Singular Elliptic Problems: Bifurcation & Asymptotic Analysis (Math Applications Series)

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OXIORD [tCTURS AND ITS ArrIICATIONS

IN tlAi111WAYIC.1 17

Singular Elliptic Problems Bifurcation and Asymptotic Analysis

Marius Ghergu Viceniiu RAdulescu

OXFORD LECTURE SERIES IN MATHEMATICS AND ITS APPLICATIONS For a full list of titles, please visit http://www.oup,co.uk/academic/science/maths/ series/OLSMA/ 10. P.L. Lions: Mathematical topics in fluid mechanics, Vol. 2: Compressible models 11. W.T. Tutte: Graph theory as I have known it 12. Andrea Braides and Anneliese Defranceschi: Homogenization of multiple integrals 13. Thierry Cazenave and Alain Haraux: An introduction to semilinear evolution equations 14. J.Y. Chemin: Perfect incompressible fluids 15. Giuseppe Buttazzo, Mariano Giaquinta and Stefan Hildebrandt: One-dimensional variational problems: an introduction 16. Alexander I. Bobenko and Ruedi Seiler: Discrete integrable geometry and physics 17. Doina Cioranescu and Patrizia Donato: An introduction to homogenization 18. E.J. Janse van Rensburg: The statistical mechanics of interacting walks, polygons, animals and vesicles 19. S. Kuksin: Hamiltonian partial differential equations 20. Alberto Bressan: Hyperbolic systems of conservation laws: the one-dimensional Cauchy problem 21. B. Perthame: Kinetic formulation of conservation laws 22. A. Braides: Gamma-convergence for beginners 23. Robert Leese and Stephen Hurley: Methods and Algorithms for Radio Channel Assignment 24. Charles Semple and Mike Steel: Phylogenetics 25. Luigi Ambrosio and Paolo Tilli: Topics on Analysis in Metric Spaces 26. Eduard Feireisl: Dynamics of Viscous Compressible Fluids 27. Antonin Novotny and Ivan Straskraba: Introduction to the Mathematical Theory of Compressible Flow 28. Pavol Hell and Jarik Nesetril: Graphs and Homomorphisms 29. Pavel Etingof and Frederic Latour: The dynamical Yang-Baxter equation, representation theory, and quantum integrable .systems 30. Jorge Ramirez Alfonsin: The-Diophantine Frobenius Problem 31. Rolf Niedermeier: Invitation to Fixed Parameter Algorithms 32. Jean-Yves Chemin, Benoit Desjardins, Isabelle Gallagher and Emmanuel Grenier: Mathematical Geophysics: An introduction to rotating fluids and the Navier-Stokes equations 33. Juan Luis Vizquez: Smoothing and Decay Estimates for Nonlinear Diffusion Equations 34. Geoffrey Grimmett and Colin McDiarmid: Combinatorics, Complexity and Chance 35. Alessio Corti: Flips. for 3-colds and 4-folds 36. Kirsch and Grinberg: The Factorization Method for Inverse Problems 37. Ghergu and Radulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis

Singular Elliptic Problems: Bifurcation and Asymptotic Analysis Marius Ghergu Vicentiu D. RAdulescu

CLARENDON PRESS OXFORD 2008

OXFORD UNIVERSITY PRESS Oxford University Press, Inc., publishes works that further Oxford University's objective of excellence in research, scholarship, and education.

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All rights reserved. No part of this publication may be reproduced. stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Ghergu. Marius Singular elliptic problems. bifurcation and asymptotic analysis I Marius Ghergu, Viccntiu D. RSdulescu. p. cm. - (Oxford lecture series in mathematics and its applications; 37) Includes bibliographical references and index. ISBN 978-0-19-533472-2

1. Differential equations, Elliptic-Asymptotic theory. 2. Differential equations, Nonlinear. 3. Bifurcation theory. 1. Radulescu, V. II. Title. QA377.G47 2008 515`.3533-dc22

2007060129

135798642 Printed in the United States of America on acid-free paper

To our families, for their patience and continuous support over the years

PREFACE The most incomprehensible thing

about the world is that it is comprehensible. Albert Einstein (1879-1955)

The development of nonlinear analysis during the last few decades has been profoundly influenced by attempts to understand various phenomena from mathematical physics. One of the beauties of the subject is the immense breadth of mathematics that has been applied in this pursuit. There is an enormous body of literature in nonlinear elliptic partial differential equations that stretches back half a century. However, we shall make almost no reference to this literature, and shall rely almost entirely upon personal re-

sults. These lecture notes are primarily intended to fill, in a substantial way, the absence of a book dealing with the qualitative analysis of some basic singular stationary processes arising in nonlinear sciences. This volume aims to offer an introduction to this subject, and also to present some research problems. The models that we analyze represent a compromise between the description of physical phenomena and analytical requirements; accordingly, our presentation is characterized by a strict interplay between mathematics and nonlinear sciences. The book is an outgrowth of our original research on the subject during the last few years, and much of the development is motivated by problems arising in applications. However, most of the proofs have been completely reworked and we are especially careful to explain where each chapter is going, why it matters, and what background material is required. Although the theory that we describe could have been carried out on differentiable manifolds even from the beginning,

we have chosen to develop it on domains on the Euclidean space. However, the techniques we develop can be extended to Laplace-Beltrami operators on Riemannian manifolds.

The major thrust of this book is the qualitative analysis of some classes of nonlinear stationary problems involving different types of singularities. Be aware,

this is definitely a research book. We are mainly concerned with the following types of problems. We first study singular solutions of the logistic equation, with a basic model that is described by the semilinear elliptic equation Du = uP, where p > 1. The research program around this equation flourished after the pioneering papers by Bieberbach and Rademacher, continued with the deep contributions of Loewner and Nirenberg in Riemannian geometry, and creating recently (because of the works by Dynkin and Le Gall) a nonlinear analogue of the classical relation between Brownian motion and potential theory. Equations of this type arise in astrophysics, genetics, meteorology, theory of atomic spec-

viii

Preface

tra, and the Yamabe problem in geometry. A first consequence of such types of nonlinearities is the possibility of the existence of a "large solution"-that is, a solution blowing up at the boundary. When the large solution is unique, it is a maximal solution and dominates any solution. In connection with the previously mentioned applications, the existence of the large solution in a ball was used by Iscoe to establish the compact support property of super-Brownian motion, demonstrating the importance of the relationship between properties of superdiffusion and the equation. Next, we are concerned with Lane-EmdenFowler equations and Gierer-Meinhardt systems with singular nonlinearity. The model problem in such cases is described by equations like --Au = u-°, where a is a positive real number. To our best knowledge, the first study in this direction is from Fulks and Maybee, who proved existence and uniqueness results by using a fixed point argument; moreover, they showed that solutions of the associated parabolic problem tend to the unique solution of the corresponding elliptic equation. Different approaches are the result of Coclite and Palmieri, respectively Crandall, Rabinowitz, and Tartar, who approximated the singular equation with regular problems, where the standard monotonicity techniques do work. Singular problems of this type arise in the context of chemical heterogeneous catalysts and chemical catalyst kinetics, in the theory of heat conduction in electrically conducting materials, singular minimal surfaces, as well as in the study of non-Newtonian fluids, boundary layer phenomena for viscous fluids, glacial advance, transport of coal slurries down conveyor belts, and in several other geophysical and industrial contents. In both cases, because of the meaning of the unknowns (concentrations, populations, etc.), the positive solutions are relevant in most situations. We intend to give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear elliptic equations, as well as their applications to various processes arising in mathematical physics. Our approach leads not only to the basic results of existence, uniqueness, and multiplicity of solutions, but also to several qualitative properties, including bifurcation, asymptotic analysis, blow-up and so forth. Moreover, because the book is concerned primarily with classical solutions, the monotone iteration processes we apply for various classes of nonlinear singular problems are adaptable to numerical solutions of the corresponding discrete processes. To place the text in better perspective, each chapter is concluded with a section on historical notes that includes references to all important and relatively new results. In addition

to cited works, the list of references contains many other works related to the material developed in this volume. The organization of the book is briefly summarized as follows. The first chapter deals with preliminary material, such as the method of sub- and supersolution, several variants of the maximum principle (Stampacchia, Vazquez, Pucci, and Serrin), and various existence and uniqueness results for nonlinear elliptic boundary value problems.

Preface

ix

Part II is composed of two chapters, which are concerned with singular solutions of logistic-type equations or systems. There are studied both equations with blow-up boundary solutions and entire solutions blowing up at infinity for elliptic systems. In all these cases, the major role played by the Keller-Osserman condition is discussed. In the third part of this book we are concerned with elliptic problems involving singular nonlinearities, either in isotropic or in anisotropic media. Chapter 4 deals with sublinear elliptic problems that are affected by singular perturbations. We distinguish between equations on bounded domains or on the whole space and we are also concerned with a related bifurcation problem. Chapters 5 and 6 are devoted to the study of a bifurcation problem in the case of linear growth for the nonlinearity. Two different situations are distinguished and a complete discussion is developed in both circumstances. The superlinear case is studied in Chapter 7, by means of variational arguments, whereas Chapter 8 is concerned with stability properties of solutions. Chapter 9 is devoted to the study of the "competition" between various terms in a singular Lane-EmdenFowler equation with convection and variable (possible, singular) potential. In the last chapter of these lecture notes, the qualitative analysis of solutions is extended to the case of singular Gierer-Meinhardt systems. We refer to the works of J.M. Ball [11,12], V. Barbu [19], L. Beznea and N. Boboc [22], H. Brezis [30], and P.G. Ciarlet [46,47] for related results and various applications to concrete phenomena.

Four appendices illustrate some basic mathematical tools applied in this book: elements of spectral theory for differential operators, the implicit function theorem, Ekeland's variational principle, and the mountain pass theorem. These auxiliary chapters deal with some analytical methods used in this volume, but also include some complements. Each problem we develop in this book has its own difficulties. That is why we intend to develop some standard and appropriate methods that are useful and that can be extended to other problems. However, we do our best to restrict the prerequisites to the essential knowledge. We define as few concepts as possible and give only basic theorems that are useful for our topic. The only prerequisite for this volume is a standard graduate course in partial differential equations, drawing especially from linear elliptic equations to elementary variational methods, with a special emphasis on the maximum principle (weak and strong variants). This volume may be used for self-study by advanced graduate students and engineers, and as a valuable reference for researchers in pure and applied mathematics and physics. Our vision throughout this volume is closely inspired by the following words of Henri Poincare on the role of partial differential equations in the development of other fields of mathematics and in applications: Nevertheless, each time I can, I aim the absolute rigor for two reasons. In the first place, it is always hard for a geometer to consider a problem without resolving it completely. In the second place, these equations that I will study are susceptible, not only to

x

Preface

physical applications, but also to analytical applications. It is using the existence theory of the Dirichlet problem that Riemann founded his magnificent theory of Abelian functions. Since then, other geometers have made important applications of the same principle to the most fundamental parts of pure analysis. Is it still permitted to content oneself with a demi-rigor? And who will say that the other problems of mathematical physics will not, one day, be called to play in analysis a considerable role, as has been the case of the most elementary of them? (Henri Poincare 1164])

May 2007

ACKNOWLEDGMENTS We acknowledge, with unreserved gratitude, the crucial role of Professor Philippe G. Ciarlet, who encouraged writing a book on this subject, but also for his constant support over the years. We are greatly indebted to Sir John M. Ball for accepting our work in his prestigious series at Oxford University Press. We are also grateful to Dr. Michael

Penn, the Oxford book program editor, and to Paul Hobson, the production editor, for their efficient and enthusiastic help, as well as for numerous suggestions related to this volume.

We extend our warm thanks to Professors Catherine Bandle, Viorel Barbu, Lucian Beznea, Bernard Brighi, Olivier Goubet, Patrizia Pucci, and Michel Willem for their useful comments and remarks during the preparation of this work.

We thank Dr. Nicolae Constantinescu for the professional drawing of figures contained in this book. Both authors acknowledge the support of grants 2-CExO6-11-18/2006 and CNCSIS PNII-79/2007. They have been also supported by grants CNCSIS AT191/2007 (M. Chergu) and CNCSIS A-589/2007 (V. RAdulescu).

CONTENTS I 1

PRELIMINARIES

Basic methods 1.1 1.2 1.3

1.4

1.5 1.6

3

A fixed point result The method of sub- and supersolution Comparison principles 1.3.1 Weak and strong maximum principle 1.3.2 Maximum principle for weakly differentiable functions 1.3.3 Stampacchia's maximum principle 1.3.4 Vazquez's maximum principle 1.3.5 Pucci and Serrin's maximum principle 1.3.6 A comparison principle in the presence of singular nonlinearities Existence properties and related maximum principles 1.4.1 Dead core solutions of sublinear logistic equations 1.4.2 Singular solutions of the logistic equation Brezis-Oswald theorem Comments and historical notes -

3 5 9

9 10 11

12 15 17 20

23 24 26 32

II BLOW-UP SOLUTIONS 2

Blow-up solutions for semilinear elliptic equations 2.1 2.2

2.3

2.4 2.5

Introduction Blow-up solution for elliptic equations with vanishing potential 2.2.1 Existence results in bounded domains 2.2.2 Existence results in the whole space Blow-up solutions for logistic equations 2.3.1 The case of positive potentials 2.3.2 The case of vanishing potentials An equivalent criterion to the Keller-Osserman condition Singular solutions of the logistic equation on domains with holes

2.6 2.7

2.8

3

Uniqueness of blow-up solution A Karamata theory approach for uniqueness of blow-up solution Comments and historical notes

Entire solutions blowing up at infinity for elliptic systems 3.1

Introduction

37 37 40 40 42 45 45 46

49 52 58 64 71

75 75

Contents

xiv

3.2

3.3

Characterization of the central value set 3.2.1 Bounded or unbounded entire solutions 3.2.2 Role of the Keller-Osserman condition Comments and historical notes

76 76 81

89

III ELLIPTIC PROBLEMS WITH SINGULAR NONLINEARITIES

4

Sublinear perturbations of singular elliptic problems 4.1

4.2 4.3 4.4 4.5 4.6

4.7

4.8 5

5.4 5.5

5.6 5.7

112

Singular elliptic problems in the whole space 4.7.1 Existence of entire solutions 4.7.2 Uniqueness of radially symmetric solutions Comments and historical notes

113 113 117

Introduction A general bifurcation result Existence and bifurcation results Asymptotic behavior of the solution with respect to parameters Examples 5.5.1 First example 5.5.2 Second example The case of singular nonlinearities Comments and historical notes

121

125 126 127 129 134 137

137 138 140 142

Bifurcation and asymptotic analysis: The nonmonotone case 6.1 6.2 6.3

6.4 6.5 7

93 93 98 99 104 107

case

Bifurcation and asymptotic analysis: The monotone case 5.1 5.2 5.3

6

Introduction An ODE with mixed nonlinearities A complete description for positive potentials An example Bifurcation for negative potentials Existence for large values of parameters in the sign-changing

143

Introduction Auxiliary results Existence and bifurcation results in the nonmonotone case An example Comments and historical notes

Superlinear perturbations of singular elliptic problems 7.1 7.2 7.3 7.4

Introduction The weak sub- and supersolution method H1 local minimizers Existence of the first solution

143 144 147 154 154

157 157 158 161 163

7.5

7.6 7.7 7.8 8

xv

Existence of the second solution 7.5.1 First case 7.5.2 Second case C1 regularity of solution Asymptotic behavior of solutions Comments and historical notes

169

Stability of the solution of a singular problem 8.1

8.2 8.3 8.4 8.5

9

Contents

Stability of the solution in a general singular setting A min-max characterization of the first eigenvalue for the linearized problem Differentiability of some singular nonlinear problems Examples Comments and historical notes

The influence of a nonlinear convection term in singular elliptic problems 9.1 9.2 9.3 9.4 9.5 9.6 9.7

9.8 9.9

Introduction A general nonexistence result A singular elliptic problem in one dimension Existence results in the sublinear case Existence results in the linear case Boundary estimates of the solution The case of a negative singular potential 9.7.1 A nonexistence result 9.7.2 Existence result Ground-state solutions of singular elliptic problems with gradient term Comments and historical notes

10 Singular Gierer-Meinhardt systems 10.1 Introduction 10.2 A nonexistence result 10.3 Existence results 10.4 Uniqueness of the solution in one dimension 10.5 Comments and historical notes

Appendix A Spectral theory for differential operators A.1 Eigenvalues and eigenfunctions for the Laplace operator

A.2 Krein-Rutman theorem

Appendix B Implicit function theorem Appendix C Ekeland's variational principle Minimization of weak lower semicontinuous coercive functionals C.2 Ekeland's variational principle

170

174 182

184

189

191 191

199 201 204 205

207 207 208 210 215 226 228 231 232 233

238 241 243 243 244 248 255 261

265 265 266 269

273

C.1

273 273

xvi

Contents

Appendix D Mountain pass theorem D.1 Ambrosetti-Rabinowitz theorem

D.2 Application to the Emden-Fowler equation D.3 Mountains of zero altitude

277 277 281 282

References

283

Index

295

PART I

PRELIMINARIES

1

BASIC METHODS If I have seen further than others, it is by standing upon the shoulders of giants. Sir Isaac Newton (1642-1727), Letter to Robert Hooke, 1675

The main purpose of nonlinear analysis is to study and describe various material systems arising in mathematical physics and other applied sciences. The language of nonlinear analysis is very close to that of functional analysis and partial differential equations. This chapter deals with some general features about the qualitative analysis of nonlinear elliptic partial differential equations. The main issues we raise in what follows concern the existence and uniqueness of the solution. The existence of solutions is essential for a prescribed model to make sense. The uniqueness of the solution is a natural requirement for many problems. In some cases, uniqueness becomes a difficult question, but related interesting questions concern multiplicity of solutions, as well as the existence of maximal or minimal solutions. That is why monotonicity properties, described by means of comparison principles, play a central role in this chapter and throughout the book. 1.1

A fixed point result

This type of result applies successfully to classical ordinary differential equations (ODE) boundary value problems. Given a second order equation y"(t) = f (t, y(t), y'(t)) subject to the Dirichlet boundary condition, we look for a solution as a fixed point of a compact map F (i.e., F(y) = y) defined on a suitable closed subset of a normed vector space. There are many results in this direction, but we restrict our attention to the following basic property, which is due to Leray and Schauder [131].

Theorem 1.1.1 (Leray-Schauder) Let C be a convex set in a normed vector space E and let U C C be an open set that contains the origin. Then each compact map F : U -* E has at least one of the following properties: (i) F has a fixed point. (ii) There exist x E 8U and 0 < A < 1 such that x = AF(x). For the proof we refer the reader to Granas and Dugundji [98], O'Regan [152], and O'Regan and Precup [153]. We next give an application of this result to a singular ODE.

Basic methods

4

Consider the Sturm-Liouville problem

0 0 such that for all 0 < A < 1 and for any solution y E C2(0,1) fl C[0,1] of (1.4), we have

W. < M. Then problem (1.1) has a solution y E C2(0,1) nCIO, 1] such that jyj,' < M.

Proof We first remark that it suffices to argue if a = b = 0. Indeed, if this is not the case, we define z(t) = y(t) - all - t) - bt, for any 0 < t < 1. Thus, z verifies

0 < t < 1,

z"(t) + g(t, z(t)) = 0 z(0) = z(1) = 0,

where g(t, x) = f (t, x + a(l - t) + bt), (t, x) E (0, 1) x R. We also observe that g satisfies the hypothesis (Fl). Thus, we can assume a = b = 0. By virtue of (1.2) and (1.3), problem (1.1) is equivalent to Y(t) = (1 - t) fo sf t (s, y(s))ds + At t (1 - s) f (s, y(s))ds.

it

(1.5)

Set U:= {y E C[0,1] : ]yj,, < M} and define the map F : U -+ C[0,1] by I It Fy(t) = all - t) + bt + (1 - t) l s f (s, y(s))ds + t I ( 1 - s) f (s, y(s))ds. JJJo

Then F is a compact operator. We claim that property (ii) in Theorem 1.1.1 cannot be true, which implies immediately that F has a fixed point. Indeed, if there exist y e W and 0 < A < 1 such that y = AFy, then y is a solution of (1.4). Thus, by our hypothesis, IyL,, < M-that is, y W. This is clearly a contradiction. So, by Theorem 1.1.1, we deduce that F has a fixed pointy e U. In view of the equivalent formulation given in (1.5), this means that y is a solution of (1.1). This ends our proof.

The method of sub- and supersolution

1.2

5

The method of sub- and supersolution

Let S2 C RN(N > 1) be a bounded domain with a smooth boundary 8Sl (for instance, we can assume that aft is of class C3) and F = fi(x, t, ) : St x R x RN R be a Holder continuous function with exponent y E (0, 1) and continuously differentiable with respect to the variables t, t; and such that

(Al) for any 11o cc St and any 0 < a 0 such that

,p(x,t,Z:)l

u, Vu)

in Q.

The method of sub- and supersolution in its classical form was developed by H. Amann [5], [6] in the framework of nonlinear elliptic partial differential equations. Roughly speaking, this method establishes the existence of a solution, provided suitable sub- and supersolutions do exist. This result is a crucial property for showing the existence of solutions to wide classes of nonlinear elliptic problems.

Theorem 1.2.2 Let u and u be a sub- and a supersolution of problem (1.6) such that u < u in Q. Then the following properties hold true:

(i) There exists a solution u of (1.6) that satisfies u < u < U. (ii) There exists a minimal and a maximal solution U and U of problem (1.6) with respect to the interval [u, u]. Let us assume that (D is defined only on the set I x (0, oo) x RN, so that 4) may by singular in 80 x {0} x W'. In this case we are looking for a solution of the problem -Du = (D(x, U, Du) in S2, u > 0

in St,

u=0

onaQ.

(1.8)

Basic methods

6

A subsolution u of (1.8) is a function u E C2(9) fl C(?l), which is positive in 52, u = 0 on 852, and fulfills (1.7)_ Accordingly, u E C2(Sl) f1C(S2) is a supersolution

of (1.8) if u is positive in 0, u = 0 on 85l, and -i

> 4D(x, U, Vu) in 0.

Theorem 1.2.3 If u and u are respectively sub- and supersolutions of (1.8) such that u < u in Sl, then (1.8) has at least one classical solution.

Proof The proof relies on the domain approximation method. In this way we avoid the possible singularities of qb(x,u,Vu) at the boundary. Let (S2k)k>1 be a sequence of subdomains of H having smooth boundaries and such that 11 CC and

S2 = U 52k. k>1

For each k > 1, consider the problem -Au = 4D(x, U, Vu)

u>0

in Q k,

in52k, on 8S2k.

u=u

Obviously, the restriction of u and u on S2k are sub- and supersolutions of problem (1.9). Thus, by Theorem 1.2.2, there exists a minimal solution Uk E C2'-'(S2k) f1 C(S2k) of (1.9) such that u < uk 1 is a sequence of continuous

functions such that

u 1 there exists C2 > 1) that does not depend on k such that IIukIIW2., (Q,) < C2(II0kIILr(Q2) + IIukIILr(Q2)) < C2[Q2I1/P(max xEQ2

I0k(x)I + Max Iuk(x)I). xEQ2

Thus, the sequence (uk)k>j+1 is bounded in W2'P(Q1). Letting p = N/(1 -y), it follows by Sobolev-Morrey's inequality that the sequence (Vuk)k>j+1 is bounded in C1(Q1). Finally, by Holder interior estimates [95, Theorem 6.2, p. 90], there exists a constant C3 > 0 independent of k such that IIukIIC2.-,(-aj) 1 is bounded in C2.7(lzj), for all j > 1.

Because the embedding C2'-'(Stj) - C2(S2j) is compact, there exists a subsequence of (uk)k>j that converges in C2(SZj). This yields u E C2(Qj), for all j > 1-that is, u E C2 (Q). Moreover, we have

-Auk = Vx, uk, Vuk)

in S2j, for all k > j + 1.

