ca(n), r E C'@(rl),

3p 2 0 such that ( c

+ pm, b + p r ) > 0.

(7)

The following result provides us the existence of principal eigenvalue of (6). The second paragraph gives a characterization of the principal eigenvalueof (6) when m = 0, i.e., an eigenvalue problem at the boundary, the classical Steklov problem. In our acknowledge this result is new, although it nearly follows by the results on Ref. [9](see Ref. [15]where a particular result is obtained.) Theorem 2.1. Assume ( m , r )> 0. Then: (1) Under condition (7), the eigenvalue problem (6) has a unique princi-

pal eigenvalue, y1 [L,B ] ,it is simple and its associated eigenfunction can be chosen strongly positive in 52. (2) If m E 0 and r > 0 , then, the principal eigenvalue exists for (6), denoted b y Xl[L, B ] , if and only if (T~[L,D] > 0. Moreover, its associated eigenfunction can be chosen strongly positive in R. Proof. The first paragraph follows with the same kind of arguments used = 0. in Theorem 2.2 of H. Amann [3] where It is clear that A1 is a principal eigenvalue of (6) with m = 0 if and only if p(X1) = 0 where p(X) := q [ L , B - Xr(z)]. We know by Proposition 2.1 that limx+-wp(X) = ul[L,D], p(X) is a decreasing and continuous function. So, it suffices to prove that limx-t+mp(X) = -w. Suppose the contrary, then limx++wp(X) = -1. Take k E B large enough such that k c(z) > 0 and k > 1 then, first part

+

179

Some Elliptic Problems with Nonlinear Boundary Conditions

of the Theorem can be applied to the eigenvalue problem

{

Lcp+ kcp = ccp in R, cp=o

on

ro,

Bcp = Xr(z)cp on rl.

Hence, there is a principal eigenvalue This is a contradiction.

+

that verifies 0 = c(i1)= p(i1) k . 0

The following result will be very useful along this work.

Lemma 2.1. Assume (7) and ( m , ~ > ) 0. 61[L,B ] > 0

Then yl[L, B] > 0

Proof. We know by Theorem 2.1 that y1[L, B]exists, and it is the unique zero of the application

[L- um, B - UT]. Since p is a decreasing function, then p(0) > 0 implies 60= yl[L, B]> 0 and the contrary. P(U) = ~1

p(u0) =

0 for

3. Bifurcation Results for Equations with Nonlinear

Boundary Consider the nonlinear equation

{ where f E Ca(n x

Lu = Xm(z)u + f (z,u)in 52, u=o Bu = XT(S)U

on ro,

+ g(z, u) on rl,

a), g E Clt"(r1 x W),

(8)

such that

f ( z , o )= o VZ E R, g(z,o) = o vz E rl,

(9)

(m,T ) > 0 and satisfy condition (7) and X is a bifurcation parameter. Remark 3.1. Due to the condition (7) we can assume, adding p m and pr to both sides of (8), that (c,b) > 0. Now, we reduce the equation (8) to a suitable equation for compact o p erators. Define = {w E : wlr, - 0 ) (analogously it can be defined and the map K1 : C;,(a) --+ by, given f , K1 (f)= u where u is the unique solution of the problem

Cflb(a) C;T(a))

P(a)

Lu = f in R, Bu = 0 on dR.

180

C. Momles-Rodrigo and A . Sudrez

We can extend this operator to Cro(0).Thanks to elliptic regularity results, this new operator, denoted again by K1, is compact as operator from Cro(0) 201 to Cro(0). We define now K2 : C(r1) -+ Crb (R) by, .given 9, K2(9) = u with u the unique solution of the problem

{

Lu = 0 in R, u=O o n r o , BU = g on Fl.

Again, it can be proved that the operator K2 : Cro(dR) -+ Cro(Q) is compact. Denote by y : C(0) -+ C(r1) the trace operator. Following the same kind of arguments that in Ref. [3],Lemma 4.1, and denoting M , R, F and G by the Nemitski operators associated to m(z)u,T(Z)U,f and g respectively, we have

+

+ K2[XR(y(u))+

Proposition 3.1. u satisfies u = K l [ XM ( u) F ( u ) ] G ( y ( u ) ) if] and only if u is a classical solution of (8).

Since we are only interested in non-negative solutions of (a), we rewrite (8) as a problem with only non-negative solutions. Let u+ = max{u, 0).

Lemma 3.1. I f u is a solution of

then u 2 0. Proof. Suppose that the problem (10) possesses solution u such that there exists a connected component R1 c 52 of the set R’ = {z E R : u(z)< 0) such that u < 0 in R1. Observe that dR1 n rl # 0 . Indeed, if R1 c R, then

Lu=O

inn,,

u=O

ondR1.

Since c 2 0, then by the maximum principle u = 0 in 521. Hence, dRlfW1 # 0 . Due to Lu 2 0 in R l and c 2 0 then, by the maximum principle, the minimum of u must be attained on dR1. As u < 0 in R l and u = 0 in dR1 n ro then, minimum must be attained on dR1 n rl, but in such points we have dU

dv = -b(z)u 2 0, contradicting Hopf’s Lemma (see Lemma 3.4 in Ref. [16]).

0

Some Elliptic Problems with Nonlinear Boundary Conditions

181

Remark 3.2. Lemma 3.1 is still true if f(z,0) 2 0 and g(z, 0) 2 0.

(a)defined by @A(.) = 21 - Kl[XM(u+)+ F(u+)]- K2[XR(y(u+)) + G(r(u+))], @I((.) = u - tKl[XM(u+)+ F(u+)]- tKz[XR(y(u+)) + G ( ~ ( u + )t) 2] , 0. Consider the maps

: Cr,

@A,

(a)

-+

Cro

Thanks to Proposition 3.1 and Lemma 3.1, u is a classical nonnegative solution of (8) if and only if @~(u) = 0 in Cro(a). Assume that lim

s-0f

fo = o unif. in a, s

lim s+o+

Finally, denote by y1 := y1 [L,B ] ,and 51 associated.

= o unif. on rl. (11) s its strongly positive eigenfunction

Lemma 3.2. Let A C R be a compact interval such that X < y1 for all X E A. Then, there exists 6 > 0 such that @ # 0IVu (. E) Cro(n)with = llull E (0,6), VX E A and W E [0,1].

(a)

Proof. Suppose the contrary, that there exist A,, t , E R and u, E Cr, such that A, -+ 1,t , -+f, JIu,)) -+ 0 and = 0. By Lemma 3.1, u, 2 0 and dividing by llunll we obtain

@k(u,)

(12) where V, = Thanks to (11) we have that the terms inside K1 and K2 are uniformly bounded in and on PI, respectively. Since K1 and K2 are compact operators, then the sequence 21, is a relatively compact in C ( Q . Therefore, we can suppose that V, + B in C@). By ( l l ) ,we have

a.

a

Passing to the limit in (12), we conclude that

+

V = T[XKi(M(B)) XKz(R(y(V)))].

Thanks to u, 2 0, llvnll = 1 and by the maximum principle, V is a strongly positive function in R. Due this fact % = y1 but this is not possible because At < y1 by the choice of the set A. 0

-

We are going to use the following notation: for R > 0, let BR = {uE Cr,(n) : llull < R}. Then, deg(@A,BR,O) stands for the degree of @ A on

182

C. Momles-Rodrigo and A. Suhez

BR with respect to 0 , and i ( @ x ,uo,0 ) denotes the index of the solution uo of the equation @x(u)= 0.

Corollary 3.1. If X < 71, then i(@px,0,O) = 1. Proof. If X > 0 consider the interval A = [0,A] in the contrary case consider A = [A, 01. Thanks to the Lemma 3.2, we know that 36 > 0 such that Vu E Cp,,(a)with llull E (0,6) we have @"Xu)# 0 , Vt E [0,1]. Therefore by homotopy invariance of the degree we obtain

i ( @ x ,0,O) = deg(@i = @ A , Bg,O) = deg(@: = I , Bg,0) = 1.

0

Lemma 3.3. Let X > 7 1 . Then, there exists 6 > 0 such that Vu E Cr,(n) With ltull E (0,6), @x(u) # T t i , VT 2 0 . Proof. Assume that there exist sequences T~ 2 0 , u, E Cro(G)such that llunll + 0 and @x(un) = T,&. Thanks to Proposition 3.1 and similar arguments that we have employed in Lemma 3.1, we have that u, > 0 is a classical solution of the problem

{

+ f(z,u,) + Y I T , ~ ( Z ) J I in R, on ro, Bun = Xr(z)u, + g(z,u,) + yiT,r(z)& on rl. Lu, = Xm(z)u, u, = 0

Since by Remark 3.1 we can assume that (b,c) > 0 , positive constants are supersolutions of ( L , B , R ) , and so by Proposition 2.1 it follows that al[L,B]> 0, and so that by Lemma 2.1, y1 > 0. Thanks to conditions (ll),we obtain

{

Lu, > Xm(z)u, - EU, in R, u, = 0 on ro, Bu, > Xr(z)u, - E U , on I'l,

Hence, u, is a strict positive supersolution of ( L - Xm(z) E , R), and then

+

6E(X) = a1 [ L - h ( ~ E , B) - X~(Z)

+ E , B - Xr(z) +

+ E ] > 0.

(13)

On the other hand, we know that y1 is the unique zero of the continuous and decreasing function &(A) = ( T ~ [ LXm(z),B- Xr(s)]. Since X > y1 then S(X) < 0. Moreover, by Proposition 2.1, we infer that exists E > 0 such that &(A) < 0 , contradicting (13). 0

Corollary 3.2. If X > 71,then i ( @ x ,0,O) = 0.

Some Elliptic Problems with Nonlinear Boundary Conditions

183

Proof. Let E E (0,s) where S is given in Lemma 3.3. Since Qpx is bounded on BE,then by Lemma 3.3, there exists a > 0 such that (Px(u) # tacl, Vu E BE,V t E [0,1]. Hence,

Let C Then,

c R x Cro(n)be the closure of the set of positive solutions of (8).

Theorem 3.1. Assume that ( m , r ) > 0, (7), (9) and (11). Then yl is a bifurcation point from the trivial solution, and it is the only one for positive solutions. Moreover, there exists an unbounded continuum CO C C of positive solutions emanating from ( 7 1 ,0 ) . Proof. The result follows by Corollaries 3.1 and 3.2 and Ref. [5], Proposition 3.5. We only remark that the uniqueness of y1 follows with the same kind of arguments as in the proof of Lemma 3.2. 0

Remark 3.3. (1) Assume that there exist constants c1, c2 E R such that

f (x, iim -= c1 unif. in R,

s+o+

s

lim s+o+

s

= c2 unif. on

rl.

Then, we can apply the above result to the problem Llu = Xm(x)u+ f l ( z , u )in R, u = 0 on I'o and ,1322~ = Xr(z)u g2(z,u) on l?l, where L1 = L - c1, ,132 = B - c2, f i ( z , u ) = f ( z , u ) - c l u and g2(x, u ) = g(z,u ) - c2u, and so f 1 and g2 satisfy (11). (2) The case that m > 0 , r = 0 (i.e., the bifurcation parameter only in the equation) can be included in the Theorem 3.1. Indeed, if b 2 0 then (7) is verified. If b < 0 or changes sign we can perform a change u = eM*v where .JI is the function that appears on Ref. [20], Proposition 3.4, and the original problem is transformed into a similar new problem where the new b, say b > 0. (3) It is also possible to cover the case m E 0 , r > 0 (i.e., the bifurcation 0 then it parameter only at the boundary). Indeed, if 01 [ L ,D]I can be proved that bifurcation from the trivial solution does not occur. Now, assume a l [ L ,D ]> 0. By Proposition 2.1 there exists p r with p enough big such that crl[L,B pr] > 0. Then, there

+

-

+

C. Morales-Rodrigo and A. Sudrez

184

exists a unique solution h > 0 in

of the problem

Lh=1 in R, {h=l on ro, (B pr)h = 0 on rl.

+

Now, we perform the change u = hv, which transforms the original problem into a new problem where the new c, E > 0. (4) A similar result can be obtained for bifurcation from infinity. ( 5 ) We have not found in the literature a general result similar to Theorem 3.1. In Ref. I271 the author studied bifurcation form infinity for a similar equation with nonlinearities asymptotically linear. In Ref. [7] the bifurcation method is studied but with nonlinearities only at the boundary. In both papers, Lu = -Au+u and b(x) 2 0. In the rest of the section we consider the problem

{

Lu = X f ( x , u ) in R, u=O on ro, Bu = g(x,u) on rl,

where f E Ca(n x W), g E C1@(rl x R). Throughout the rest of the section we assume the following conditions (c,b) > 0 , (9) and

-

f(x,3)

lim -= +oo uniformly in R, s+o+ s lim

8-+0+

s

= 0 uniformly on

rl.

(16)

We have that u is a classical nonnegative solution of (14) if and only if @A(.) = 0 in Cr,(n), where QA : Cr,(Q -+ Cro(n)is defined as Qx(u) =

- Ki(XF(u+)) - Kz(G(r(u+))).

Consider Qi(u)= u - tKl(XF(u+))- tKz(G(r(u+))).

Lemma 3.4. If X

< 0 then there exasts b > 0 such that Vu E Cro(fi) with

l l ~ l l q f l )= llull E (0,d) we have

Qi(4# 0 , vt E [O, 11.

Proof. Suppose the contrary, then there exist sequences t n E I%, U n E -+ t, llunll -+ 0 with Qkt,”(u,) = 0. Dividing by IIu,ll, we obtain

Cr,(II) such that t ,

Some Elliptic Problems with Nonlinear Boundary Conditions

185

fi.

where vn = Since X < 0, the fact that IlUnll 2 llunllrl and using (15) and (16) we get that v, + 0 in C(n),a contradiction because llvnll = 1.

Lemma 3.5. If X > 0 then there exists 6 > 0 such that Vu E Cr,(n) with I(u(I E (0,d) and V r 2 0 we have Qx(u) # rp1, where cp1 is a positive eigenfunction associated to 01 [L,B].

Proof. Let us assume that for some sequence un E Cr,(n) with llunll + 0 and numbers 7, 2 0, Qx(un)= T,(PI. It is clear, by the maximum principle, that u, > 0 and it is a classical solution of the problem

{

Lu, = X f (z, u,) u, = 0 Bun = g(z, u,)

+

01

[L, B]T,(P~ in R, on ro, on r l .

Take E > 0, and M > 01 [L, B+E]. Since 01 [L, B]> 0 and due to u, + 0 in C(n) we have, using (15) and (16), that there exists no such that V n 2 no

{

+

Lu, = X f (z, u,) 01 [L, B]~,pl> Mu, in R, u, = 0 on ro, on rl. Bun = g(z,~ n >) -EU,

(17)

+

Therefore, u, is a positive strict supersolution of ( L - M , B E , a),,then 0 01 [L- M ,B E ] > 0, and so M < 01 [L, B E ] , a contradiction.

+

+

Theorem 3.2. Under conditions (c,b) > 0 , (9), (15) and (16), X = 0 is a bijhrcation point from trivial solution and it is the only one f o r positive solutions. Moreover, there exists an unbounded continuum Co of positive solutions emanating from (0,O). Proof. It is possible, thanks to Lemmas 3.4 and 3.5, reasoning as Theorem 3.1 to prove that there exists an unbounded continuum CO. We only need to prove uniqueness of bifurcation point. By Lemma 3.4 we can prove that bifurcation from the trivial solution does not occur for points of the form (Xo,O), XO < 0. Let us assume that there exists a sequence (X,,ux,) E R x Cro(n)verifying (X,,UX,) + (X0,O) in R x Cr0(n) with XO > 0. Then

Lux, = X,f (z,ux,) > XnMux,,

Dux, = g(z,ux,)

> -EUx*.

At this point we only need to follow the reasoning of Lemma 3.5 to obtain a contradiction.

C. Morales-Rodrigo and A. Sudirez

186

Remark 3.4. (1) A similar result is obtained under the condition

f

(z,

= m @ ) f(s),

with m E C"(a), m ( s ) 2 0, and non-trivial, f E P ( R ) and = +oo, instead of (15). lim,,o+ (2) Similar results still are true for equations of the form

{

Lu = h ( z , u ) in 52, u=O on ro, Bu = Xi(s,u)on rl,

where h and i play the same role as g and f , respectively. 4. Stability, Uniqueness and a-priori Bounds

In this section we present (without proofs) some results concerning to the stability, uniqueness and a-priori bounds of the solutions of the problem

{

Lu = f ( z , u ) in R, .U=o on ro, f3u = g(z,u ) on rl,

(18)

where f and g are regular functions. Let u a non-negative solution of (18). For the study of the stability of u,we linearize (18) around u and consider the eigenvalue problem:

(m=O

+ +

Lw = f z L ( u)w z, y(u)w in R, on ro, Bw = gu(z,U ) W y(u)won rl.

(19)

Thanks to Theorem 2.1, we know that the eigenvalue problem has a unique principal eigenvalue n ( u )= yl[L- f u ( z , u ) B , - gu(x,u)].

Theorem 4.1. Let u a nonnegative solution of (18).

If y1(u) > 0 , then u is linearly asymptotically stable (1. a. s.). (2) I f n ( u ) < 0, then u is unstable. (1)

In general, determinating the sign of n ( u ) is not easy. Due this fact, we give the following characterization using the following related problem:

{

Lw = fu(x,u)win 0, w=O on ro, Bw = gu(z,u)w on rl.

(20)

Some Elliptic Problems with Nonlinear Boundary Conditions

187

Using Lemma 2.1 and Proposition 2.1, we get

Theorem 4.2. n ( u ) > 0 (resp., -yl(u) < 0 ) i f and only i f the problem (20) admits a positive strict supersolution (resp., subsolution). With respect to the uniqueness, we have:

Theorem 4.3. Assume f E C1(ax [O,+oo)) and that

t w -f(x,t ), t w -g(x, t, t t

and g E C1(rl x [O,+oo))

are nonincreasing functions in t > 0,

and at least one of t h e m is a decreasing function. admits at most one positive solution.

Then, problem (18)

We assume f E C(ax [0,+oo)), g E C1@(r1 x [0,+m)) and there exists p E (1, q E (1, that verifies

B),

uniformly in

&)

a with h E C(n) a positive function and

uniformly on rl with i E C1>O(n)a positive function.

Theorem 4.4. Let u E C2(52)nC1(a) a nonnegative solution of the problem (18). Suppose that one of the following conditions is satisfied:

# 29 - 1; (2) The maximum of u is attained on 352, (22), (21) is satisfied f o r any function h and p < 29 - 1.

(1) (211, (22) and P

Then, there exists C ( p ,q, 52) a positive constant depending o n p , q and 52 such that f o r all z E

a

4.) I C(P79,W. Remark 4.1. (1) The condition p # 29 - 1 appears in other papers, see Ref. 1121 and it is necessary to apply a Gidas-Spruck argument.

C. Morales-Rodrigo and A . Sudmz

188

(2) The proofs of Theorems 4.1, 4.3 and 4.4 can be found in Ref. [23]. Theorem 4.1 complements and improves Theorem 5.6.2 of Ref. [25] and Theorem 3.1 of Ref. [28]. Theorem 4.3 is proved in Ref. [24] where other uniqueness results can be found. Finally, a-priori results have been shown in Ref. [30] with nonlinearities only at the boundary (see also Ref. [12] for systems) and Ref. [14] for particular nonliiearities in the equation and on the boundary. 5. Some Applications In this section we are going to study some equations with nonlinear boundary. The first equation is

{ where T

in R, U.=o on ro, Bu = a(x)ur on I'1 Lu=Xu

> 1 and a E C1ya(l?l).

Theorem 5.1. (1) Assume that a < 0. Then, (23) has a positive solution i f and only if o ~ [ LB] , < X < o ~ [ L , D ] .Moreover, if the solution exists, it is unique and 1. a. s. (2) Assume that a > 0. If u is a positive solution of (25') then, X < 01 [L, B]. If 1 < r < then there exists at least a positive solution of (23) for all X < 01 [L, B]. Moreover, all positive solutions of (23)

&

are unstable. Proof. (1) Assume a < 0. Suppose u is a nonnegative solution of (23), then, by the strong maximum principle, u is strongly positive, and so, X = al[L, B - a(z)uT-l]. Applying Proposition 2.1, we obtain (TI [L,B]

< X = 01 [L,B - a(z)ur-'] < 01 [L,D].

Now, we construct a sub-supersolution for the problem (23). Fix XO E (al[L, B],ol[L,D]). By Proposition 2.1 there exist Icl and Ic2 such that m [ L ,B k2] > Xo > ol[L, B Icl]. Now, the pair = E ( P ~and E = M ( P ~ with E small and M large enough, and (pi a strongly positive eigenfunction associated to o ~ [ LB , k,], is a sub-supersolution of (23). Uniqueness follows by Theorem 4.3. For the stability we use Theorem 4.2. Choose 'ii = u with u solution of (23) then, (L- A)E = 0 in R,

+

+

+

Some Elliptic Problems vvith Nonlinear Boundary Conditions

189

and

u = o on ro,

(a - a(x)rur-l)a > Bu - a ( x ) u r = 0

on

rl,

i.e., u is a positive strict supersolution of the linearized problem around u, ( L - A, B - ra(x)ur-l, a). (2) Assume now that a > 0. If u is a nonnegative solution of (23) then X = al[L,B - a(x)uT-'] < crl[L,B ] . By Theorem 3.1, there exists a unbounded continuum CO emanating from (01 [L,B ] ,0). Its direction is subcritical by the limitation of X and, under condition 1 < T < Theorem 4.4 proves us that the projection of CO on X-axis Px(C0) = (-m, q [ L ,B ] ) . Positive solutions of (23) are unstable because if u is a positive solution then,

A,

01

[ L- A, B - ra(x)ur-'] < a l [ L- A, B - a(x)ul--'] = 0.

0

Remark 5.1. In the case a < 0 and thanks to the subsolution that we have built, it could be proved that for K a compact subset of \ I'o, lim

x+ol[L,D]-

minux = +m. K

5.1. Elliptic equation with a logistic term at the boundary From the results obtained of the equation (23), we can deduce results for the equation (3).

Theorem 5.2. ( 1 ) If a1 [L,D ] 5 0, then (3) does not have positive solutions. (2) Assume [L,D ] > 0.

(a) If a < 0 , then (3) has positive solutions i f and only if p

> Xl[L,B].

Moreover, if the positive solution exists, is unique and 1. a. s. (b) If a > 0 and there exists a positive solution of (3), then p < X1[L,B ] . Moreover, under condition 1 < T < there exists at least one positive solution if p < XI [L,B ] . Furthermore, positive solutions of (3) are unstable.

A,

Proof. We only need to put X = 0 and x ( x ) = b(x) - p in Theorem 5.1. If a < 0, (3) possesses a positive solution if and only if 01 [L,B - p] < 0 < a l [ L ,D ] . By the definition of X1[L,B ] ,the result follows. Analogously the case a > 0. 0

C. Momles-Rodrigo and A . Su6re.z

190

5.2. A sublinear-superlinear equation

Now, we study the equation (4). Theorem 5.3. (0,O) is the unique point of bifurcation f r o m the trivial solution, and there exists a n unbounded continuum CO of positive solutions emanating from (0,O). Moreover, (1) Respect bifurcation direction:

(a) If p < r (resp., p > r ) then the bifurcation direction is supercritical (resp., subcritical). (b) If p = r then the bifurcation direction is supercritical (resp., subcritical) for Is21 > [as21 (resp., IR( < las2l).

(2) If p = r and Is21 5 Ids21, (4) does not have positive solutions for XLO. (3) If p < 2r - 1, (4) does not have positive solutions for X large enough. (4) If p I r and X 5 0 every positive solution is unstable. (5) If p < 2r - 1 and r < then every positive solution is bounded in Lbo norm. (6) If p > 2r - 1, there exists solution for all X 2 0.

fi

Proof. Due to Theorem 3.1, we have a unbounded continuum COof positive (4) emanating from (al[-A,N] = 0,O). We study the bifurcation direction. Consider A, + al[-A,N] and its solutions associated u,. Then, multiplying the equation by 91 = c > 0, the eigenfunction associated to the eigenvalue al[-A,N], we obtain

< r , multiply (24) by IlunII-Pand taking into CW) --t 9 1 in C(a) (see the proof of Lemma 3.2), it follows

Assume for example that p account that

S g ( a l [ - A , N ] -A,)

= Sg

(s, -

d+'dX

),

hence, al[-A,N] < .A, All results related to local bifurcation can be proved by the same way. Let u a positive solution of (4) with p = r . Then, if we multiply the equation (4) by l/uT,and integrating by parts, we get

Some Elliptic Problems with Nonlinear Boundary Conditions

Then, paragraph ( 2 ) follows. Assume that the problem (4) has a positive solution for every X Consider the parabolic problem ( wt

-Aw

191

> 0.

= -wP in R x (O,T),

- = 'w

on dR x (O,T),

w(z,O) = w o

in R.

(25)

We know by Ref. [6], Theorem 2.3, that if p < 2r - then all positive solutions of ( 2 5 ) blow-up in finite time for w o with large La norm. Take ux a solution of (4),if we prove that ux is supersolution of ( 2 5 ) for large A, then ux(2) > w(z,t)for all t E (0,T)which is a contradiction. In order to prove this, we only need that ux > WO. It is clear that for X > 0 ux is supersolution of the problem

( -AW [;=0

= xu - u p in R,

on dR.

As solutions of ( 2 6 ) are, for X > 0 , A1/(P-') then ux > Al/(P-l)*

Now, there exists X > 0 large enough such that llwoIloo < X1/(P-') < ux, this concludes paragraph ( 3 ) . Let u a positive solution, we are going to prove that under condition p 5 T this solution is unstable. For that, thanks to Theorem 4.2, we have to show that 01

[-A- X + pP-l, N - TU'-']

< 0.

(28)

For this fact we choose as subsolution, g = u p , where q will be fixed later. We have that

and in R,

+

(-A - X + pP-')g = q ( 1 - q)UQ-21Vu12 XuQ(q- 1)

+ ~ ~ + ~ - ' (-pq).

Choosing q such that p 5 q 5 T , it follows ( 2 8 ) , so that paragraph (4). By ( 2 7 ) , u attains its maximum on dR. So, paragraph (5) follows from Theorem 4.4.

192

C. Momles-Rodrigo and A. Suctrez

For the last paragraph we only need to find a sub-supersolution of (4) for every X 2 0. We choose as subsolutiona u = Ee-641

where ~ , > 6 0 can be chosen later and 41 is the positive eigenfunction associated to n1 = al[-A, D]with 114111~= 1/2. After some calculations we obtain

0%= -ESe-641 061, Ag = - - ~ 6 e - ~(-6Iv41)~ ~' +Ah). Thanks to Hopf's Lemma, it follows that

Since, $1 = 0 on d o , we only need to verify on the boundary that

6C1 5 Er-l.

(30)

In the equation we must check that

-621~411z- 60141

+ E*-1e-6+1(p-1)

5 A.

(31)

Observe that if X > 0, we only need to choose E and 6 positive and small enough for that (30) and (31) hold. So that, we are going to study the case X = 0. From (30) we choose 6 = so that (31) transforms into

s,

Now, due to $1 = 0 on dR, but on the boundary d41/dn < 0, there exist some constants CZ,C, > 0 such that

lV41I 2 CZ en R1 := {x E 52 : 41(z) 5 CS}.

(33)

In this way, in 521 for that the condition (32) must be fulfilled we need that p - 1 > 2(r - 1) thus p + 1 > 2r. On the other hand, in 52 \ a1 we need that p - 1 > r - 1. The supersolution follows by Ref. [22], it was used also in Ref. [18], in both cases for the particular case X = 0. We choose

E := hfA[2- (1 - 41)B]C, aThis subsolution appears on Ref. [18] for the particular case X = 0.

Some Elliptic Problems with Nonlinear Boundary Conditions

193

where M > 0 will be chosen large and A, B and C will be fixed later. Observe that VTi = BCMA[2- (1- 41)B]C-1(1- $1)B-1V41, ATi = BCMA[2- (1- 41)B]C-2(1- $1)B-2* {(C- 1)(1- 41)BIV4112 [2 - (1 - 41)BlA41(1- 41)12 - (1- 41>BI(B- 1)IV41I2}. Taking into account (29), on the boundary it must be verified (observe that $1 = 0 and so that (2 - (1 = 1): 1 -BCMA(l-T) > _. (34)

+

-

9

For the equation, we need that X 5 -BC(C - 1)[2 - (1- 4 1 ) ~ ] - ~-( cj1)2(B-1)IV$1)2 1 -BC[2 - (1 - $l)B]-lA+l(l - &)B-' +BC(B - 1)[2 - (1- 41)B]-1(1- +1)B-21V+112 +MA(P-1)[2- (1 - $l)B]C(P-l).

(35)

Take A > 0, C = -1/C1 and B = Mb,with b to be fixed later. With this choice, for condition (34) it will be needed

b

+ A(1-

T)

2 0. (& b > 0).

(36) Now, we study the term (35). First term in the right hand tends to --CQ or zero (the term (1- $ ~ ) ~ ( ' - l )can tend to zero). Second term is similar, we remind that -A41 = 01 [-A, D]$l. Third term tends to --oo with order M2' and the last one to +oo with order MA(p-l), so we have to impose A ( p - 1) > 2b. This last inequality and (36) are possible because p+ 1 > 2 8

Remark 5.2. Except paragraphs (2) and (3), Theorem 5.3 is true for more general operators L and B. 5.3. The concave-convex equation Finally, we consider (5). Assume the following conditions c

> co > 0 in w, with ~0 E R.

(37)

We distinguish two different cases: a negative and positive. Theorem 5.4. Assume that a < 0. The problem (5) has a positive solution vx i f and only af X > 0. For X > 0, it is the unique positive solution, it is 1. a. s. and

194

C. Momles-Rodrigo

and A. Sudrez

Proof. Thanks to the maximum principle ( 5 ) does not posses nonnegative solutions for X 5 0. By Theorem 3.2, there exists a continuum CO emanating from (0,O) supercritically. On the other hand, ?i = M p l is, for M large enough, supersolution of ( 5 ) where ' p l is the positive eigenfunction associated to c1[L,N]. This is true because e l [L,N] > 0, which follows by (37). Since M can be chosen large enough that M p l > vx for X > 0 small, where vx denotes the solution of the problem founded by bifurcation. Then, we have a family of supersolutions such that a solution belonging to the continuum is smaller than the supersolution. Now, adapting the proof of main theorem of Ref. [13],we have that there exists at least a positive solution for all X > 0. Finally, for X > 0

o = c1 [ ~ - ~ m ( z ) v : -~l -, a ( z ) v : - ' ] < el [ ~ - ~ q m ( z ) v ; i, -Nl - r a ( ~ ) v : - ~ ] , so the stability follows. Uniqueness follows by Theorem 4.3.

0

Theorem 5.5. Assume that a > 0. (1) From (0,O) emanates supercritically an unbounded continuum CO of positive solutions. Moreover, it is the unique bifurcation point from the trivial solution. (2) There exists A* > 0 such that for X > A* problem (5) does not have positive solutions. (5') There exists 6 > 0 such that there exists at most a positive solution ux of (5) such that I I u ~ l56. l~ (4) Moreover, i f L is self-adjoint and r < then:

(a) Px(CO) = ( - m , A ] , f o r some A < +co. (b) There exists at least two positive solutions in (0,A). (c) There exists a unique positive solution in (0,A) 1. a. s. Proof. Since the proof follows the same lines that Theorem 6.9 in Ref. [ l l ]we , only sketch it. The existence of the continuum CO follows by Theorem 3.2. We prove now that the bifurcation direction is supercritical. Assume that there exist A, 5 0 and ux, E C(n), ux, 2 0 such that (Xn,UX,) -+ (0,O) in R x C(n). Then, for n 2 no we get

where aM = maxm a. On the other hand, since u1 [L,N] > 0 then, for E > 0 small, el [L,N - E ~ M >] 0, and applying the strong maximum principle we obtain that ux, 3 0, a contradiction.

Some Elliptic Problem with Nonlinear Boundary Conditions

195

Now, we are going to prove paragraph (2). Suppose that there exists positive solution ux of (5) for all A, in particular for X > 1. Let w1 be the unique positive solution of

Since u x / X is supersolution of (39) for X > 1, then ux > Xu1 for X the other hand, since u,+ is a positive solution of (5), we get

> 1. On

0 = q [ L- Xm(z)ui-1,N- a ( z ) u y ] < +,N- X-la;-lw;-l], where a0 = minan a(.).

This is an absurdum. Indeed, since r > 1, we have

Now, define

A := sup{X

ER :

(5) has a positive solution}.

We have proved that 0 < A < +oo. Moreover, it is not difficult to prove the existence of a minimal solution ux for all X E (0,A). The following result shows properties of the principal eigenvalue, denoted by yl(X), of the linearized around the minimal solution ux, i.e.,

or equivalently, the unique zero of the map

p(0) = ol[L- X q r n ( ~ ) u i --~u,N

-

ra(z)uL-'

-

4.

Lemma 5.1. (1) If ux is the minimal solution of (5), then yl(X) L 0. (2) If yi(Xo) > 0 , for some Xo > 0 , then the set of positive solutions of (5) can be parametrized in a neighborhood of ( X 0 , u o ) by a regular and increasing function o n A. (3) If yl(Xo) = 0 , f o r some Xo > 0, then the set of positive solutions of (5) can be parametrized by a new parameter s E ( - - E , E ) , such that ( X ( s ) , u ( s ) ) is a positive solution of (5) and X(s) = X0+s2X2+o(s3),

u ( s ) = U ~ ~ + S % - , + S ~ ~ O + O ( S ~ )(41) ,

C. Momles-Rodrigo and A . Sudrez

196

where

J,

(PO

is the positive eigenfunction associated to ~ ( X O and )

(PoQo = 0 . Moreover,

SS(X’(4) = Sdn(u(.))).

(42)

Finally, if L is self-adjoint, A2

where

Xp

< 0,

(43)

is defined in (41).

Remark 5.3. Except the first paragraph, the result is true for any positive solution u not necessarily being the minimal. Proof. (1) Assume that y1 < 0 and denote by $1 the positive eigenfunction associated to 71. It is not difficult to show (see Ref. [ll])that ux - a$1 is supersolution of ( 5 ) , for a > 0 small. Since ux > vx, where vx is the unique positive solution obtained in Case 1 (a(.) < 0), and ox is subsolution of ( 5 ) , it follows the existence of a solution u < ux of ( 5 ) , an absurdum because ux is the minimal solution. (2)-(3) Except (43), these two paragraphs follow by el Propositions 20.6, 20.7 and 20.8 of Ref. 111. Using (41) and the definition of ( P O , we get

To determine the sign of X p , we use the Picone’s identity, see for instance Lemma 4.1 in [19]. Taking Q ( t ) = t2,w = (PO and ZG = ux,, we get

whence it follows that

Xp

< 0.

0

As an easy consequence we obtain:

Corollary 5.1. Assume L self-adjoint and let (XO, U O ) be a positive solution of (5) with X = XO > 0 , such that -yl(Xo) = 0 Then, there exists E > 0 such that for each X E (A0 - E , X O ) , (5) has two positive solutions, one of them 1. a. s. and the other one linearly unstable. Moreover, there exists a neighborhood of (X0,uo) such that (5) does not have a positive solution f o r

x > Xo.

197

Some Elliptic Problems with Nonlinear Boundary Conditions

Now, we will prove paragraph (3) of Theorem 5.5. Assume that there exists a second solution w=ux+w where w > 0 and llwlldo < 6. Consider w1 the solution of (39). Then, there exists p

0 = 01[L- m(z)wy-l,N] < p

> 0 such that

< 01[L- qm(z)w;-l,N].

(45)

We claim that

< 0,

0 1 [ L- qm(z)w?-',N - aMr6'-'1

(46)

which is an absurdum with (45). In order to prove (46), it suffices to prove that w is a positive subsolution of (L-qm(s)vy-l,N-aMpGp-l, 0). Indeed, since w is solution of ( 5 ) and by the concavity of the map tQ,it follows that

Lw 5 Xm(z)qzlQx-lw. But, since X1/('-Q)wl

is subsolution of (5), then ux

> X1/('-Q)wl, and hence

( L - qm(z)wy-l)w < 0. On the other hand, a(z)[(ux

+ w)'

-

u:] 5 aMrdT-lw, whence

The following result shows that all the positive solution of (5) are unstable for X 5 0. Lemma 5.2. If u is a positive solution of (5) for X 5 0, then u is unstable.

Proof. It suffices to prove that cr1[L- Xqm(z)uQ-',N - a(z)rur-']

< 0.

First, observe that the first eigenvalue is well defined because minnu > 0. It remains to find a positive subsolution of ( L - Xqm(z)uQ-l,N a(z)ru'-l, 0). It is hot hard to show that g = u: with 1 < p 5 r is the desired subsolution. 0 We are going to finish the proof of the Theorem. Since T < Theorem 4.4, we have that Px(C0) = (-m,A]. We consider the set

r := {(x,Ux) : x > o,yl(x) > 01.

A,by

198

C.Momles-Rodrigo and A . Sudrez

We claim that A = s u p r . By Lemma 5.1, the uniqueness of solution with small norm and Corollary 5.1, it follows that s u p r = i> 0. It is clear that 5 A. Assume that < A, then there exists Xo > isuch that uxo is supersolution of (5) for all X 5 XO. Since we always can build small subsolutions, then there exists a solution ux for X < XO. Since ux is built by the sub-supersolution, then yl(ux) 2 0, and so, by Lemma 5.1 and Corollary 5.1 we have that there exists 5; > 1such that Tl(U’;)

> 0.

Now, we can continue this solution to the left, we call

ro:= { ( x , u ~: x) < X} to this new set. It can occur four possibilities. First, there exists X2 6 (0,i) such that uxz = u x z ,which is not possible because around a 1. a. s. solution there is not another solution. Second, there exits Xs such that ux3= 0. Recall that the unique bifurcation point is X = 0, so this is not possible. Third, there exists ux for all X 5 XO, a contradiction with Lemma 5.2. Finally, there exists A4 such that yl(ux4)= 0, which is impossible by Lemma 5.1 and Corollary 5.1. This proves that ?; = A. With a similar reasoning it can be proved the uniqueness of 1. a. s. positive solution. For the existence of two positive solutions for all X E (0, A) it is used the fixed point index respect to the positive cone. Basically, the total index is zero and the index of ux equals one: other positive solution must exist, see Refs. [Ill and [17].