Thus, taking the limit of the subsequence converging in C2(-Qj), we conclude that

u satisfies -Au = 4)(x, u, Vu) in the whole Q. Furthermore, from u < u < u in S2 we obtain u(x) = 0, for all x0 E 0Q, so u E C(! ). It remains to prove u E C2°^'(52), which follows easily from the interior regularity theory of elliptic equations. This finishes the proof of Theorem 1.2.3. Consider the problem

-©u = F(x, u)

in (2,

u>0

in n,

u=0

on t9Q.

(1.13)

Assume that gi : 11 x (0, oo) -> (0, oo) is a Holder continuous function with exponent y (0 < -y < 1) on each compact subset of Sl x (0, oo) and it satisfies the following assumptions:

Basic methods

8

(A2) lim supt_ 0 ±( 't) < Al := A1(-A, 52) uniformly for x E 52. (A3) limt\o ±42!I = co uniformly for x E Q. Note that may be singular at the origin with respect to the second variable. The existence of a classical solution to (1.13) is given by Theorem 1.2.5.

Theorem 1.2.5 Assume 4) satisfies hypotheses (Al), (A2), and (A3). Then -problem (1.13) has at least one positive solution u E C2.7(52) fl C(S2).

Proof For any positive integer k, consider the approximated problem

-Au = 4i(x,u)

in 52,

u>0

in Q,

u=

on 852.

(1.14)

Let co be the normalized positive eigenfunction corresponding to the first eigenvalue al of the problem

in c,

Au = Au u=0

on 852.

(1.15)

Using (A3), there exists c > 0, which is small enough and ko > 1, which is large enough, such that Vk._

-CW1+

1

is a subsolution of (1.14) for all k > ko. Without loss of generality, we assume that ko = 1. To provide a supersolution, let us fix 0 < A < Al such that lim

t-oo Then there exists

{x, t)

t

< A < A,,

uniformly for x E Q.

E C2(S2) such that

Ae > . >0

in 52,

on 852.

We now choose M > 0 large enough such that S := MC is a supersolution of (1.14) for all k > 1 and vk < C in Q. There exists ul E C2(52) a solution of (1.14) in Q. Now, ul is a supersolution of (1.14) with k = 2 and v2 < U1 in Q. Hence, there exists u2 E C2 (E2) a solution of (1.14) (with k = 2) such that v2 < u2 < u1 in Q. Repeating the previous process, we obtain a sequence (uk)k>1 such that

with k = 1 such that v1 < ul
0 for all v E H1(11), v > 0. Then u > 0 in Q.

Proof Let u^ = max{-u,0}. Then u- E H'(Sl) and Vu = -Vu on the set where u < 0 and Vu- = 0 on the set where u > 0. Because Lu > 0, it follows

that a(u, u-) > 0. On the other hand, a(u, u ) = -a(u-,u-) and, from the coercivity of a, we deduce that

a11u 11H1(-) 0 and Lv > 0 in i, then w = min{u, v} also verifies Lw > 0 in Q. We refer to the book of Stampacchia [183] for further details. 1.3.4

Vazquezs maximum principle

We recall that the standard version of the maximum principle asserts that if u E C2(1l) n C(Ti) is a superharmonic function such that u > 0 on aI, then the following alternative holds: Either u > 0 in 1 or u = 0 in Q. Stampacchia extended this result for linear perturbations of the Laplace operator and showed

that the same conclusion holds if u > 0 on asl and -Au + an > 0 in Q, provided -A + aI is coercive in H'(Sl), where a E LO°(Sl). A natural question is to determine whether a similar result holds true when the Laplace operator is affected by a suitable nonlinear perturbation. In [190], J. L. Vazquez established an important extension of the maximum principle for semilinear and quasilinear problems, in close relationship with the behavior of the nonlinearity around the origin.

Theorem 1.3.11 (Vazquez's maximum principle) Let Sl be a bounded domain in RN, N > 3, and let u E C2(c) n C(i) be such that u > 0 in 11 and Au < f (u)

in Sl,

(1.18)

where f : [0, oo) -+ R is a continuous and increasing function such that f (0) = 0 and f satisfies the integral condition dt

F(t)

Jo

=00 >

with F(t) :=

J0

t f (s)ds.

(1.19)

Then the following alternative holds: Either u > 0 in 52 or u =_ 0 in D.

Proof We start with the following auxiliary result. Lemma 1.3.12 Consider the Sturm-Liouville problem

f v"=Klv'+K2f(v)

j

v(0) = 0,

in (0,ri),

(1.20)

v(rl) = vl,

where K1i K2, rl, and vI are positive real numbers. If f (0) = 0 and f is increasing, then problem (1.20) has a unique solution. If, moreover, f satisfies condition (1.19), then the unique solution v of (1.20) satisfies v'(0) > 0 and 0 < v < vi in (0, r1)

Proof We first observe that v := 0 is a subsolution and v := C is a supersolution of (1.20), provided C is large enough. Thus, problem (1.20) has at least a

Comparison principles

13

solution. To argue that this solution is unique, let us assume that V1 and V2 are solutions of (1.20) and set v := V1 - V2. Then

f v" = Kl v' + K2 (f (Vi) - f(V2)) v(0) = v(ri) = 0.

in (0, r1)

(1.21)

We claim that V1 < V2 in (0, ri). Indeed, if not, there exists xo E (0, r1) such that v(xo) = supZE(o,ri) v(x) > 0. Hence, v'(xo) = 0 and v"(xo) v"(xo) = Klv'(xo) + K2 (f (Vi (xo)) - f(V2(xo))) > 0, which is a contradiction. Thus, V1 < V2 in (0,ri) and, after changing the roles of V1 and V2, we deduce that Vl = V2, which shows that problem (1.20) has a unique solution. Let us now assume that f satisfies condition (1.19) and let v be the unique solution of (1.20). Set

ro := sup{O < r < ri : v(r) = 01. Then 0 < ro < ri and v(ro) = 0. We show in what follows that ro = 0, which is enough to conclude the proof of the lemma. Arguing by contradiction, we assume

that ro > 0. Then v'(ro) = 0 and v' > 0 in (ro,ri). Because v cannot have local maxima in (ro, ri), it follows that the mapping v : [ro, ri] -+ 10, vl] is a bijection. Multiplying (1.20) by v' and integrating, we find

w' = 2K1w + 2K2(F(v))' ,

(1.22)

where w := vi2. After multiplication in (1.22) by e-2K1r and integration on [ro, r] we deduce that

e-2K1rw(r)

- e-2K1T0w(ro) = 2K2

rr e-2K1s

(F(v(s)))'ds.

Tp

Because w(ro) = 0, this relation yields e-2K1Tw(r) = 2K2

rr

e-2K1s

Jr,

(F(v(s)))'ds

< 2K2 f C 21,r0 (F(v(s)))'ds = 2K2 er02K1r,, (F(v(r)) - F(v(ro) )) . Hence,

v'(r) F(v(r))

0 in 1 it is enough to show that u > w > 0 in Q. Because f E C1 [0, oo), we have 2

t\O F(t)

2t

t\o f (t)

2

f'(0) >

0,

(1.34)

which implies immediately that fo dt/ F(t) = oo. By Vazquez's maximum principle (Theorem 1.3.11) we conclude that w > 0 in Q.

Existence properties and related maximum principles

21

We now prove that u > w in Q. To this aim, fix e > 0. We claim that

w(x) < u(x) + e(1 +

1x12)-1/2

for any x E Q.

(1.35)

Assume the contrary. Because u = w on t9S2, we deduce that max{w(x) - u(x) - e(1 +

1x12)-1/2}

zEn

is achieved in Q. At this point we have 1x12)-1/2)

O >A (w(x) - u(x) - e(1 +

=11p11-f(w(x)) - p(x)f (u(x)) - c (1 + IxI2)-1/2 1x12)-3/2 ?p(x) (f (w(x)) - f (u(x))) + e(N - 3)(1 + +3-,(l

IX12)-5/2 > 0,

which is a contradiction. Because e > 0 is chosen arbitrarily, inequality (1.35) implies u > w in IL. SECOND PROOF: Because 0 is not identically zero, there exists x0 E S2 such that

u(xo) > 0. To conclude that u > 0 in IL, it is sufficient to prove that u > 0 in B(xo,T), where F = dist (xo, aQ). Without loss of generality we can assume xo = 0. By the continuity of u, there exists r E (0, f) such that u(x) > 0 for all x with Ixl < r. So, p := mini.=, u(x) > 0. We define

M :=

P+1

n := fp

dt

f(t)

_

rP+1

and E

dt

f(t) for 0 < s < p.

It remains to show that u > 0 in A(r,T), where A(r,T) :_ {x E RN : r 0 small enough, the problem 1-AV = M

in A(r,T),

v =

aslxl = r,

V = v(e)

aslxl = T

(1.36)

has a unique solution, which is increasing in A(r,T). Proof By the maximum principle, problem (1.36) has a unique solution. More-

over, v is radially symmetric in A(r,f)-namely, v(x) = v(r), r = 1x1. The function v satisfies

r0

(1.39)

on 09.

Moreover, v > 0 in Q. Note that u is a supersolution for (1.39) and a standard D argument shows that u > v > 0 in Q. Dead core solutions of sublinear logistic equations The positiveness of the solution in Theorem 1.4.1 follows essentially by the assumption f E C'[0, oo). We show in what follows that if f is not differentiable at the origin, then problem (1.33) has a unique solution that is not necessarily positive in Q. However, in this case, the positiveness of the solution may depend on c and on the geometry of S2. Indeed, let us consider the problem 1.4.1

r AU =

in 11,

u>0

in!Q,

u=c

on ad,

(1.40)

where c > 0 is a constant. The existence of a solution follows after observing that u = 0 and u = c are respectively sub- and supersolutions for our problem, whereas the uniqueness follows from Theorem 1.3.17. The following example illustrates that in certain situations, the unique solution of problem (1.40) may vanish.

Example 1.1 Let Q = B(0,1) C R', w(x) = alx14 with c < a < (4N + 8) -2. We have

r Ow = (4N + 8)alxI2 < vjx12 = v Sl

w=a>c

in d, on alt.

This means that w is a supersolution of (1.40). Hence, problem (1.40) has a

-

solution u such that 0 a >

1

(4N + 8)2

Then w satisfies

1'Lw=(4N+8)alx-xoI2> Sl

in St,

/ix-xoI2=V,,W-

w=aIx-xoI4 w(x) > 0, for any x E St`{xo}. If diam 9 < 2R < 2R,, then there exists two points, xo and x1, such that 11 can be included in each of the balls B(xo, R) and B(xi, R). Using the previous conclusion we have u(x) > amax{Ix - x014, Ix - x114} > a

xl - xe

4

2

>0.

Choosing a = c/R4, Ixl - xo I = 2R - diam S2, and R = R, we find

(2R- diam U)4 =

di2mRS2)4

R4

>0

in 0.

Hence, u is positive solution of (1.40). Singular solutions of the logistic equation We now illustrate how Theorem 1.4.1 can be applied in the qualitative analysis of a class of singular solutions for the logistic equation. We are first concerned with the nonlinear problem 1.4.2

I Au = up

u>0 U=00

in B1, in B1, on 8B1,

(1.41)

where p > 1 and B1 C RN is the unit ball centered at the origin. The notation u = oo on 8B1 signifies that limy,/l u(x) = oo. A function u E C2(B1) satisfying (1.41) is called blow-up boundary (explosive) solution of problem (1.41).

Theorem 1.4.5 Assume that p > 1. Then problem (1.41) has a unique solution u. Moreover, u is infinitely differentiable in B1, u is radially symmetric, and

- r 2(p + 1 */'1

(p - 1)2

1/(p-ll

Existence properties and related maximum principles

25

Proof Using Theorem 1.4.1 we deduce that for any positive integer n, there exists a unique function u,z E C2(B1) such that

f Dun = un un > 0

in B1, in B1,

un=n

(1.42)

onrB1.

By uniqueness we deduce that un has radial symmetry. Standard elliptic regularity implies that un E C°O(B1). Next, by the maximum principle, un < U"+1 in B1, for any n > 1. We now observe that u(x) := C(1 - r2)-2/(p-1) is a supersolution of problem (1.42), provided C is large enough. This shows that for any n > 1, un < u, hence (un) is locally bounded in B1. Thus, for any x E B1, there exists u(x) := llmn un(x) E R, which is a solution of (1.42). Schauder and Holder regularity arguments imply that u E C°O(B1). To deduce the blow-up rate of u near the boundary, we observe that the differential equation fulfilled by the solution is

N-1 u, = up

in (0, 1).

(rN-lu') = rN-lup

in (0, 1).

Uit + Therefore

r

(1.43)

After multiplication by rN-lu' in (1.43) and integration, we find p+1

ui2(r) = 2

P+1()

2N

-2

+'

t2N-1,up+1(t)dt

(p + 1)r2 N Jo

I

(1.44)

At this stage, it is enough to argue that

B(r) = o(A(r))

`

as r /1,

where

B(r) := r-2N f rt2N-1up+1(t)dt.

A(r) = uP+i (r) A(r)

Jo

Indeed, relations (1.44) and (1.45) yield

1-

p+ l u'(r) 2uP+1(r)

= 1 - 1 - B(r)

A(r) )

1/2

ti B(r) = o(1) as r / 1. 2A(r) -

Integrating this relation between r and 1 we deduce that (1-r)-

roo Ju(r)

p+ldt=o(1) 2tP+1

which implies the desired asymptotic estimate.

asr/1,

(1.45)

Basic methods

26

To prove (1.45), let us fix E E (0, 1/2). Thus, for any r E (1 - e, 1),

B(r) Ar

1

r

r

( ts

2N-1

,u t

P+1

u(r)

dt

(!q,,) p+1 +

= J 1_E +1( ) f(x,llull.) u Ilull. Hence f (x, u) > -Mu in SZ for some positive constant M. Thus, u satisfies -Au + Mu > 0 in fl. By the maximum principle (Theorem 1.3.5) we obtain the conclusion. Brezis and Oswald [37] established the following general result.

Theorem 1.5.2 Assume that f satisfies (f 1) - (f3). Then, problem (1.46) has solutions if and only if ao < 0 < a00. Moreover, in this case problem (1.46) has a unique solution.

Proof We divide the proof of Theorem 1.5.2 into three steps. Step 1: Necessary condition. Let u E HH(Q) n L0(Q) be a solution of (1.46). Multiplying by u in (1.46) and then integrating by parts we obtain J IVul2dx = j f(x,u)udx < J ao(x)udx.

s

n

This implies that A0 < 0. Set

b(x):= f(x, lull. + 1) E L°O S) Hull. + 1

Basic methods

28

and let (µ, ,O) be the first eigenvalue and eigenfunction of the linear operator

-A - b(x). That is, in 0,

1-AO - b(x)O = AO

0 >0

in Q,

1i = 0

on aa.

Because ac,, b(x)u by virtue of (f 1), which also yields u fn uV;dx > 0. Thus, ) > it > 0. Step 2: Sufficient condition. Consider the energy functional E : HO'(Sl) - (-oo,oo], defined by

E(u) = 2 jIVuI2ds_jF(x,u)dx

for all u E

1

where F(x, t) = ff f (x, s) ds and f (x, s) is extended to be f (x, 0) for s < 0. Remark that E E C' because F(x, t.) < c(t2 + Iti)

for all (x, t) E St x IR,

(1.47)

by virtue of (f3). In the statement of the next auxiliary result, we need the notion of lower semicontinuity. We recall that if S is a topological space, then a functional E : S -> R U {+oo} is lower semi continuous if and only if for any a E R, the sublevel set Sa := {u E S : E(u) _< a} is closed. If S is a metric space, this condition is

equivalent to E(u) < lim infra-a, E(un) whenever un - u. Lemma 1.5.3 The following properties hold true: (i) E(u) - oo as l1u11rii -+ oo. (ii) E is lower semi continuous. (iii) E(w) < 0 for some w E Ho (1).

Proof (i) Assume by contradiction that there exists (un)n>1 C HH(Sl) such that IIun11H,, -F oo and E(un) < C. From the definition of E and (1.47) we obtain 2

J

IVun12dx

< M j(un + 1)dx.

(1.48)

Brezis-Oswald theorem

29

Set to = 1Iun112 and vn = un/tn. From our assumption and (1.48) we have to -> oo,

and (vn)n>1 bounded in Ho(1l).

11Vn112 = 1

Therefore, one can assume that

vn-v vn -> v

weakly in Ha(Q)as n->oo, strongly in L2(Q) as n - oo,

vn - v

almost everywhere in Sl and 11v112 =1.

We claim that F(x, tnyn)

lim sup / n- oo

t2 S2

ao0u2dx .

dx < 1 2

to

(1.49)

{v>0}

Indeed, we have

F(x, tnvn)dx =

J J

F(x, tnvn )dx (1.50)

F(x,tnvn)dx. {v 0 for Ix[ sufficiently large. An example of function p that satisfies both (p2) and (pl)', with p vanishing in every neighborhood of infinity, is provided here:

p(r)=O for r= Ix[ E [n-1/3,n+1/3], n> 1; 00

p(r) > 0 in lR+ \ U [n - 1/3, n + 1/31; n=1

p c C1 [0, oo)

and

_

max

2

p(r) n2(2n + 1) Obviously (pl)' is fulfilled for Stn := B(0, n+ 1/2). On the other hand, condition n

1

(p2) is also satisfied because

I

n+l

00

00

rp(r) dr =

rp(r) dr

J

n=111n n+l

00

n=1 n

=E 00

1 n2

2r

n2(2n+1)

dr

w in RN and w(x) --+ oo as This will also imply that U is positive. Set rt

t1-N

I

J0

sN-1 (s) ds dr

for all r > 0.

Obviously, 4) E C2[0, oo). Furthermore, we have the following lemma.

Blow-up solutions for semilinear elliptic equations

44

Lemma 2.2.4 iim 4P(r) is finite if and only if ff rq (r)dr is finite. r-.oo

Proof For any r > 0 we have

N 1 2 (10" t-0(t) dt - rlv-Z f'tN-10(t)dt)

fi(r)

(2.11)

< N1

Jr t(t) dt.

2

f

On the other hand, J

fr

r

N-2

Jff

-

(rN_2 tN-2) to(t)dt r_2 tN_10(t)dt = rrN-2 - (r/2)N-2

fr/2

rN-2

>

The last inequality combined with (2.11) yields

1V 1 2

f'tO(t)dt > I(r) >

1-

(11/2)ZN-2 fr/2

to(t)dt.

The conclusion follows now by letting r -> oo in the previous estimates. This finishes the proof. By virtue of hypothesis (p2) and Lemma 2.2.4 it follows that L := 1imr- o (D(r) is finite. Then, z(r) := L - 0 is positive and verifies

Az=0(r)

z(x)0

r:=lxj >0,

(2.12)

asIxI-4oo.

According to Lemma 2.2.1, the mapping ( 0 , 0 0)

3t!

->

100 f(s)

E (0, oo)

is decreasing and bijective. We implicitely define w as for all x E RN.

(2.13)

w (z)

Note that by (2.12) we have w E C2(RN) and w we have

-Az(x) = Hence,

Aw(x) T(W)

oo as IxI -+ oo. Furthermore;

(w } IOw12 fZw

in I[ '

Blow-up solutions for logistic equations

Ow > p(x)f (w)

45

in RN.

We claim that w _ 1. Obviously this inequality is true on BSln. Using the same arguments as in the proof of inequality (1.35) in Theorem 1.4.1 (with Q replaced with Stn), we obtain that for any e > 0 and n > 1 we have

w(x) < un(x) +,_(I + Ix12)-112

in Stn.

Passing to the limit with e --p 0 we derive w < un in Q. Consequently, U > w in RN and, by (2.13), w(x) -' oo as IxI - oo. This completes the proof. 2.3 Blow-up solutions for logistic equations In this section we discuss the existence of blow-up solutions to the following semilinear elliptic equation: Au + Au = p(x) f (u)

in Q.

(2.14)

We assume that St C RN (N > 3) is a smooth bounded domain, A E R, and f E C' [0, oo) is a nonnegative function that satisfies (f 2) and

(f3) the mapping (0, oo) 3 t -->

ff (t) is increasing.

We also assume that p E C°'7(St) (0 < y < 1) is nonnegative and p $ 0 in Q. Notice that we allow p to vanish in St (or even on (9Sl).

Remark 2.3.1 If f satisfies (f2) and (f3), by Lemma 2.1.3 and l'Hospital's rule we deduce that limt.< f(t)/t = limt.. f'(t) = cc. Remark 2.3.2 Typical examples of nonlinearities satisfying (f 2) and (f 3) are

(i) f (t) = et - 1. (ii) f (t) = tp, p > 1. (iii) f (t) = t[In(t + 1)]P, p > 2. The case of positive potentials We discuss here the existence of blow-up solutions to (2.14) under the additional hypothesis p > 0 in U. More generally, we consider the problem 2.3.1

Au + a(x)u + b(x) = p(x) f (u) in St

(2.15)

where a, b E C°'7(St), b > 0. We first need the following result.

Proposition 2.3.3 Assume that p > 0 in SI and f satisfies (f2), (f3). Let 0 E C°''r(8St) be a nonnegative function such that 0 # 0. Then the boundary value problem

Au + a(x)u + b(x) = p(x) f (u)

u>0 u=4 has a unique solution u E C2 (U).

in St, in St,

on 8Q

(2.16)

46


Proof The uniqueness follows directly from Theorem 1.3.17. We will focus on the existence part. Applying Theorem 1.4.1, the boundary value problem in 52,

Au = ((a((.u + ((p((oof (u) U>0

in 52,

u=O

(2.17)

on On

has a unique solution u E C2(52). Clearly u is a subsolution of (2.17), To provide a supersolution of (2.17), let us remark that for all n > 1, the mapping

n(x, t) := a(x)t + b(x) - p(x) f (t) +

-,

(x, t) E 52 x (0, oo),

satisfies the hypotheses of Theorems 1.2.5 and 1.3.17. Hence, there exists a unique solution u E C2(SZ) of the problem

Au + a(x)u + b(x) = p(x) f (u) - n

in 52,

U>0

in 52,

u=0

on W.

By Theorem 1.3.17 we deduce 0 < u < un+1 < un in 52. If u(x) := limn,, un(x), x E 52, by standard elliptic regularity arguments we derive that u E C2(52) is a 0 solution of (2.16). This finishes the proof. Under the assumptions of Proposition 2.3.3, we obtain the following theorem. Theorem 2.3.4 There exists a positive blow-up solution of equation (2.15).

Proof Let 4D(t) := pof (t) - ((a((,,t - b, t > 0, where po = info p > 0 and b := supo b + 1 > 0. Let also r be the unique positive solution of the equation 4?(t) = 0. By Remark 2.3.1 we deduce that limt_,,, 4)(t)/ f (t) = po > 0. By (A2), we further derive that the mapping (0, oo) t ----> 4i(t + r) satisfies the assumptions of Theorem 2.1.2. Hence, there exists a positive blow-up solution of Av = 4b(v + r) in Q. Thus, U(x) := v(x) + r, x E 52, satisfies

AU+{(a((.U+b=pof(U)

in 52,

and blows up at the boundary of 52.

Let un be the unique solution of (2.16) with 0 - n. By Theorem 1.3.17, un < un+l < U in fl. It follows that u(x) := limn. un(x), x E 52, exists and defines a positive blow-up solution of (2.15). Moreover, every positive blow-up solution v of (2.15) satisfies un < v in 52, which yields u < v in fl. This means that u is a minimal blow-up solution of (2.15). This concludes the proof. The case of vanishing potentials We assume in this section that p vanishes in 52. Set 2.3.2

520 := int {x E 52 : p(x) = 01

and suppose that 520 C 52 and p > 0 in 52 \ 520.

(2.18)

Blow-up solutions for logistic equations

47

Let L be the unique self-adjoint operator associated with the quadratic form 0(u) = fo jouj2 dx on HD(S2o) := {u E Ho(S2) : u(x) = 0 for almost everywhere x E 9\ Sto}.

If aSto satisfies the exterior cone condition then, according to Alama and Tarantello [4), Hb(Sto) coincides with Ho (SZo) and L is the classical Laplace operator with the Dirichlet condition on %2o. Let A,,,1 be the first Dirichlet eigenvalue of G in S2o. Set f := limt\o f (t)/t. and denote by Ao,1 the first eigenvalue of the operator -A + fp(x) in Hp (S2). Consider the following nonlinear Dirichlet problem: Au + Au = p(x) f (u)

in 52,

u>0 u=0

in 0,

(2.19)

on as2.

Alama and Tarantello [41 established the following necessary and sufficient condition for the existence of solutions to problem (2.19).

Theorem 2.3.5 Assume f satisfies assumption (f 3). Then problem (2.19) has a solution if and only if A0,1 < A < A,,,,,1. Moreover, in this case the solution is unique.

A related interesting problem is to establish a necessary and sufficient condition for the existence of blow-up boundary solutions of equation (2.14). The following result asserts that such a solution exists for small values of the real parameter A, up to a certain range. However, if the variable potential is positive, then (2.14) has a blow-up boundary solution for any value of A. We observe that our assumptions on the potential p allow that it may vanish on 852. This is the most interesting case, because it corresponds to the "competition" between a vanishing potential and a blow-up nonlinearity.

Theorem 2.3.6 Assume that f satisfies conditions (f 2) and (f 3). Then problem (2.1.4) has a positive blow-up solution if and only if A E (-oo, A"C',1). Proof NECESSARY CONDITION. Let u be a blow-up solution of equation (2.14). As we have already argued, u is positive in Q. Suppose A,,,),1 is finite and let us assume by contradiction that A > A,,.,1. Fix A0,1 < ti < and denote by v a pos-

itive solution of problem (2.19) with A = µ. If M := max {maxi v/ mina u; 1}, then we have A(Mu) + A,,.,1(Mu) < p(x) f (Mu) in S2, in S2, Mu > v Mu = 00 on 852. Hence (v, Mu) is an ordered pair of sub- and supersolution of problem (2.19) with A = A,,,1. Thus, problem (2.19) with A = A.... 1 has at least one positive solution (between v and Mu), which is a contradiction. So, necessarily, A E (-oo, A.,1).


48

SUFFICIENT CONDITION. We first need a similar result to that in Proposition 2.3.3.

Proposition 2.3.7 Assume that p > 0 on 9Q and let ¢ E C°'7(& I) be a nonnegative function such that 0 0 0. Then the boundary value problem

IAu + Au = p(x) f (u)

in 9,

u > 0

in S2,

u=q

onl1l

(2.20)

has a solution if and only if) E (-oo, a(,,,,). Moreover, in this case, the solution is unique.

Proof The first part follows exactly in the same way as in the proof of the necessary condition in Theorem 2.3.6. For the sufficient condition, fix A < AQ,,1. By Theorem 1.4.1 there exists a unique classical solution u of the problem

f Du = jA u + IIpII.f (u) I

in 52,

u>0

in 0,

u=0

on 8Q.

(2.21)

Obviously u is a subsolution of (2.20). Let A.,, > A > max {A, Ao,1} and u, be the unique solution of (2.19) with A = A.. Let S2i (i = 1, 2) be two subdomains of 52 such that S2°CCS21CCS22CCS2

_

and S2 \ S21 is smooth.

Because p > 0 in SZ \ Stl, by Theorem 2.3.4 there exists U E C2(i \ fll), a positive solution of (2.14) in ci \ 521. We define v E C2(Q) as a positive function in SZ such that v = U in S2 \ 522

and u = u, in 521. Using Remark 2.3.1 and the fact that info,\n, p > 0, it is easy to check that u := Cv satisfies Du + Au < p(x) f (u) U > max q5

an

00

in S2,

in Sl,

(2.22)

on 8i,

provided C > 0 is large enough. It is clear that u is a positive supersolution of (2.20) and u < maxan 0 < u in Q. Therefore, by the sub- and supersolution method, problem (2.20) has at least a solution u E C2 M) such that u < u < u in Q.

An equivalent criterion to the Keller-Osserman condition

49

Now let -oo < A < A,,.1 and ul, u2 be two solutions of (2.20). Because p may vanish in Q we cannot apply Theorem 1.3.17 directly. However, we can adapt the arguments in the proof of Theorem 1.5.2 to obtain

f(oi)l

2

2

\f (u2) This yields ul = u2 on the set {x E 51 : p(x) > 0}-that is, p(x)f(uj) = u1

U2

p(x) f (u2) in Q. Furthermore, by the strong maximum principle we obtain ul =

u2 in ft In the same manner we obtain that problem (2.19) has at most one solution. The proof of Proposition 2.3.7 is now complete.

Proof of Theorem 2.3.6 completed. Fix A E (-oo,

In our further

analysis, two cases may occur: CASE 1: p > 0 on 851. Denote by uf, the unique solution of (2.20) with = n. For 0 = 1, let u and u be the solutions of problems (2.21) and (2.22), respectively. The sub- and supersolution method combined with the uniqueness of the solution

to (2.20) shows that u < un < U,, 1 1 there exists a unique vn E C2(1l) such that Avn + AYln =

((x)+)f(v)

in 51,

vn>0

in 51,

vn = n

on 051.

By Theorem 1.3.17, the sequence (vn)n>1 is nondecreasing. Moreover, (vn)n>1 is uniformly bounded on every compact subdomain of Q. Indeed, let K cc 1) be an arbitrary compact set and d := dist (K, 812) > 0. Choose S E (0, d) small enough such that 'L C Ca, where Ca = {x E SZ : dist (x, 812) > b}. Because p > 0 on 8C5, by the results in Case 1 there exists a positive blow-up solution V of (2.14) for 11 = Q. Using Theorem 1.3.17 in C5, we deduce vn < V in C5, for all n >_ 1. Hence, (vn)n>1 is uniformly bounded in K. By the monotonicity of (v,,),,>, we

conclude that up to a subsequence vn - v in C(K). Finally, standard elliptic regularity arguments show that v is a positive blow-up solution of (2.14). This concludes the proof.

2.4 An equivalent criterion to the Keller-Osserman condition Our aim in this section is to supply an equivalent criterion to the KellerOsserman condition (f 2). We assume that f satisfies the following assumption:

(f4) f fulfills (f 3) and there exists limi_,a, (.E)' (t) := a. Consider the set 9 of all functions g : (0, oo) -+ R satisfying the hypotheses


50

(i) there exists 6 = d(g) > 0 such that g E C2(0, S) and g" > 0 on (0, 5); (ii) limt\o g(t) = co; (iii) there exists limt\,o 9(t) g" M Remark first that g is nonempty. Indeed, let 9 E C2(0, oo) be a convex function such that limtNO 9(t) = oo. Because 9' is nondecreasing, it follows that

limt,,o 9'(t) = -oo. Thus, (9'(t))2 + 9"(t)

[9 (t)I

0

as t

o,

which proves that es E t

Remark 2.4.1 ego

g,({t)

=

Jiimo

g"(t}

0 for any function g E

Indeed, if g E G is chosen arbitrarily, then limt\,o g(t) = -oo. Hence, the mapping V(t) = In Ig'(t)) is decreasing in the neighborhood of the origin and by (ii) there exists limt,,o 0'(t). Because limt\p r/0 (t) = oo, it follows that lim ifi'(t) _ -oo. Hence, lim g"(t) = lim Ifi'(t) _ -00, t-"o t'-'O g'(t)

and then by ('Hospital's rule we find g(t) = lim ge(t) = 0. t\,o g'(t) t1-o g"(t) lim

Lemma 2.4.2 Assume that f fulfills (f4). Then, the following hold:

(i) a > 0; (ii) a < 1/2 provided that (f2) is fulfilled.

Proof (i) Suppose that a < 0. Then, there exists ti > 0 such that .f

r {t) < 2 < 0

for any t > ti.

Integrating this inequality over (ti, oo) we obtain a contradiction. It follows that

a>0.

(ii) Using the definition of a we find lim too

F(t) f'(t) = 1 f 2(t)

a.

By Remark 2.3.1 and l'Hospital's rule we obtain tl-'.00

F(t) _ 1 _0 l 2f'(t) f2(t) t-oo

An equivalent criterion to the Keller-Osserman condition and

o

F(t)/ f (t) lim _< t-. ° oo ft, ds/ F(s)

F(t) P(t)

- 2 + tl 1

f 2{t)

-

1

2

51

- a.

( 2 . 23)

This concludes our proof.

Lemma 2.4.3 Assume that f fulfills (f 4). Then the Keller-Osserman condition (f2) holds if and only if (Ag)

lim tf (g(t)) = oo

t\,o

for some function g E 9 .

g (t)

Proof NECESSARY CONDITION. Because (f2) holds, we can define the positive function g as follows:

J

°O

ds

t)

F(s)

= t" for all t > 0,

(2.24)

where t9 > 3/2 is arbitrary. Obviously, g E C2 (0, oo) and limt\,o g(t) = oo. We claim that g E 9 and condition (Ag) is fulfilled. For this purpose, we divide our argument into three steps.

/

Ie(t)

Step 1: i\,o

79

t"

I a - 2)'

(t)) = We derive twice relation (2.24) and we obtain

g'(t) = -t9t"-1 F(g(t)) g"(t) =

t9 _

1

(2.25)

,

g'(t) + t2_2f(g(t))

t 2t2,9-2 f(g(t))

(226) .

2(t9 {

92

1) t25-,(t) f(g(t))

+ 1)

.

By (2.23) and (2.25) we find

g'(t)

t\0

t21-1 f (g(t))

-t9ti9-I

= lim

t,,o

= lim -19

to

F(g(t))

t21- If (g(t))

F(g(t))/ f (g(t))

f (t) ds/ F(s) = 71-00 lira -t9 F(u)/f (u) = t9 (a - 1 f, ds/ F(s) 2 Step 2: g" > 0 in (0, 5) for b small enough. Because a > 0, by the result obtained in Step 1 we find 2(t9 - 1)

t\o

192

g'(t) _ 2(t9 - 1) / - 1\ a 2J t21-If (g(t)) t9

In view of (2.26), the assertion of this step follows.

1 > -1.

(2.27)


52

(g(t)) _ 00 lim t f g"(t) Taking into account (2.26) and (2.27) we find Step 3: t\o lim gg(t) (t) = 0

lim

and

t1,o

2t

g'(t)

0 g (t} -

g'(t)

_

1

z92 t2,1 -If (g(t)) 2(

1) t2 v

+1 g'(t) f(9(t))

°,

and, for any t E (0, 8) where 5 > 0 is given by Step 2, we have tf (9 M) =

g"(t)

tf (g(t))

tf (g(t)) lg'(t) + 22t2s-2 f(g(t)) >

2

792t2'0-3'

Sending t to 0, the claim of Step 3 follows. SUFFICIENT CONDITION. Let g E 4 be chosen so that (A9) is fulfilled. By 1'Hospital's rule we find (g'(t))2 =

"o F(g(t))

2

tNp f (g(t))

0.

We choose 5 > 0 small enough such that g'(t) < 0 and g"(t) > 0 for all t E (0, 5). It follows that dt f9(0150)

F(t)

lira

t\0

f ){s) s(t)

ds

(S

ca

-g'(s) ds F(g(s)) -g'(t} < b sup F(g(t)) tE(0,5)

= lim t\O Jt

< 00.

Hence, the growth condition (A2) holds. This ends the proof.

2.5 Singular solutions of the logistic equation on domains with holes Denote by V and R. the boundary operators

Du := u

and

Ru := 8 +'3(3;)u,

where n is the unit outward normal to 852, and,3 E Cl>_'(8St) is a nonnegative function. Hence, V is the Dirichlet boundary operator and 7Z is either the Neumann boundary operator, if 3 = 0, or the Robin boundary operator, if Q 0. Throughout, B can define any of these boundary operators. We are concerned in this section with the following boundary blow-up problem:

Du + Au = p(x) f (u)

in Il,

on BQ, Bu = 0 (2.28) on BSlo, u = 00 where Sto is the interior of the zero set of p as defined in (2.18). We assume that p > 0 on all, 00 CC 52 is nonempty, connected, and has a smooth boundary.

Singular solutions of the logistic equation on domains with holes

53

In (2.28) condition u = 00 on fflo means that u(x) -> oo as x E i \ Sto and d(x) := dist (x, Sto) - 0. Because of the mixed boundary condition imposed on asi, a suitable comparison principle is needed. To this aim, we first provide the following result. Lemma 2.5.1 Assume that p > 0 in 1Z \ SZo, f satisfies (f3), and let u1.. u2 E C2 (P \ S2o) be such that

(i) Dul + Au1 - p(x)f (ul) < 0 < Lu2 + AU2 - p(x)f(u2) in 9 \ Po; (ii) u1, u2 > 0 in 1 \ SIo and Bul > 0 > 13u2 on aQ; (iii) lim supdist (x,8R,,) .o (u2 - u1) (x) < 0.

\Qo. Proof If 13 = D, the conclusion follows by Theorem 1.3.17. We next assume that 13 = R. Let 01, 02 be two nonnegative C2 functions in Il \ Q0 vanishing

Thenul >U2 in

near o Do. Multiplying the first inequality in (i) by 01, the second one by 02, and applying integration by parts together with (ii), we deduce that

-

j(Vu2 ' ©02 - v2L1 - V 1) dx -

i

Jasp 13(x)(u2g2 - ul01) du(x)

(229)

p(x)(f(u2)q52-f(ul)01)dx+A(u141-u202)dx, Ln

where 1 := S2 \ Sto. Let £1 > E2 > 0 and denote fl+(£1, E2) := {x E

: u2(x) + £2 > u1(x) + £1}. vi := (ui + £i)-l ((u2 + £2)2 - (ul + _1)2)+ , i = 1, 2.

Because vi can be closely approximated in the H1 fl L°° topology on Sl \ S2o by nonnegative C2 functions vanishing near a52o, it follows that (2.29) holds for vi taking the place of qi. Because vi vanishes outside the set SZ+(£1, 162), relation (2.29) becomes

-J S2+(El, Ea}

J

(vu2 vv2 - Vu1 Vvj) dx - f O(x)(u2V2 - u1v1) d r(x) re f p(x)(f(u2)v2 - f(ul)vl)dx + a J (u1v1 - u2v2)dx. (2.30)

A straightforward computation shows that the first integral in the left-hand side of (2.30) equals

)n+(1.2)

vu2 -

U2 + £2 u1 + £1

2

Vul

2) +61 vu1 - u1 + vu2 dx 0, the second term on the left-hand side of (2.30) converges to 0. Also, the first term in the right-hand side of (2.30) converges to

54


f(u2) U2

f(oi)l (u2-ui1) dx ul )

whereas the other term converges to 0. Hence, we avoid a contradiction only in

the case that SZ+(0, 0) has measure 0, which means that ul > u2 in Q. This concludes the proof.

We start the study of (2.28) with the following auxiliary result.

Proposition 2.5.2 Assume that (f2), (f3) hold and p > 0 on %1. Then, for any positive function 0 E C2'7(8920) and A E ]R, the problem

in 0,

Au + Au = p(x) f (u) Bu = 0

on 892,

u = q5

on 8920

(2.31)

has a unique positive solution.

Proof By Lemma 2.5.1 we find that problem (2.31) has at most one positive solution. To prove the existence of a positive solution to (2.31) we use the suband supersolution method. Let w cc 920 be such that the first Dirichlet eigenvalue of (-0) in the smooth domain 920 \ w is greater than A. Let q E C°°7(92) be such that q(x) = p(x) q(x) = 0 q(x) > 0

in Sl \ 920,

in NO \ w,

in w.

By virtue of Proposition 2.3.7, the problem

Av +.v = q(x)f(v)

v=1

in 11,

on O1

has a unique positive solution v. Let us choose 921 and S22 as two subdomains of 52 such that 920 cc 921 C C 922 C C 52.

Define w E C2(Sl \ 92o) so that

w=1inS2\922i w=vinSi1\920, and let m := min,-ono w > 0. We claim that Mw is a supersolution of problem (2.31), provided M > I is large enough. We first remark that for all M > 1 we have

-A(Mw) = )Mv - Mq(x) f (v) > A(Mw) - p(x) f (Mw)

(2.32) in S21 \ 920.


55

Let a := supn\n, (Aw + tlw). By Remark 2.3.1, there exists c > 1 such that

f (mM) >

Ma

min?j,nl p

for all M > c.

Therefore, for all x E I2 \ S2i and M > c we have

p(x) f (Mw) > (min p) f (mM) > M(Aw + ow), Q\nl

which can be rewritten as

-A(Mw) > A(Mw) - p(x) f (Mw) for x E I l \ aj and M > c.

(2.33)

By (2.32) and (2.33) it follows that

-O(Mw) > A(Mw) - p(x) f (Mw) in 0 \ Sto, for any M > c. On the other hand,

B(Mw) > Mmin {1, min 3(x)} > 0 on 5 1, for every M > 0. xEaQ

Now the claim follows by taking M > max {maxan O/m; c}. In view of Theorem 1.4.1, the boundary value problem 1©u = IA(u + IIpIIoof (u)

u>0 u=0

in 52,

on 090,

(2.34)

onOco

has a unique nonnegative solution u such that u > 0 in I2 \ S2o. Because u = 0

on OR we find that Ru = 8u/8n < 0 on 852. It is easy to see that a is a subsolution of (2.31) and it < Mw in S2\S2v for M large enough. The conclusion

of Proposition 2.5.2 follows now by the sub- and supersolution method. This ends the proof.

Corollary 2.5.3 For m > 1 sufficiently large, set

Q,:= {x E Q: d(x) < 1/m}.

(2.35)

If io is replaced by Ilm, then the statement of Lemma 2.5.2 holds.

Proof The proof is very easy in this case. The construction of a subsolution is made as before. As a supersolution we can choose any number M > 1 large enough.

We are now ready to prove the main result of this section.

Theorem 2.5.4 Let (f 2) and (f4) hold. Then, for any A E R, problem (2.28) has a minimal (respectively a maximal) positive solution UA (respectively Ua).


56

Proof We first prove the existence of the minimal positive solution for problem (2.28). For any n > 1, let un be the unique positive solution of Au + Au = p(x) f (u) Bu = 0

u=n

in 52,

on 852, on r3S2o.

By Lemma 2.5.1, un(x) increases with n for all x E 52\S2o. Moreover, the following

result holds true. Lemma 2.5.5 The sequence (un(x))n>1 is bounded from above by some function V (x), which is uniformly bounded on all compact subsets of fl \ S2o.

Proof Let q be a C2 function on 0 \ S2o such that

0 d(x) is a C2 function in the set Qs = {x E S2 : 0 < d(x) < S}. Then, for x e S2b we can define t

jd(x)

q(x) = J 0

min b(z) ds dt.

Jo

d(z)>s

Let g E 9 be a function such that (As) holds. The existence of g is guaranteed

by Lemma 2.4.3. Because q(x) --p 0 as d(x) \ 0, by virtue of hypothesis (f 1) and Remark 2.4.1, there exists b > 0 such that for all x E SZ with 0 < d(x) l we have g'(q(x)) inf (Aq(x))+A g(q(x)) (2.36) q(x)f(Mg(q(x))) > sup 1Vq(x)I2+ Mg"(q(x)) 9"(q(x)) g"(q(x)) \00 SE\c2o

Here, 8 > 0 is taken sufficiently small so that g'(q(x)) < 0 and g"(q(x)) > 0 for all x with 0 < d(x) < J. For no > 1 fixed, define V* as follows: (i) V* (x) = uno (x) + 1 for x E Si and near 852;

(ii) V*(x) = g(q(x)) for x satisfying 0 < d(x) < b; (iii) V* E C2 (?j \ ao) is positive in 52 \ Sta.

We show that for M > 1 large enough the upper bound of the sequence (un(x))n>1 can be taken as V := MV*. Because on851 and

lim (un(x) - V(x)) = -oo < 0,

d(x)\0


57

to conclude that n, < V in 52 \ lio it is sufficient to show that

-AV -AV +p(x)f(V) > 0

in S2\Sto.

(2.37)

For x E Q satisfying 0 < d(x) < 5 and M > 1 we have

-AV -,1V +p(x)f(V) =M( -Og(q(x)) - Xg(q(x)) + p(x)f (g(q(x))) )

> Mg"(q(x))Oq(x) (-::(x) ) - JVx)12 g(q(x))

-

grr(q(x))

f(Mg(q(x))) q(x)

Mg"(q(x))

By virtue of (2.36) we thus derive

-AV- AV + p(x) f (V) > 0

for all x E 5l \ Qo with 0 < d(x) < b.

For x E n satisfying d(x) > b we also have

-AV - AV + p(x)f (V) = M

(_v-

AV* + p(x)

\

f (V *) I > o,

M

//

for M > 0 sufficiently large. It follows that (2.37) is fulfilled provided M is large 0 enough. This finishes the proof of the lemma. We now come back to the proof of Theorem 2.5.4. By Lemma 2.5.5, UA(x) :_ u, (x) exists, for any x E 12 \ S1o. Moreover, Ua is a positive solution of problem (2.28). Using Lemma 2.5.1 once more, we find that any positive solution

u of (2.28) satisfies u > u on Sl \ 1o, for all n > 1. Hence U,, is the minimal positive solution of problem (2.28). To achieve the existence of the maximal positive solution to problem (2.28), we use essentially the same argument that we use earlier.

Lemma 2.5.6 If S'1o is replaced by defined in (2.35), then problem (2.28) has a minimal positive solution provided that (f2) and (f3) are fulfilled.

Proof The argument used here, which is easier because p > 0 on S2 \ Q,,,, is similar to that in the proof of Lemma 2.5.5. The only difference that appears here (except the replacement of Qo with St,,,) is related to the construction of V*(x) for x near BST,,,. For this purpose, instead of Lemma 2.4.3 we use Theorem 2.3.6,

which says that for any A E R there exists a positive blow-up solution u, of (2.14) in the domain S2 \ 52,,,. We next define V * (x) = u, (x) for x E ) \?!,,, and near 49Q,,,. For M > I and x E S2 \ 52,,, near on,, we have

-AV - AV + p(x) f (V) = -MOV* -,MV* + p(x)f (MV*) = p(x) (f (MV*) This completes the proof of our lemma.

- Mf (V*)) > o. El


58

Let v,,,, be the minimal positive solution of the problem considered in the statement of Lemma 2.5.6. It is easy to see that vm > vm+1 > u in 1l\S1,n, where u is any positive solution of (2.28). Hence U),(x) := lim,n_,, vm(x) > u(x), for all x E S2\S2o. A regularity and compactness argument show that U,\ is a positive blow-up solution of (2.28). Consequently, Ua is the maximal positive solution. 0 This concludes the proof of Theorem 2.5.4.

2.6

Uniqueness of blow-up solution

We establish in this section a general uniqueness result for problem (2.28) under

suitable conditions on p and f. Assume that there exists a positive increasing function k E C' (0, So) for some bo > 0 such that p satisfies the following hypotheses: (p3)

p(x) = c for some constant c > 0. lim d(x)\G k(d(x))

(p4) K(t) _

jo

k(s} ds

Also assume that there exist ( > 0 and to > 1 such that

(f 5) f (gt) < '+5 f (t), for all (f6) The mapping (0,1]

E (0,1) and all t > to /e-

l; H A(C) = limt,,,.

f {£t}

is continuous and posi-

tive-

Remark that limt-,,,, f (t)/tl+S exists and is positive because the mapping t r-a f (t)/t1+S is nondecreasing in a neighborhood of infinity. Thus, the hypotheses (f 1) and (f5) imply (f 2). By (f6) we derive that A can be continuously extended in the whole (0, oo) by setting A(1/i;) = for all l; E (0, 1). Moreover, we have the following lemma.