References 1. H. Amann, SIAM Rewiew, 18,620 (1976). 2. H. Amann, Israel J. of Maths. 45,225 (1983). 3. H. Amann, “New Developments in Diff. Eqns” (Eckhaus, W., ed.), Math. Studies, 21, North-Holland, Amsterdam, 43 (1976). 4. H. Amann and J. Lbpez-G6mez, J. Diff. Eqns. 146,336 (1998). 5. A. Ambrosetti and P. Hess, J. Math. Anal. Appl. 73, 411 (1980). 6. F. Andreu, J. M. Mazbn, J. Toledo and J. D. Rossi, Nonl. Anal. 49,541 (2002). 7. J. M. Arrieta, R. Pardo and A. Rodriguez-Bernal, submitted (2004). 8. S. Cano-Casanova, On the Positive Solutions of the Logistic Weighted Elliptic BVP with Sublinear Mixed Boundary Conditions. In this Volume. 9. S. Cano-Casanova and J. Lbpez-Gbmez, J. Diff Eqns. 178,123 (2002). 10. M. Chipot, M. Fila and P. Quittner, Acta Math. Univ. Comenianae, 60,35 (1991). 11. M. Delgado and A. Sukez, Houston J. of Math. 29,801 (2003).

Some Elliptic Problems with Nonlinear Boundary Conditions 12. 13. 14. 15. 16.

17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

199

J. Fernhdez Bonder and J. D. Rossi, Adv. Dig. Eqns. 6, 1 (2001). J. L. G h e z , Nonl. Anal. 28,625 (1997). J. Garcia Azorero, I. Peral and J. D. R m i , J. Diff. Eqns., 198,91 (2004). J. Garcia-Melibn, J. D. Rossi and J. C. Sabina de Lis, submitted (2004). D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer Verlag (1983). R. G6mez-Refiasco and J. Lbpez-Gbmez, J . Dig. Eqns., 167,36 (2000). A. W. Leung and Q. Zhang, Math. Meth. Appl. Sci. 21,1593 (1998). J. L6pez-G6mez, Comm. Partial Diflerential Equations, 22, 1787 (1997). J. Lbpez-G6mez, Adv. Di8. Eqns. 8,1025 (2003). J. Lbpez-Gbmez, V. Mkquez and N. Wolanaski, Revista Un. Mat. Argentina 38,196 (1993). W. Mingxin and W. Yonghui, Chin. Ann. of Math. 16B,371 (1995). C. Morales-Rodrigo, DEA Memory, University of Sevilla, (2004). C. Morales-Rodrigo and A. Sukez, submitted (2004). C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum, New York, (1992). P. Quittner, Comment. Math. Univ. Carolinae, 34,105 (1993). K. Umezu, J. Math. Anal. Appl., 267,651 (2002). K. Umezu, Nonl. Anal. 49,817 (2002). K. Umezu, Communications in Applied Analysis, 8,533 (2004). M. Zhu, J . Diff. Eqns., 193, 180 (2003).

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DYNAMICAL SYSTEMS METHOD (DSM) AND NONLINEAR PROBLEMS

A.G. RAMM Mathematics Department, Kansas State University, Manhattan, K S 66506-2602, USA Email: rammomath. ksu. edu http://www. math. ksu. edu/-ramm

The dynamical systems method (DSM), for solving operator equations, especially nonlinear and ill-posed, is developed in this paper. Consider an operator equation F ( u ) = 0 in a Hilbert space H and assume that this equation is solvable. Let us call the problem of solving this equation ill-posed if the operator F'(u) is not boundedly invertible, and well-posed otherwise. The DSM for solving linear and nonlinear ill-posed problems in H consists of the construction of a dynamical system, that is, a Cauchy problem, which has the following properties: (1) it has a global solution; (2) this solution tends to a limit as time tends to infinity; (3)the limit solves the original linear or non-linear problem. The DSM is justified for: (a) an arbitrary linear solvable equations with bounded operator, (b) for well-posed solvable nonlinear equations with twice Frkhet differentiable operator F , (c) for ill-posed solvable nonlinear equations with monotone operators, (d) for ill-posed solvable nonlinear equations with non-monotone operators from a wide class of operators, (e) for ill-posed solvable nonlinear equations with operators F such that A := F'(u) satisfies the spectral assumption of the type II(A+sI)-lII 5 c/s, where c > 0 is a constant, and s E (0, SO), so > 0 is a fixed number, arbitrarily small, c does not depend on s and u; and (f) for some monotone operators which are not Frkhet differentiable and for some unbounded, closed, densely defined F . In Newton-type schemes the main difficulty is to invert the derivative of the o p erator. A novel scheme, based on DSM, allows one to avoid this inversion. Global convergence theorem is obtained for the regularized continuous analog of Newton's method for monotone operators. Global convergence means that convergence is established for an arbitrary initial approximation, not necessarily the one which is sufficiently close to the solution. A general approach to constructing convergent iterative schemes for solving wellposed nonlinear operator equations is described and convergence theorems are obtained for such schemes. Stopping rules for stable solution of ill-posed problems with noisy data are given.

201

A . G. Ramm

202

1. Introduction The aim of this paper is to present in a self-contained way a series of the results obtained in [lo]-[22], where the DSM (dynamical systems method) in its current scope was developed. Among related ideas is the old method of steepest descent. One of the first papers, where the DSM, specifically, its version corresponding to a continuous analog of the Newton’s method for well-posed equation F ( u ) = 0 was developed, is [5). In [9]iterative processes for solving nonlinear well-posed equations in finite-dimensional spaces are presented, and in [4] iterative methods for solving some ill-posed problems are discussed. In [1]-[3], [7], [26] some results which are obtained by the DSM method are given, and in [24]and [25] the efficiency of the DSM method is illustrated in the problem of stable numerical differentiation of noisy data. The DSM method can be considered as a general method for solving operator equations, especially ill-posed, nonlinear. At first, one may think that solving the Cauchy problem (4) of the DSM in order to solve equation (1)is replacing a simpler problem (1) by a more complicated problem (4). However, if problem (1) is ill-posed in the sense that (3) fails, then there are no general approaches to solving (l),and the DSM provides such an approach for a very wide class of equations (1). It also provides a general approach to construction of convergent iterative schemes for solving (1). Finally, numerical solution of (4), after a discretization, amounts to solving a Cauchy problem for a system of ODE. This is an area of numerical analysis which was much studied and is well developed. Let us now describe the idea of the DSM. Let H be a real Hilbert space, F : H t H be an operator. One wants to solve an equation

F(u)= 0

(1)

and one assumes that there is a y, possibly nonunique, such that F ( y ) = 0. Let u g E H be an element, and B(uo,R) := {u : ~ ~ u I - R}. u ~Assume ~ ~ SUP IIF(j)(u)ll I Mj(R), UEB(U0,R)

1Ij 52,

where F ( j ) is the FrCchet derivative. We call problem (1) well-posed, if

(2)

Dynamical S y s t e m Method (DSM) and Nonlinear Problems

203

Otherwise we call (1)ill-posed. The dynamical systems method (DSM) for solving (1) is the method consisting of the following steps: a) finding a map @(t, u ) such that the problem u = @(t,u), u(0) = 210;

.

du

u := -,

dt

(4)

has the following properties: 3!u(t) W

> 0;

3u(00) := lim u(t);

b) solving (4), then taking t -t

t-oo

00,

F(u(co))= 0,

(5)

and finding the solution as the limit

> .(. Note that u(00) = U ( O O , U O ) if (1) has more than one solution. How does one find a? There are many ways [13] to find @ such that ( 5 ) holds. We assume that @ is locally Lipschitz with resepct to u and continuous with respect to t. This implies local existence and uniqueness of the solution to (4). Our aim is to review some of the results obtained recently (101-[22]. These results demonstrate the power of the DSM, both as a theoretical tool for proving the existence of a solution to equation (l),proving that F is a global homeomorphism (under suitable additional assumptions), deriving convergent iterative schemes for solving (l),and developing numerical methods for solving a very wide class of linear and nonlinear operator equations, especially ill-posed. There is a large literature on linear ill-posed problems (e.g., see [6]), and a less extensive one on nonlinear ill-posed problems (e.g., [27], [lo]). Let us describe briefly the scope of the results obtained by the DSM in [lo]-[22], assuming (2) unless otherwise stated: (1) Every solvable well-posed equation (1) can be solved by the DSM method which converges exponentially fast. (2) Every solvable ill-posed linear equation (1) with a bounded operator A can be stably solved by the DSM. Here F ( u ) := Au - f = 0, and stably means that if the noisy data f6 are given, llfs - f l l 5 6 , then there is a t b such that lim6,o Ilu(t8) - yII = 0. (3) Every solvable ill-posed equation (1) with monotone operator F , i.e., ( F ( u )- F(w),u - w) 2 0, can be stably solved by the DSM. (4) Every solvable equation (1) can be stably solved by the DSM pro:= *i 2,:= F’(y), maps the set vided that the operator

A . G. Ramm

204

{u : llull 5 T } , where p := {u: 0 < llull 5 1).

T

> 0 is sufficiently small, into the set

(5) If F is monotone, hemicontinuous, defined on all of H , but (2) is not assumed, then if equation (1) is solvable, possibly nonuniquely, then its minimal-norm solution y can be stably found by the DSM. (6) If F = L g r where L is a closed, linear, densely defined in H operator, which has a bounded inverse, g is a nonlinear operator satisfying (2), and equation (1) is solvable, then its solution can be stably found by the DSM, provided, for example, that SUPu~B(uO,R) 11 [I + L-lg’(u)]-’ 11 5 m(R) and s’PR>O = O0. (7) A sufficient condition for the surjectivity of a map F is suPR>O = OO. (8) A sufficient condition for the map F to be a global homeomorphism is 11 [F’(u)]-’II 5 h( IIuII), where $j = 00, h(s) > 0 is a continuous function on [O, m). (9) A method for constructing convergent iterative schemes for solving equation (1) is given. (10) A method for constructing a Newton-type method without inverting the derivative operator is developed.

+

&

&

s”

2. Well-posed problems

Assume (2) and (3). Take @ = -[F’(u)]-lF in equation (4):

ti = -[F’(u)]-’F(u),

~ ( 0= ) UO.

(6)

This is a continuous analog of the Newton’s method. From (2) and (3) it follows that @ is Lipschitz, so ( 6 ) is locally solvable. It will be globally solvable, i.e., solvable Vt > 0, if sup,,o ((u(t)(( 5 00. Let us prove this estimate. Let IIF(u(t))ll:= g ( t ) . Then g g = -g2 by ( 6 ) , so g ( t ) = g ( 0 ) e - t . From ( 6 ) one gets 11ti11 I g ( O ) e w t m ( R ) . Thus Ilu(t) - u(0)II I g ( O ) m ( R ) as long as g ( O ) m ( R ) L R,

that is, as long as the trajectory u(t)stays in the ball B(uo,R) for all t Assume (7). Then 3u(m),

(7)

> 0.

and from ( 6 ) , as t -+ 00, one gets F ( u ( m ) )= 0. We have proved that equation (1) can be solved by the DSM (6).

Dynamical Systems Method (DSM)and Nonlinear Problems

205

Theorem 2.1. If (2), (3) and ( 7 ) hold, then equation ( 6 ) has a unique global solution, and (5) holds. Moreover, estimates ( 8 ) hold, i.e., u(t) converges to the solution u ( m ) at the exponential rate and the trajectory u(t) stays in the ball B(uo,R) for all t 2 0 . Remark 2.1. Many other choices of Q, are discussed in [13],[14]. These choices include continuous analogs of the modified Newton’s method, Gauss-Newton method, gradient method, method of simple iterations, etc. 3. Surjectivity of F Theorem 3.1. Assume (2), (3) and let the following condition hold:

R m(R)

sup -= 00. R>O

(9)

Then F is surjective. Proof. The proof of Theorem 2.1 shows that if (9) holds, then, for any fixed uo, (7) holds for some R > 0. Thus ( 1 ) is solvable. The same argument holds for the equation F(u)-f = 0, for any f E H . Theorem 3.1 is proved.0 4. When is F a global homeomorphism?

Theorem 4.1. Assume (2), (3) and let h : R+ continuous function such that

Then F : H

-+H

-+

R+, R+

:= ( O , o o ) ,

be a

is a global homeomorphism.

Proof. E’rom (3) it follows that F is a local homeomorphism. To prove that F is a global homeomorphism one has to prove that: a) F is surjective, and b) if F ( u ) = F(w) then u = w. Let us prove a). As in Section 2, one gets

dllull

Il4I i h(llUII)gOe-t, go dt I where the inequality 5 ((iL(1was used. Thus

:=

do),

(11)

A . G. Ramm

206

This and (10) imply S U ~ Ilu(t)II ~ > ~ < 00, so h(llu(t)ll)< 00, and (11) yields lliLll 5 ce-t. Thus Ilu(t)- uoll 5 c, u(00)exists, and, as in Section 2, F(u(00))= 0. Since this argument remains valid if F ( u ) is replaced by F ( u ) - f with an arbitrary f E H , the surjectivity of F follows. Thus, a) is proved. Below we consider the equation F ( u ) - f = 0. Let us prove b). If lluo - wall is sufficiently small, and sup IIU(t,UO) - u ( t , ~ o ) I I I clluo - wall,

(12)

t>O

where c > 0 stands for various constants, then b) follows. Indeed, let u := u(00,uo) = limt+- u(t,U O ) , F ( u ) = f = F(w), where w = limt,, w ( t , WO). Define w(s) := U O + S ( W O - ~ O ) . If s is sufficiently small, then Ilw(s) - uoll is as small as one wishes, and Ilu(t,w(s))- u(t,uo)ll is as small as one wishes uniformly for all t > 0. Thus llu(00,w(s))u(oo,u~)ll is small, F(u(00,uo))= f = F(u(oo,w(s))).Since F is a local homeomorphism, it follows that u(m,w(s)) = u(00,uo) = u. Finitely many small steps are needed to get to w(1) = wo and conclude that w = u(m,w(1)) = u(m,uo) = 21. To complete the proof, let us check (12). Let u = - [ F ’ ( u ) ] - l ~ ( u )U, ( O ) = uo, v = - [ F ’ ( w ) ] - ~ F ( wW(O) ) , = wo, $J := u(t)- w ( t ) . Then

4= -([F’(u)]-’ - [F’(w)]-l)(F(u)- f) - [ F ’ ( w ) ] - ~ ( F (-u F) ( w ) ) . (13) Let g ( t ) := Il$(t)ll. Multiply (13) by $, use the formula F ( u ) - F(w) = F’(v)(u- w) K , 11K1 l I ~ l l $ J l 1 2 and , get

+

gg

I -g2

+ cg3 + cg2e+,

(14)

where the estimate IIF(u(t))- f l l I ce-t was used. Since g ( t ) 2 0, one can consider instead of (14) the following one: g

I -g

+ cg2 + cge-t,

g(0) = lluo - ~011.

+

(15)

Let g = e-th, so that go := g(0) = h(0). Then h I c(h2 h)e-t. This implies In I In c, i.e., I s e C ,where go = Ilu(0)- v(0)ll. Therefore, if go is sufficiently small, so that go < then h(t) 5 ch(0) = cg(0). Therefore g ( t ) 5 ce-tllu(0) - w(O)ll, so (12) follows. Theorem 4.1 is proved.

&

%+

&

A,

Dynamical Systems Method (DSM) and Nonlinear Problems

207

5. Linear ill-posed equations Assume

F ( u ) = AU- f = 0,

(16)

where A is a bounded linear operator, Ay = f , y is the minimal-norm solution, B := A*A, h := A * f , llfg - f J l 5 6, E ( t ) > 0 is a monotonically decreasing function, limt-,m E ( t ) = 0, E(s)ds = 00. A solvable equation Au = f is equivalent to

Bu = h,

B = A*A 2 0.

(17)

Consider the equation 4 = -U

+ [B + ~ ( t ) ] - ' h ,

~ ( 0= ) UO, h = A * f .

(18)

Theorem 5.1. Any solvable linear equation (16) with J J A J 0 and is unique. The solution to (18) is: u(t)= uoe-t+J~e-(t-s)[B+E(s)]-lhds,and, since h = Bu, it is easy to check that limt+m u ( t ) = y . Indeed, lim,+o[B+~]-~Bu= y , where u = u ( t ) and E = e ( t ) , and limt+m s," e-(t-s)g(s)ds= g ( o 0 ) provided there exists g(m) := limt+m g ( t ) . Thus ( 5 ) holds and the DSM (18) is justified. If fg is given in place of f , then hg will replace h in (18). Without loss of generality one may assume that l(hg- hi( 5 6. Let u g ( t ) solve (18) with h replaced by hg. Let us prove: Claim: There exists t g , limg+o t g = 00, such that

lim

6-+0

I l u g ( t g ) - yII = 0.

Toprove the claim, note that IJug(tg)-yll 5 Ilug(tg)--~(tg)ll+jlu(t6)-yII. Since 1img-o 11u(tg) - yJI = 0, it is sufficient to prove limg+o I I u g ( t g ) u ( t g ) 1 1 = 0. Let ug(t) - u ( t ) := v ( t ) ,hg - h :=p . Then V

= -W

+ (B + &(t))-'p,

~ ( 0= ) 0.

Thus

= 0 is suitable Therefore any t g such that limg+o t g = 00 and limg+o 6 4ta) for the claim to hold. Theorem 5.1 is proved. 0

A . G. Ramm

208

6. Nonlinear equations with monotone operators

Assume that F is monotone in the sense

( F ( u )- F ( w ) , u- W ) 2 0 VU, w E H , (2) holds, and

F ( Y )- f

= 0,

(19)

where y is the minimal-norm solution to (19). Note that if F is monotone and (2) holds, then N ( F ) := {u : F(u)- f = 0 ) is a closed and convex set. Such sets in Hilbert spaces have a unique minimal-norm element, as is well known. Consider the problem U

where A, := F'(u) E

+

= -A;~[F(u)

EU

- f],

~ ( o =) uo,

(20)

+ €1,I is the identity operator,

=E(t)

> 0,

C(t)

< 0,

lim ~ ( t=)0 ,

t+m

161 -

€

I -.1 2

Theorem 6.1. If (2) and (21) hold, and uo E H is arbitrary, then (5) holds (with F(u) - f in place of F ( u ) ) and u(00) = y. If f6 replaces f in (19) and (20), llfs - f l l I 6, then there exists t s , 1ims-o t s = 00, such that lims+o I I u g ( t 6 ) - yII = 0 where ug(t) solves (20) with fs in place o f f .

Proof. Consider the equation

+

F(V) €(t)V- f = 0.

(22)

It is known (e.g., [13]) that if F is monotone and (2) and (21) hold, then there exists a unique solution V = V ( t )to (22) and

Since Ilu(t)-yll I Ilu(t)-V(t)II+IIV(t)-yll,it follows that limt,, yII = 0 provided that limt,, Ilu(t)- V(t)II= 0. Let

Ilu(t)-

g(t) := Ilu(t)- V(t)ll, w := u(t)- V ( t ) . We want to prove the relation: limt+mg(t) = 0. From (21) and (20) one getszir = - V - A A , ' [ F ( ~ ) - F ( V ) + E W ]Moreover, . F(u)-F(V) = A w + K , where IlKll I %$, by Taylor's formula. Thus 2ir = --w - A;lK - V .

209

Dynamical Systems Method (DSM) and Nonlinear Problems

Multiply this by w and get g 5 -g

+ &g2 + Ilglly.

Let

:= q,

J J y := J J c1. Then

It follows from (24), (21), and Example 7.1 in the next section, that g(t) 5 ~ ( t--+)0 as t -+ 00. Theorem 6.1 is proved.