Lemma 2.6.1 The function A : (0, oo) -* (0, oo) is bijective, provided that (f5) and (f6) hold.

Proof Because A is continuous and 1/A(C) for all t; > 0, the surjectivity of A follows at once if we show that limt\o A(C) = 0. To this aim, let l; E (0, 1) be fixed. Using (f 5) we find

fW)

t

which yields A(C) < 6C. Because C E (0, 1) is arbitrary, it follows that 0.

A(C) _

Uniqueness of blow-up solution

59

We next show that the function f ,) A(l) is increasing on (0, oo), which concludes the proof of our lemma. Let 0 < tt < C2 < oo be chosen arbitrarily. Using assumption (f5) once more, we obtain

Mit} = f 1

2t < (

f (62t)

2}

for all t > to S1

It follows that

AGO Slf(t)

0,

(2.39)

which shows that h" is positive in (0, 61) for some bl > 0. This concludes our

0 Theorem 2.6.3 Assume that conditions (f 1) through (f6), a # 0, (p3), and proof.

(p4) hold. Then for any A E R, problem (2.28) has a unique positive blow-up solution Ua. Moreover, UA(x)

d(x)N0 h(d(x)) _ o> where h is defined by ds Fh(t)

= ft

2F(s)

k(s) ds

for all 0 < t
1. (ii) f (t) = tp ln(t + 1), p > 1. (iii) f (t) = tp arctan t, p > 1. (iv) f (t) = t lnv(t + 1), p > 2.

Remark 2.6.5 Assume that f satisfies (f3) and (f5). Then equation (2.14) with A = 0 and p - 1 has a unique blow-up solution ii. Moreover, u has the following asymptotic behavior:

u(x)

lim

dist (x,812)-.0 r(dist (x, ml))

= 1,

where F is the function defined as ds J

Ji(t)

2F(s))

=t

for all t > 0.

Let 521 CC S2 be a connected subdomain with a smooth boundary and such that SZo C 01. A direct consequence of Theorem 2.6.3 is seen in the following corollary.

64


Corollary 2.6.6 Let (f 6) be added to the assumptions of Remark 2.6.5. Then, for any A E R, problem (2.28) with p - 1 on 8521 and no replaced with S21 has a unique positive solution Ua. Moreover, U,, behaves on 8521 exactly in the same manner as is on 812-that is, lim Uj (x) =1 dirt (x,aSZ,)-'o F(dist (x, 8121))

Proof We use the argument of Lemma 2.5.6 to deduce the existence of a positive solution for the problem considered here. Concerning the uniqueness, let us

remark that conditions (p3) and (p4) are fulfilled by taking c = 1 and k - 1 on (0, oo). It follows that h defined by (2.40) coincides with F. Notice that

F'(t) _ - 2F(I'(t)) and F"(t) = f (F(t)) for any t > 0. Thus, we obtain r E G (without calling Lemma 2.6.2) and II(C) = A(C), for all C > 0. So, by Lemma 2.6.1, II : (0, oo) -* (0, oo) is bijective. From now on, we proceed as in the proof of Theorem 2.6.3, which only requires us to replace h with r and go El with 521. This concludes the proof.

2.7 A Karamata theory approach for uniqueness of blow-up solution In this section we provide a different approach for the uniqueness of a positive blow-up solution of the logistic equation. We are concerned with the following boundary blow-up problem: Du + Au = p(x) f (u)

in 52,

u>0

inc,

2u=oo

(2.45)

on 852,

where A E R, p E C°'7(12) (0 < y < 1), such that clo := int {x E SZ : p(x) = 0}

is smooth (possibly empty), S2° C 52, and p > 0 in Sl \ 52o. The uniqueness of the solution to (2.45) will be achieved using the Karamata regular variation theory, originally introduced by Karamata [1131. We start this section with the following definition.

Definition 2.7.1 A positive measurable function f : [m, co) -+ (0, oo), m > 0, is called regularly varying (at infinity) with index p E R if lim f (it) t-.c f (t)

= l;p

for all Z; > 0.

When the index of regular variation p is zero, we say that the function f is slowly varying.

A Karamata theory approach for uniqueness of blow-up solution

65

We denote by IIBp the class of regular varying functions with index p E R. The canonical p-varying function is f (t) = tp, t > 0. The functions

ln(1 + t), In ln(e + t), exp{(lnt)a} (0 < a < 1), vary slowly, as well as any measurable function on [m, oo), m > 0, with a positive limit at infinity. Furthermore, any function f E IR, can be written in terms of a slowly varying function. Indeed, set f (t) = tpg(t). From the previous definition we conclude that g varies slowly. The basic properties of regular and slowly varying functions are summarized next. For further details, we refer the reader to Seneta [180].

Proposition 2.7.2 We have the following: (i) For any slowly varying function g : [m, oo) -' (0, oo), m > 0, and any q > 0 we have t9g(t) --+ oo and t-4g(t) --* 0 as t --+ oo. (ii) Any positive function g E C'([m, oo)) satisfying tg'(t)/g(t) -> 0 as t -9 00 is slowly varying. Moreover, if the previous limit is p E R, then g E R. (iii) Assume f : [m, oo) - (0, oo), m > 0, is measurable and locally integrable. Then f varies regularly if and only if there exists q E R and Q := lim

t--.oo

t4+1 f(t)

fm sa f (s)ds

> 0.

In this case, f E lRp with p = t:- q - 1; (iv) If f E Rp is Lebesgue integrable on each finite subinterval of [m, oo), then

for all q > -p - 1 we have e+1 f(t)

lim

= q + p + 1. f,t sa f(s)ds

We also have the following equivalence for Cl functions.

Lemma 2.7.3 Let f E C' 10, oo) be a nonnegative function that fulfills (f 3).

t

Then the following conditions are equivalent:

(i) f' E Rp for some p E R. {iz)

There exists ?9

iim

(iii) There exists a := limt-,,,

tf'(t} < 00. f (t)

Ff )j

(t) > 0.

Moreover, p > 0 and a = 1/(p + 2) = 1/(r9 + 1).

Remark that if f' E R. with p > 0, then limt, f (t)/tP = oo, for all 1 0. Also notice that (f 2)

may fail when p = 0. This is illustrated by f (t) = t or f (t) = t ln(t + 1), t > 0.

Blow-up solutions for sernilinear elliptic equations

66

Proof

Let f E RP. Then f'(t) = tPg(t) where g varies slowly. If p < 0 then, by Proposition 2.7.2 (i) and 1'Hospital's rule we find limt_,"' f (t) /t = limt-,,,, f'(t) = 0, which contradicts (f 3). Fence p > 0 and by Proposition 2.7.2 (iv) (with q = 0) we obtain limt_,,,. t f'(t)/ f (t) = p + 1. (ii)=(i). Using the assumption (f 3) we have t f'(t) > f (t) for all t > 0, which yields 19 > 1. By Proposition 2.7.2 (iii) we now obtain f E Iti9_1. By l'Hospital rule we deduce 1imr ,o F(t)/(t f (t)) = 1/(1+19) and thm00(.f),=1-tlm°°Ff2(/}t)

F(t) tf'(t) = 1 - lim t- o tf(t)

f(t)

1

1+r9 Hence or = 1/(1 -{-19).

(iii)=(ii). From limt_,, (F/ f )' (t) = a > 0 we have a < 1 and tli}m 00

F

f2{t) t)

= 1 - a.

(2.46)

Let us choose to > 0 such that (F/ f )' (t) > a/2, for all t > to. Then,

F(t) > (t - to)a + F(to) f (t)

2

for all t > to.

f (to)

Passing to the limit with t -> oo, we find limt-. F(t)/f (t) = oo. By l'Hospital's rule we deduce tl moo

tf(t) _

1

F(t)

a

(2.47)

Multiplying (2.46) and (2.47) we obtain limt , t f'(t)l f (t) = (1-a)/a. According to Proposition 2.7.2 (iii), it follows that f' E R p, where p = (1 - 2a)/a. This completes the proof. 0 Inspired by the definition of a, we denote by K the Karamata class, consisting of all positive, increasing C' functions k defined on (0, v), for some v > 0, which satisfy f t k(s)ds l i m ( ° k(t) 2Z , i = 0,1. We observe that Po = 0 and Li E [0, 11, for every k E K. Our next result gives examples of functions k E K with limt\o k(t) = 0, for every f, E [0, 1].

Lemma 2.7.4 Let f Cr C' [m, oo), m > 0, be such that f' E RP, for some p > -1. Then we have the following:

(i) If k(t) = exp{- f (1/t)}, 0 < t < 1/m, then k E K with f, = 0.


67

(ii) If k(t) = 1/f(1/t), t < 1/m, then k E K with tj = 1/(p+2) E (0,1). (iii) If k(t) = 1/ In f (1/t), 0 < t < 1/rn, then k E K with el = 1. Proof Because p > -1, from Proposition 2.7.2 (i) and (iv) we have

lim tf'(t) = 00

t-.oo and

tf,(t) = p+ 1 > 0. f(t)

lim

(2.48)

Therefore, in any of the cases (i), (ii), or (iii) we have limt\,o k(t) = 0 and k is a Cl increasing function on (0, v), for v > 0 sufficiently small. (i) Using (2.48) we have

\

_

tk'(t)

1/tf'(1/t) = _(P+1).

to k(t) In k(t) to \,

t f (1/t)

Thus, by ('Hospital's rule we obtain

tk(t)

tk(t)/ In k(t) fo k(s)ds (tk'(t) + k(t)) In k(t) - tk'(t)

= lira urn tNO In k(t) fo k(s)ds tNo t\O

_

k(t) In2 k(t)

( tk'(t) 1 tNo t k(t) In k(t) + In k(t)

tk'(t)

- k(t) In2 k(t)

_ -(p+ 1). Hence,

lim

t\a

(fo k(s) ds) k'(t)

-iim In k(t) f,) k(s)ds tk'(t) t\o

k2(t)

tk(t)

k(t)

1,

which implies

el = 1 - t\o

(f o k(s)ds) k(t) k2(t)

= 0.

(ii) It is easy to see that tk(t)) two

=

By 1'Hospital's rule we obtain

_ 21-1

1/f(1/t)t) t\o

t\o

=p+

1.

fo k(s)ds/(tk(t)) = 1/(p + 2). Thus,

ff k(s)ds tk'(t) kt() tk(t)

P+2


68

(iii) In a similar manner we have limt\o tk'(t)/k2(t) = p + 1. By 1'Hospital's rule, limtNo fo' k(s)ds/(tk(t)) = 1. Thus, £1 = 1 -limo

-

tk'(t)

fo k(s)ds t

k2(t) = 1.

0

This concludes the proof.

As a consequence of Proposition 2.7.2 and Lemma 2.7.4 we obtain the following corollary.

Corollary 2.7.5 Let f E Cl [m, oo), m > 0. Then f E RF with p > -1 if and only if there exist q > 0, C > 0, and B > D such that

f(t)=Ctgexp{f' ytt)dtI

for alit

B,

(2.49)

111

where y E C[B, oo) satisfies limt_,,,,, y(t) = 0. In this case, f' E R,, with p = q- 1. The core result of this section is the following.

Theorem 2.7.6 Let f E C' [0, oo) be such that (f3) holds and f E R p with p > 0. Assume that p - 0 on 892 satisfies p(x) = c k2(d(x)) + o(k2(d(x)))

as d(x) \,0,

for some constant c > 0 and k E 1C. Then, for any A E (-oo,A,,), problem (2..45) has a unique blow-up solution UA. Moreover,

U,\ () d()NO h(d(x))

0 =

Jh(t)

p) I

,

(2.50)

+

and h is defined by

/°°

\ t/p

2

when eo = G (2

ds

2F(s)

I

= f t k(s)ds

for all t E (0, v).

(2.51)

By Corollary 2.7.5, the assumption f E Rp with p > 0 holds if and only if f satisfies (2.49) with q = p + 1 and for some B, C > 0. If B is large enough (for instance, if y > -p on [B, oo)), then f (t)/t is increasing on [B, oo). Thus, to obtain the whole range of functions f for which Theorem 2.7.6 applies, we have only to "paste" a suitable smooth function on [0, B] in accordance with condition

(f3). A simple way to do this is to define f(t) = tp+'exp{ fo z(s)/sds}, for all t > 0, where z E C[0, oo) is nonnegative and such that limtNo z(t)/t E [0, oo) and limtN,,. z(t) = 0. Clearly, f (t) = t", f (t) = t' ln(t+ 1), and f (t) = tQ arctan t (t > 1) fall into this category. Lemma 2.7.4 provides a practical method to find functions k which can be considered in the statement of Theorem 2.7.6. Here are some examples: k(t) =


69

exp {-1/ta}, k(t) = exp {- ln(1 + 1/t)/ta}, k(t) = exp {- [arctan (1/t)] /tQ }, k(t) = -1/ In t, k(t) = to'/ ln(l + 1/t), k(t) = ta, for some a > 0. As we have already seen in the previous section, the uniqueness lies upon the crucial observation (2.50), which shows that all explosive solutions have the same boundary behavior.

Proof Fix A E (-oo,.1,1). By Theorem 2.3.6, the problem (2.45) has at least one blow-up solution.

The uniqueness will follow at once if we prove that (2.50) holds for any solution u), of (2.45). Indeed, if ul and u2 are two arbitrary blow-up solutions of (2.45), then (2.50) yields limd(x)\au1(x)/u2(x) = 1. Hence, for any 0 < e < 1, there exists S = S(E) > 0 such that (1 - E)u2(x) < u1 (x) < (1 + E)u2(x),

(2.52)

for all x E S2 with 0 < d(x) < S. Choosing a smaller S > 0 if necessary, we can assume that 1 o C C5, where Ca :_ {x E 12 : d(x) > S}. It is clear that ul is a positive solution of the boundary value problem in Cs, on 8Cs.

J AO + Aq _ p(x)f (0)

4=u1

(2.53)

By (f3) and (2.52), we see that (1 - e)u2 is a subsolution and 0+ (1 + 6)u2 is a supersolution of problem (2.53). By the sub- and supersolution method, (2.53) has a positive solution 01 satisfying 0- < ¢1 < ¢+ in Ca. Because p > 0 in Ca \ 12o, by Lemma 2.3.7 we derive that (2.53) has a unique positive solution, (i.e., ul - 01 in C5). This yields (1 - -)u2 (x) < ul (x) < (1 + '-)U2 (X) in C5, so that (2.52) holds in Sl. Passing to the limit with e -* 0, we conclude

that ul - u2. To establish (2.50) we first state some useful properties about h.

Lemma 2.7.7 We have (i) h E C2(0, v) and limt-,o h(t) = oo; (ii) for all

> 0 there holds limt\,o

k2(t) f (h(t)t;)

p+l 2 + pl ,

so, h" > 0

on (0, 26), for S > 0 small enough;

(iii) limt\,o h(t)/h"(t) = limt\o h'(t)/h"(t) = 0. Proof (i) Follows directly from (2.51). (ii) Because f E Rp+1, it suffices to prove the claim for C = 1. Clearly h'(t) = -k(t) 2F(h(t)) and

h (t) = k 2(t) f (h(t))

(1_2

k'(t) (fo k(s)ds)

k2(t)

F(h(t))

f(h(t))

(t f 00)

F(s)ds

)

'

(2.54)


70

for all 0 < t < v. It is easy to be seen that limt-,,,,

F(t)/ f (t) = 0. Thus, from

1'Hospital's rule and Lemma 2.7.3 we infer that

t-lim'I f (t) f' F(s)-1/gds

2

p 2(p + 2)'

-a

(2.55)

Using (2.54) and (2.55) we derive (ii) and also

F(t) h'(t) = -2(2 + p) . lim fo k(s)ds lim k(t) t-->oo f(t) f too t\o t\o h"(t) 2 + tip F(s)ds lim

(2.56)

-peo =0. 2+tlp

From (i) and (ii) we derive that limt\,o h'(t) = -oo. So, l'Hospital's rule and (2.56) yield limtNa h(t)/h'(t) = 0. This and (2.56) lead to limt\,o h(t)/h"(t) = 0, which proves (iii). This completes the proof of the lemma. Let us come back to the proof of Theorem 2.7.6. We fix e E (0, c/2). Because p =- 0 on aQ, we may take 6 > 0 such that

(i) d(x) is a C2 function on the set {x E RN : d(x) < 25}; (ii) k2 is increasing on (0, 28);

(iii) (c-e)k2(d(x)) 0, and g(t) _

bkt)'

if t > 0-

j=1

(ii) f (t) = (1 + t2)a/2 and g(t) = (1 + t2)a/2 for t E R, with a, 6 > 0 and

a,3 0 such that a/3 0, we find

Entire solutions blowing up at infinity for elliptic systems

82

ul (r) < a+

J0

r

(0) +

ti-N f isN-lp(s)g(b(s)) ds dt

jt1j 0

s(p+q)(s)(f +g)((s))dsdt

=,O(r)Thus, ul < 9fi. It follows that i

vl(r) 0 and k > 1.

Thus, {(uk, Vk)}k>i converges and (u(r), v(r)) := limk-0(uk(r), vk(r)), r > 0 is a radially symmetric solution of (3.1) with central value (a, b). This completes the proof. A direct consequence of Lemma 3.2.4 is the following corollary.

Corollary 3.2.5 If (a, b) E 9, then (0, a} x (0, b) C g.

Proof The process used before can be repeated by taking uk(r) = ao + vk(r) = b0 +

J0

r ti-N r ti_ N

J0

t SN-ip(s)g(vk_i(s)) ds dt

for all r > 0,

t $N-iq(s) f (u (S)) ds dt

for all r > 0,

J0 J0 where 0 \n(x)g (u 2 v)

in RN.

On the other hand, ((x) -f oo as jxj -> R and u,,v E C2(RR). Thus, by the weak maximum principle, we conclude that u + v < Sin BR. But this is impossible O because u(O) + v(0) = a + b > ((0). This finishes the proof.

Lemma 3.2.7 F(G) C G. Proof Let (a, b) (=- F(G) and no > 1 be such that no min {a, b} > 1. We claim

that

(a - 1/n, b- 1/n) E G

for all n> no.

Indeed, if this is not true, by Corollary 3.2.5 we find

D := (a - 1/n, oo) x [b - 1/n, oo) C (R+ x R+) \ G, for some n > no. Hence, we can find a small ball B centered in (a, b) such that B CC D; in other words, B n G = 0. But this contradicts the choice of (a, b).

84


Consequently, for all n > no there exists (un, vn), a radially symmetric solution of (3.1) with the central value (a -1/n, b -1/n). Thus, for any n > no and r > 0 we have

+ J ti_N J

un(r) = a -

t

sN-1p(s)9(vn(s)) ds dt,

r0

vn(r) = b -

n

+

Jo

r r tl-N J t sN-1 q(s)f (un(s)) ds dt. 0

We observe that (un)n>n0 and (vn)n>n0 are nondecreasing sequences. Next, we prove that (un)n>n0 and (vn)n>n0 converge in RN. To this aim, let xo E RN be arbitrary. Because t is not identically zero at infinity, we may find R > 0 such that x0 E BR and rl > 0 on MR. Because o, = lim inft.,,,, f (t)/g(t) > 0, we find T E (0,1) such that

f (t) > Tg(t),

-

b f o r all t > a +

1

no

2

Therefore, on the set where un > vn we have 2 f(un)>f un+vn\

Tg(un+yn\ 2

lJ

Similarly, on the set where un < vn we have

>Tgfl\ un+yn 9(vn)>9 un+ynl 2 ) 2 /J Now it is easy to see that for all x E l[PN we have ©(un + vn) > Tr](x)g

Un + vn 1 2

On the other hand, by Theorem 2.2.2 there exists (E C2(BR) such that Ot; = TTl (x)9 G

(>0 (

oo

J

in BR, in BR, as IxI -r R.

The weak maximum principle yields un + vn < ( in BR. So, it makes sense to define (u(xo),v(xo)) := limn-,,(un(xo),vn(xo)). Because xo is arbitrary, the functions u, v exist on RN. Hence, (u, v) is a radially symmetric solution of (3.1) with central value (a, b)-that is, (a, b) E G.

For (c, d) E (R+ x R+) \ 9, denote by R,,d the supremum over r > 0 such that there exists a radially symmetric solution of (3.1) in B(0, r) so that (u(0), v(0)) _ (c, d).

Characterization of the central value set

85

Lemma 3.2.8 For all (c, d) E (R+ x R+) \ 9 we have 0 < Rc,d < 00-

Proof Because v > 0 and p, q E C[0, oo), there exists E > 0 such that (p + q)(r) > 0 for all 0 < r < E. Let 0 < R < E be arbitrary. Hence, there exists a positive radially symmetric solution of

ay'R = (p+q)(x)(1 + g) (OR) in BR, which blows up as jxj -+ R. Furthermore, for any 0 < r < R we have r

t

f s'-'(p + q)(s)(f + 9)(OR(s)) ds dt.

OR (r) = OR(0) + r tl-N 0

Obviously, 0R > 0. Thus, we find

0R (r) = rl-N J r sN_1 ('+ q)(s)(f + 9)(OR(s)) ds < C(f + g)(OR(r)), 0

where C > 0 is a positive constant such that f, ,(p + q)(s) ds < C. Because f + g satisfies the hypotheses (H1) and (H2), by Lemma 2.2.1 (ii) we derive. °°

dt

< oo.

J1

(f + 9)(t)

Therefore,

d

_

ds

dr +Grz(r) (f + 9)(s)

OR jr)

(f + 9)(')R(r))

C

for all 0 < r < R.

Integrating in the last inequality and taking into account the fact that VR(r)

oo as r / R, we obtain ds fo""O(O)

(f + 9)(s)

< CR.

Now, we let R \ 0 in the previous relation and we have ds lint W =0 R%04"'(0) (f + g)(s)

This implies that l'R(0) -+ oo as R \ 0. Thus, there exists 0 o are nondecreasing, and for all k > 1 there holds max{uk(r),vk(r)} < Op (r)

for all 0 < r < p.

Thus, for any 0 < r p > 0. By the definition of R,,d we also derive lim u(r) = oo

lim v(r) = oo.

and

r/&,d

rJ'Rte,d

(3.24)

we conclude that Rc d is finite. This

On the other hand, because (c, d) completes the proof.

U

The main result in this section is the following.

Theorem 3.2.9 Assume that v > 0, rl is not identically zero at infinity and (Hi), (H2), and (3.6) hold. Then any entire radially symmetric solution (u, v) of (3.1) with (u(0), v(0)) E F(CQ) blows up at infinity.

Proof Let (a, b) E F(G). By Lemma 3.2.7, (a, b) E G so that there exists (U, V), a radially symmetric solution of (3.1) with (U(0), V(0)) = (a, b). Obvi-

ously, for any n > 1, (a + 1/n, b + 1/n) E (R+ x R+) \ 9. By Lemma 3.2.8, Rn := Ra+1/n,b+l/n is a positive number. Let (Un, Vn) be the radially symmetric

solution of (3.1) in BR, with the central value (a + 1/n, b + 1/n). Thus, for all 0 < r < R, (Un,V,,,) is a solution of the system of integral equations r

U(r) = a +

t

+ Jl t1I SN-1p(S)g(Vn(s)) dsdt, n °jr

V(r) = b +

+

r(3.25)

t1-N J sN-1q(s)f (Un(s)) ds dt. o

In view of (3.24), we have

lira Un(r) = lim VV(r) = oo

rZRn

TZRn

for all n > 1.