+

as an approxWe have used in (20) the operator A;' := (A imation to the inverse of the operator A := F'(u), where E > 0 tends to zero. This is natural because A L 0 is a bounded selfadjoint operator. For such an operator one can derive the relation lim,,o A;' Au = u, using the spectral theorem for selfadjoint operators, and assuming that u E D(A), u IN ( A ) := {u : Au = 0}, and otherwise u is arbitrary. For non-monotone F , and also for monotone F , one may use other approximations for the inverse of A when this inverse is unbounded or does not exist. If A is selfadjoint, but not necessarily non-negative, or if A is normal, or, more generally, when A is a spectral operator, then one can use spectral theory to approximate the inverse of A. For instance, if A is selfadjoint, then one may use a function &(A) such that lim,,oqb,(A)Au = u for all u E D(A), u IN(A). When A is non-selfadjoint, one may use +,(A*A) to approximate the inverse of A, when this inverse is unbounded or does not exist. The regularized Gauss-Newton method is based on such an approximation. A motivation for this is the formula (A*A)-lA* = A-l valid for a boundedly invertible linear operator A. One can derive a stopping rule for problem (20) as in Section 5 . If llfa - f l l 5 6 , and ug(t) solves (20) with fa replacing f, then the error E can be estimated as follows:

E := IIua(t) - YII I Ilus(t> - va(4ll + IIva(t) - V(t>II+ IIV(t>- Yll, where Va(t) solves (22) with fs replacing f . One has IlV(t)-yll := a ( t ) -+ 0 as t -+00. Using monotonicity of F , one derives the estimate

If wg(t) := ua(t) - &(t) and gs(t) := Ilws(t)ll, then, as in the proof of Theorem 6.1, one derives the inequality similar to (24):

A . G. Ramm

210

where the last new term comes from the estimates

I4 6 llvsll I Tllvsll, llvsll I IlVll + *

El

and

IlVll I IlYll.

Applying Theorem 7.1, one gets the inequality g6(t) 5 E ( t ) ,where c > 0 is a constant, provided that t I t a , where the stopping time t 6 is found from the equation ha = E ( t ) , for a &xed 6 and 0 < a < 1. This equation has a unique solution t a because E decays to zero monotonically. Moreover, ta -+ 00 as 6 -+ 0. The error estimate is: 6 E I ga(ta) - a(t6) 0 as 6 0, 4ta) -+ 0 as because, due to the inequality 0 < a < 1, one has = 6 -+ 0. A novel discrepancy principle for nonlinear equations with monotone operators is formulated and justified in [15].

+

+

-+

-+

&

7. A differential inequality

Theorem 7.1. Let g ( t ) 2 0 and i ( t ) 5 -7(t)g

+ 4 ) g 2+ P(t),

d o ) = go,

t 2 0.

(25)

Assume that y, a, and p are continuous nonnegative functions, and there exists a p(t) > 0 limt,oo p ( t ) = 00, such that

Then every solution g ( t ) 2 0 to (25) exists for all t > 0 and

Proof. Let h := geJl y ( s ) d s . Then (25) implies

h

I a(t)h2+ b(t),

h(0) = go,

where a = ae-

Consider the equation

JJ y d s ,

b = Pe.f; y d s .

Dynamical Systems Method (DSM) and Nonlinear Problems

211

Its solution is (cf. [8]):

,sag o .

&

From a comparison lemma it follows that h 5 u < CL Theorem 7.1 is proved.

0

Example 7.1. Consider an example of applications of Theorem 7.1. This example deals with inequality (24). Let p = X = w n s t . Then f =

&,

- 0, leaves invariant and transforms into Thus, one can choose v sufficiently large to satisfy the second inequality (29) and = CE, where also the third one. By Theorem (25) one gets 0 5 g ( t ) < 1 c := 5; > 0.

y

9

ig.

8. A spectral assumption and nonlinear equations

Theorem 8.1. Let (2) hold, E = const > 0, and assume

IIAFl(u)ll 5

;,C

A(u) := A := F’(u), A, := A + &I.

(30)

Then there exists a solution to the equation: F(u)

+ EU = 0.

(31)

A . G. Ramm

212

Proof. Consider the problem .iL = -A;'(u)(F(u)

+

EU),

~ ( 0=) UO.

(32)

This problem is uniquely locally solvable, because (2) and (30) imply that the right-hand side in (32) is a Lipschitz function. Let

9 ( t ) := IIF(u(t>> +E

M l l l

where u(t) solves (32). Then

gcj = -(AEG,F ( u ) + E U ) = -g2. Thus g ( t ) = goe-t, and (32) and (30) imply 11.Li 11 5 T e - t . Thus limt.+m u(t) := u(00)exists. Passing to the limit t -+ 00 in (32) one proves that u(00)satisfies equation (31). Theorem 8.1 is proved. 0 9.

A singular perturbation problem

In this Section we want to give sufficient conditions for the solution u, to equation: F(UE)

to have a limit lim,,ouE element.

+ & ( U , - w)= 0

(33)

= y, which solves (1). Here w E H is some

Theorem 9.1. Assume:

a) b) c) d)

(2) with j 5 3 holds, 3y : F ( y ) = 0, (30) holds for all E E (0,E O ) , EO > 0 is a small fixed number, 3w such that y - w = Av, where A := F'(y) and llvll < [2Mzc(l+ c)]-', where MZ is the constant from (2) and c is the constant from (30).

Then (33) has a solution u, for any E E (O,EO), and IIy - uEll = O(E)as -+ 0. In a suftciently small ball { u : IIu-yII 5 R}, R = O(E),the solution uE is unique. E

-

Remark 9.1. If R(A) = H , then there exists a w such that y - w = Av, 0

11v11 is arbitrarily small. In fact, such a w exists if & n B(0,r)#

0 , where

0

B(0,r ) := {u : 0 < \lull 5 r } , R b stands for the closure.

:= { v : v = Au,

llull 5 b } , and the overbar

Dynamical Systems Method (DSM) and Nonlinear Problems

213

Proof of Theorem 9.1. Let z, := u, - y, zo = 0, F ( y + z,) := q5(zE). Then (33) can be written as +), where IlKll I Then

w.

Z, =

+

EZ,

+ EAZI= 0. Write 4, = Az, + K ,

-AFIK - EAF’Av := T(z,).

The map T is a contraction on B(R) := { z : llzll Indeed, llTzll 5 ~ l l ~ l l l I ~ ( 1c)llvll that

+ 5911~11~ + &

R = -(1cM2

(34)

I R}, where R = O(E).

+e

R 2 5 R, provided

p),

(35)

where p = J1 - 2M2c(l+ c)IIwII. Here we have used the estimate 1 IIA,1A4 I 1141 + &IIA2VII I (1 + ~ ) 1 1 4 . If 1141 < 2Mzc(l+c)’ then 0 < p < 1, R = O(E),and T B ( R ) c B(R). Furthermore, IITz - Tpll = 11 - AF1[r-f(z)- K(p)]ll,where z , p E B(R). By Taylor’s formula one gets K ( z )= (1 - s)F”(y S Z ) Z Z ~ S .Thus

so

+

One has

Thus

Therefore T maps B(R) into itself and is a contraction on B ( R )where R is defined in (35). Indeed

if E is sufficiently small. For such E equations (34) and (33) are uniquely solvable and llzEll I O ( E )as claimed. Theorem 9.1 is proved. 0 Remark 9.2. Theorem 9.1 holds if w = 0 and y = Av, .)]-I.

llzlll < [2M2c(l+

A . G. Ramm

214

10. Nonsmooth monotone operators

Assumption A): Assume that F is hemicontinuous (that is, (F(u+Xh),w) is a continuous function of X in a neighborhood of X = 0 for any u, h and w),monotone: ( F ( u )- F(w),u - w) 2 0 Vu, w E H , defined on all o f H , and 3y : F ( y ) = 0 , where y is the minimal-norm solution to (1). It is known that under Assumption A) the set NF := { z : F ( z ) = 0) is closed and convex, and such a set in H has a unique element of minimal norm, so y is uniquely defined if (1) has a solution. Consider the problem: u = -[F(u)

+€4,

u(0) = U O ,

E

= wnst

> 0.

(38)

Lemma 10.1. Problem (38) has a unique global solution if A ) holds. &

This lemma is known (see e.g., 0.

[lo]) and holds also for any integrable

= &(t) 2

Lemma 10.2. The solution to (38) satisfies the estimates sup Ilu(t))II c < 00; g ( t ) I g(0)e-Et,g ( t ) := Ilu(t+h)-u(t)ll, t 2 0, (39) t>o

where c = wnst > 0 and h = const > 0 is arbitrary.

+ h) - u ( t ) , llzll := g ( t ) . From (38) one gets ) u(h)- UO. i = -[F(u(t + h ) ) - F ( u ( t ) ) ]- €2, ~ ( 0=

Proof. Let z := u ( t

(40)

Multiply this equation by z , use the monotonicity of F and get gg 5 -Eg2. So g 5 g(0)e-Et.Let us prove the first inequality (39). Denote u ( t ) - y := p , llpll := q. Then zj = -[F(u) - F ( y ) ]- E(U

- y) - EY.

(41)

Multiply (41) by p , use the monotonicity of F and the inequality q 2 0, and get

Q I -EQ

+ EIIYII,

6

Q(0) = l b o

- Yll.

(42)

This implies q ( t ) 5 q(0)e-Et+EIIyII e-E(t-S)ds.Thus q ( t ) I IIuo-yll+llyll, 0 and the first inequality (39) is proved. Remark 10.1. We claim that the first inequality (39) holds also if E = ~ ( t>) 0, where E ( t ) is a locally integrable function. Indeed, since q 1 0, write (42) as Q I -Eq ellyII. Let q = Q ( t ) e - J i E ( S ) d SThen . Q I llylleJtE(S)dyE(t), so Q(t)I Q(0) llyll JiE(s)e=f:EdTds I Q(0) llylleJo'E(S)dS. Thus q ( t ) 5 q(0) Ilgll, and the claim follows.

+

+

+

+

Dynamical Systems Method (DSM) and Nonlinear Problems

Theorem 10.1. If A) holds and lim,+o

Et,

215

= 00, then

lim IIu(te)- 911 = 0. S+O

Proof. If A) holds, then equation (31) is uniquely solvable. Denote by K its solution. Then V, satisfies the first and the third relation (23). Let u(t)- V, := 7, where u(t) solves problem (38). Then (38) implies ?j= - [ F ( u ) - F ( V )

+ €(U - V ) ] .

(43) Multiply this equation by 7, let llqll := cp, use the monotonicity of F , and get +(p 5 -e(p2. Thus, cp(t) 5 cp(O)eVEt.If t = t, and lim,,oEt, = 00, then lim Ilu(t,) - VSll I cp(O)

E-tO

!% e-Stc = 0.

(44)

Since

I l W - YII I IlU(t> - Kll + IlK - Yll,

(45) one concludes from (44) and (23) that limE+o IIu(te)- yII = 0. Theorem 10.1 is proved. 0 Assumption B: Assume that the function E ( t ) > 0 is continuous, differentiable, monotonically decaying to zero as t + 00, and

E(s)ds = m,

I4)l lim -

t-tm € ( t ) 2

- O.

Theorem 10.2. If A) and B) hold, then equation u = -F(u) - € ( t ) U , u(0) = UO, has a unique solution, (5) holds, and u ( m ) = y.

(47)

Proof. Lemma 10.1 holds for problem (47) as well. If B) holds, then (23) holds with V = V,(t).Thus, by (45), if one proves lim Ilu(t)- V(t)II= 0,

(48)

t+cn

then Theorem 10.2 is proved. Let 7 := u(t)- V ( t ) ,llqll := cp(t).From (47) one gets as above, @ I --E(t)cp Filyll. Thus

+

cp(t>I cp(0)e- S,"E ( 5 ) d 5

+ e- S,"

llpllds + 0 as t

--f

m,

(49) by (46), as one checks using L'Hospital's rule, for example. Theorem 10.2 is proved. 0

A . G. Ramm

216

11. Unbounded operators

Consider equation (1)with F = L + g , where L is a linear, densely defined, closed operator, and g is a nonlinear operator satisfying (2). For example, a semilinear boundary-value problem

-Au+g(u) = f

in

u (s=O

D,

(50)

is of this form, L is the Dirichlet operator -A, H = L 2 ( D ) .

Theorem 11.1. Assume that IIL-lll sup

I m and

11 [I + L-lg’(u)]-l 11 I m1,

zLEB(u0$3

+

((uo L-lg’(.uo)((mi5

Then (5) holds for problem (4) with Q, := - [ I

R.

(51)

(52)

+ L-’g’(t~)]-~[u + L-lg(u)].

Theorem 11.2. Assume that L = L* 2 0, F = L+g, g’(u) 2 0 Vu E H , g satisfies (2), equation (1)is solvable, y is its minimal-norm solution, L , := L €1,and Q, := Q,, := - [ I L ; l g ’ ( ~ ) ] -[u ~ + L;lg(u)], E = const > 0. Suppose (51) holds with L replaced by L,, and ml = m l ( ~>) 0. Then (5) holds for (4) and

+

+

lim llv, - yII = 0,

E’O

(53)

where v, solves the equation LEvE+ g(ve)= 0. Proof of Theorem 11.1. Equation (1) is equivalent to

u + L-lg(u) = 0.

(54)

Assumptions (51), (52), and Theorem 9.1 imply the conclusion of Theorem 2.1. 0

Proof of Theorem 11.2. Consider the equation

+

LEU g ( u ) = 0,

21

= 21,.

(55)

By Theorem 11.1 equation (4) with Q, = Q,, satisfies (5). Let u,(co):= v, solve equation (55). Let us prove (53). Let v, - y := w,. Then Lw, +EV, g(uE)- g(y) = 0. Multiplying by W , and using the monotonicity of g and L , one gets ~(v,, w,) 5 0. Thus

+

IlVEII

5 IlYll.

(56)

Dynamical Systems Method (DSM) and Nonlinear Problems

-

Thus, there is a weakly convergent subsequence w, := v,, v, - y w o := wo - y as E, + 0 , and limsup Ilwnll 5

Il~ll,

217

2

wo, w,

liminfllvnll n+oo L 11v011.

:=

(57)

Let us check that vo E D ( L ) and pass to the limit E, -+ 0 in (55). Since L is selfadjoint one has

where = lim n-00

Lv,.

(59)

This strong limit exists because of (55). Indeed, limn+, E,W, is bounded (see (56)). Let us check that

=0

since w,

If (60) holds then limn,,g(vn) = g(w) by the continuity of g, and (59) follows from (55). To prove (60), we first prove that wo = y. Assuming this for a moment, one has from (57): llyll 5 liminf,,, llvnll 5 limsup,,, 11v,(( 5 ((y((. Therefore (60) holds with vo = y. Let us prove vo = y. One has by the monotonicity:

(Lv, +gun

where t > 0 and z E D ( L ) is arbitrary. Since v, one gets from (61) as n 4 00 the following:

(-Lvo Let t

-+

-

- L ( v ~-t z ) -g(vo -t z ) --E(w~ - t z ) , W,

-vO

+t Z ) 2 0

(61)

vo and t = const > 0 ,

+ tLz - g(w0 - t z ) ,z ) 2 0.

(62)

0 in (62). Then

(-Lvo

- g(vo), z ) L 0

vz E D(L).

(63)

Since D ( L ) is dense in H , it follows from (63) that

Lwo

+ g(v0) = 0.

(64)

However, llvoll 5 Ilyll, and y is the minimal-norm solution to (64). Since such a solution is unique, it follows that vo = y, as claimed. The uniqueness of the minimal-norm solution to (64) follows from Lemma 11.1.

Lemma 11.1. If F is closed, hemicontinuous, and monotone, and D ( F ) is a dense linear set, then NF = {u : F ( u ) = 0) is a closed and convex set.

A. G. Ramrn

218

Proof. First let us prove that NF is closed. Let F(un) = 0, un -+ u. Then, since F is closed, one gets F ( u ) = 0. So NF is closed. Now let us prove that NF is convex. Note that under the assumptions of Lemma 11.1 one has: 'u,

E NF

( - F ( v ) , 0 - U) 2 0

v't) E D ( F ) .

(65)

Indeed, if u E NF then the monotonicity of F implies inequality (65). Conversely, if inequality (65) holds, then set IJ = u + t z , where t = const > 0 and z E D ( F ) is arbitrary, and get - ( F ( u + t z ) , z ) 2 0. Pass to the limit t -+ 0 using the hemicontinuity of F and get - ( F ( u ) , z ) 2 0 Vz E D ( F ) . Since D ( F ) is dense in H , this implies F ( u ) = 0. Lemma 11.1 is proved. 0 In Hilbert spaces any convex and closed set has a unique minimal-norm element. From our argument and from (55) it follows that LVn -+r], 't)n-+ y, so Ly = r] because L is closed. Theorem 11.1 is proved. 0 12. Equations in Banach spaces Consider equation (31) in a Banach space X . Assume that E = const > 0 , F : X -+ X, (2) and (30) hold, and consider problem (32). Let r] E X* be arbitrary, cp(t) := ( F ( u ( t ) ) m(t),r]), g ( t ) := IIF(u(t)) au(t)II = sup^^^^^ 0, then

The proof of Theorem 9.1 remains valid for equation (33) in Banach spaces. 13. Iterative processes for well-posed problems

In this section we prove that any solvable well-posed problem can be solved by an iterative process with constant stepsize and the process converges exponentially fast.

Dynamical Systems Method (DSM) and Nonlinear Problems

219

ii) ll@(t,u)llI g2(t)llF(u)ll,Vu E B(uoR);and iii) IIF(u0)JJ G(t)dt I R, where g1 and 92 are positive integrable functions, G ( t ) := gz(t)e-Jiglds E L1(O,cm), and (4) has a unique local solution. Then ( 5 ) holds for problem (4).

so"

Proof. Since there is a unique solution to (4),this solution exists globally if S U ~ Ilu(t)/I ~ , ~ < 00. Let g ( t ) = IIF(u(t))ll. Using (4) one gets jrg = (F'c,F) = ( f ' @ , F ) 5 -g1(t)g2. Since g 2 0, one obtains g ( t ) Ig(0)e;-figldS,g(0) = IIF(uo)ll. From (4) it follows that 11Cll 5 gz(t)g(O)e-Jo g l d s = g(O)G(t)E L1(O, cm). Therefore u(cm)exists,

Remark 13.1. If g1 = c1 = const > 0, and g2 = cz = const > 0, then Gds = iii) takes the form ~ ~ F (:= ~r I oR, )and~ one ~ has ~

so"

2,

Ilu(t) - u(m)II 5 re-c1t,

IIF(u(t))llI IJF(uo)IJe-C1t.

(66)

Theorem 13.1. Assume (2), (3), \l@'(u)llI L1 Vu E B(uo,R), and let F ( y ) = 0. Let the conditions i), ii), and iii) of Lemma 13.1 hold with g = cj, j = 1,2, g = const > 0, and uo is suficiently close to y. Then there exists an h > 0 such that the iterative process un+l = u n

+ h@(un),

(67)

uo = U O ,

converges to y:

11% - YII

-chn

I re , IIF(udll I I l F ~ l l e - ~ ~IIFoll ~, where c = const, 0 < c < c l , and T := 2IIFoll.