We claim that the sequence (Rn)n>1 is nondecreasing. Indeed, if (uk)k>1 and (vk)k>1 are the sequences of functions defined by (3.23) with c = a + 1/(n + 1)

andd=b+1/(n+1), then for all k > 1 and 0 < r < Rn we have uk(r) < uk+1(r) < U,(r),

vk(r) < vk+1(r) < V,(r).

(3.26)

This implies that (uk(r))k>1 and (vk(r))k>1 converge for any 0 < r < R. vk) is a radially symmetric solution of (3.1) in BR with the central value (a+1/(n+ l ), b+l/(n+l)). By the definition of Rn+1, it follows that Rn{1 > Rn for any n > 1. Moreover, (Un+1, Vn+1) :=

Characterization of the central value set

87

Set R := Rn and let 0 < r < R be arbitrary. Then, there exists nt .n1(r) such that r < Rn for all n > n1. From (3.26) we derive that Un+1 n1. Thus, for all 0 < r < R we can define and Vn+1

U(r) := lim Un(r) n-.oo

V(r) := Iim Vn(r). n-roo

and

(3.27)

Because UU (r) > 0; from (3.25) we find

Vn(r) < b + _ + f (Un(r)) fo"O t1-N r tsN_lq(s) ds dt. n

0

This yields

for all 0 < r < R,

VV(r) < C1Un(r) + C2 f (Un(r))

(3.28)

where C1 is an upper bound of (V(0) + 1/n)/(U(0) + 1/n) and C2 =

f 0o t1-N [ sN-lq(s) dsdt
0. It is easy to see that h satisfies the hypotheses of Lemma 2.2.1. Hence, we may define

r(t) = /

forallt>0.

ds)

Notice that Un verifies

AU, = p(x)g(Vn)

in BR.,

which combined with (3.28) implies AU,, < p(x)h(UU)

in BR,,.

A simple computation yields

Ar(Uu) = r`(Uu)AUu + rl"(Un)IvUnl2 h(UU)

AU,

IVUn12

+ (h(Un))2

> h(I p(x)h(Un)

= -p(x)

in BR,,.

Therefore,

[ rN-1 L r(Um)

Ir

>

-rN-ip(r)

for any 0 < r < R,

88


Fix 0 < r < R. Then r < Rn for all n > n1, provided n1 is large enough. Integrating the previous inequality from 0 to r we deduce

drr(Un) > -r 1-N J0r sN-1p(s)ds

for all 0 < r < Rn.

Integrating the last inequality over [r, Rn - s] we obtain

-e

s)) - r(Un(r)) > -

Jr

ti-N f t sN-lp(s) dsdt

for all n > nl.

0

Recall that UU(Rn - s) -> oo as s --+ 0 and r(t)

0 as t -+ oo. Therefore, passing to the limit with s -> 0 in the last inequality we deduce that rRrt

r(Un(r)} _
ni.

0

By letting n - co in the previous relation, we obtain R

r(U(r)) < f t1-N fo s N-lp(s) ds dt

for all 0 r-1 (J R tl-N f t sN-lp(s) ds dt) r

for all 0 < r < R.

0

Passing to the limit as r / R, and using the fact that limt,,o r-1(f.) = oo, we deduce R

lim U(r) > lim r-1

r/R

r/R

J$Np(s) ds dt = oo.

t1-N

r

But (U, V) is an entire solution of (3.1), so that we conclude R = oo and limr__,,,, U(r) = oo. Using (3.6) and the fact that V'(r) > 0 we find 00

U(r) < a + 9(V (r)) fo t1-N f'SN- ip(s) ds dt

< a + g(V(r)) N

1

2

J0

tp(t) dt,

for all r > 0. This last estimate leads to limr_. V (r) = oo. Consequently, (U, V) 0 is an entire blow-up solution of (3.1). This concludes the proof.

Comments and historical notes

89

3.3 Comments and historical notes We have provided in this chapter a characterization of the central value set of radially symmetric solutions to a nonlinear elliptic system of logistic type. Our study has pointed out the role played by the Keller-Osserman condition in this characterization. First, if the nonlinear terms f and g have almost sublinear growth, then the central value set consists of entire R+ x R+, and the existence of radially symmetric solutions blowing up at infinity depends on the growth of potentials p and q. In turn, if f and g are comparable, in the sense that lim inft, f(t)/g(t) E (0, oo), and satisfy the Keller-Osserman condition at infinity, then the central value set is bounded. Furthermore, any element on the positive boundary of c-that is, on 8q fl (R+ x R+),--is a central value of an entire blow-up solution. The results in this chapter are the work of Cirstea and Radulescu [52]. We also refer to Lair and Shaker [126] for the particular case of pure powers in the nonlinearities.

In Part II of this volume (Chapters 2 and 3) we have been concerned with boundary blow-up solutions of some classes of nonlinear elliptic equations and systems. Our approach was essentially based on the maximum principle, in combination with other ingredients, such as elliptic estimates, regularity arguments, and differential equations techniques. In all the results we have established in these two chapters, we studied only positive solutions, especially because of the physical meaning of the corresponding unknowns. A different approach was developed by Aftalion and Reichel [1], who argued the existence of multiple boundary blow-up solutions of the problem

Au = f (u)

in 1?,

where Q is a bounded and convex domain. Taking into account the growth of f, they distinguished two distinct situations: (i) f is a sign-changing nonlinearity. In such a case there is both a positive and a sign-changing blow-up boundary solution. (ii) inf f > 0. In this case, Aftalion and Reichel [1] considered the bifurcation problem .

Au = A f (u)

in cZ

(3.29)

and showed that there is some critical value A* > 0 such that problem (3.29) has blow-up boundary solutions if and only if 0 < A < A*.

PART III

ELLIPTIC PROBLEMS WITH SINGULAR NONLINEARITIES

4

SUBLINEAR PERTURBATIONS OF SINGULAR ELLIPTIC PROBLEMS In mathematics the art of proposing a question must be held of higher value than solving it. Georg Cantor (1845-1918)

From now on, we are interested in the qualitative analysis of solutions to semilinear elliptic equations or systems involving singular nonlinear terms, such as u_a (with a > 0) or gradient terms like IVula (with 0 < a < 2). We are concerned with existence and uniqueness properties, but also with the asymptotic behavior of solutions. A special feature will be played by the influence of one or several real parameters, which usually are referred as bifurcation parameters. The term bifurcation is one that is also used in topological dynamics and catastrophe theory. In such cases, it is usually considered a family of functions or vector fields dependent on parameters. The associated bifurcation values of parameters are those in which the system is not actually stable, in the sense that small variations of the parameters change the topological nature of the collection of orbits. Our setting in this volume is quite closely related to the study of stability phenomena, by means of the first eigenvalue of the associated linearized operator. Singular problems arise in the study of non-Newtonian fluids, boundary laver phenomena for viscous fluids, chemical heterogeneous catalysts, as well as in the theory of heat conduction in electrically conducting materials. The associated singular stationary or evolution equations describe various physical phenomena. For instance, superdiffusivity equations of this type have been proposed by de Gennes [60] as a model for long-range Van der Waals interactions in thin films spreading on solid surfaces. Singular equations also appear in the study of cellular automata and interacting particle systems with self-organized criticality (see [39]), as well as to describe the flow over an impermeable plate (see 138]). 4.1 Introduction This chapter is devoted to the study of elliptic problems of the following type:

1-Du = 4)(x, u, A)

in Sl,

u>0

in D,

U=0

on %I

(4.1)

94

Sublinear perturbations of singular elliptic problems

in a smooth domain Sl C RN (N > 1), where : Si x (0, oo) x 1W -+ R is a Holder continuous function. The main feature of this chapter, as well as of the following

ones, is that we allow 4i to be unbounded near the origin with respect to its second variable-that is, 4P is a singular nonlinearity. Therefore, the singular character of our problem lies not in the prescribed behavior of the solution at the boundary, as we required in Chapters 2 and 3, but in the nonlinearities that govern problems of type (4.1). To make these arguments more transparent, let us present a simple example. Consider the problem in Q, I Du = up

u>0

in 9,

U=+00

on 992.

If p > 1, by Theorem 2.1.3, the previous problem has a solution u. By the change of variable u = 1/v we derive that v satisfies

-AV = v2-p -

2 1ov12

in 9,

v>0 v=0

in Q,

on00.

This equation contains both singular nonlinearities, like v-1 or v2-p (if p > 2), and a convection (gradient) term, denoted by 1 Vv12. The influence of the gradient term in problems like (4.2) will be emphasized in Chapter 9.

By classical solution of problem (4.1) we mean a function u E C2(S1) n C(SC) that satisfies pointwise (4.1). Because of the presence of the singular term D (x, u, A), solutions to (4.1) are not C2(SZ) functions. However, in some particular cases, which will be discussed herein, the regularity of classical solutions to (4.1) can be improved up to the class u E C1'7(SZ) and Au E L1(Sl). The simplest model that falls within the theory we develop in this chapter is

Au = g(u) u>0 u=0

in Q,

in 1, on BSl,

where 1 C Rl'(N > 1) is a smooth bounded domain and g c- C1(0, oo) is a decreasing function such that limt\o g(t) = no. Existence of a classical solution to (4.3) follows directly from Theorem 1.2.5. In particular, if SZ = (0, 1) C IW then there exists a unique y E C2(0,1) n C[O,1] such that -y"(t) = g(y(t)) 0 < t < 1, y(t) > 0 0 < t < 1, y(0) = y(1) = 0.

Introduction

95

A surprising result is that boundary estimates for the solution it of (4.3) in higher dimensions are expressed in terms of y. More exactly, setting d(x) dist(x, &S2), we have the following result.

Theorem 4.1.1 Problem (4.3) has a unique solution it E C2(11) n C(a) and there exist c, m, M > 0 such that my(cd(x)) < u(x) < My(cd(x))

for all x E Sl,

(4.5)

where y is the unique solution of (4.4). Moreover, if g E L1(0, 1), then the solution of problem (4.3) verifies cid(x) < u(x) < c2d(x)

for all x E 9

(4.6)

for some positive constants ci and c2.

Proof Because y is concave, there exists y'(0+) := limt\o y(t) E (0, oo]. Thus, we can find 0 < a < 1 such that y' > 0 in (0, a). That is, y is strictly increasing on (0, a). From Corollary A.1.3 in Appendix A, there exist two positive constants, Ci and C2, such that Cicpi(x) < d(x) < C2Soi(x)

in Q.

Let us take c > 0 such that cC2coj < a/2 in Q. We first prove that there exists M > I large enough such that u := My(ccpi) is a supersolution of (4.3). Indeed, we have

-Llu = Mc2]DVij2g(y(ccpi)) +)iMcpiy'(ccpi)

in 11.

By virtue of the strong maximum principle, there exist 11o CC S2 and 6 > 0 such that (4.7) IVcpi1>6 inc\SZ0. Thus, we can choose M > 1 such that Mc]Qcpil > 1

in SZ \ 11

(4.8)

.

Therefore,

-Au > g(y(ccpi)) ? g(My(ocpi)) = g(u)

in S2 \ Slo.

(4.9)

Because cpiy'(ccpi) is bounded away from zero in 110, we can take M > 1 such

that Mcaicpiy'(cpi) > g(y(cpi))

in Q o.

(4.10)

Hence, (4.11) -©u > g(y(coi)) ? g(My(ccPi)) = g(u) in 1o. From relations (4.9) and (4.11), it follows that u = My(cpi) is a supersolution of problem (4.3), provided M > 1 satisfies (4.8) and (4.10). In a similar way


96

we deduce the existence of a constant 0 < m < 1 such that u := my(WI) is a subsolution of (4.3). Thus, u < u < u in S2 and (4.5) follows.

We claim that if g E L'(0,1) then y'(0+) < oo and y'(1-) < oo-that is, y E C1[0,1]. To this aim, we multiply by y'(t) in (4.4) and we integrate over [e, t], where 0 < e < t < 1. We find -y,2(t) + y/2(E) = 2

rt JE

V(t)

g(y(s))y'(s)ds = 2

J y(E)

g(r)dr,

for all 0 < e < t < 1. Because f0 g(r)dr < oo, we can let E --- 0 in the previous equality and we obtain y'(0+) < oo. In the same manner we derive y'(1-) < oo. Hence, there exists cl, c2 > 0 such that

cit1 is weakly convergent in HH(Sl) to some v E HH(Sl). Because (uk)k>1 converges pointwise to u in Q, we conclude that u = v E H o (Il). Assume now a > 3 and that problem (4.12) has a solution u E Ho (Q). Taking

into account estimate (4.13) and the fact that 2(a -1)/(a+l) > 1, by Corollary A.1.3 (ii), it follows that

fc

ul-adx = oo.

(4.16)

Let (Wk)k>1 C CC (S1) be such that wk -* u in Hol(Sl) as k --> oo. For each k > 1 set wk = max{wk, 0}. Then (wk)k>1 C Hfl (f1) and Vwk = VwkX{,Uk>o} . Passing to a subsequence if necessary, we may assume that (wk )k>1 converges to u almost everywhere in Q. By (4.16) and Fatou's lemma we find

wk u-adx = oo.

lim

k--oo

SE


98

Then,

f Vu Vwtdx =

- f wudx =

/ wka-adx.

It follows that

L

IVuf 2dx

= lim J Vu Owk dx = oo, k-oo

which is a contradiction. Hence, u V Ho (1) and the proof of Theorem 4.1.2 is now complete.

4.2 An ODE with mixed nonlinearities To see more clearly the nature of the problems we are dealing with, let us consider the following ODE:

I -y"(t) = y(t) + Y, (t) y(t) > 0

- 1 < t < 1, - 1 < t < 1,

(4.17)

Y(-') = y(1) = 0, where 0 < a, p < 1. The existence of a solution to (4.17) follows by virtue of Theorem 1.2.5. To achieve the uniqueness, we effectively construct a solution y of (4.17) such that y" E L1 (-1, 1). Then, by Theorem 1.3.17, we derive that y is the unique solution of (4.17). We assume that y' is odd, which yields y'(0) = 0

and y(-t) = y(t), for all -1 < t < 1. Multiplying by y'(t) in (4.17) and then integrating over [0, t], 0 < t < 1, we obtain y(0)

(y')2(t) = 2 jy(t) f (s) ds

for all 0 < t < 1,

(4.18)

where f (t) = t-a+tP. Because fo f (t) dt < oo, it follows that y'(1-) = limt\l y'(t) is finite. Similarly, we obtain y'(-l+) E ]R so that y E C'[-l, 1]. This yields

cl(1-It]) 0 such that fi(y)
l xl - yl then, by Lemma 4.3.4 (i), we have -yl-N+1d-Y(y)

Igx(x1,y)Id-1(y) 0

as a\0.

Furthermore, by Schauder estimates, (ua)o 0 and u* > 0 such that problem (4.38) has at least one solution in .6 if A > A. or p, > p*; problem (¢.38) has no solution in £ if 0 < A < A,, and 0 A or p. > p,R, then (4.38) has a maximal solution in £, which is increasing with respect to A and p. The diagram of dependence on A and a in Theorem 4.5.1 is depicted in Figure 4.3.

At least one solution

p.

No solution

A

Ficuitu 4.3. The dependence on A and p in Theorem 4.5.1.

Proof We split the proof into several steps. Step 1: Existence of a solution for large A. Let ti > 0 be fixed. Repeating the arguments in the proof of Theorem 4.3.2 we find that for all A > 0 there exists a unique solution vau, e £ of the problem


108

in 1, in 9,

-Ov = A f (x, v) + µk(x) v>0

v=0

(4.39)

on M2.

Obviously, ua,,, := vas, is a supersolution of (4.38). The main point is to find a subsolution uau of (4.38) such that uar, vau, in 0. For this purpose, let 4i : [0, 00) -+ [0, 00),

4) (t) t

ds.

1

2 fo g(r)dr

0

Remark first that 4? is well defined, because g E L1(0, 1). Indeed, there exists m > 0 such that g(s) > m, for all 0 < s < 1. This yields 1/2

/ r8

1

_ 1, it follows that

-Au,,, +p(x)g(ux,,) - MIVW1J2)g(h(Wi)) + MAiWih'(Wi) in Q.

(4.45)

From relations (4.43) through (4.45) we have

-1\u aµ + p(x)g(ua,,) < -g(h(Wi)) + MAicplh'(Wi) < 0

in S2 \ Qo.

Hence,

+ p(x)g(uxw) < Af (x, u,\u) + pk(x)

in Q \ S1o.

(4.46)

Furthermore, from (4.45) we have -4uau + p(x)g(uxi,) < IEplloog(h(Wi))IQWi l2 + MAiWih'(wwi)

in S1.

(4.47)

Because Wi > 0 in Sto, we can choose A > 0 such that

A min f(x, h(Wi)) > max{JIpjI.g(h(Wi)) + MAiWih'(Wi)}. xESEo

(4.48)

xE?Yo

Combining (4.47) and (4.48) we derive -Duau, + p(x)g(uau,) < ,\ f (x, u,N,) + p,k(x)

in S1o.

(4.49)

From (4.46) and (4.49) we note that uau is a subsolution of (4.38), and the proof of our lemma is now complete.


110

To end the proof in this step, it remains to show that uAu _< u,\,, in P. This follows immediately by using Theorem 1.3.17 for the mapping

t) _

Af (x, t) + p.k(x), (x, t) E Sl x (0, oo). Thus, problem (4.38) has at least one solution, ua., such that Mh(cpi) < u,\, < va, in ft. The regularity of u,\,, and the fact that ua,, e E follows in the same manner as in the proof of Theorem 4.3.2. Note that from the previous inequality we derive the same estimates as in (4.27), which are necessary to achieve the regularity ua,, E C1,1-7(St)

Step 2: Existence of a solution for large µ. The proof is the same as that shown earlier. We have only to replace (4.48) with

p min k(x) > max{IIpll.g(h(w1)) + xEi o

zE12n

Step 3: Nonexistence for small A and p.. Let m = inft>o{tA1 + p*g(t)}, where P. = minxE) p(x) > 0. Because g is positive and decreasing on (0, oo), we have m > 0. Denote by ( E S the unique solution of (4.39) for A = p = 1. Because infxE-a k(x) > 0, there exists c > 0 such that f (x, () < ck(x) in 0. We claim that (4.38) has no solutions for A and p in the following range:

0 0, problem (4.38) has at least one solution in £}. Let A := inf A. From the previous steps we have A # 0 and A > 0. Let A, E A, p > 0, and U.,,,, be the maximal solution of (4.38) for A = A1. We prove that (A1, oo) c A. Indeed, if A2 > At, then Ub,,, is a subsolution of (4.38) with A = A2.

On the other hand, if v,\,,, is the solution of (4.39) with A = A2, then Ova2u + 4'a2/,(x,vA,2M) < 0 < AUa,,, + vA2 ,, Ua11 > 0 VA2w ? U,,,

U,\,,,)

in Q,

in 5i,

on

,

©U,x,,,, E L1(0)

By Theorem 1.3.17 we deduce Ua,,, < vat,, in U. Therefore, by the sub- and supersolution method, problem (4.38) has a solutionnua2u E £ such that u')2j1
0 : for all A > 0, problem (4.38) has at least one solution in £}.

Let u := inf B. The conclusion follows in the same manner as just demonstrated. El The proof of Theorem 4.5.1 is now complete.


112

4.6

Existence for large values of parameters in the sign-changing case

In this section we study problem (4.21) assuming that the potential p changes sign in 0. We shall obtain the existence of solutions in the class £ for large values of parameters A and it. Let p* := ma_xp(x) XEn

and p* := minp(x). XJ

Theorem 4.6.1 Assume that p* > 0 > p*. Then there exists A*, p* > 0 such that (4.21) has at least one solution uAK E E if A > , or it > p.. Moreover, for A > A. or p > it., ua,, is increasing with respect to A and p.

Proof Because p* < 0, Theorem 4.5.1 implies that there exist A*, p* > 0 such that the problem

-Au = p*g(u) + of (x, u) + pk(x)

u>0 u=0

in St,

in P,

one

has a maximal solution U,\,. E £, provided A > , or p > p.. Then U,\,, is a subsolution of (4.21). Moreover, Ua,, is increasing with respect to both A and p. Consider the problem

-Av = p*g(v) + Af (x, v) + pk(x) v>0

v=0

in S2,

in n,

(4.51)

onl3l.

Because p* > 0, by Theorem 4.3.2, problem (4.51) has a unique solution vau, E E. Obviously, va , is a supersolution of (4.21). Note also that 4)A,,, (x, t) = p*g(t) + A f (x, t) + pk (x), (x, t) E 1 x (0, oo), fulfills the hypotheses of Theorem 1.3.17. According to this result, Ua,, < va, in Q. Hence, by Theorem 1.2.2, there exists

a minimal solution uau, of (4.21) with respect to the ordered pair (U,\, ,v,\,,). Furthermore, using Theorem 1.3.17 it is easy to see that uau is minimal in the class of solutions u E E of (4.21) that fulfill Uaµ A2 > a*, p > 0 and let u,\,,,, ua2u E £ be the solutions of (4.21) for A = Al and \ = A2, respectively, that we obtained earlier. Note that ua,, is a supersolution of (4.21) with A = A2. Moreover, because Uaµ is increasing with respect to A > a*, we obtain uA14 > Ua1M > Ua2µ

in Q.

Using the minimality of ua2,, from the previous relation we deduce UA,u >

uA2u

in 9 and, by the strong maximum principle, we deduce that uA12 > ua2, in Q. The dependence on p follows in the same way. This completes the proof of Theorem 4.6.1.

Singular elliptic problems in the whole space

113


4.7

In this section we extend the study of singular elliptic problems to the case where the domain St is the whole space RN. We shall be concerned with problems of

the type

1-Du = 4i(x, u) u>0

u(x) - 0

in RN, in RN,

(4.52)

as jxj -> oo,

where (: RN x (0, oo) - [0, oo), N > 3, is a Holder continuous function having a singular behavior at the origin with respect to the second argument. Existence of entire solutions We are concerned here with the existence of classical solutions to (4.52)-that 4.7.1

is, functions u E CZ(RN) (or even u E Cj , (RN) for some 0 < ry < 1) that verify (4.52) pointwise in RN. A solution of (4.52) is often called a ground-state solution. Roughly speaking, such a solution achieves a minimal level of energy as a result of the prescribed condition at infinity.

A natural way to construct solutions to (4.52) is to provide a monotone sequence of positive functions that are solutions of similar problems to (4.52), but in bounded domains. The main point is to supply a supersolution w of (4.52) that must satisfy w(x) - + 0 as lxi oo. Then, standard elliptic arguments will imply that the pointwise limit of the previously mentioned sequence is, in fact, a genuine solution of (4.52). This procedure clearly fails in the absence of a comparison principle for problems like (4.52) in bounded domains. Bearing these points in mind, we start out the study of existence by considering first the case where 4* in (4.52) is monotone with respect to its second variable. This particular situation will help us to construct in an easier way the monotone sequence with a limit that is the solution of (4.52). Let us consider the problem

-du = p(x)g(u) U>0

u(x)-'0

in R v, in RN,

(4.53)

asjxi -'oo,

where g E C1(0, oo) satisfies g > 0, g' < 0 in (0, oo) and limt\o g(t) = oo. We shall also assume that p : RN -> (0, oo) is a Holder continuous function. Let

O(r) := maxp(x) and O(r) := min p(x), r > 0. IzI=r

Izl=r

Theorem 4.7.1 The following properties are valid: (i) If f O° oo, then problem (1.55) has no classical solutions. (ii) If f000ro(r)dr < oo, then problem (4.53) has a unique classical solution.


114

Proof (i) Assume that (4.53) has a solution u E CZ(RN) and let U be the spherical average of u defined as

U(r) =

N_1

wNr

f

u(x)du(x)

IxI=r

for all r > - 0,

where wN is the surface area of the unit sphere in RN. Because u is a positive ground-state solution of (4.52), it follows that U is positive and U(r) -- 0 as r -* oo. Let m := minxERN g(u(x)). Because u(x) -4 oo as lxi - oo, it follows that m > 0 is finite. With the change of variable x = ry, we have

f

U(r) = -1

WN

u(ry)dc(y)

for all r > 0

u1=1

and

U'(r) =WN1 f' 1=1 Vu(ry) ydo(y)

for all r > 0.

Hence,

U,(r) =

1

WN

8u (ry)du(y)

( 1711=1

Oar

f

_' (x)do, (x ) 1 = WNrN-1 I.,=r Or

.

That is,' 1

U1(r) = WNTN-1

1

This yields

_(rN_1U'(r))' = J

1

d(

wN d r

`

Irk 0,

which finally leads us to

-(rN-1U'(r))' > mrN-1 O(r)

for all r > 0.

(4.54)

Integrating in (4.54) we find

U(0) - U(r) > m

jr t1-N

ft sN-10(s)dt

for all r

0.

(4.55)

a

Because fo rib(r)dr = oo, according to Lemma 2.2.4 we deduce that the righthand side in (4.55) tends to infinity as r -> oo. Thus, passing to the limit with r - oo in (4.55) we obtain a contradiction. Hence, problem (4.53) has no solution in this case.


115

(ii) We first apply Theorem 1.2.5 for Bn := {x E RN : Ix I < n}. Thus, for all n > 1 there exists un E C2'Y(B) fl C(B,ti) (0 < -y < 1) such that

-dun = p(x)g(un)

in Bn,

un>0

in B.,

un = 0

on f3Bn.

We extend un by zero outside of B. Because g is decreasing, we deduce

U1 < u2 < ... < Un < un+1 < ...

in RN.

We next focus on finding a supersolution w of (4.53). To this aim, define

((r) := rl-iv jtN_1(t)df

for all r >_ 0.

(4.56)

By Lemma 2.2.4, and that fact that f oo rq5(r)dr < oo, we find f ow rc (r)dr < oo. Consider now fi(x) ((t)dt for all x E RN. llfxl

Then

satisfies

{

-d6 = O(Ixl)

in RN,

t; > 0 fi(x) -> 0

in RN, as lxl -> oo.

Because the mapping [0, oo) D t H fo ds/g(s) E [0, oo) is bijective, we can implicitly define w : RN --> (0, oo) by a'(x)

dt = fi(x) g(t)

for all x E RN .

It is easy to see that w E C2(RN), w > 0 in RN, and w(x) -> 0 as Ixl -> oo. Furthermore, we have

-off = g'(w) I©w12

g(w) = c(IxI)

in RN.

Because g' < 0, it follows that -Llw > O(Ixl)g(w) > p(x)g(w)

in RN.

(4.57)

Therefore, w is a supersolution of (4.53). Now, it is easy to deduce that un < w in

RN. By standard elliptic arguments we find that u(x) := limn" un(x), x E RN, satisfies u E Coy (RN) for some 0 < y < 1, and -Du = p(x)g(u) in RN. Because u,, < w in RN, we obtain that u < w in RN, which yields u(x) -> 0 as jxl - 00. Hence, u is a solution of (4.53). Finally, if ul and u2 are two solutions, by the maximum principle and the fact that ul - u2 tends to zero at infinity we derive ul =- u2-that is, problem O (4.53) has a unique solution. This concludes the proof.

116


Next we study the existence of classical solutions for the problem

-Au = p(x) (g(u) + f(u)) U>0

u(x) -s 0

in 1[ N,

in RN,

(4.58)

as 1xI --> 00.,

where g and p are as considered earlier and f : [0, oo) - [0, oo) satisfies the sublinear conditions (f 1) and (f2) introduced in the beginning of this chapter. As a result of the lack of monotonicity, the uniqueness is not so obvious as in the study of problem (4.53). This matter will be discussed in the last part of this section for radially symmetric solutions.

Theorem 4.7.2 Assume that foo r¢(r)dr < oo. Then problem (4.58) has at least one classical solution.

Proof We use the same approach as in Theorem 4.7.1 (ii). By Theorem 1.2.5, for all n > 1 there exists un E C2°"r(Bn) n C(B,) such that Aun = p(x)T (un)

un>0

in Bn, in B.,,,

onBBn, where W(t) = g(t) + f (t), t > 0. We have Aun+1 + p(x)%P(un+i) < 0 < iu,i +p(x)qf(un)

in Bn,

un = 0 < un+1 on OBn. Notice that Au,,+1 E L'(B,,), because un+i is bounded away from zero in B,,. Thus, by Theorem 1.3.17 it follows that un < un+1 in B. We extend u, by zero < un < un+1 < . . . in RN. It remains to outside of B. Then 0 < ul < find an upper bound for (un)n>1. By Theorem 4.7.1 with g replaced with g + 1, there exists v E C2(RN) such that

-Av = p(x)(g(v) + 1)

v>0 V(X) -} 0

in RN,

in R"',

(4.59)

as jxI - oo.

Because f is sublinear and v is bounded in RN, we can find M > 0 such that M > f (Mv) in RN. Hence, w := My satisfies -iw = p(x) (g(w) + 1 (w))

in RN ,

W>0 W(X) -. 0

in lE$N

As seen earlier, we find un < w in RN and u(x) is the solution of (4.58). This finishes the proof.

as jxI -* oo.

un(x), for all x E RN,


117

Uniqueness of radially symmetric solutions We discuss in the sequel the uniqueness of a radially symmetric solution to (4.53). More precisely, we shall be concerned with the uniqueness of classical solutions 4.7.2

to the problem

-u"(r) - Nr 1u'(r) =p(r)*(u(r))

in [0>00),

u>0 u'(0) = 0,

in [0, oo),

u(r)-f0

asr -goo,

where p : (0, oo) -> [0, oo) and IF functions such that

:

(4.60)

(0, oo) -> (0, oo) are Holder continuous

is decreasing; (Al) the mapping (0, oo) D t (A2) there exists to > 0 such that %P is decreasing on (0, to) and increasing on [to, oo);

(A3) there exists

to r (t) E (0, oo) for some a > 0. i o Notice that these hypotheses are quite natural (see Figure 4.4). Typical examples

of nonlinearities encountered so far, such as '11(t) = t--a+t', a > 0, 0 < q < 1, or W(t) = t-a+ln(l+t), a > 0, satisfy (Al) through (A3). Obviously, any solution u of (4.60) provides a radially symmetric solution u(x) = u([xI), x E 1[l;^`, of problem (4.52) with t (x, t) = p(I xI)'1(t).

(0,0)

to

t

FIGURE 4.4. The shape of function T.

Theorem 4.7.3 Assume that (Al) through (A3) hold. Then, problem (4.60) has at most one solution.


118

Proof Assume that there exist u, v E C2[0, oo) two distinct solutions of (4.60). By virtue of Theorem 1.3.17 we easily deduce the following useful result.

Lemma 4.7.4 If there exists R > 0 such that u(R) = v(R), then u - v in [0, R).

By virtue of Lemma 4.7.4, one of the following three situations may occur: 1. u > v in [0, oo). 2. u(0) = v(0) and u > v in (0,co).

3. There exists R > 0 such that u =- v in [0, R) and u > v in (R, oo). We shall discuss these three situations separately. CASE 1: u > v in [0, oo). Let us notice that u and v verify

-(rN-lu'(r))' = rN-1p(r)'y(u(r)),

(4.61)

-(rN-lv'(r))' = rN-1p(r)`y(v(r)),

(4.62)

for all r > 0. We multiply the first equality by v', the second one by u', and then subtract. We obtain

(rN-1(u'v - UV'))' = rN-1p(r)uv(%P(u) u

- %F(v)I v

for all r > 0.

"' (v)}dt

for all r > 0.

An integration over [0, r], r > 0 yields

frtN-1P(t)UV(T(u) rN-1(u'v -7GV') = l

U

v

-

Therefore, u'v - uv' > 0 in [0, oo), which implies

v is increasing and

v > v'

in (0, oo).

(4.63)

Hence, there exists t:= limr_,oo u(r)/v(r). Because u(0) > v(0), it follows that

1 0 such that

1'(u(r)) > T(v(r)) for all 0 < r < a, '1(u(a)) = ql(v(a)), I'(u(r)) a. Proof Let us first remark that 41(u(0)) >''I(v(0)). If we would have the converse inequality, then, by (A2) and u(0) > v(0), we obtain v(0) < to. It follows that for all r > 0, W(v(r)) > f(u(r)), which yields 00

00

tN-1p(t)W(u)dt < J 0

tN-lp(t),Q(v)dt.

0

Again by 1'Hospital's rule we find

1 < t = lim

u(r)

r-+oo v(r)

Jim

=

u'(r) _ fa tN-lp(t)I1(u)dt < 1, oo r-+oo v'(r) (v)dt ftN-lp(t)°

which is a contradiction. Hence, 1F(u(0)) > W(v(0)). Furthermore, because u > v

and u(t) --> 0, v(r) -> 0 as r - oo, for r > 0 large enough we obtain f(u(r)) < W(v(r)). Thus, by continuity arguments, we can find-a > 0 such that xk(u(a)) W(v(a)).


120

Assume that there exists two real numbers al > a2 > 0 such that 41(u(ai)) _

W(v(ai)), i = 1, 2. Then we must have u(ai) > to > v(ai), i = 1, 2. Because al > a2, we deduce u(al) < u(a2), and by (A2) we further obtain IP(v(al)) = W(u(al)) < T(u(a2)) = W(v(a2)).

Because v(ai) < to, i = 1, 2, the last equality implies v(al) > v(a2), which in turn implies a1 < a2, but this is clearly a contradiction. Now, the rest of the proof follows from the uniqueness of the number a > 0 with W(u(a)) ='I'(v(a)). By ]'Hospital's rule we have found °° tN-lp(t)W (u)dt P

- f °O tN-lp(t)I(v)dt

Thus, fort 0 < E < 1 there exists ro > a large enough such that J

r tN-lp(t)41(u)dt > (B - r) JOrtN_1P(t)(V)dt

for all r

ro.

0

By Lemma 4.7.5 we have '(u(t)) < T(v(t)), for all t > a, which yields

a tN-lp(t)w(u)dt + J0

Ja

rtN-lp(t)'F(v)dt > (t - E) J r 0

for all r > ro. This leads us to ftN_1p(t)(W(u) - W(v))dt > (e - 1 - E) f r tN-1p(t)T(v)dt

for all r > TO.

Passing to the limit with r -> oo, and because 0 < E < 1 was arbitrarily chosen, we obtain

fa

tN-1p(t)(T(u) - IF(v))dt > (e - 1)

J0

tN-lp(t)T (v)dt.

That is, aN-1 (v' (a) -

u'(a)) > (t - 1) J r tN-1p(t)WY(v)dt. 0

On the other hand, from (4.63) we have u'/v' < u/v < t. This yields v'(a) - u'(a) = v'(a) I 1 - -I(a) J
0.

(4.69)

0

We claim that u(0) = v(0) < to. If u(O) = v(0) > to, then u(t) > v(t) > to for t > 0 small enough, and by the hypothesis (A2) on %F we deduce *(u) > %k(v) in a small neighborhood of the origin. But using (4.69) this yields w(t) < 0 for some t > 0, which is a contradiction. Hence, u(0) = v(0) < t0. Then

v(r) < u(r) < u(0) < to

for all r > 0,

which implies T(u) - W(v) < 0 in (0, oo). Now, to obtain a contradiction we look at (4.69). The left-hand side tends to zero as r oo whereas the right-hand side tends to a positive quantity as r -+ oo. This is clearly a contradiction.

CASE 3: There exists R > 0 such that u = v in [0, R) and u > v in (R, 00). We proceed exactly in the same manner as we did in Case 2 for ii(r) = u(r + R) and ii(r) = v(r + R), r > 0. This finishes the proof of Theorem 4.7.3.

4.8


Singular elliptic problems arise in various branches of mathematics (see [38], [56], [134], and [160] for more details). To our best knowledge, the first study in this direction is from Fulks and Maybee [81] and have been intensively studied after the pioneering work by Crandall, Rabinowitz and Tartar [57].

In their work [57], a similar problem to (4.3) has been studied for more general differential operators. The existence of a classical solution to (4.3) has been obtained by considering the perturbed problem

122


-Du = g(u +,-), 0 < e < 1

in S2,

u>0

in S2,

U=0

on 852.

(4.70)

Then (4.70) has a unique solution uE E C2 (?I), which is decreasing with respect to e. If u(x) := limb\o u,(x), x E S2, then u E C2 (Q) n C(S2) is the unique solution of problem (4.3). In the case of pure powers in nonlinearities and p = 0, problem (4.21) was studied by Shi and Yao [182] and was then generalized by Ghergu and Radulescu in [83]. For more results concerning singular elliptic equations in bounded domains we refer to [40], [42], [48], [55], [61], [67], [84], [85), [86], [96], [101], [127], [174], [201], and [202] as well as to recent surveys (102], [175]. If a < 0, problem (4.12) is called the Lane-Emden-Fowler equation and arises in the boundary-layer theory of viscous fluids (see Fowler [78] and Wong [197]). As we have already mentioned, as a result of the meaning of the unknowns (concentrations, populations), the positive solutions are relevant in most cases. But nonnegative solutions may also arise in some situations. In fact, the presence of the singular term g(u), which is not locally Lipschitz near the origin, may give rise to dead core solutions-that is, solutions identically zero on subdomains of S2 with positive measure. For more results on dead core solutions involving singular nonlinearities, we refer the reader to Davila and Montenegro [59], where the following problem was considered: Au = X{u>o} u=0

(-u-a

+ AuP)

in S2, (4.71)

on 852,

with 0 < a!, p < 1. It is proved in [59] that for all A > 0, problem (4.71) has a maximal solution ua. Moreover, there exists A* such that for all A > A* the maximal solution ua is positive and stable. Another direction regarding the study of singular elliptic problems refers to sign-changing solutions for such problems and was considered by McKenna and Reichel [137]. In this respect, the precise asymptotic behavior of a solution at the boundary where it changes sign plays an essential role in the study of existence. The approach in obtaining the C" (11) regularity of the classical solution to (4.21) follows the method in Gui and Lin [99] in the case A = u = 0, g(t) = t-a,

and p(x) behaves like d(x)°, 0 < a < 1 near 852. The method in [99] can be extended to more general nonlinearities than those we dealt with in this chapter. As a conclusion, Cl"'Y(S2) (0 < 7 < 1) is the best regularity of solutions we can

expect for these kinds of problems. We have seen in Theorem 4.1.2 that if g decays fast in the neighborhood of the origin, the solutions may not be in Cl (S2) or, even more unexpected, they do not belong to the usual Sobolev space Ho (52). In other words, it may happen that a classical solution is not a weak solution. We will come back to this issue in Chapter 7.


123

The growth condition (g2) on g is a natural assumption. This allowed us to show that if problem (4.38) has a classical solution u E C2(f) f1 C(Sl), then u E E. We shall see in Chapter 9 that condition g E L1(Q) is necessary to have the existence of a classical solution. Condition (4.40) is often called the Keller-Osserman condition around the origin. As proved by Benilan, Brezis, and Crandall [23], condition (4.40) is equiv-

alent to the property of compact support. That is, for every k E L'(RN) with compact support, there exists a unique u E W"' (RN) with compact support such that Au E Ll(IRN) and

-Du + g(u) = k(x)

almost everywhere in RN.

Theorem 4.5.1 leaves open the question of multiplicity. This is clearly a delicate issue even in simpler cases. In this sense we refer the reader to Ouyang, Shi, and Yao [156], who studied the existence of radially symmetric solutions of the problem -Au = A(ur - u-°) in B, u>0 in B, (4.72)

u=0

on 49B,

where B is the unit ball in RN, (N > 1), 0 < a, p < 1 and A > 0. Using a bifurcation'theorem of Crandall and Rabinowitz, it has been shown in [156] that there exists Al > AO > 0 such that problem (4.72) has no solutions for A < A0, one solution for A = \o or A > A1i and two solutions for al > A > .10. Finally, the results obtained for singular elliptic problems in bounded domains allowed us to extend the study to the case when the domain is the whole space. The first results in this sense concern problem (4.53) and have been obtained by Kusano and Swanson [122] and Edelson [74]. Condition f rq5(r)dr < oo in the statement of Theorem 4.7.1 was first supplied by Lair and Shaker [124] (see also [125] also [200]), where the existence of a ground-state solution of (4.53) is obtained under a weaker assumption on the potential p, namely, p satisfies the assumption (pl) in Section 2.2. That is, p is nonnegative and any zero of

p is surrounded by a region where p > 0. In the case '(x, t) = p(x)(t,-° + tp), 0 < p, a < 1, the uniqueness of a radially symmetric solution to problem (4.60) has been obtained in [199].

5

BIFURCATION AND ASYMPTOTIC ANALYSIS: THE MONOTONE CASE A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. Godfrey H. Hardy (1877-1947)

In the current chapter and in the following one, we are concerned with bifurcation problems of the type Au = Af (u)

in Il,

u>0

in Q,

u=0

on 852,

where 0 is a smooth bounded domain in R N (N _> 1), A is a positive parameter, and f : [0, oo) -> (0, oo) is a continuous function. We study the existence and the qualitative properties of solutions u E C2(Sl) n C(Sl) (provided such solutions exist), as well as their asymptotic behavior with respect to A. To give an elementary example, let us consider the class of functions f with

linear or superlinear growth-that is, there exists a > 0 such that f (t) > at, for all t > 0.

(5.2)

We observe that if f satisfies condition (5.2), then there exists 0 < A0 < Ao < 00 such that (i) problem (5.1) has at least one solution if A E (0, Ao); (ii) problem (5.1) has no solution if A > Ao. Of course, natural questions are the following: Is Ao = Ao?

If not, what happens if A E (Ao, Ao)? Assertion (i) follows by the implicit function theorem (Theorem B.1 in Appendix 13). To argue (ii), let \i/a. An important feature in our analysis concerns the notion of stability, which is viewed in terms of a related linearized problem. More precisely, to any solution (u, A) of the nonlinear problem (5.1), we associate the linear eigenvalue problem

Aw - Af'(u)w = uw

in Q,

W=0

on 852.

0 and f is asymptotically linear. That is, lim f-(t) = a E (0, 00).

1-400

t

A first existence and bifurcation result concerning problem (5.4) is given by Amann's bifurcation theorem in Section 5.2 in a more general framework. Notice that because f is convex, by (5.5) we have limt_,00 f'(t) = a, and the mapping g : [0, oo) --* [0,-00), g(t) = f (t) - at is nonincreasing. Hence, there exists E := tliim g(t) E 1-00, 00).

The sign oft plays a crucial role in the analysis of (5.4). The current chapter is concerned with the case t > 0. Thus, we study here the monotone case on f

A general bifurcation result

127

f(t}t

t

FIGURE 5.1. The monotone case on f.

because the mapping t --> f (t)/t is nonincreasing, so that a = inft>o f (t)/t (see Figure 5.1).

In turn, if f < 0 then m := inft>o f (t)/t > a, and both numbers a and m will play an important role in the study of (5.4). This fact will be emphasized in Chapter 6. For a E L°° (1l) we denote by Al (a) and co, (a) the first eigenvalue and the first eigenfunction, respectively, of the operator -A - a in Ho (S2). By means of the variational characterization of the first eigenvalue of a linear operator, AI (a) is defined as

At(a) =

inf

J (IDu12 - a(x)U2)dx.

uEHo(A),IiulI2=1 n

(5.6)

Definition 5.1.1 A solution u of (5.4) is called stable if Ai(Af'(u)) > 0, and, is unstable otherwise.

5.2 A general bifurcation result The main result in this section is the following.

Theorem 5.2.1 (Amanri's bifurcation theorem) Assume that f : R -* R is a C2 function that is convex, positive, and such that f'(0) > 0. Then there exists A* E (0, oo) such that the following holds: (i) If A > A*, then problem (5.4) has no classical solutions. (ii) If 0 < A < A*, then problem (5.4) has a minimal solution u;k E C2(S2) such that (iil) ua > 0 in &1; (112) the mapping (0, A*) E) A ua(x) is increasing, for every x E S2; (ii3) ua is the unique stable solution of the problem (5.4) for all 0 < A < A* .

Proof We divide the proof of Theorem 5.2.1 into several steps.

Bifurcation and asymptotic analysis: The monotone case

128

Step 1: Definition of At. To apply the implicit function theorem (see Theorem B.1) to our problem (5.4), define F : Co'a(S2) x R -> C°,a(Sl) by

F(u, A) = -Au - o f (u),

(u, A) E C2,' (Sl) x 1EF,

where 0 < a < 1. It is clear that F verifies all the assumptions of the implicit function theorem. Hence, there exists a maximal neighborhood of the origin I and a unique map u = ua that is the solution of problem (5.4) for all A E I and such that the linearized operator -A - )if'(ua) is bijective. In other words, for every A E I, problem (5.4) admits a stable solution, which is given by the implicit function theorem. Let A* := supl < oo.

Step 2: The mapping (0, A*) E) A --v u,\ (x) is increasing and ua > 0 in 0. For x E i, denote by v,$x) the derivative with respect to A of the mapping (0, A*) D A 1

) u (x). We differentiate in (5.4) with respect to A and we obtain

(-A - Af'(ua))va = Af (u,$ va = 0

in 92,

on acl.

Note also that the operator -A-A f'(ua) is coercive. Therefore, by Stampacchia's maximum principle (see Theorem 1.3.10), either VA = 0 in Sl or va > 0 in 9. The first variant is not convenient, because it would imply that f (uA) = 0, which is impossible, by our initial hypotheses. Hence, va > 0 in i, which then yields

ua>0in0.

Step 3: ua is stable. Set O(A) := Al(Af'(uA)). Thus, Vi(0) = Al > 0. Furthermore, by the implicit function theorem the linearized operator -A-A f'(ua) is bijective, which yields i1(A) # 0 for all A < A*. Now, by the continuity of the mapping A i--- A f'(ua), it follows that 0 is continuous, which further implies that 0 > 0 on [0,A*).

Step 4: A* < oo. Because f is convex, it follows that f'(ua) _> f'(0) > 0, which leads to A1(Af'(0)) > Aj(Af'(ua)) > 0 for every A E I. Notice that A1(Af'(0)) = Al - Af'(0). This yields Al - Af'(0) > 0 for any A < A*-that is, A* < Al/f'(0) < oo. Step 5: Problem (5.4) has no classical solutions provided A > A*. Assume that there exists IL > At and a corresponding solution v to problem (5.4). We first prove that for every A < A* we have ua < v in Q. Indeed, by the convexity of f it follows that

-A(v - u>,) = lf(v) - Af(ua) > A(f(v) - f(ua)) > A f'(ua) (v - u,\)

in n.

Hence,

(-A - A f'(ua))(v - ua) > 0

in 11.

(5.7)

Existence and bifurcation results

129

Because the operator -A - \f'(uA) is coercive, by Stampacchia's maximum principle, we deduce v > u,\ in 0 for all A < A*. Therefore, ua is bounded in LOO(S2) by v and there exists u* = lima/. ua in n and u* = 0 on 852. We claim that \,(A*f'(u*)) = 0. We already know that Aj(A*f'(u*)) > 0. Assume by contradiction that Ai(A* f'(u*)) > 0. In other words, the operator -A-A* f'(u*) is coercive. By applying the implicit function theorem to F(u, A) = -Au-,\f (u) at the point (u*, A*), there exists a curve of solutions of the problem (5.4) passing through (u*, A*), which contradicts the maximality of A*. Hence, al(A*f'(u*)) = 0, which implies the existence of 0 in ci such that cpl = 0

on 8i and -Awe

-

0

(5.8)

in 92.

Passing to the limit with A / A* in (5.7), we find

(-A - A* f'(u*))(v - u*) > 0

in Q.