:= IIF(uo)ll,

(68)

Proof. For n = 0 the inequalities (68) hold (see (66) with u ( m )= y). We prove that if they hold for n then they hold for n 1. The assumption that uo is sufficiently close to y ensures that y = u(cm). Let w,+l(t) solve (4)for t > t,, wn+l(t,) = u,, t o = 0 , t, = nh. By Lemma 13.1 one has c2 IIw,+l(t) - yII I -IIFnlle-C1t 5 re-chn-clt tn < t I tn+l, (69) c1

+

7

A . G. Ramm

220

where F, := F(u,). Also,

and

From (70) and (71) one gets

provided that h is so small that

Let us check the second inequality (68) with n

+ 1 in place of n:

One has

where (71) was used. From (76), (74), and (75) one gets:

+

IIF(u,+l)ll I IIFOlle-Chn(M1L1h2c2 e-clh)

I IIFOlle-ch(n+l),

(77)

provided that (78)

If h is sufficiently small, then (73) and (78) hold. Theorem 13.1 is proved.0

Dynamical Systems Method (DSM) and Nonlinear Problems

221

14. Iterative process for ill-posed problems with monotone operators Assume ( 2 ) , but not (3). Let A := F’(u) 2 0 Vu, A, := A + d , and consider an iterative process

+

un+l = u n - hnAil[F(un) &nun],

uo = UO,

(79) where A, := A(u,) &,I, and uo E H is arbitrary. Let y be the minimal norm solution to (1).

+

Theorem 14.1. Under the above assumptions one can choose E, and h, so that

Proof. Let V, solve (22) with E ( t ) = E,. Let z, := u, - V,, llznll = g,. Then IIu, - yII I g, IlV, - 911, and we know that limn+oo IlVn - yI1 = 0 if limn+W E, = 0. Thus (80) holds if limn+W gn = 0. Let us prove the last relation. Let b, := IIV,+l - V,ll, limn--roob, = 0. Rewrite (79) as

+

zn+l

= (1 - hn)Zn - hnAilK(zn)- (K+I- Vn),

(81)

where we have used the Taylor formula:

F(un)

+

+

= F(un)- F(Vn)

EZ,

= Anzn

+ K(z,),

From (81) one gets

Let En

= 2cg,.

Then (82) yields:

hn 0 < a, = - < 0.5. (84) 2 Theorem 14.1 follows from Lemma 14.1, which is stated and proved be1ow.n gn+l

I (1 - an)gn + b,,

Lemma 14.1. If b, 2 0, limn--roob, = 0 , 0 < a, M

n- 1

I

and

A . G.Ramm

222

then (84) implies

Proof. From (84) one gets by induction: n-1

gn+l < b n + C b k k=l

n n

(87)

(l-aj)+gl.

j=k+l

n,”=k+l

If a 2 0 , then 1- a < e-a. Thus (1 - a j ) 5 e(85) and (87) imply (86). Lemma 14.1 is proved.

Therefore

cy=k+l

0

Remark 14.1. One can always choose a, such that (85) holds. Indeed, let aj = logpj = logp Vj, where 1 < p 5 &. Then e-cy=k+l a ’ - p - n + k , and limn+m CtI,’b,p-n+k = 0 if limrc+m bk = 0. J

15. Newton-type methods without inverting the derivative In using Newton-type methods the most difficult part of solving numerically problem (1). This part consumes major computer time and leads to numerical errors. Let us consider a DSM of the form U

= -QF,

Q = -TQ

~ ( 0= ) UO,

+ A*,

Q(0) = Qo,

(88) (89)

where A := F’(u), T := A*A. Therefore, instead of an unknown u we are looking for a two-component vector

(G) ,

where Q is an operator-function which plays the role of

[F’(u)]-lin the usual Newton-type method. We prove that problem (88)(89) has a unique global solution and (5) holds under suitable assumptions. First, we need a lemma which is a Gronwall-type lemma for operator equations.

Lemma 15.1.

Q = -T(t)Q + G ( t ) ,

Q(0) = Qo,

(90)

where T , G and Q are bounded linear operator-functions in H . Assume that (T(t)h,h) L e(t)llh112

Vh E H ,

(91)

223

Dynamical Systems Method (DSM) and Nonlinear Problems

where ~ ( t>)0, ~ ( tE )L:,, ( 0 , ~ )Then .

IlQ(t)II I a-l(t)llQoll

I”

+ a-l(t)

a(s)llG(s>llds, a ( t ) := e j i E ( S ) d S(92) .

Proof. Let g := Q(t)h, b(t) := Gh, h E H is arbitrary, h does not depend on t. Then (90) implies g = - T g b(t). Multiply this equation by g, let (g,g) := p 2 ( t ) , and get: p@ 5 -&(t)p2 Ilb(t)llp. Since p ( t ) 2 0, one gets p ( t ) I a-l(t)p(O) a-l(t) s,” Ilb(s)Ila(s)ds. Now take the supremum with respect to h, llhll = 1, and get (92). Lemma 15.1 is proved.

+

+

+

Let us state the main result of this section.

Theorem 15.1. Assume (2), (3), and let uo and QO be suficiently close := [F’(y)]-l, where F(y) = 0. Then problem to y and, respectively, to (88)-(89) has a unique global solution

(;!))),

11u(t)- yII + 0 exponentially fast as t -+

3400):= y, F(y) = 0 , and

00.

+ +

Proof. Let w := u(t)- y and llwll := g. Then F ( u ) = Xw K, 11K11 5 o.5M2g2 := cog2, and (88) can be written as ti^ = -Q[Xw K ] . Let A := I - QX. Then w = -w Aw - QK. Multiply this equation by w and get

+

gg = -g2

Since T ( t )2 c

+ (Aw, W) - (QK,w).

(93)

> 0 , c = const, Lemma 15.1 and equation (89) imply

where M I is the constant from (2). Thus

II(QK,w)l( i cicog3 := k3. We prove below that SUP IlA(t)II I A < 1. tpo Thus (94) yields: gI -yg+kg2,

If Icg(0) < 1, then (97) implies:

0 < y := 1- A

< 1.

A. G. Ramm

224

Inequality kg(0) < 1 holds if kJJu0- yII < 1. This is the “closeness of uo to y” condition. Inequality (98) shows that u(t) + y exponentially fast. The trajectory {u(t>}t>o E B(uo,R ) if IlClldt = Ilt5lldt I R. Using (88), and the estimate: IIF(u)II = llF(u)- F(y)II I Mlg, one gets llzirll I 11QIlllFll I clM1g = c2re-Tt, where c2 := clM1. Thus llzirlldt 5 y r 5 R if r is sufficiently small, or (see (98)) if uo is sufficiently close to y. To complete the proof, one has to verify (96). One has

Jr

A = -QZ

=

TQZ - A*Z = -TA + A * ( A- 2).

(99)

Lemma 15.1 and (99) imply

where C := M1M2 ciently small, then

-

s,” e-ct+(c-’)sds.

If llAol1 and lluo - yII are suffi-

IlAoll + C r := X < 1. Theorem 15.1 is proved.

(101) 0

Remark 15.1. In the ill-posed case, when (3) fails, one can prove a similar result (see [23]). 16. Equations with non-monotone operators Let us assume (2), but not (3), let y solve equation (I), and let A := F’(u), T := A*A, T, := T &I, A0 := F’(y), and TO:= A;Ao. Choose @ = -T;’[A*F(u) E(U - G O ) ] , where E = E ( t ) satisfies conditions (21). Assume that 60 is such that

+

+

y - 60 = TOZ,

l l ~ l 1, which also cannot occur in the separated case). Clearly, when deg(Rx) # 0 , (12) has a solution for all h E L1(0,2n). On the other hand, X can also lie in a subinterval on which deg(Rx) = 0, in which case it is not known in general if there is h E L1(O, 27r) such that (12) has no solution. When k = n = 1 in the example there are exactly two simple halfeigenvalues in E f , of the form A:(€) = f 4 ~ / 3 o ( E )and , it is shown in [l] that deg(Rx) = 0 for all A E (AT(€), AT(€)) and, for any y E (0, l), there exists E ( Y )> 0 such that, if (€1 < E ( Y )and if

+

E

(-4YE/3,4YE/3) C (&(e)lXf(f)),

then (13) has no solution u. This provides a partial answer to the above question in this special case, where the jumping term ~ a u +is small. The next theorem extends this result to the general case, when Cf consists of exactly two simple half-eigenvalues.

B. P. Rynne

236

Theorem 5.3. Suppose that Cf = { A r x , A?'.}, and these halfeigenvalues are simple (and so distinct). Then there exists an open interval A C (A?., AT-) and h E L1(O, 27r) such that if A E A then (12) has no solution u E H . Proof. We use the constructions in the proof of Theorem 2.1, in paxticular, the function i k . Since half-eigenvalues are critical points of x k , the hypotheses here imply that the 27r-periodic function 1 , has exactly one local maximum and one local minimum on the interval [0,27r), while on P, x k is strictly monotonic on the intervals between its maxima and minima. The Priifer equations for (12) are now Tt

= T < ( t , e, A)

Ot = q(t,8, A)

- 2h cose,

+

(14)

h sin 8.

(15)

T - ~

However, instead of solving these with t = 0 as the initial time, as in the proof of Theorem 2.1, we will for now use an arbitrary time t = t o ( t o will be chosen specifically below). That is, for any ( t o , T O ,a, A) E R4 and h E L1(O, 27r), let r ( t ,tO,rO,a,A,h ) , e(t,tO,ro,a,A, h), t E R, be the solutions of (14), (15) satisfying the initial conditions

+o, t o , T o , a, A, h) = T o ,

q t o , t o , To, a, A, h)

= a.

(16)

For fixed h, the continuity and differentiability properties of 0 are wellknown, see Chapter I11 of [5]. Note that we will now require the solutions of (14), (15), to be defined on domains other than [0,27r],so we extend the domains of the coefficients q, a, b, h to R by 27r-periodicity. As before, for (to, a ) E P2, we let X k ( t 0 , a)be the unique solution of 8(2T

+to, to, To,

(Y,

X k ( t 0 , a ) ,0) = a

4-2kn

(17)

(we note that when h = 0 the functions 0 and X k do not depend on To). Clearly, X k ( 0 , a ) = &(a),a E R, and as before, i k : ~2 -+ R is ~ 1 and , is 27r-periodic with respect to each variable. Also, by varying t o we can 'slide' the graph of i k ( t 0 , along B (while distorting it) - the following lemma gives a more precise statement. a)

Lemma 5.1. For any t l , t o , T O , a E R,

-

-

A ( t o , a ) = x(tl,~(tl,tO,rO,a, A(to,a),O)).

Proof. The definition of

x(t0,a ) is

equivalent to

e(t + 27r, to, a,I(to,a ) )= e(t,to, a,x ( t o , a ) )+ 2kr, vt

(18)

237

Some Recent Results on Periodic, Jumping Nonlinearity Problems

By Lemma 5.1, we can choose t o E [O,27r), po E

Having now fixed t o and PO, we let t f := t o Tf(T0, +(TO,

(AT'", AT")

such that

+ 27r, and

a , h) := d t f ,t o , T o , a ,Po, h), a, h) := q t f , t o , T o , 0, Po, h).

-

If a = 0 or a = x,we have PO = A k ( t O , a ) , so by (17) +(To,

a,0) = e ( t f ,t o ,r0,a, i k ( t Oa, ) ,0) =

+2 k ~ .

(20)

hrthermore, aXk/ba(to,0) < 0, aXk/aa(to,7r) > 0, so from (11) Tf(TO,O,O)

< To,

Tf(T0,7r90)> To

(21)

(TO = 1 in ( l l ) , but by (14), T ~ ( T o , Q , O ) = r g ~ f ( l , a , O ) so , (21) is clear). Now, from (17), (19), (20) and the monotonicity of 8 with respect to a and A, we have Bf(ro,K ,0) = (2k 1)7r and

+

(2k (2k

+ 1). > a,0) > a + 2kx, + 1)x < 8f(ro,a,0) < a + 2kx, +(To,

a E (0, T ) , a E ( T , 27r).

(22) (23)

Also, by (20) and (21) we can choose 6 > 0 sufficiently small that, Tf(TO,a,O) Tf(To,

< To,

a,o) > T o ,

lef(To,a,o)

Ief(To,

- 2lc4

+

< 1, a E [-6,61,

a,o) - (21~ i ) r ~< 1,

E [r- 6,

+ 61.

(24)

For E > 0, let t , := t f - E , let h, be the characteristic function of the interval [tE, t f ] extended , to R by 2x-periodicity, and let (12), denote

B. P. Rynne

238

equation (12) with h = h, and X = po. We will show that if E is sufficiently small then (12), has no 2n-periodic solution. We now regard T O , a,po as fixed, and let g(t,h) := O(t,to,ro,a,po, h), F(t, h) := r ( t ,t o , T O ,a,po, h). By definition, +(To,

a,h ) = g ( t j , h ) , g(t(te,he) = g(tC,o),

with similar results for

F.

Lemma 5.2. Suppose that po is not a half-eigenvalue of problem (4), defined on the interval [ t o , t f ] , with Dirichlet boundary conditions at t o , t f . Then there exists € 1 > 0 such that if E < €1 then any 2n-periodic solution u of (12), satisfies r( t ) 2= u(t)’ ut(t>’ > C ( E ) > 0, for t E [to,t f ] .

+

Proof. Suppose there exists a sequence E , + 0 such that, for each n, (12),,, has a 2n-periodic solution u, with r,(t,)2 = 0. It is clear that t , E [t,,tf],and JunJ1= O ( E ) .Defining w, := Un/E,, n L 1, a relatively standard convergence argument shows that wn + w, # 0 in Co[to,t f ] with , w m ( t o ) = wm(tf) = 0, and that w, satisfies the differential equation (4), on the interval [to,t f ] with , X = po. This contradicts the hypothesis of the lemma, and so proves the result. For now we suppose that Lemma 5.2 holds. Since we are only interested in 2n-periodic solutions this ensures that we need not consider T going to zero in (15). We consider the following cases.

+

E [n S,2n - S]

In this case, Bf(r0,a,0) satisfies (23), so if E is sufficiently small g(t,O) also satisfies (23), for t E [t,,tj]. Thus, by (15), the fact that h, > 0 on [t,,t f ] ,and the sign of sin g(t,h,), (Y

a

+ (2k - 2)n < (2k + 1)n < g(t,h,) < g(t,0 ) < + 2kn < (2k + 2). (Y

(we note that if O(., h,) = nn, n E Z,then 7 = 1 in (15), so g(., h,) cannot decrease through nn - this observation will also be used several times below). Hence, putting t = t f in this inequality shows that (12), has no 2n-periodic solution in this case. aE

[S,n - 61 In this case, by (22), if E is sufficiently small and t E [t,,t f ] , 2kn

< a + 2kn < g(t,0) < (2k + 1)n < a + (2k + 2)n.

+

Thus, if @,he) < (2k 1). on [t,,tf],then, by (15), g(tf,h,) > $ ( t j , O ) and so (12), has no 2n-periodic solution.

Some Recent Results on Periodic, Jumping Nonlinearity Problems

239

+

Now suppose that $(t(t,, he) = (2k l)n, for some tl E [t,,t p ] , and so $(it, h,) > (2k l)n for t E (tl,tp]. Suppose also that E is sufficiently small that any solution of the initial value problem

+

dt

= v(t, d, Po),

+

d(t2) = (2k: + 1h,

t2

E [t€, tpl,

-

satisfies $(t) < (2k 2)n, t E [t,,t p ] . Now, by (15), while t E (tl,tr] and $(t,h,) < (2k 2)n, we have Ot < 17, so

+

ct

+ 2ks < (2k + 1)s < 8(t,h,) < (2k + 2)7r < + (2k + 2)s. (Y

+

Hence, the condition g(t,h,) < (2k 2). in fact holds on [tl,tp], so the preceding inequality shows that again (12), has no 2n-periodic solution.

a E [-6, 61 By (24), if E is sufficiently small then F ( t E 0) , < TO and $(t,, 0) is close to 2kn. Then, by (14), F(t,hE)< F(t,O), for t > t, so long as cos$(t, h,) > 0. If this remains true for all t E [t,,t p ] then F(tp, h,) < T O , and so (12), has no 2x-periodic solution in this case. Now suppose that $(., h,) decreases through (2k - $)n on the interval [t,,t p ] , so that cosg(., h,) becomes negative. We know that g(.,h,) cannot decrease through (2k - 1)n, but also, while g(.,he) < 2k7r the h, term acts to decrease g(.,h,), so a similar argument to that in the case a E (6, n - 6) shows that if E is sufficiently small then h,) cannot return to the interval (2kn - 6,2kn 6). Thus, again (12), has no 2n-periodic solution. Similarly, if $(-,he)increases through (2k + $)n then, while 2kn < &(.,he)< (2k l)n, the h, term acts to increase $(.,he),and so, if E is sufficiently small, g(., he) cannot return to the interval (2kn - 6,Zkn + 6). Furthermore, if g(-,h,) crosses (2k + l)n, a similar argument shows that h,) cannot reach the interval ((2k + 2)n - 6, (2k + 2)n 6). Thus, again (12), has no 2n-periodic solution. $ ( a ,

+

+

+

$ ( a ,

+

a E [n- 6,n 61

This is similar to the previous case.

Finally, the above arguments hold if PO is replaced by any X sufficiently close to PO. Since the Dirichlet half-eigenvalues in Lemma 5.2 are discrete (see [7]), we can find an open interval of X values close to PO for which Lemma 5.2 and the above arguments hold, and hence for which the conclusion of Theorem 5.3 holds. This completes the proof. 0 Theorem 5.3 is still only a partial result. One would like to know whether or not the non-existence result holds for all X E (XF'",XF"). Furthermore, nothing is known about non-existence if X lies in an interval with deg(RA) = 0 when CF consists of more than {XF"", A?'"}. It seems possible that the non-existence result may not be true in such a case.

B. P.

240

Rynne

6. The FuEik spectrum

In this section we show that the structure of the FuEik spectrum (defined below) can be considerably more complicated in the periodic case than in the separated case, even for a ‘generic’ periodic problem. We remark that it is shown in [3] that the structure of the FuEik spectrum of the Dirichlet Laplacian, in a bounded region R c Rn with n > 1, can be more complicated than that of the separated Sturm-Liouville problem. The results below show that even for the 1-dimensional periodic problem the FuEik spectrum can be ‘complicated’. We first recall the definition of the FuEik spectrum (many more details are given in [l]and [7], and the references therein). Consider the problem

Lu = au+ - pu-,

(25)

with (a,p) E R2, and in the separated case let

EF := { ( a , @E) R2 : (25) has a solution u E Si},

C F :=

u

Cf;

kZO

in the periodic case we replace 3; with S i k . The set C F is the FuEa7c spectrzlm of L. A point (a,p) E R2 can be regarded as a pair of constant functions, so it follows from the definitions that

C f ( L ) = {(a,/?)E R2 : 0 E C,H(L,a,/3)). In general, the set CF is ‘trivial’, and consists of a pair of known horizontal and vertical lines (in both the separated and periodic cases). Thus we ignore this set and consider only k 2 1. As with E f , the structure of E f , k 2 1, differs significantly in the two cases, so we again consider them separately.

6.1. The separated case For any k 2 1, it is shown in [7] that Ef consists of a pair of C1, strictly decreasing curves,:-I passing through the point (pk,p k ) , and having vertical and horizontal asymptotes. For each point ( a ,p) E,:?I equation (25) has a unique normalized solution uf ( a ,p) E Sf, and

C F ( L ) = {(a,P> E R2 : x:(L;CY,p) = 0 ) . The curves corresponding to different values of k do not intersect, so the curves lie ‘above’ the curves.:?I The curves I?; may intersect at points other than (pk,pk). Indeed, for all odd k we have)&(:-I = I?;(&).