Multiplying this inequality by cpl and integrating over S2, we obtain

- J (v - u*)Av,dx - A* J f'(u*)(v - u*)cpldx >_ 0. According to (5.8), the previous relation is in fact an equality, which implies that -A(v - u*) = A* f'(u*)(v - u*) in Q. It follows that pf(v) = A* f (u*) in Q. But p > A* and f (v) > f (u*) so that f (v) = 0, which is impossible. Hence. problem (5.4) has no solutions for A > A*.

Step 6: ua is the minimal and the unique stable solution of (5.4). Fix an arbitrary A < \* and let v be another solution of problem (5.4). We have

-A(v - ua) = of (v) - Af(ua) ? Af'(ua)(v - u,\)

in Q.

Again, by Stampacchia's maximum principle applied to the coercive operator -A - Af'(ua), we find that v > ua in 92. Now let w be another stable solution of (5.4) for some A < A*. With the same reasoning as applied earlier to the coercive operator -A - A f'(w), we derive that 0 ua > w in S2 and, finally, uA = w. This concludes the proof.

5.3 Existence and bifurcation results By Amann's bifurcation theorem we have the following information about problem (5.4): (i) There exists A* E (0, oo) such that (5.4) has solution when A E (0,,\*), and no solution exists if A E (A*, cc). (ii) For .1 E (0, A*), among the solutions of (5.4) there exists a minimal one, say ua. (iii) The mapping A'--p ua is a C' convex and increasing function.

Bifurcation and asymptotic analysts: The monotone case

130

(iv) u,\ can be characterized as the only solution u of (5.4) such that the operator -0 - A f `(u) is coercive; that is, ua is the only stable solution of (5.4).

Assuming now (5.5), we aim to discuss in an unified way some natural questions raised by (5.4), namely what can be said when A = A*? what is the behavior of ua when A approaches A*? are these results still valid if f is unbounded around the origin? is there other solution of (5.4) excepting ua? if so, what is their behavior? In the rest of this chapter we try to answer these questions. The approach we present in the following is the result of the work by Mironescu and Radulescu [142], {143]. We are first concerned with the monotone case, corresponding to

2 := tlirn (f (t) - at) > 0, where a:= limt.(,,, f (t)/t E (0, oo).

Theorem 5.3.1 Assume f > 0. Then we have (1) A* = Ai/a; (ii) The problem (5.4) has no solution for A = A*; (iii) ua is the unique solution of (5.4) for all 0 < A < A*; (iv) lima/A' ua = oo uniformly on compact subsets of Q. The bifurcation diagram in the monotone case is depicted in Figure 5.2.

A=At/a

it

FIGURE 5.2. The bifurcation diagram in Theorem 5.3.1.

Existence and bifurcation results

131

Proof (i)-(ii) If 0 < A < Al/a, then Af satisfies the hypotheses of Theorem 1.2.5. Hence, problem (5.4) has at least one solution for all 0 < A < Al/a, so A* > Ai/a. We claim that problem (5.4) has no classical solutions for A > Ai/a. Assume that there exists A > Al/a such that (5.4) has a solution u. Because the mapping s H f(t)/t is nonincreasing, it follows that A f (u) > Al u in Q. Then, multiplying by Cpl in (5.4) and integrating by parts, we have

Al I ulpldx = A f f(u)Vldx > Ai

r

J

This yields A = Ai/a and f (u) = au; that is, f (t) = at for all 0 < t < maxxC?j u(x). But this is a contradiction because f (0) > 0. Therefore A* = At/a and (5.4) has no solutions for A = A*. (iii) As we have argued in the previous section, the fact that f is convex and fulfills (5.5) implies that g : [0, oo) -> [0, oo), g(t) = f (t) - at is nonincreasing. Then ua satisfies -Aua - Aaua = Ag(ua) in Q. Let vA be another solution of (5.4). According to the minimality of UA, we have ua < va in Q. Thus, w := VA - UA > 0 satisfies -Ow - Aaw < 0 in S1. Because

as < Al, the strong maximum principle yields w = 0 in Q and the uniqueness follows.

(iv) Because of the special character of our problem, we will be able to prove that, in certain cases, L2 boundedness implies Ho boundedness! Assume by contradiction that (ua)o 0 does not depend on A. From the previous estimates it is easy to see that (wa)o u*

of (i)-(ii). We first multiply by Aa sz

si

f

uMoidx

for all 0 < A < A*. (5.16)

st

Passing to the limit in (5.16) with A J' A*, by virtue of Lebesgue's theorem on dominated convergence we find

ai

fz

u*(Pi =

A1

a

*

f ('a )cpidx..

Hence, f (u*) = au*, which contradicts the fact that f (0) > 0. This shows that lima/a. uA, = oo uniformly on compact subsets of Q, and the proof is now complete.

0

134


Remark 5.3.2 The results in Theorem 5.3.1 hold in a little more general context -namely, the mapping (0, oo) 9 t 0 1 f (t)/t is nonincreasing, f satisfies (5.5), and f (0) > 0. By Theorem 1.2.5 and the strong maximum principle we may show that (i)-(iv) in Theorem 5.3.1 hold. However, the convexity assumption on f provides a complete description of the unique solution ua to problem (5.4). We will see in Chapter 6 that the hypothesis f convex is strongly needed when the issue of multiplicity arises.

Example 5.1 Let us consider the one-dimensional problem

-u" = a

u2 + u + 1

in-(0,7r),

u(0) = u(7r) = 0.

(5.17)

Clearly, f (t) = 02 -+t + 1, t > 0, obeys the monotone case described in this chapter. In this case AI = 1, and by Theorem 5.3.1 we have that (5.17) has a solution if and only if 0 < A < 1. Moreover, for all 0 < A < 1, problem (5.17) has a unique solution u),, which is stable, and lim. fi u), = oo uniformly on compact subsets of (0, 7r). Using the collocation method we have computed the solution ua of (5.17) when A approaches A1. The solution is plotted in Figure 5.3.

FIGURE 5.3. The unique solution of problem (5.17).

5.4 Asymptotic behavior of the solution with respect to parameters In this section we discuss the asymptotic behavior of the unique solution ua of (5.4) with respect to A. We have seen that limA fx. ua = oo uniformly on compact subsets of 9.

Asymptotic behavior of the solution with respect to parameters

135

We consider the normalized sequence (w,\)0 -Ow in D'(SZ). On the other hand, we have

-Owa = A k((A))

i n Q.

(5 . 19)

We claim that f (uA)/k(A) -> aw almost everywhere in fl as A / A*. If x E Q is such that w(x) > 0, then, by (5.5), we obtain lira

'\/A

f (k(A)wa(x)) k(A)

= lira f (k(A)w,\ (x))wa(x) = aw(x). ala k(A)w,\(x)

If w(x) = 0, let e > 0 and 0 < Ao < A* be such that w,\(x) < a for all Ao < A < A*. Because f (t) < at + b for all f (k(A)WA(x)) k(A)

< ae +

b

k(A)

0 (with b = f (0)) we deduce for all Ao < A < A*.

This yields f(wa(x))/k(A) -* 0 as A / A*, which proves the claim. Passing to the limit with A / A* in (5.19) we obtain -OW = A1w, W E HH(fI), w > 0, IIwI12 =1.

But this means exactly that w = cp1. Moreover, (wa)o 1/3.

alal

Proof We first need the following auxiliary result.

Lemma 5.5.4 There exists eo > 0 and two positive constants c1, c2 > 0 such that 1

c1I lnel
e} P1

Proof Consider c > 0 such that cd(x) < cp1 < 1 d(x) c

Let b > 0 and Ab be as in the proof of Theorem 5.4.2. Define 4; : Ab -+ t9

x (-b, 5)

and

W : 8St x (-S, b)

Ab

by

4i(x) = (7r(x), (x - ir(x), n(7r(x))))

Then 4, T are smooth and

and

$(xo,e) = xo +En(xo).

= I -1. Replacing b if necessary, by a smaller

number, we may assume that there exist C1, c2 > 0 such that 0 < c1 < IJ(WY)I < c2

in 8!2 x

(5.24)

where J(W) is the Jacobian of W. Let Eo

min{ inf cp1, c5}.

Now, for 0 < E < Eo we decompose

f

dx = w1>E} Y1

1 dx + J{fGwi <eo} W11 dx.

J{t't>eo} 1

Note that by (5.23) we have {E/c < d(x) < ceo} C {e < cp1 < Eo} C Ice < d(x) < eo/c}.

Hence,

(5.25)

Examples

139

I

1 dx 0 be fixed. Because (A1 A)2/3(k(A)wa + 1)2 -->

as A / A1i we can find 0 < Ao < Al such that

(Al - A)2/3(k(A)wa + 1)2 < c2(cp2 + e)

for all AO < A < A1.

Then (5.29) yields

L Because the limit of the left-hand side is c, we deduce

c >C2-

'r1

J11'P2+S

dx

for all F > 0.

Letting s \ 0, by Lemma 5.5.4 we obtain c = oo, the desired contradiction.

140


(ii) We argue again by contradiction. Thus, there exists a > 1/3 such that up to asubsequencewehave (A1 - A)1-a in (5.20) we obtain

(.11 - A)alplk(A)dx = A in (.\1

-

A)2c,-j3(k(A)w,\ +

1)2dx.

(5.30)

The limit of the left-hand side is c E (0, oo]. To estimate the right-hand side, we have

r

Cpl dx 0<J{w(A1 - A)1-a(k(A)wa + 1)2

ft(A1-'\1 A)1dx

A)aII I -40

as A / A*.

By Lemma 5.5.4 we also obtain

0<J

X01

C(A1

ai-a} 'p1

where C = sup0 0 satisfies (f 1) and (f2), is not convex, and f = oo (see Figure 5.4). The convexity assumption of f is needed in the framework of Amann's theorem and provides the stability of the minimal solution. In our context we prove the uniqueness of a classical solution whereas the stability of the solution will be achieved in a more general setting in Chapter 8.

The case of singular nonlinearities

141

f(t)

FIGURE 5.4. The singular monotone case on f with t < oo respectively f = 00.

Theorem 5.6.1 Assume that condition (f 1) and (f2) are fulfilled. (i) If A > At/a, then problem (5.4) has no classical solutions. (ii) If 0 < A < Ai/a. then problem (5.4) has a unique solution ua such that (iii) ua E C2(S2) n C1,1-- (P); (ii2) lima/A. ua = oo uniformly on compact subsets of Q. Proof Existence and nonexistence of a classical solution as well as the uniqueness follows in the same manner as in the proof of Theorem 5.3.1. The regularity of the solution in (iii) is similar to that obtained in Theorem 4.3.2. For the proof of (ii2) we claim that the arguments used in Theorem 5.3.1 (iv) still work here. Indeed, we have only to show that the sequence wa = UA/Ilux112 (0 < A < Ai/a) is bounded in Hol (S2). Using the assumptions (f 1) and (f2), there exists b, c > 0

such that f (t) < at + bt_a + c for all t > 0. Then, by Holder's inequality we may write

jIvwAI2 = - fwiw < _ A*a

(A)

k(a)

J waf (U,\)

J wa(aua + bua- + c)

fn

w2 + A*b

A*c r A*b W1_0, wa + a Jo k(A) J k(a) Iq(1+a)/2 + A*c I911/2.

< A*a + k(A)1+a

k(A)

From now on, we follow the proof of Theorem 5.3.1 line to line. This completes 0 the proof.

Finally, let us remark that the same result holds for bifurcation problems of the type


142

U

10

(0,0)

A

1 = A,/ a

FicuRE 5.5. The bifurcation diagram for the singular problem (5.31).

-Du = g(u) +.\f (u)

u>0 u=0

in Sl,

in Q, on B1,

(5.31)

where g is a singular nonlinearity that satisfies (f 2) and f is a C' function having a sublinear growth. The bifurcation diagram is depicted in Figure 5.5.

5.7 Comments and historical notes The constant a` in the statement of Theorem 5.2.1 is also called extremal value (or the frank-Kamenetskii constant in the combustion literature). The class of functions satisfying assumption (5.2) includes the family of convex mappings f : 10, oo) --+ (0, oo), which are increasing in a neighborhood of 00.

The important example corresponding to f (t) = et is related to the celebrated Liouville-Gelfand problem

Du=Ae"

in St,

u=0

on 0"0.

(5.32)

This problem was initially studied by Liouville [132) if N = 1 and, subsequently,

by Gelfand [82]. The solutions of (5.32) arise as steady-state solutions of the nonlinear evolution problem (called the solid fuel ignition model)

vt = Av + A (I -

v=0

in d, on 811,

within the approximation e at for all t > 0, where a :_ limt . f (t)/t E (0, oo), was studied in Theorem 5.3.1. The proof of this theorem relies essentially on the maximum principle. The complementary nonmonotone case is more difficult and more rich in information. This different setting is analyzed in this chapter by means of combined arguments, including variational methods like the mountain pass theorem of Ambrosetti and Rabinowitz.

(0,0)

t

FIGURE 6.1. The nonmonotone case on f. More precisely, we study the bifurcation problem (5.4) if the nonlinear term f obeys the nonmonotone case (see Figure 6.1)-that is, f : (0, oo) - (0, oo) is a Cl convex function such that f`(0) > 0, there exists a := limt- 3 f (t)/t. E (0, oo), and

144

Bifurcation and asymptotic analysis: The nonmonotone case

t:= tl (f (t) - at) E [-oo, 0). The basic hypothesis 2 < 0 will change-radically the study of (5.4). At the same time, new and different methods are needed to deal with the multiplicity of solutions to (5.4).

Auxiliary results Lemma 6.2.1 Let a E L°O(Q) and w c- HH(St) \ {0}, w > 0, be such that 6.2

A1(a) < 0 and -Ow > aw in 9. Then the following properties are valid: (i) A, (CO = 0.

(ii) -Ow = aw in Q. (iii) w > 0 in St.

Proof Let us multiply the inequality -Ow > aw by col(a) and then integrate by parts. We obtain

acpl(a)wdx+A1(a) fz

f

cpi(a)wdx >

r

Jn

a(pl(a)wdx.

This implies that A, (a) = 0 and -Ow = aw in Q. Because w > 0 and w O_ 0, we deduce w = cV1(a) for some c > 0, which concludes the proof. 0

It is .obvious that A f satisfies the hypotheses of Theorem 1.2.5 for all 0 < A < Al/d. This yields A* > Al/a. Lemma 6.2.2 The following properties hold true: (i) If problem (5.4) has a solution for A = A*, then this solution is necessarily unstable.

(ii) Problem (5.4) has at most one solution when A = A*. (iii) u,, is the only solution u of problem (5.4) such that A,(Af'(u)) > 0. Proof (i) Suppose that (5.4) with A= A* has a solution u* with A1 (A* f'(u*)) > 0. Thus, by the implicit function theorem applied to the mapping G : CQ'1/2(?) x l1 , C°°1/2(52),

G(u, A) = -Du - Af(u),

it follows that problem (5.4) has a solution for A in a neighborhood of A*, contradicting by this the definition of A*. (ii) Let u be a solution of (5.4) for A = A*. Then u is a supersolution of (5.4) for all 0 < A < A*. Using the minimality of ua, it follows that u > ua in 92 for all 0 < A < A*. This shows that ua (which increases with A) converges to a certain u* in L'(Q). Let us multiply by ua in (5.4) pand integrate by parts. We obtain

f

IVuaj2dx = A J f (ua)u. dx < A* o

Hence, (ua)o<j 0. Then

-Aw = A* (f (u) - f (u*)) > A* f'(u*)w

in Q.

(6.2)

Because A, (A* f'(u*)) < 0, by Lemma 6.2.1 it follows that either w - 0 or

w > 0,

Al (A* f'(u*)) = 0, and - ©w = A* f'(u*)w in 9.

Assuming w $ 0, by (6.2) we deduce that f is linear in all the intervals of the

form [u*(x),u(x)], x E Q. It is easy to see that this forces f to be linear in [0, max.-u]. Let a, f3 > 0 be such that f (u) = au + ,Q and f (u*) = au* + /3. We have

0 = A1(A*f'(u*)) = A1(A*a) = Al - A*a,

that is, A* = Al/a. Hence, u satisfies

-Du = A zu + A1-

in

ft ,

in ft, onoSl,

U>0

u>0

which is a contradiction. Thus u* = u. (iii) Suppose that problem (5.4) has a solution u # ua with A,(Af'(u)) > 0_ By the strong maximum principle we derive u > ua in Si. Let w = u - ua > 0. Then -Aw = A(f (u) - f (ua)) < A f'(u)w in ft. (6.3)

Next, we multiply by cp := ccl(Af'(u)) in (6.3) and we integrate by parts. We find

AJ f'(u)cpwdx + A,(Af'(u)) n

J

cpwdx < A f

s?

f'(u)cpwdx. t

Thus, A1(Af'(u)) = 0 and in (6.3) we have equality that is, f is linear in [0, maxi u]. Let a, /3 > 0 be such that f (u) = au +,3, f (ua) = aua + /3. Then 0 = A1(Af'(u)) = A1(Af'(ua)),

which is a contradiction. This finishes the proof.

Lemma 6.2.3 The following conditions are equivalent: (i) A* > Al/a. (ii) Problem (5..¢) has exactly one solution for A = A*. (iii) The sequence (ua)5/ 1 there exists /f n E Ho (f l), II'n II 2 = 1 such that

J IVVnl2dx
1 is bounded in HO(SE). Let V) E Ho (SE) be such that, up to a subsequence, 7l' - ' in Ho (S2) and Yin --> ip in L2(1). Furthermore, up to a subsequence, (V)n)n>1 is dominated in L2(l). This yields IIb1I2 = 1 and

j

J1n f'(un)Vindx -* Af'(u)zb2dx

as n -> oo.

(6.7)

By (6.6) and Fatou's lemma we obtain

L We have thus obtained the existence of 0 E HH(SE) such that 110112 = 1 and fsi I©/I2dx < f0A f'(u)02dx-that is, u is unstable. This finishes the proof of 0 our lemma. 6.3 Existence and bifurcation results in the nonmonotone case In the framework of the curent chapter, existence and bifurcation results are completely different to those presented in Theorem 5.3.1 in the monotone caseFurthermore, the uniqueness no longer holds in a neighborhood of the bifurcation point A*. This will be proved by means of the mountain pass theorem of Ambrosetti and Rabinowitz (see Theorem D.1.3). Let m := mint>o f (t)/t. As a result of (6.1) we have m > a. Theorem 6.3.1 Assume that f verifies (5.5) and (6.1). Then the following hold: (i) A* E (A1/a, Al/m).


148

(ii) Problem (5.4) has exactly one solution u* for A = A*;.

(iii) lima/, . ua = u* uniformly in N. (iv) ua is the unique solution of (5.4) for all 0 < A < Ai/a. (v) For all Al/a < A < A*, problem (5.4) has at least an unstable solution va. For any solution v,, 0- ua we have (vi) lima\al/a va = oo uniformly on compact subsets of Il; (vii) limA fa. va = u* uniformly in U.

The bifurcation diagram in the nonmonotone case is depicted in Figure 6.2.

FIGURE 6.2. The bifurcation diagram in Theorem 6.3.1.

Proof We first prove that A* _< Al /m. To this aim, it suffices to show that problem (5.4) has no solution for A = Al/m. Suppose the contrary. Let u be a solution of (5.4) for A = Al/m. Then, multiplying in (5.4) by cpl and integrating by parts we find Al

J

cprudx = A f cot f (u)dx.

(6.8)

Furthermore,

Al J cc1ndx =

in

1m

J ,1f(u) ? Al f o1udx, fn n

which forces f (u) = mu. This clearly contradicts the fact that f (0) > 0. The remaining parts of (i), (ii), and (iii) are equivalent in view of Lemma 6.2.3. In this sense, it is enough to prove that A* > Al/a. Assume by contradiction that A* < Al/a. By Lemma 6.2.3 we derive that A* = Al/a. Then, problem (5.4) has no solution for A = A*. Indeed, if u* would be such a solution, then by Lemma

Existence and bifurcation results in the nonmonotone case

149

6.2.2, u* is necessarily unstable. On the other hand, because f'(u*) < a we obtain

0 < a1(A* f(u*)) < a1(A*a) = 0. Hence, A (A* f'(u*)) = 0-that is, f'(u*) = a. We now proceed as in the proof of Lemma 6.2.2 (ii) to obtain a contradiction. Hence, (5.4) has no solution for A = )11/a. With the same arguments as in the proof of Lemma 6.2.3, we derive limA/A. ua = oo uniformly on compact subsets of Q.

From (6.8) we have

0=

Js

Jn

o1 [A1uA - Af (ua)] dx

cpl [(A1 - aA)uA - A(f (uA) - auA))]dx

-J

0

(6.9)

W i If (ua) - au,\] dx.

Because lima/a. ua (x) = oo for all x E 9, it follows that f (ua (x)) - aua (x) e as A / A*, for all x E Q. Passing to the limit in (6.9) we obtain the contradictory inequality 0 > -DA* fn cpi > 0. We have proved that Al/a < A* < Aj/m, and by Lemma 6.2.3, we have that problem (5.4) has a solution when A = A*. This shows that )1* < Ai/m. (iv) Taking into account Lemma 6.2.2 (iii), it is enough to prove that for all 0 < A < Ai/a, any solution u of (5.4) verifies a1(Af'(u)) > 0. Because f'(u) < a we obtain A1(Af'(u)) > A1(Aa) > 0 for all 0 < A < aI/a. (v) We want to find solutions of (5.4) different from ua, which are critical points other than u\, of the energy functional J : Ho (12) -4 R,

J(u) = 2

J

[Dul2dx

-

fF(u)dxw

F(t) = A f f f (s)ds. In the following, for each Al/a < A < A* we will take ua as uo in the mountain pass theorem of Ambrosetti-Rabinowitz. Clearly, J E C1(Ho (S2), R) and uo is a local minimum for J. To apply the Ambrosetti-Rabinowitz theorem, we transform uo into a local strict minimum by modifying J. Let E E (0, 1) and define

JE:Ho(se)-R, JE(u)=J(u)+2f IV(u-uo)I2dx. Then JJ E C1(Hfl (S2), R) and for

u, v E Hp (I) we have

JE(u)v = j Vu Vvdx - A J f (u)vdx + E in O(u - uo) Ovdx. t

insz

s2

Moreover, uo is a local strict minimum for JE provided e > 0 is small enough.

Lemma 6.3.2 Let Eo = (Aa - )11)/(2x1). Then there exists vo E Ho (1) such that JJ(vo) < JE(uo)

for all 0 < E < Eo.


150

Proof Because JE is nonincreasing with respect to e, it suffices to prove that

-oo.

lim JeO (tcpl)

1C'O

Notice that JE(tcci) = 21 t2 + 2 ait2 - exalt in (pluodx

+ 2 j IVuoI2dx a

f F(t5o1)dx.

in

Let a = (3a.\ + )1)/(4A). Because a < a, there exists a E R such that f (t) > at +,0 for all t > 0, which implies that F(t) > as/2t2 + J3At when t > 0. Then (6.10) shows that

i--

t2 A (tto1) 0.

0

The following result states that the Palais-Smale condition (see Appendix D) on Je is satisfied uniformly in E.

Lemma 6.3.3 Let (ur)n>1 C HO '(Q) and 0 < en < eo be such that and

is bounded in R

J(un) -3 0 in H-1(52).

(6.11)

Then (ur)n>i is relatively compact in Ho(c2).

Proof It suffices to prove that (ur)n>1 contains a bounded subsequence in Ho(c2). Indeed, suppose we have already proved this claim. Then, there exist 0- < e < eo and u E Ho (52) such that, up to a subsequence and as n - oo we have en -* a and

un - u

weakly in HQ (52),

un -* u

strongly in L2(52), almost everywhere in ft

un

u

Because JE, (un) -> 0 in H-1(92) we deduce

-Dun - A f (un) - En0(un - uo) - 0

in D'(52) as n

oo.