Some Recent Results on Periodic, Jumping Nonlinearity Problems

241

6.2. The periodic case For any k 2 1, the set C r ( L 0 ) consists of a single C1, strictly decreasing curve I’k(LO), passing through the point (pk, pk), and having vertical and horizontal asymptotes. The curve rk+l( L O )lies strictly above the curve r k ( L 0 ) . If ( a , P ) E r k ( L 0 ) then the half-eigenvalues X Y ( L 0 ; a,

p) = XpyLo; a,p) = 0.

Also, these half-eigenvalues are not ‘regular’ (see [l]),since ‘regularity’ would imply that the corresponding set of normalized half-eigenfunctions is discrete, as in the separated case, whereas for the periodic problem this set consists of a circle in L1(0,2n); this is discussed in more detail in [l]. For a general operator L the structure of C F ( L ) has not been described. max / min We can obtain a partial result as follows. For any y E R, let A, (7)1 k 2 1, denote the half-eigenvalues constructed in Theorem 4.1 for the problem

Lu = yu+

+ xu = (y + X)u+ - xu-

(using u = u+ - u-),and let

Clearly, rFm/ min ( L ) c C r ( L ) , and by the continuous dependence and monotonicity results in Theorem 2.3 in [l],these sets are continuous, decreasing curves in R2,and C ; ( L ) lies between these curves. In fact, such ‘upper’ and ‘lower’curves (in a weak sense) were constructed in [9]. As mentioned in Section 6.1, in the separated case C F ( L ) consists of exactly two (possibly coincident) curves, but little is known about the periodic case in general. We will now use a perturbation method similar to that in Section 4 of [l]to construct an example in which the ‘non-regular’ F u E i k curve F k ( L 0 ) ‘splits apart’ under a suitable perturbation, giving rise to an arbitrarily large number of other curves between these upper and lower curves. Fix k 2 1 and (a,P) E CF(L0). We consider the problem

Lou = EQU + au+ - pu-

+ xu,

(27)

with q E L1(O, 27r) and E E R. When E = 0 we can construct a non-trivial solution 1c, of (27), having 2k nodes, by placing a positive and negative sinusoid, of total length 27r/k, adjacent to each other, and extending the resulting function to R by 27rlk-periodicity. Clearly, for any 8 E R the translated function defined by w ( O ) ( t ) := @(t- 8 ) is also a solution of (27)

B. P. Rynne

242

(with E = 0), and the mapping 8 -+v(0) is 27rllc-periodic and gives a closed 27r) (NB, 6 no longer denotes Priifer angle). curve of solutions in L1(O, Now, to construct solutions of (27), with E # 0, we define the functional

J ( e ) := (qv(e),v(e)), e E R. The following theorem is similar to Theorem 4.2 in [l], but the perturbation in (27) has a different form to that considered in [l],so we sketch the differences in the proof.

Theorem 6.1. Suppose that all the critical points of J are non-degenerate, and let &$, i = 1 , . . . ,m } denote the critical points. There exists EO > 0, and C1functions X i : (-€0, €0) -+ R, i = 1,.. . ,m, such that, if I E ~ < EO then Xi(€) = -EJ(&) O ( E ) and C f ( L 0 - q,a,p) = {Xi(€) : i = 1,.. . ,m}. If these half-eigenvalues are all distinct, they are all simple.

+

Proof. For any 0 E R define the function

{

a, if v(e)(t)> 0,

x(O)(t):= p, if .(e)(t) < 0, 0,

otherwise,

for t E [0,27r], and for any u E L1(0,2n) let E ( u) := au+ - pu-. Then, by these definitions,

and so the linear operator L(0) := Lo - x(e) has dimN(L(0)) = 2, with

N ( L ( 8 ) )= s~an{v(Q),ve(e)}. Let P(0) denote the orthogonal projection from L1(O, 27r) to N ( L ( 0 ) ) . Now choose an arbitrary point 00 E R, and let PO:= P(Oo),QO:= I-PO, Wo := R(Q0). We look for solutions u of (27) near to v(OO), having the form u = v(e) w, with w E WO,and X near to 0. Putting u = v(0) w and X = €1 into (27) and using (28) shows that (27) is equivalent to the pair of equations

+

+ w e ) + w) - q ~ ( e ) ) +] EQO[(q+ X)(v(e) + 41 = 0, P ~ [ - L ~ w+ qq)+ W) - q ~ ( e ) )+] EPo[(q + X)(v(e) + w)l = 0.

Q~[-L~w

+

(30) (31)

Some Recent Results on Periodic, Jumping Nonlinearity Problems

243

Clearly, ( 8 , w , i , e ) = (80,0,0,0) is a solution of (30)-(31). Now, for any (8, w, 1,E ) E R x WOx R2, we define H ( 8 ,w,i, E) to be the left hand side of (30). The derivative D,H(Bo, O , O , 0 ) : WO-+ WOis

D,H(Bo, O , O , 0)s= -QoL(e0)q

tij

E Wo,

and is non-singular, so equation (30) has a local solution w(O,i,e), for (8,i, E) near to (O0, O , O ) , with w(8, i, 0 ) = 0 for (8,i) near to (80,O). Defining G(8, i, e ) := w(8, i, e ) / ~ for , e # 0, we have

qq)+ ei.z(e,i, €1) - q q ) ) = ex(e)G(e,i, E ) +O(E) (in L*(O,2 n ) ) , so denoting the left side of (31) by EG(8,?;,e ) (defining G by continuity when e = 0 ) , (31) becomes

X,€1 = po[(-L(e)+ x(e))G+ ~ ( i+)(q + i)(u(e) + €4= o (with G = G(8,i, e ) ) . Also, by definition, PO[-L(Bo) + ~ ( O O ) = ] 0 , so G(e,

W O , ' j ; , O ) = POKQ+ X)4eo)l. The argument now follows that in [l](from (4.8) onwards in [l]).

0

Lemma 6.1. Suppose that a # p. Then, f o r any integer n 2 1, there exists a function qn E Co[O,2x1 such that the corresponding functional Jn has only distinct, non-degenerate critical points, and at least 2 n of them. Proof. Choose 2 n points 0 < 81 < 82 < . < 8Zn < 2 x / k . The functions (w(8i))2, i = 1,.. .,212, are C", except at their zeros, where they are C' but not C 2 ,and these zeros are distinct, so these functions must be linearly independent. Hence, there exists a function ijn E Cw[O,27r] such that a

It follows that the corresponding functional Tn has at least 2 n critical points, and a genericity argument shows that we can choose a nearby function qn E Co[O,2x1 such that the corresponding functional Jn has only distinct, non-degenerate critical points, and at least 2 n of them. 0 The above results show that if e # 0 is sufficiently small then there are at least 2 n distinct, simple half-eigenvalues X i ( € ) , i = 1 , . . . ,2n. That is, for suitable ui (e ) , we have

LOU~(E) = EqnUi(e) + ~ u + ( E PUT(€) ) +x~(E)u~(E) = eqnui(E) ( a X i ( e ) ) ~ + ( e) (P x i ( e ) ) ~ ~ ( ~ ) ,

+ +

+

B. P . Rynne

244

and hence,

+ A ~ ( E ) ,+PXi(€))

(a

E

c;(Lo

- q,),

i = 1,. . . ,2n.

Thus, there are at least 2n points in CC(L0 - E q n ) on the diagonal line D(,,p) := {(a s , p s) : s E W} through the point (a,P). Now, for each i the half-eigenvalue Xi(€) is simple, so the implicit function theorem can be applied to equation (27), at (A, u)= (Xi(€), ui(~)) (this is, essentially, the definition of a 'regular' half-eigenvalue in 111, which includes 'simple'; see [l]for details). This enables us to construct a C1curve of solutions with a parametrisation of the form

+

+

sE

(-46)

-+

(Cyi(S),Pi(S),V i ( S ) )

+

(for some b > 0), with la:(O)l IPi(0)I respect to s) and

> 0 (here, ' denotes derivative with

(ai(o),Pi(o),vi(o)) = (a+A i ( E ) , P + Az(c),ui(f)) (in fact, we could choose either a or P as the parameter). The projection of this curve onto R2,given by the parametrisation s -+ (ai(s),Pi(s)), lies in Cc(L0 - E q , ) . In addition, it is easy to show that, locally, this curve is strictly decreasing. The calculation is well-known, but we sketch it here for completeness. The above parametrisation satisfies the equation

L,u(s) = a(s)u+(s) - P(s)u-(s),

s E (-6,b)

(writing L , = Lo - €4, and omitting the subscript i for simplicity). Differentiating this equation with respect to s, at s = 0, yields - b S ( O )

= a s ( O ) u + ( O ) - Ps(O)u-(O)

+ '"(O)XO+"s(O) + P(O)xOus(O),

where xof. is the characteristic function of {t E [0,27r]: fu(O)(t) > 0}, and the derivative of the mapping u -+ &u* at u(0)is the mapping -+ xofv; see Lemma 3.1 in [l]for a more precise statement. Clearly, x:u(O) = *u*(O). The above two equations now yield

+

0 = ('ZLs(O), L€U(O) - a(O)u+(O) P(O)u-(O)) = (.S(O),

IL, - a(O)xO+- P ( O ) X O l ~ ( O ) )

= ([L€ - '"(O)XO+ - P ( O ~ X O l ~ S ( O ~ , ~ ( O ) ) = a s ( O ) b + ( O ) , .+(O))

+ P s ( O H u - ( O ) , u-(O)),

from which we see that as(0)Ps(O)< 0, which proves that the FuEik spectrum curve is strictly decreasing near the diagonal line D(,,p).

Some Recent Results on Periodic, Jumping Nonlinearity Problems

245

We conclude that for any n 2 1 there are at least 2n distinct, strictly decreasing curves in the FuEik spectral set C [ ( L , - ~4~), near the line D(a,p). Since these curves are constructed by using the implicit function theorem, we can also conclude that they persist (at least locally) under small perturbations of the coefficient functions. Thus, these complications are not a ‘non-generic’phenomenon and can occur in a relatively ‘large’set of problems. We know nothing about the global structure of these curves. They may, of course, link up outside this neighbourhood. References 1. P. A. Binding and B. P. Rynne, Half-eigenvalues of periodic Sturm-Liouville problems, J. Differential Equns. 206 (2004), 280-305. 2. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York (1955). 3. E. N. Dancer, Some results for jumping nonlinearities, Topol. Methods Nonlinear Anal. 19 (2002), 221-235. 4. J. Dieudonnk, Foundations of Modern Analysis, Academic Press, New York (1969). 5. E. J. McShane, Unified Integration, Academic Press, New York (1983). 6. W. Reid, Ordinary Differential Equations, Wiley, New York (1971). 7. B. P. Rynne, The FuEik spectrum of general Sturm-Liouville problems, J . Differential Equations 161 (2000), 87-109. 8. B. P. Rynne, Non-resonance conditions for semilinear Sturm-Liouville problems with jumping non-linearities, J. Differential Equations 170 (2001), 215227. 9. M. Zhang, The rotation number approach t o the periodic FuEfk spectrum, J . Differential Equations 185 (2002), 74-96.

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SOME REMARKS ABOUT THE CUBIC SCHRODINGER EQUATION ON THE LINE

L. VEGA* Universidad del Pais Vasco Apdo. 644, 48080, Bilbao, Spain. E-mail: mtpvegolOlg. ehu. es

1. Introduction

We will consider the equations iatu

+ a:u f 1u12u= 0

xER t

E R,

(1)

and more in particular the case when

In this note we will survey the motivations to study that particular inital value problem, the analytical questions that naturally arise trying to solve it and some recent results about it. Also let us recall a classical result by H. Brezis and A. Fkiedman [l] which states that

has no solution. Here the notion of solution is more restricted whether general functions or just positive are considered. Recently, J. Aguirre [2] 1 proved that if in (3) u(z,0) = bvp - then there exists a1 least one solution X for all b. As far as I know this is just an existence result and nothing has been proved about uniqueness or continuous dependence. *Partially supported by a MECyT grant 247

L. Vega

248

2. Some Physical Models

As it is well known cubic NLS is a canonical or universal model [3] which describes dispersion and nonlinear interaction. In this case the dispersion relation is quadratic and the nonlinearity can be focussing or defocussing depending on the choice of the sign in (1). In that spirit NLS appears in nonlinear optics where the temporal variable denotes a privileged direction in the physical variables. Take for example a standing wave W

that has y as the preferent direction of propagation. Also assume that the velocity c is a small perturbation of a constant co > 0. Therefore we fix the ansatz

to obtain that as a first approximation

_-

2 2 “ti

4

ti

has to be a solution of

+ k2u rt 2ikdyti + dy”u+

@ti

= small.

Finally if we assume that the variation of ti in the y-variable is much smaller than k (i.e., suppu^(x,.) is contained in a ball of radius small w.r.t. k) we can neglect dzu to obtain the free Schrodinger differential operators

f2ikdyti

+ a;ti

= small.

Then the cubic term in (1)appears when one considers that the variation of the speed c around co depends on the intensity 1 2 ~ 1 ~ In . this model the key dimension is two (x E EX2) and explicit blow-up solutions of (1) with the positive sign can be easily given. However our main interest comes from the geometric flow given by

X t = X , A X,, = c b.

(4)

Here X ( s , t ) denotes a curve in EX3, s is the arc length parameter, c is the curvature and b the binormal. Making the identification of the normal plane at a given point of the curve with a copy of the complex numbers at that point, we can always write b = in so that (4) becomes

x,= icn, and therefore the relation to Schrodinger equation is clearly established. Moreover using Hasimoto transformation 9 ( s , t ) = c(s, t)ei J”

Some Remarks about the Cubic Schrodinger Equation on the Lane

249

we get [4]that

i&P

+ d2P + 51 (1Ql2 + A ( t ) )P = 0.

(5)

In the case P(s,O) = cod we get 151

X(s,O)=

A+s, s > 0 A - s , s < 0,

1 for some A + , A- of unit length, and P(s, 0) = cvp - is related to logarithS

mic spirals [6]

X ( s , O )= s

'a1

(ezT

3

(bl

+ i b z ) , b3)

+ +

with bf bi bg = 1. The binormal flow appeared for the first time in a work by DaRios [7] as a crude approximation to the evolution of a vortex filament within Euler equations. In this setting one of the possible scenaries for the formation of singularities [a] is the development of a corner as the one of example (6). Also it is interesting to recall a model given by R. Klein, A. Majda and K. Damodaran [9]which describes the interaction of a family of vortex filaments in R3.Each of the filaments is given by (P(s,t ) ,s) E R3 with P a complex function. The system is precisely as follows

Here rj is the strength of the j-filament and aj is a constant related to the DaRios approximation. Therefore the dispersion term given by the second derivative measures the interaction of the each j-filament with itself while the interaction of different filaments is given by a Coulombian-like potential. Also notice that the cubic term appearing in DaRios approximation is neglected. Recently C. Kenig, G. Ponce and L. Vega [lo] have proved the orbital stability of two and three equilateral helical vortices with equal strength using this model.

3. Conservation Laws It is well known that (1)is a completely integrable system. However among the infinitely many conserved quantities we are interested in the following ones:

L. Vega

250

(ii)

1

(v)

J r ( z ,t)dz = J

Iu(z,t)I2dz'F

T(z, ~

)dz,

( ~ 1 ~, - 1 1 2 1 ,

with r the torsion of the X-curve in the binormal flow. In the above list we include whithin brackets the Sobolev H s and Lebesgue Lp spaces which are invariant under the same scaling of the corresponding quantity. Because of (2) we are mainly interested in L 1 - H112 scalings which are the ones of the (iv) and (v) quantities. 4. Symmetries

Among the transformations which leave invariant the set of solutions of (1) we have the following ones. (i) (z,t ) translations; (ii) Dilations: ux = Xu(Xz, X2t) is a solution if so is u; In terms of the initial condition we have ux(z,O) = Xuo(Xz).

Therefore the solutions which are invariant with respect to sealing should be related to initial data which are homogeneous of degree -1: uo = a6

+ bp.v. -.X1

(iii) Multiplication by a constant of modulus one; (iv) Galilean transformations: given N E R then U N ( 5 , t ) = e - i t N 2 + i N y z - 2Nt, t )

(7)

Some Remarks about the Cubac Schrodinger Equation on the Lane

251

is a solution if u is a solution. The corresponding initial conditions are related by the identity uN(z,

0) = eiNzu(x,0).

(8)

Therefore the solutions which are invariant with respect to the Galilean transformations should verify that eiNzu(x,0) = u(z,0) for all N . Hence u(x,O) = a6. In fact if 'llN =?J.

for all N we can differentiate in N in both sides to conclude that i r p ( Q , t ) - 2tdqu(77,t ) = 0.

And from (1) and some simple manipulations [ll]we get U

u(x,t ) = -exp

4

1

i&

ze

4t converges as t goes to zero to a constant times the But &function. As a consequence u, has no limit at t = 0 and the IVP:

{

iatu

+ a:u * J U ) % = 0

u(x,O) = a6

(9)

either has no limit or has more than one solution [ll]. 5. Stability

Recall that the equation coming from the binormal flow (5) has an extra term A(t) with respect to equation (9). In fact u(S,t)= a-e

4

is2/4t

,

solves

This suggests that the natural problem related to the &function as initial datum has to be modified as in (10). In fact with S. GutiCrrez [6] and G. Perelman [12] we have proved that (9) has no solution within the set of self-similar solutions according to scaling.

L. Vega

252

At the same time a natural question is if the explicit solution (10) of (11) is stable in any sense. The problem suggests different possibilities that we shall try to explain. The first issue we want to consider is which is the minimal regularity needed to solve the problem

The question is solved if uo is in the Sobolev space H 3 , s 2 0 (131 [14] [15]. The way of doing it is to find by a fixed point argument a solution to the corresponding integral equation

Hence one has to look for good estimates for the solutions of the linear problem + A , ~ = F z€IWn,~ { i v tw(z, vo(z). 0) =

E R

(14)

The key estimate in n = 1 to solve (12) is

I CllvOllL: + IlFlILy

11~11LB,,

(15)

with v the solution of (14). The smallness condition necessary to obtain a fixed point is obtained by taking the time of existence T small enough. and from the L2-conservation law the local result is But T = T(lluoll~z), extended to all t > 0. It is easy to see using the Galilean transformations [ll],that H 3 with s < 0 is not very well adapted to solve (12) ( H 3 , s < 0 , is not invariant under translations of Fourier space cf. (8)). Instead of using Sobolev spaces in a joint work with A. Vargas [16] we proposed to use the space x = {uo s.t. eita2=uo E L & , ~ ~ L ; ) . (16) This space is not easy to handle. A sufficient condition to belong to X is for example to assume that GO E el and

with a > 1/6. Then we prove [16] that if uo E X then (12) is locally well posed in time. Notice that if uo = a6 then uo verifies (17) with (Y = 0.

Some Remarks about the Cubic Schrodinger Equation on the Line

253

We also obtain a global result following some previous ideas developed by J. Bourgain [17]. For this result we need to be able to decompose uo as uo = f N llfN11L2

+gN;

f N E L2,

gN E

L N ; llgNllx L N - s

x,

P > 1,

(18)

for all N > 1. We find [16] such a decomposition by imposing some extra properties. I think that to find good ways of achieving (18) it is a very interesting question in Fourier Analysis. Another approach to “decrease” the regularity is to look at iut

+ a; f IZLJYU = 0,

(19)

and to look for selfsimilar solutions with respect to the corresponding scaling. The question is then which is the minimal y so that those solutions can be constructed. In a joint work with T. Cazenave and M. C. Vilela we improve previous results obtained by T. Cazenave and F. Weissler [18] and later extended by F. Planchon [19] to the classical H S , s 2 0. We extend [20] Planchon’s result to s = 0 in dimensions one and two, and define some alternative spaces which behave as H”, s < 0, in all dimensions. In n = 1 8 the lowest y obtained by this method is y = -, still far from the cubic 3 equation y = 2. The spaces introduced in [20] are quite simple:

Y P = (210 : Go E L P ( W ) } 1 5 p 5

00.

Interestingly A. Grunrock [21] has seen that this type of spaces are quite well adapted to deal with the cubic (and others) non-linearity. He has announced the result of local wellposedness of (12) with voEYP

p1

4%1) = Wl(S),

E ~ ( s This ) . amounts to solve

--ift

2a + E,, + 2 a-'tR e ~ + --ERRE + -1 1 ~ 1 ~0 ~ t t ;=

E(S,

From that we obtain

1

And therefore

IIE(t)llL2

=- 4 2

t

5

1) = E1(S).

(23)

1

Re d m E .

ll-(1)llL2*

On the other hand given EO = E O ( t 0 ) E L2 we define the operator

and use (15) in the classical way to obtain a fixed point of (24), and therefore a solution of (23) for 1 < t < T and all T. Going back to (22) we easily obtain the conservation law:

=

] [lw,(1)12

- (lw(1)I2 -a2)'] ds.

Some Remarh about the Cubic Schrodinger Equation on the Line

255

It is also easy to prove that if v8(1) E L2 then v8(t) E L2. But from Gagliardo-Niremberg’s inequality

Hence for u2 < 1/2 we get

This can be understood as an orbital stability result of the the solution = a, a < 1/2, because just IlaSvllpremains small. Therefore the main remaining question is to identify the limit

21

References

4.

H. Brezis, A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983) 73-97. J. Aguirre, Self-similarity and the singular Cauchy problem for the heat equation with cubic absorption, Appl. Math. Letters 14 (2001) 7-12. G.B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, A Wiley Interscience Publication 62 (1974). H. Hasimoto, A soliton in a vortex filament, J. Fluid Mech. 51 (1972) 477-

5.

485. S. Gutikrrrez, J. Rivas, L. Vega, Formation of singularities and self-similar

1.

2. 3.

vortex motion under the localized induction approximation, Comm. Part. Diff. Eqns. 28 (2003) 927-968. 6. S. Gutikrrrez, L. Vega, Self-similar solutions of the localized induction approximation: singularity formation, Nonlinearity 17 (2004) 2091-2136. 7. L.S. Da Rios, O n the motion of an unbounded fluid with a vortex filament of any shape, Rend. Circ. Mat. Palermo 22 (1906) 117. 8. A. Majda, A Bertozzi, Vorticity and incompressible flow. Cambridge texts in Appl. Math., Cambridge University Press (2002) 9. R. Klein, A. Majda, K. Damodaran, Simplified equations f o r the interaction of nearly parallel vortex filaments, J. Fluid Mech. 288 (1995) 201-248. 10. C. Kenig, G. Ponce, L. Vega, O n the interaction of nearly parallel vortex filaments, Comm. Math. Phys. 243 3 (2003) 471-483. 11. C. Kenig, G. Ponce, L. Vega, O n the ill-posedness of some canonical nonlinear dispersive equations, Duke Math. J. 106 3 (2001) 716-633. 12. G. Perelman, L. Vega, Selfsimilar solutions f o r modified K d V equation. Some applications. Preprint .

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13. J. Ginibre, G. Velo, Scattering theory in the energy space for a claass of nonlinear Schrodinger equations, J. Math. Pures Appl. 9 64 (1985) 807836. 14. Y. Tsutsumi, L2-solutions for nonlinear Schrodinger equations and nonlinear groups, Funkcial. Ekvac 30 (1987) 115-125. 15. T. Cazenave, F. B. Wiessler, The Cauchy problem for the critical nonlinear Schrodinger equation in H S , Nonlinear Anal. 14 (1990) 807-836. 16. A. Vargas, L. Vega, Global well-posedness f o r I d non-linear Schrodinger equation for data with a n infinite L2 norm, J. Math. Pures Appl. 80 10 (2001) 1029-1044. 17. J. Bourgain, Refinements of Strichartzz’s inequality and applications t o 2DNLS with critical nonlinearity Internat. Math. Res. Notices 5 (1998) 253283. 18. T. Cazenave, F. B. Wiessler, Asymptotically self-similar global solutions of the nonlinear Schrodinger and heat equations, Math. 2. 228 (1998) 83-120. 19. F. Planchon, O n the Cauchy problem f o r Besov spaces f o r a nonlinear Schrodinger equation, Comm. Cotemp. 106 3 (2001) 716433. 20. T. Cazenave, L. Vega, M. C. Vilela, A note o n the nonlinear Schrodinger equation in weak Lp spaces Comm. Contemp. Math. 3 1 (2001) 153-162. 21. A. Griinrock, Abstract in Mathematisches Forschungsinstitut Obenuolfach report 50 (2004).

SOME REMARKS ON THE INVARIANCE OF LEVEL SETS IN DYNAMICAL SYSTEMS

JOSE M. VEGAS Departamento de Matemcitica Aplicada. Facultad de Matemciticas Universidad Complutense. 2804 0-Madrid. Spain

To the m e m o r y of J e s h Esquinas Conditions of Liapunov type are given on a function V : Wn -+ W in order that a “sublevel set” {z E Wn : V(z) 5 c} be positively invariant under the flow generated by a C1 ODE x = f(z) in Wn. The necessity and sufficiency of the weak Lyupunov condition V ( z ) := VV(z) . f(z) 5 0 holding only on the level set {V = c} are given in terms of the size of the set of critical points on V lying on {V = c } . The case in which V is not differentiable is also studied by means of several extensions of the concept of V. Higher order derivatives of V with respect t o the flow are also used.

1. Introduction Liapunov’s direct method is a very powerful technique in the qualitative analysis of dynamical systems generated by autonomous ordinary differential equations x = f (z) defined in (open subsets of) Rn.It is based on the crucial observation that the time evolution of a state function V(z) along the trajectories 4(t,z o ) of the system can be studied without the explicit knowledge of the trajectories themselves, and this is done by means of the Lie or Liapunow derivative V ( z ) := V V ( z ) f (x)which only involves the vector field f . Traditionally, one imposes the Liapunow condition “V 5 0” on a subset in phase space, which forces V ( $ ( t , x o ) ) to decrease as t increases, tending toward some local minimum of V. Extra properties of V usually imply that the trajectory $ ( t, xg ) must approach the point or set where V reaches the minimum. The applicability of this method thus depends heavily on having V 5 0 on a large region. On the other hand, under the strong Liapunow condition V < 0 holding only on a level hypersurface {V = c} (necessarily a manifold, and thus a zero measure set), one can interpret geometrically this condition and 3

257

258

J. M. Vegas

observe that VV(z) f(z) < 0 means that vector f(z) “points toward” the “inner region” {V < c } and then, so it seems, trajectories starting inside may never reach the boundary {V = c } at an future time. It is easy to check analytically that this is indeed the case, but a rather surprising result is also true: if c is not a critical value of V, the weak Liapunov condition V 5 0 on {V = c } is enough to reach the same conclusion, even though the vector field f(z) is tangent to the hypersurface { V = c } at some points and it would be conceivable that the trajectory $(t,zo)might escape the region { V < c} at those points. This problem of determining “(positively) invariant regions” of phase space via state or “Liapunov” functions arises naturally in areas like p o p ulation dynamics, biology, mathematical economics and many others. The aim of this paper is to anayze in some detail this problem trying to understand under exactly what circumstances the weak condition implies the positive invariance of the inner region {V < c } . A generalization of the nondegeneracy condition mentioned above is given in terms of the “size” (Hausdorff measure) of the set of critical points lying on the level set (no longer a manifold) {V = c } . The second part relies on some basic ingredients of the general theory of invariance of closed sets, like Nagumo’s theorem (see Section 2), and uses some techniques introduced by Yorke 1191 to deal with nondifferentiable (even discountinuous) Liapunov functions: this is essential to our purposes, since we will have to work with odd-order roots of functions, thus losing the necessary regularity to apply Liapunov’s method. Finally, we use higher-order Liapunov derivatives of V in order to strengthen the results obtained in the previous section.

2. Statement of the problem

The following notation and terminology are used in this paper: 1x1 denotes the euclidean norm in R”, and u.u is the inner product in R”, also written in matrix form uTu when necessary. clA, intA and dA denote the closure, interior and boundary of a set A c R”. p(z;A) = inf{ Iz - yI : z E A} is the distance from z E R” to A c Rn.It will also be written p ~ ( z ) . The words “increasing” and “decreasing”will be used in the wide sense; otherwise, we will say “strictly increasing” or “strictly decreasing”. For a given function V : 2) := domV -+ R, and a real number c, the “level sets” {z E D : V(z) < c } , {z E D : V(z) 5 c } , {z E D : V(z) = c}, which play a fundamental role in the sequel, will be simply denoted by {V < c } , {V 5 c } , {V = c } . The set {V = c } will also be denoted by Vc.

Some Remarks on the Invariance of Level Sets in Dynamical Systems

259

Observe that, if V is continuous and V C W n is open, {V < c } is open both in R” and in V ,and {V 5 c } is closed in V ,but may not be closed in R”. In the sequel, V c Rn denotes a fixed open set, f : V + Rn is a fixed C1function (or vector field), whose associated differential equation

x =f ( x ) induces a local flow 4 ( t , x o ) on V ,where $(t,zo) stands for the unique solution of the initial value problem { k = f (x), z(0) = ZO), defined on an open interval I(x0) := (a(zo),b(x0)) containing 0. The positive semiorbit of xo E V is y+(xo) := { 4 ( t , x ) : 0 5 t < b ( x o ) } , the negative semiorbit is y-(xo) := { $ ( t , x ) : ~ ( 2 0 )< t 5 0) while the (full) orbit is y(z0) = { + ( t , x ) : t E I(x0)). If B c V satisfies “xo E B =+ y+(xo) c B” (resp. y-(xo) C B ) then B is said to be positively invariant (resp. negatively invariant). B is invariant if it is both positively and negatively invariant, that is, if ‘(20E B =+ y(x0) c B”. B is strongly positively invariant if q5(t,xo) E int(B) for zoE B , t > 0. The following important results relating positively invariance with boundary behavior will be used (see, e.g., Amann [l]for proofs): Proposition 2.1.

c V is (strongly) positively invariant if and only i f f o r every x E d M there exists p > 0 such that 4(t,x) E M ( i n t ( M ) ) for 0 < t 5 p. (2) If a set M c V is positively invariant, so are c l ( M ) and i n t ( M ) . (3) If G C V is open, then G is positively invariant if and only if cl(G) is positively invariant. (1) A closed set M

An important result, due to Nagumo [15](and rediscovered fourteen times since, according to Aubin [3]) is the following: Theorem 2.1. (NAGUMO) Let Q c R” be closed and Q c V. Then Q is positively invariant i f and only i f for eve y x E dQ the vector f (x) is subtangential to Q at x, that is, i f lim inf PQ(x -k h f h+O+ h

=0

where P Q ( Z ) = p(x;Q) := inf{lz - qI : q E Q } is the distance from x to Q.

J . M. Vegas

260

Remark 2.1. This condition is equivalent to the following statement: “There exist sequences hk 4 0 , Zk + 0 such that z hk[f(z) Zk] E Q”. If Q is positively invariant, the solution itself q5(t,zo) can be written as tf(z0) o(t),which implies the necessity of the condition. The sufficiency is much more difficult, and Nagumo proved it even without uniqueness assumptions, that is, for f merely continuous; in that case, one can only show that at least one solution of x = f(z)with z(0) = zo lies in Q for t 2 0 in its domain of definition (such sets are sometimes called “weakly positively invariant” or “positively semi-invariant”). Similar results (with important variants) were given by Wazewski [17],Hartman [ll]and Martin [13]. Br6zis [4]gave a different proof for locally Lipschitzian f , and obtained the important fact that liminf could be substituted by lim . Nagumo’s theorem has been extended even to multivalued differential equations (Clarke [ 6 ] ,Haddad [lo]) and other fields. Very complete surveys of this topic can be found in Motreanu and Pave1 [14] and C5rj5 and Vrabie [5].

+

+

+

Remark 2.2. The tangent cone T,(Q) (rather, one of the many tangent cones, see, e.g., Clarke et al. [7]) at z E 6’Q is defined as the set of subtangential vectors to Q at z, that is,

or, equivalently, u E T,(Q) if and only if there exist sequences hk + 0, zk + 0 such that z hk(u zk) E Q. The Nagumo condition may thus be stated as: “f(z)E T,(Q) for all z E aQ”.

+

+

If V : V 4 R is a C1function, the (Lie) derivative of V along f (or 4) is the function d V ( z ) := -V(m(t,z))l = DV(z)f(z= ) V V ( z ). f (z) (4) dt t=O for each x E D. DV is the gradient VV written as a row vector, that is, DV(X) = VV(z)T. V is said to be a Liapunov function for f (or 4) on a subset A c V if V ( z ) 5 0 for all z E A . The essential point of Liapunov’s direct method is the well-known fact that V can be found without the explicit knowledge of the solutions of the differential equation, and that if 4(t,30)E A for all t on some interval [a,/3],then the composite function t H V($~(t,zo)) is and, in particular, V(4(p,zo))I decreasing (in the wide sense) on [a,@] V($(cr,zo)). In general, if I C R is an interval (closed or not), V($(t,zo)) is decreasing on I .

Some Remarks on the Invariance of Level Sets in Dynamical Systems

261

Remark 2.3. Although the definition of V involves an inner product, it is intrinsic since it actually involves the differential DV rather than the gradient VV, and is invariant under smooth changes of variables. An immediate and well-known consequence of this property is the following: If V 5 0 on V , the (perhaps empty) level sets {V < c } and {V 5 c } are positively invariant. If V 5 0 on a smaller subset A c V ,then one can show the following:

Proposition 2.2. If V : 23 + R satisfies V 5 0 o n A c V ,and c E R satisfies cl{c - E < V < c } c A for some E > 0, then the level sets {V < c} and cl{V < c } are positively invariant. Proof. (Sketch) Let V(x0) < 0 and assume that V($(tl,xo)) 2 c for some tl > 0. Then there exist t* E (O,tl], E > 0 and 6 > 0 such that V($(t*,xo))= c and c - E < V($(t,xo)) < c (and therefore $(t,xo) E A ) for 0 5 t* - 6 < t < t*, which is impossible since V($(t,xo))decreases on [t*- 6, t*]. Proposition 2.1 implies that cl{V < c } is also positively invariant. 0 Remark 2.4. If {V 5 0 ) # cl{V < 0}, it may not be positively invariant, since V 3 0 on { V = O}\a{ V < 0 ) for any vector field f . In the sequel, we will take c = 0 for simplicity, and denote V o = {x E 2)

: V(x) = 0). We say that the closure hypothesis holds if

(CHI

Vo is nonempty and closed in R"

(5)

Unless stated otherwise, this hypothesis will always be assumed to hold. A simple hypothesis implying the hypotheses of Proposition 2.2 is to have the strong Liapunov condition on {V = 0}, given in the following

Proposition 2.3. Assume that the strong Liapunov condition

(SJW

V(x)
0 imply that V(+(t,zo)) goes from negative to positive at t = 0.

Is it possible to “fill in” the gap between V I 0 and V < 0 (on Vo)? The fact that it can be done in a “generic” sense is somewhat surprising: Theorem 2.2. If V contains no critical points of Vo (that is, i f 0 is a Tegular value of V) then

( I ) V I o on VO (2) V = o on

VO

* {V 5 0) is positively invariant; ++

{V = 0) is invariant.

Proof. Let us give just a sketch (see, e.g., Amann [l],16.9): Assume V is C 2 .The derivative VEof V along the perturbed flow j: = f(z)- &2VV(Z)

(8)

satisfies VE(z)= VO(z)- E~ (VV(z)I2 < 0 on Vo, since, by assumption, Vo(z) 5 0 and lVV(z)l > 0 on Vo (Vo represents V , for E = 0). By the previous results, {V 5 0) is (strongly) positively invariant for the perturbed flow, hence, by taking the limit as E -+ 0, {V 5 0) is positively invariant for the case E = 0. In order to prove (ii), it is enough to apply (i) to both V and -V to show positive invariance of both {V 5 0) and {V 2 0) and hence of their intersection. To prove invariance, change f to -f to obtain negative invariance of this set. On the other hand, if V # 0 at some 20 E Vo then +(t,zo)enters {V < 0) or {V > 0) for t > 0 small, against the invariance of vo. a

Remark 2.5. This proof requires V to be C2. A more sophisticated approximation process (substituting VV by a Lipschitz “approximate gradient”) gives also the result if V is just C1. This theorem allows for the analysis of general oscillators of Lienard type

2 + g(z, j:)k + z = 0

(9)

Some R e m a r h on the Invariance of Level Sets an Dynamical S y s t e m

or k = y , y = -z - g ( z , y ) y . The total energy function E = 4(zz satisfies

263

+ y2)

E = -dz, Y)Y2

(10)

if the damping coefficient g ( z , y ) is positive or zero on the circle CR = {z2 y2 = R 2 } ,hence E 5 0 on the circle, yet there are always points with E = 0. However, the preceding extension enables us to conclude that the circle together with its interior is positively invariant, since E contains no critical points on CR.

+

As pointed out above, it is clear that some condition must be assumed on the criticality of V" : For any W E C1,V(z) := W ( Z ) ~has the same 0-level sets as W, yet O is always a critical value of V (in fact, all points in V" are critical). In order to "desingularize" V one would have to take odd roots W = V1I3 thus destroying the differentiability properties which enable us to compute V directly from VV and the vector field f. That is a real problem that must be dealt with on a very general setting (some ideas are presented in Section 3). 2.1. The main theorem

If 0 is a critical value of V, but the critical set is not too large, the same conclusion can be reached: Theorem 2.3. Assume that V" c cl{V < 0) and V 5 0 o n Vo. Let K = {V = 0, VV = 0) be the '%riticalset". If K has zero n - 1 dimensional Hausdorff measure 7 P - l (that is, if the Hausdorff dimension of K is strictly less that n - 1 ) then {V I 0 ) is positively invariant. If V" C cl{V < 0) n cl{V > 0), V = 0 o n V" and 'H"-l(K) = 0 , then Vo is invariant.

Proof. Assume first that V < 0 on V"\K, and suppose that {V I 0) is not positively invariant. Let z o satisfy V(z0) = 0 and V(q5(t,zo)) > 0 for t E (0,6), for some 6 > 0. Obviously, V ( z 0 ) 2 0, hence zo E K . Since zo is not an equilibrium point, Lefschetz's Lemma (see Arnold [2], 7.1) guarantees the existence of a change of variables (as smooth as f) that transforms f into the constant vector field f (z) = ( 1 , 0 , . . . ,O)T on a rectangle

%, rTZ := ( ( ~ 1 ~ 2. .2. ,,z),

E R" : 1x1I I TI,

I(m,. . .,$,)I

I r2)

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J. M. Vegas

Then V = D,,V and $(t,z)= ( t + z l , 2 2 , . . . ,2,). Denote y = ( 2 2 , . . . ,xn), let IT : R" + R"-l be the projection II(z1,y) = y, and let 6, E > 0 satisfy V(6,y) 2 ,u > 0 for IyI I E . Let Rs,€ denote the open rectangle {I211 < 6, IYI < E l . The assumption {V = 0) C cl{V < 0) implies that R s ,n~{V < 0) is nonempty and open, hence its projection I T ( R h , , n {V < 0)) (which will be denoted by P ) is a nonempty open set contained in &(O) c IW"-l. Every jj E P satisfies V(6,y) 2 ,u and V(zy,y) < 0 for some z; E (-6,6). Therefore, there is some 51 E (-6, 6) such that V(21,jj) = 0. In other words, II-l(P) n Vo is a nonempty subset of Vo. The theorem would be proven if we could show the existence of at least one jj E P for which every associated 31 (that is, V(21,jj) = 0) is not a critical point of V. Indeed, in that case the function ~ ( 2 1 := ) V(q,y) would go from negative to positive, yet every 51 with u(&) = 0 would satisfy ~ ' ( 5 1 )< 0, which is obviously impossible. In other words, we are looking for some jj E P such that II-'(jj) n K = 0 , or, equivalently, jj E P\II(K). And here is where the size of K comes into play, since Hausdorff measures do not increase by orthogonal projections (see, for instance, Federer [9], 2.10.11):

7P-l ( I I ( K ) )I 3-In-l(K) and this is zero by assumption. Therefore, II(K) cannot cover the open set P c BE(0),since X"-'(P) > 0. Thus there exists jj E P\II(K) mentioned above, and this concludes the proof under the strong Liapunov condition "V < o on vO\K". In the general case, one applies the perturbation argument given in the proof of Theorem to show that {V I 0) is positively invariant under the perturbed flow

d = f(2)- &2VV(Z)

(11)

since K is the same for every E > 0. The final result is obtained by taking the limit as E --+ 0. In order to prove the final part, we only have to apply the previous I7 result to V and -V, and then change f to -f. 2.2. A n application (first p a r t)

Let us apply this result to a particularly important case. Assume V = VlV2, where 0 is a regular value for both V1 and V2. The critical set of V in Vo is contained in the set S = {Vl = 0) n {V, = 0).

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If VV1 and VV2 are linearly independent, then S is a ( n - 2)-dimensional manifold, hence 'FI"-'(S) = 0 and the theorem above holds. Taking into + VlV2, this means that in order to have V 5 0 on account that V = Vo we need

{

V ~ ; O ifv2=0, ~

$

0

V ~ $ Oi f & = O , ~

$

0

(12)

which implies, in particular,

Vl = o = ~2

on

s = {v1= o = ~

2 )

(13)

If conditions (12) hold, then

{V 5 0 ) = {Vl 5 0 ,

v 2

1 0 ) u {vi L 0,

v 2

50)

(14)

is positively invariant, and so is its interior {V < 0). Now,

{V < 0) = {Vl < 0,

v 2

> 0) u {vi > 0, v, < 0)

(15)

is the union of two disjoint open sets, each of which must be positively invariant by connectedness, and so must their closures {Vl I 0, V2 L 0) and {Vl 2 0, VZ 5 0). Furthermore, we can show that S = {Vl = 0) n {VZ = 0) is invariant by the following argument: S is an ( n - 2)-dimensional manifold whose tangent space at a point 20 E S is the intersection of the tangent spaces Tzo(Vp)and T,,(V;), and can be described as

T,,(S) = { u E Wn : VVl(z0). u = 0, VV2(XO) nu = 0)

(16)

But, by (13), f(zo)satisfies both conditions: V v i ( X 0 ) . f ( X 0 ) = i.;(zo)

= 0,

VV2(XO) . f(X) = V2(XO) = 0

Therefore, f ( ~ E) T,,(S) and the Nagumo condition (Theorem 2.1) is satisfied. We may thus state the following result. Theorem 2.4. Let V1, fi : D + W be C1 functions which have 0 as a regular value. Assume that for every z E S := {I4 = 0, V2 = O ) , VK(2) and Vfi(x) are linearly independent. If conditions (12) are satisfied, then the ('quadrants" {Vl 5 0 , V2 >_ 0) and {Vl >_ 0 , V2 5 0) are positively invariant, and their intersection S is invariant.

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266

Remark 2.6. Another approach to dealing with “sectors” {Vl 5 0, V25 0 ) is to use the function W ( x )= max{K(x),Vz(z)). W is not C 1 ,but k can be defined in a generalized sense. We will come back to this later. Remark 2.7. Observe that some condition on the gradients is necessary, since K = Vz gives us V = V f , the critical case we have found already, and S = {K = 0) is an ( n - 1)-dimensional manifold. If Vj and V2 are functionally independent but the gradients are not linearly independent at some xo E S, then S is no longer a manifold, but its tangent cone (see, e.g., Aubin and Frankowska [3]) at any x E S is still defined, and is contained in the intersection of the tangent cones (spaces in this case) T,(Vp) and Tz(V:). We know that f(x) is contained in this intersection, but some condition must be added in order to make sure that, in fact, f(x) E Tz(S) (which might be reduced to (0)). In general, if V is (real) analytic, it can be factored as V = Km1V2m2 . . . KmrW, where V , = 0 and W # 0 on V o . If some mi is greater than 1, then V , = 0 is contained on the singular set K , and then IHn-l(K) > 0. On the other hand, if every mi = 1, the use of techniques from real-analytic geometry enable us to considerably relax the hypotheses on the rank of the Jacobian matrix of {Vl,. . . ,Vr}.This is a complex question that will not be discussed here (see, e.g., Federer [9], 3.4.10). 3. Nondifferentiable Liapunov functions

In order to make progress and deal with critical cases like V = 1xI2 mentioned above it is necessary to consider nondifferentiable (even discontinuous) Liapunov functions. In such cases, V must be understood in a generalized sense, and several choices are possible, all of them useful. For the purposes of this paper we will introduce special names which makes them easier to remember:

Definition 3.1. Let V : D

0

Y,V(x) :=

liminf

t-+O+,

t

R be an arbitrary function and x E V.Then:

+

+

V(x tf(x)

12(-+0

tZ) -V(x)

t

will be referred to as the Lie (or Liapunov), the Nagumo and the Yorke derivatives of V at 3: (along f), respectively.

Some Remarks on the Invariance of Level Sets an Dynamical S y s t e m

267

It is obvious that Y f V 5 NjV, and Y f V 5 CfV can be easily checked by observing that 4(t,x) = tf(z) + O(t2)= tf(z) + to(t).All inequalities can be strict (see Yorke [18] for examples), but there is an important case in which all three derivatives coincide [21]: Proposition 3.1. (YOSHIZAWA) If V is locally Lipschitz, then C f V = N f V = YfV. Proof. Take, for simplicity, x = 0, V ( 0 )= 0 and denote fo = f(O), K = Lipschitz constant around 0. Then

V ( t f 0 )- V(tf0+ t z ) I IV(tf0)- V(tf0

+ tz)l I K It1 IzI

which implies

v(t.fo) t

5

+ +-K It1 IzI t

V(tfo t z ) t

and then

+

NfV(0)5 Y j V ( 0 ) 0 This, together with the reverse inequality Y f V I N f V ,which always true, gives Y f V = N f V. By substituting z = +(t,0 )- t f o in the expressions above one proves Y f V I C f V , hence Y f V = CfV. 0 The differences between the three derivatives thus appear when V is not Lipschitz. Many pathologies arise in that case, since Jlffand Y f cannot be directly related to derivatives of V along solutions of (1). However, Yorke has proven the following essential fact: Theorem 3.1. (YORKE) Let W : D

+

R be continuous and V

:

D

+

B lower semicontinuous. Then the following are equivalent: (1) Y f V ( x )5 W(z) for all x E D.

(2) ~ ( 4 (x)) t , -v

( ~ o ,I st”, w(+(s, x>)ds for all t E [to,b ( x ) ) .

Remark 3.1. In particular, if Y f V 5 0 on a set A c D , then V decreases along orbital arcs lying entirely in A and conversely. Proposition 3.2. If {V 5 0) is positively invariant, then Y f V 5 C f V I 0 on V o .

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Proof. If zo E V oand {V 5 0) is positively invariant, then V(q5(t,zo)) 5 0 for t 2. 0, which obviously implies LjV(z0) 5 0. The property YjV(z0)5 LjV(z0) is always valid. 0 Yorke’sresult extends Liapunov’s method to lower semicontinuous functions, via the Yorke derivative. By the previous proposition, the same extension is obviously true for the Lie derivative, but this one is almost never computable in the non-Lipschitz case without an explicit knowledge of the solutions of (1). For Nagumo’s derivative, the converse result is false, as shown by the following example, based on a similar one given in Yorke [19]. Consider f ( 2 1 , z z ) = (zz, -q), so that

Let V ( z )= ([.I2 -1 sign(lz1’ - 1). Then YjV(1,O)5 LjV(1,O) = 0 but Nj V ( 1, 0) = 1, as can be easily shown. Yet, V = 0 (hence decreasing) along the orbit y(1,O). This very example shows that this proposition cannot be extended to Nagumo derivatives, since both { V = 0 ) and { V 5 0 ) are invariant sets. On the other hand, these concepts allow for a substantial generalization of the nondegeneracy condition “0 is not a critical value of V”. It is Yorke’s “positive gradient condition” relating the growth of V toward the region {V > 0) and the distance function p(z) = p(z; {V 5 0)).

Definition 3.2. V is said to have a positive gradient at zo E V oif there exist r > 0 and k > 0 such that V ( z )2 kp(z) for z E B,(zo) n {V > 0). It has positive gradient in V oif this condition holds for every Vo. Proposition 3.3. Assume that V has positive gradient on V o . Then (1) {V 1.0) is positively invariant if and only if YjV 5 0 on V o .

(2) If V is C1, the weak Liapunov condition ‘i/ 5 0 on V0” is necess a r y and s u f i c i e n t for {V 1. 0 ) to be positively invariant.

(3) If V is C1and both V and -V have positive gradient on V o ,then the condition “V = 0 on Vo” is necessary a n d sufficient for V o to be invariant. Proof. (1) If the Nagumo condition (Theorem 2.1) fails at some zo E a{V 5 0 ) C V o ,and since the distance function p is Lipschitz, the Nagumo

Some Remarks on the Invariance of Level Sets in Dynamacal Systems

269

and the Yorke derivatives coincide:

+

and there exists 6 > 0 such that [ p ( z ~ hf(z0)+ h z ) / h 2 1 - E > 0 for h E (0, S), IzI < 6. In particular, $0 h f ( z 0 ) hz belongs to {V > 0) and also to B,-(ZO) (perhaps for a smaller 6’). Therefore, V(z0 h f ( z 0 ) hz) 2 kp(z0 h f ( z 0 ) h z ) for h E [0,6’). But then, for every h E (0, 6’)

+

+

+

+

+

+

which implies Y f ( V ) ( z o 2 ) k(l - E ) > 0, contradicting the hypothesis. (2) If V is C’, then V = C f V = Y f V and part (1) can be applied. (3) As before, apply (2) to V and -V, and then change f to -f.

Remark 3.2. (1) is a variant of Yorke [18],proved for “limsup” instead of “lim inf ” and for the Nagumo-type derivative. The “positive gradient condition” is assumed in Yorke [18]to hold for a uniform k throughout V o . Remark 3.3. Even under the positive gradient condition, N f V 5 0 need not hold on V o if V is not Lipschitz, as the example given above shows. It does, though, if we have an upper bound similar to the lower bound, that is, if V(z) 5 K p ( z ) on BT(z)n {V > 0). Indeed, in that case, V ( z ) 5 min{O,Kp(z)} on B T ( z ) ,which implies N f V ( s ) = min{NfO,Nf(Kp)(z)} = 0 by the Nagumo theorem. Remark 3.4. Interestingly, a condition of Lojasiewicz type “V(z) 2 k p ( ~ for ) ~some power m > 1” (related to analiticity properties of V )is not enough, as V ( z )= 1zI2 again shows. In that case, one would need to have Yf[V’lm]5 0 on V o ,but V’/” is not C’ in general and the Yorke derivative will be hard to compute. We will come back to this later. 3.1. A n application (second p a r t )

Going back to the situation analyzed previously, assume that V1, V2 : D +

lR are C’, that 0 is a regular value for both and that VK(z0) and VV2(zo) are linearly independent at each 20 E {K = 0 , V2 = 0). Let us define

J. M. Vegas

270

This is not a C’ function, but it is locally Lipschitz. One can show (see, e.g, LaSalle [12], 8.3) that, with K,V2 of class C1,

L ~ W ( Z=)

{

if ~ ( z>) ~ ( z )

%(z)

V2(z) if ~ ( z

Let us denote the three pieces of the boundary a{W I 0) as follows: = {vl = o,v2 < 0),22 = = 0,K < O),Z12 = {K = 0,vz = 0). Then,

{vz

21

IVSO

on{W=o)-

{

% S O onz1={K=O,Vz 0) u { h > O } . If &(x) > 0 but V2(z) 5 0, then W ( z )= K(z) 2 klp(z,{K = 0 ) ) = klp(z,{W = 0 ) ) if z is close enough to {V, = 0 ). Similarly for VI(Z) I 0 , Vz(z) > 0. Finally, in a neighborhood of { W = 0 ) one has P(5,

w I 0 ) L k3P(Z)

and W has positive gradient. 3.2. Desingularization 2

We will see now that critical functions like V = 1x1 which contain powers can be “desingularized”. Before getting there we need to adapt some wellknown some concepts of stability theory.

Hahn’s classes Hahn’s function classes are very important in the application of Liapunov’s direct method to the stability of equilibrium solutions. We now present the notations for Hahn’s classes and introduce a slight generalization: their odd extensions.

Definition 3.3. Let 0

Ic:

T

> 0 (including m).

= {a : [O,r) --+ a(0)= 0 ) Ic, = {a : ( - T , T ) increasing}

R : a is continuous, strictly increasing and 4

R : a is continuous, odd and strictly

Usually there is no need to specify T , and then we use simply Ic+ or

K.

Remark 3.5. These functions, sometimes also called wedges, are very important as “comparison gauges” in stability theory. Of course, the most important are the positive powers a ( s ) = s P , p > 0 . F’unctions in Ic, have well-defined continuous inverses which belong to the class IcT’ for T’ = lims+, a ( s ) .

J. M. Vegas

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The reason for the usefulness of these function classes is the following important technical result, which states that a “positive definite” p : [O,r) + R is bounded below by a function of class K+ (hence the denomination “wedge”): Lemma 3.1.

p : [O,r) + R be continuous and satisfy p(0) = 0 , p(s) > 0 (resp. p(s) < 0) for 0 < s < r. Then there exists a E K, such that p(s) 2 a ( s ) (resp. p(s) 5 -a(.)) for all s E [0,r). (2) L e t p : ( - r , r ) -+ R be continuous and satisfyP(0) = 0, p(s) < 0 for -r < s < 0 and p(s) > 0 for 0 < s < T . Then there exists a E Kr such that p(-s) I a(-s) and p ( s ) 2 a ( s ) for 0 5 s < T . (1) Let

Part (a) is well known (see Rouchk and Mawhin [IS]). Part (b) is an obvious application of (a) to p ( s ) := -p(-s). In stability theory, a “positive definite” V ( t ,x) satisfies V ( t ,x) 2 a ( l ~ 1 ) for some a E K+, and V ( t , x )has an “infinitesimal upper bound” if the opposite inequality 0 L V ( t , x )I P(lx1) holds for some p E K+. In this paper we will use similar results for the function V whose zero level sets we are studying; in particular, we now proceed to prove that positive invariance of { V 5 0) implies the existence of a “desingularizingrescaling” of V if the set V o does not contain “recurrent” orbits in the following sense: Proposition 3.4. Let V : 2) + R be C’,let S = (50E V o : there exists \ 0 such that 4 ( t k , xo) E vo). If {v5 0 ) is positively invariant, then:

tk

K , Lf[ao V ]5 0 (and therefore Y f [ ao V ]5 0) on VO. (2) For every xo E Vo\S there exist a ,/3 E K such that Y f [ a o V ] ( x o I) L f [ ao V ] ( Z OI) -1 and Y[Po V ] ( Z O=)Lf[po V ] ( Z O=)-m. (1) For every a E

Conversely, if S is positively invariant and for every xo E Vo\S there exists a E K such that L f [ ao V ] ( Z O.Then {V 5 0 ) is positively invariant if and only if f o r every

Some Remarks on the Invariance of Level Sets in Dynamical Systems

275

E Vo\M the first nonvanishing derivative is odd-order and negative. Given this condition, {V 5 0) is strongly positively invariant if and only i f M is empty. I n this case, if V o is compact, there exists an odd integer k such that Lf [V1Ik]= -m on Vo.

xo

Remark 4.1. If both V and f are analytic, the condition of existence of a nonvanishing derivative on Vo\M is automatically satisfied.

Remark 4.2. The perturbation argument presented in the proof of Theorem 2.2 implies that the above conditions need only be checked at critical points of V. References 1. 2. 3. 4. 5.

6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17.

H. Amann, Ordinary Differential Equations. Walter de Gruyter, 1990. V. I. Arnold, Ordinary Differential Equations. MIT Press, 1980. J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, 1990. H. BrBzis, On a characterization of flow-invariant sets, Comm. Pure Appl. Math., 23 (1970), 261-263. 0. Ciirj5, I. I. Vrabie, Differential equations on closed sets, in Handbook of Differential Equations 11, A. Caiiada, P. Drabek, F. Fonda Editors, Elsevier for Science, in print. F. H. Clarke, Generalized gradients and applications, Trans. Amer. Math. SOC.,205 (1975), 247-262. F. H. Clarke, Yu. S. Ledyaev, R. J . Stern, P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, 1998. M. G. Crandall, A generalization of Peano’s existence theorem and flowinvariance, Proc. Amer. Math. SOC.,36 (1972), 151-155. H. Federer, Geometric Measure Theory, Springer, 1969. G. Haddad, Monotone trajectories of differental inclusions and functional differential equations with memory, Israel. J. Math., 39 (1981), 83-100. P. Hartman, On invariant sets and a theorem of Waiewski, Proc. Amer. Math. SOC., 32 (1972), 511-520. J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. R. H. Martin Jr., Differential equations on closed subsets of a Banach space, Trans. Amer. Math. SOC.179 (1973), 399-414. D. Motreanu, N. H. Pavel, Tangency, Flow Invariance for Differential Equations and Optimization Problems, Marcel Dekker Inc., 1999. M. Nagumo, Uber die Lage der Integralkurven gewohnlicher Differentialgleichungen, Proc. Phys. Math. SOC. Japan, 24 (1942), 551-559. N. RouchB, J. Mawhin, Equations diffgrentielles ordinaires, vol 2., Masson, 1973. T. Waiewski, Sur un principe topologique de l’examen de l’allure asymptotique des integrales des Bquations diffhrentielles ordinaires, Ann. SOC.Polon. Math. 20 (1947), 279-313.

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18. J. A. Yorke, Invariance for ordinary differential equations, Math. Systems Theory, 1 (1967), 353-372. Correction, Math. Systems Theory, 2 (1968), 381. 19. J. A. Yorke, Extending Liapunov’s second method to non-Lipschitz Liapunov functions. Bull. Amer. Math. SOC.74 (1968), 322-325. 20. J. A. Yorke, A theorem on Liapunov functions using V . Math. Systems Theory 4 (1970) 40-45. 21. T. Yoshizawa, Stability theory by Liapunov’s second method, Math. SOC. Japan, 1966.

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