(6.12)

Note that f' < a implies F

if (un) - d (u)) < aIun - ul

In 0,

which yields f (u,,) --+ f (u) in L2(Q) as n --+ oo. Thus, from (6.12) we have

-Du-Af(u)-e0(u-uo) = 0.

(6.13)


151

Multiplying (6.13) by u and integrating over Il we obtain

(1+E)i IVul2dx-AJ of(u)dx.-eAJ of(uo)dx=0. in n in

(6.14)

From (6.11) and the boundedness of (un)n>1 we also deduce JE(un)un

0

as n -+ oo.

Therefore, as of -+ oo we have

(1 + En) in I Vun12dx - A in unf (un)dx - EnA in unf (uo)dx - 0.

(6.15)

Combining now (6.14) with (6.15) we obtain un -f u in HO'(Q) as n -+ oo. We now establish that (un)n> 1 is, up to a subsequence, bounded in HI(Q)To this aim, it suffices to show that passing to a subsequence (un)n>1 is bounded in L2(1). Then, by virtue of (6.11) we deduce that (un)n>1 is bounded in Ha(1). Let un = knwn with IIwn112 = 1, kn > 0 and assume by contradiction that up to a subsequence we have kn -* oo as n -p oo. We also may assume that En --+ e. By (6.11) we derive JE,. (un) k2n

-40

asn ->oo.

That is,

2f

J Vwn 12dx -

kn

f

2

F(un)dx + 2

dx -+ 0

st

(6.16)

as n --+ oo. Notice that n IV

2J

(Wn

- !!-0) kn

2

dx = 2 J IVwnl2dx sa f En 2 uo dx + 2n kJ

Af - En kn J

w.f(uo)dx,

so that (6.16) produces l m.

1+2En

('

J JVwn12dx - k2 12

f F(un)dx = 0.

(6.17)

2

Because F(un) = F(knwn) < Aaknw2/2+Abknwn (where b = f(0)), from (6.17) we deduce that (wn)n>1 is bounded in Ha(ft). Let w E HH(S1) be such that up to a subsequence and as n - oo we have

w,, - w wn

w

wn -+ w

weakly in Ho (1), strongly in L2(St), almost everywhere in ft


152

We claim that -(1 + E)Aw = Aaw.

(6.18)

Indeed, dividing (6.11) by k,,, we have

hm {( 1 + -,,. )

j

V wn - V v dx -

A / .f(u") v dx }=0 , kn

lS2

ffff n

ll

(6 . 19)

JJJ

for each v e Ho (1). This implies (1

+ E) in Vw . V v dx =

A h 1 f (unn) vd x 0

for all v E HQ (0) .

(6 . 20 )

With the same arguments as in the proof of Theorem 5.4.1, we derive that (f (u,,)/kn)n>1 converges almost everywhere to aw in Q. Moreover, because f is asymptotically linear and up to a subsequence (wn)n>1 is dominated in L2(St), we obtain f (un)/kn - aw in L2(0) as n - oo. Now, from (6.20) we obtain

(6.18). Furthermore, from (6.18) we deduce that w = '1 and Aa/(1 + E) = al, which contradicts the choice of Eo. This completes the proof.

0

Let us come back to the proof of (v) in Theorem 6.3.1. By the AmbrosettiRabinowitz theorem, for all 0 < E < Eo there exists ve E Ho '(!Q) such that Ef (vv) +

1

+Ef (uo)

in Q.

(6.21)

Set Ce = Jelve)

Because JE increases with E, we have co < cE < cep-that is, is bounded. Furthermore, by Lemma 6.3.3 there exists v E Ha (f) such that, up to a subsequence, vE - v in H01(9) as e -> 0. Now (6.21) implies -AV = A f (v) in Sl and, by standard regularity arguments, v is a classical solution of (5.4). To conclude the proof of (v) it remains only to show that v # uo = uA. This will be achieved by Theorem 5.2.1 and the fact that v is an unstable solution of (5.4). It can be readily seen that uo is the minimal solution of (6.21). Therefore, by Theorem 5.2.1 applied to the mapping

RDt,--ti

A

'\S

+

Ef(t)+l+Ef(uo),

it follows that vE is unstable in the sense that Al

(-_.j'V) < 0.

By virtue of Lemma 6.2.4 we derive that A, (A f ` (v)) < 0 and thus uo completes the proof of (v).

v. This


153

(vi) Suppose the contrary. With the same arguments as in the proof of Theorem 5.3.1 there exist (An)n>1 C (0, oo) such that An \ Al/a and vn E C2(SI) fl C(Q) a solution of (5.4) for A = An which is unstable and the sequence

(vn)n>1 is bounded in Li (SI). We first claim that (vn)n>1 is unbounded in Ho (52). Otherwise, let w E Ho '(Q) be such that, up to a subsequence, vn -k w weakly in H01(9) and strongly in L2(1). Then

-Avn -* -Aw in D'(SI) as n -> oo and

f (vn) -+ f (w)

in L2(SI) as n -4 oo,

which yields -Aw = Al f (w)/a. Thus, w E C2(SI) fl C(SI) is a solution of (5.4) for A = Al/a. From Lemma 6.2.4 we also deduce that A1(A1 f'(w)/a) < 0-that is, w u,\,/a,, which contradicts (iv) in the statement of our theorem. Because of the nature of our problem, the fact that (vn)n> 1 is unbounded in L2(SI) implies that (vn)n>1 is also unbounded in Ho(SI). Let vn = knwn with 1Iwn I12 = 1, kn > 0 and, up to a subsequence, k, -4 oo as n -* oo. This yields

-Awn = " f (vn) - 0 n

in Lio(I) as n --+ oo.

Also remark that the previous convergence holds.in the distribution space D'(Q), and (wn)n>1 is bounded in H01(Q) with an already provided argument. If w is a limit point of (wn)n>1 in H0 1(Q), we obtain -Aw = 0 in SI and 11w112 = 1, the desired contradiction. (vii) As before, it is enough to prove that (VA)a oo. If we write again vn = kvi,, then -Awn = A f (un) in Q. (6.22)

The fact that the right-hand side of (6.22) is bounded in L2(l) implies that (wn)n>1 is bounded in Ha (Q). Let to be such that, up to a subsequence, wn --> w weakly in Ho (SI) and strongly in L2(1). A computation already done shows that

-Aw = A*aw, w > 0 in SI and 11w112 = 1, which forces to v1 and A* = A,/a. This is clearly a contradiction because O A* > Al/a. This concludes the proof of Theorem 6.3.1.

With the same arguments as in Section 5.4 we may describe the asymptotic behavior of the unstable solutions as A \ Al/a.

Theorem 6.3.4 Let f : [0, oo) -+ (0, oo) be a Cl convex function that satisfies hypotheses (5.5), (6.1), and f'(0) > 0.

154


(i) For any unstable solution va of (5.4) we have lim (A1 -- aA) IIvx 112 = L.

a\,Aila

(ii) If va = k(A)w,\ with IIW)112 = 1 we have wA -> W in Cl(l2) as A \ Al/a, and the quotient Cpl /w,, is uniformly bounded when A approaches Al/a.

6.4 An example Example 6.1 Let f (t) = t + 2 - t -+I, t > 0. Then lim (A -AZ)ztlva1I2

Al

j3f2) z

X01

Proof Multiplying by W, in (5.4) and integrating over lZ we obtain (Al - A)

Jn

cplvadx =

Aj(2 -

v + 1)(pldx.

v,\co1dx to both sides of the previous equality and then multiplying by A - Al we obtain Adding A fn

f c,1(A - A1) v),(A - (A - Al) va)dx = 2A(A - Al) fo Wjdx (6.23) - A ( P1(A - A1)(

va + 1 -

va dx.

Jsl

Let vA = k(A)w,\, where k(A), wa are as usual. We first prove that limsupa\,\, (A - A1)2k(A) < oo. Supposing the contrary, up to a subsequence, we have (A-A1)zk(A) -+ oo as A \ A1. Then the right-hand

side of (6.23) tends to zero whereas the left-hand side goes to -oo as A \ A1, which is a contradiction. Furthermore, from (6.23) we have

Cpl v (A - (A - A1) v,)dx < 2A r p,dx.

(6.24)

If lim inf,NA, (A-A1)2k(A) = 0 then the left-hand side of (6.24) goes to oo; which

is again a contradiction. Now let c E (0, oo) be a limit point of (A - al)zk(A) as A \ A1. From (6.23) we deduce c = (A1 f cpi/zdx)z and the proof is now complete.

D

6.5 Comments and historical notes We have seen in Theorem 6.3.1 that A is a returning point from which unstable solutions va begin to emanate, which exist for any A E (Ai/a, A*). It has been proved that any such solution va has the same behavior, in the sense that


lim

,\\,\I /a

va = oo

155

uniformly on compact subsets of S2

and

lim va = u*

uniformly in S2,

where u* is the (unstable) solution corresponding to A = A*. An interesting open problem is to establish whether there is a unique unstable solution vA, for any A E (a;/a,A*). Returning points exist in many bifurcation problems. An interesting phenomenon is related to the Liouville-Gelfand equation (5.32) corresponding to f(t) = et, provided f C IRN is a ball (see, for instance, [111]). In such a case the bifurcation diagram strongly depends on N, as depicted in Figure 6.3. These results can be described as follows:

CASE 1: 1 < N < 2. Then there exists a unique solution for A = )A* and exactly two solutions for any A E (0, A*). Moreover, the unstable solution v., tends to oo as A \ 0, uniformly on compact subsets of St. This is a result of the work by Liouville [132] (if N = 1) and Bratu [29] (if N = 2). CASE 2: 2 < N < 10. Then there exists a continuum of solutions that oscillate around the line A = 2(N - 2). This result was found by Gelfand [82]. CASE 3: N > 10. Then A* = 2(N - 2) and there is a unique solution for any A E (0, 2(N - 2)). This result is a work by Joseph and Lundgren [112]. If N < 9 then the extremal solution u* corresponding to A* is smooth. If N > 10 then u* is unbounded and, in this case, u*(x) = In (1/1x12) provided fl is the unit ball in RN. In this situation, u* fails to be a classical solution but it is the unique weak solution of the Liouville-Gelfand problem in the limiting case A = A*.

FIGURE 6.3. The bifurcation diagram for the Liouville-Gelfand problem.

7

SUPERLINEAR PERTURBATIONS OF SINGULAR ELLIPTIC PROBLEMS The art of doing mathematics consists in finding that special case which contains all the germs of generality.

David Hilbert (1862-1943)

7.1

Introduction

In this chapter we are interested in the study of singular elliptic problems in the presence of smooth nonlinearities having a superlinear growth at infinity. Our analysis, which includes existence, regularity, bifurcation, and asymptotic behavior of solutions with respect to the parameters, will concern the model problem

1-Au = u-a + AuP

u>0 u=0

in Q, in Q,

on00

in a smooth bounded domain 12 C RN, where A > 0, 0 < a < 1. As a result of the continuous (or even compact) embedding of HH(dl) into

the standard LP(1l) spaces, we will be concerned with the case N _> 3 and

1 0. Furthermore, if p = I then problem (7.1) has solutions if and only if A < A1, and in this case the solution is unique. Uniqueness

of the solution has been obtained so far via Theorem 1.3.17. In our setting the assumption (HI) in Theorem 1.3.17 is not fulfilled, so that the issue of multiplicity of the solution to (7.1) is raised. The first task in this chapter is to investigate the existence of weak solutions to problem (7.1) in the following sense.

st

Definition 7.1.1 We say that a E Ho (Q) is a weak solution of (7.1) if u > 0 in S2 and

J VuVodx =

j(u

+ AuP)idx

for all 0 E H(St).

(7.2)

Superlinear perturbations of singular elliptic problems

158

In contrast to the case 0 < p < 1, a different approach is needed here because the hypotheses of Theorems 1.2.5 and 1.3.17 are not satisfied. However, we are able to show that for small values of A, problem (7.1) has at least two weak solutions, and no solutions exist if A is large. As usual in this case, the existence is proved by considering the energy functional

JA(u) = 2 / IVnI2dx Jsi

1 1

a

f ui-adx - p + I

up+idx,

u E Ha (52).

1

The main difficulty in the study of (7.1) here consists not only of the fact that Ja is not differentiable, but also of the presence of the unbounded term u_a combined with the presence of the superlinear term ur. The existence of the first solution is obtained by the sub- and supersolution method in the weak sense presented in the next section. Moreover, a direct analysis in the Hi neighborhood of the solution reveals the fact that this solution is also a local minimizer with respect to the Hi topology. The second solution is then obtained by Ekeland's variational principle. More precisely, the following result holds true.

Theorem 7.1.2 Assume that 0 < a 0 such that (1) for all 0 < A < A*, problem (7.1) has at least two weak solutions ua and va such that UA < VA in 0; (ii) for A = A*, problem (7.1) has at least one weak solution ua; (iii) for all A > A*, problem (7.1) has no weak solutions. If a is small enough, then any weak solution of (7.1) is in fact a classical solution. This will be proved in Section 7.6. In Section 7.7, the problem of asymptotic

behavior as p \ 1 will be addressed both for the bifurcation point A* and for solutions of (7.1). This matter will point out new and interesting features of problem (7.1). For instance, we shall see that if a < 1/N, then (7.1) possesses a minimal solution (which is in fact uA in the statement of Theorem 7.1.2), and any other solution of (7.1) blows up in the LO3 norm as p \ 1. 7.2 The weak sub- and supersolution method In this section we present an existence method for weak solutions that is similar to that described in Theorem 1.2.2 for classical ones. To begin with, we introduce the concept of sub- and supersolution in the weak sense. For our convenience,

let us set f,\(t) = t-a + At for A, t > 0. Definition 7.2.1 We say that u E Ho (S1) is a weak subsolution of problem (7.1)

ifu>0 in52 and n

VuVOdx < f fa(u)¢dx, s1

for all 0 E Ho '(Q), §6 > 0.

Reversing the sign in the previous inequality, we obtain the definition of a supersolution in the weak sense. The main result of this section is the following.

The weak sub- and supersolution method

159

Theorem 7.2.2 Let u and u be a weak subsolution respectively a weak supersolution of problem (7.1) such that u < u in Q. Then there exists a weak solution

uE Ho(1) of (7.1) such that u pt)

(u-" - u. ")wdx.

The last integral in the right-hand side can be evaluated as

7.5)

H' local minimizers

f

161

(u-° - u-')(u), - u)dx

(u-a - uaa)1P dx = f u

u

f+

u,+e 5>a}

(u-a - udx

(u-° - ua °)q5dx

a - a11011oo f

uA+Ek>U}

(u-

Thus, we obtain AE E

>

f

(u-° - I-a)dx.

V(ux - u)VOdx - 11011,, f

Because the Lebesgue measure of the set {u,\ + aO >- ù} tends to zero as a -# 07

the previous inequality leads us to Ae/e > o(1) as e -* 0. Similarly, we obtain Be/a < o(1) as a --+ 0. Therefore, from (7.5) we find

J (VuAVO

fa(ua)Q)dx > o(1)

as a - 0.

Replacing 0 with -0 in the previous relations and letting a -i 0, we deduce

/o (Vuxv¢ - fa(ua)4i) dx = 0

for all 0 E Ho (0).

Hence, ux is a weak solution of (7.1). The proof is now complete.

7.3 H' local minimizers An interesting property of the weak solutions obtained by the method described in Section 7.2 is that these are H1 local minimizers for the energy functional Jx. To be more specific, a function u E Ho (S2) is called a local minimizer for JA in the H1 topology if there exists r > 0 such that JA(u) < J,\ (v)

for all v E H01(9) with liv - ujjH1

)

, C HO '(Q) such that un --i ux in Ho(52) as n -> oo and JA(un) < Jx(ux). Define

wn :_ (un - u)+,

wn :_ (un - u)

Let also En = supp wn and Fn = supp wn. We first need the following result.


162

Lemma 7.3.2 lim£_o IEnI = lime,o IFnI = 0.

Proof Lets > 0. Then there exists J > 0 such that IQ \ Stag < E/2, where 5t6 = {x E St : dist(x, 80) > b}. By Corollary 1.3.8 we have

c dist(x, 8S2) > 2 > 0 for all x E Q8/2, Then,

2) ua) > -a rl Cb

-0(u - u,\) >

-1-a (u - u,\)

in Q6/2,

Again by Corollary 1.3.8 we obtain

u - ua > cldist(x, 8126/2) > C2 > 0

for all x E Sta.

Hence,

IEnI < ISt\S16l+IEnn9tal s

1

2

c2

E

1

< 2+

f f

>,nn6

fsz(un - ua)Zdx.

c2

Because un -a ua in Ho (St), for n sufficiently large, the previous estimate yields IEnI < s. Thus, limE.._,o IEnI = 0, and similarly we obtain lim,o IFFI = 0. This finishes the proof. 0 Because (wn)n>1 and (wn)n>1 are bounded in Ho (St), by Lemma 7.3.2 we have

and wn --> 0

7lln --> 0

in Ho (1)

as n -> oo.

(7.6)

Set vn := max{u, min{un, u}}. Then vn E M, vn = u in E, and vn = u in F. Furthermore,

Ja (un) = J,\(v.,,) + An + B,

(7.7)

where An

4.

ÌvunI2

2 A

p+1 B.

2

J

1

aJ

E

-TLI-«)dx

(IunI1-a

(IunIP+1 -2LP+11dx, 1

E

(Ivunl2

P+1

- IVul2)dx - 1

- IDUI2)dx - 1

(IunIP+1

(lunI1-a

1

- uP+11dx.

a IF,.

-

u1-a}dx

Existence of the first solution

163

Because ua is a minimizer of J., restricted to M, it follows that

J(un) > Ja(ua) + An + Bn. To evaluate An, we use the fact that un = u + wn in En. By the mean value theorem we have

A,,='f (IV(u +wn)f2 2

p+l,fE

- IVu(2)dx

(Iu+wnI1-a-ut-a)dx

(I1L+'t11nir+1 -UP+1)dx

>2IlwnIIH1(S2) + L,. ViVwdx

fE,. f(u+0w)wdx,

for some 0 < 0 < 1. Because u is a weak supersolution of (7.1), we deduce 2I10nIIHo(c)

An >

-

L.

(u-a + AP)wndx

+

((V + Bwn)-a + A(u + Own)P) wndx

? 2IIWnIIHi

A

JE

> 2 IIwnIIH (sz} - cA

((u + 0wn)P - uP Jwndx ll

(uP-t +

-1)wndx

E

By (7.6), Lemma 7.3.2, and Sobolev's inequality, for n large enough we derive

An > -IIWfIIH,i(n) - (L,, uP+1dx)

l

IIwnIIH) -IIH)

= 2IIWiIIHi() - o(1)IIwnhIHi(f) >- 0,

for some positive constant C > 0. In the same way we obtain Bn > 0. Hence, by

(7.7) it follows that J(un) > JA(ua), which contradicts our assumption. This finishes the proof.

7.4 Existence of the first solution By means of the weak sub- and supersolution method described in Section 7.2 we establish the existence of the first solution of (7.1).

The core of this section is the following result, which is the first step in proving Theorem 7.1.2.

Theorem 7.4.1 There exists 0 < A* < oo such that for all 0 < A < A* problem (7.1) has at least one weak solution ua, and no solutions exist if A > A*.


164

Because the embedding Hol (Sl)y L7(1) is compact for 1 0: problem (7.1) has at least one weak solution} and set A* := sup A.

Proposition 7.4.2 The set A is nonempty and A* is finite. Proof By Sobolev's and Holder's inequality we derive JA(u) > 2IIUIIHo(si) - CIIUIIHO'(cI) -

for all u c Ho (S2). Thus, we can choose A > 0 small enough and r, 8 > 0 such

that A

2IIuIIHo(O) 1

2

p

+ 1 IIuIIP+i > 26 A

p+ I IIuIIP+i j 0 JA(u)>b

for all u E BB,.,

for all u e Br, for alluEc3B,.,

where B,.{u E Ho(1l) : IIuIIH, (ci) < r}. Set

C:= inf Ja(u). uE Br

Because 0 < 1 - a < 1, for all u E B,. \ {0} there exists t > 0 small enough such that JA(tu) < 0. This yields c < 0. Let (un)n>1 C B, be a minimizing sequence for c. Then there exists u E B, such that, up to a subsequence, we have

asn --goo

un - it

weakly in H,1(0)1

'ILn -> u strongly in L9 (Q) for all 2 < q < (N + 2)/(N - 2), un -> it almost everywhere in Q. Because JA(un) = JA(Iunl), replacing if necessary un by un we may assume that un > 0 in 0, for all fn > 1. By Holder's inequality we have as n - oo

J un-c'dx < f ul-`rdx + J Iun -

uIl-adx

< J ul-adx + CIIun - U1121-c' on.

=

In

-a

In the same manner as shown earlier we obtain

Existence of the first solution

J ul-adx < j unâdx + 0

z

Thus,

i

J in

165

Iun, - ull-adx = j u'-o'dx +O(1). t

ul-adx = 1 un-adx + o(1)

as n --4oo.

n

(7.10)

A similar relation holds in the Lp norm, even in a more general setting.

Lemma 7.4.3 (Brezis-Lieb) Let 1 1 C LL(D) be a bounded sequence such that u" -+ u almost everywhere in SZ as n -> oo. Then u E LP(Q) and hull' = ;-lim (Ilunllp - Ilun - ullp)

Proof Let f > 0 and M := supn>l Ilunlip. Let us first notice that there exists CE > 0 such that

Ilt+l1'-Itl1'-1I <eltl"+Cf

for alit ER.

(7.11)

This comes from the fact that lim

It + lip - Itlp - 1

Fti-00

- 0.

Itlp

From (7.11) we derive

Ila+ blp - Ialp - IbiPI 0) which means that the set {u - u > 0} has the Lebesgue measure zero. Hence, u 1 C A be an increasing sequence such that An / A* as n - oo and let un be a solution of (7.1) for A = An. Then 1 f un-°dx - An JA(un) =21 f Iounl2dx - 1 -aJJSa P+1

and

un -« dx - A

z

f

S2

j un+1dx < 0

un+ldx = 0.

S2

From the previous relations it is easy to see that (un)n>1 is bounded in HH(Sl). Thus, there exists u* E Ha (Sl) such that, passing to a subsequence, we have as

n-aoo

un - u* weakly in Ho (Q), un --b u*

almost everywhere in Sl.

As before, we derive that un > v in Q. Because un is a weak solution of (7.1) with A = A, we have

L

Vu, VCbdx =

J A(un)0dx

for all q5 E Ho (Sl).

Passing to the limit in the previous equality and using Lebesgue's theorem, we O deduce that u* is a weak solution of (7.1) for A = A*. This ends the proof. 7.5 Existence of the second solution We have seen so far that there exists A* > 0 such that problem (7.1) has at least one solution ua if 0 < A < A*, and no solutions exist if A > A*. As a result of the superlinear character of (7.1), we are able to show that there exists another solution va of (7.1) provided 0 < A < A*. Moreover, we have 0 < ua < va in Q. We use Ekeland's variational principle, as stated in Theorem C.2.1. Define

A:= fu EHa(Sl) : u>uA inSl}. Because ua is an H1 local minimizer of JA, there exists 0 < ro < IIuA IIH (O) such

that JA(u) > JA(UA)

for all u E H1()) with IIu - UAIIH1(o) < ro.

In our further analysis, the following two cases may occur: CASE 1: inf{JA(u) : u E A, IIu - ual1Ho(O) = r} = JA (u.\), for all 0 < r < ro. CASE 2: There exists 0 < r1 < ro such that inf{JA(u) : U E A, IIu - U,\ 11H'(0) = r1} > JA(ua).

We shall discuss these two situations separately.


170 7.5,1

First case

Proposition 7.5.1 Let 0 < A < A' . Then, for all 0 < r < TO there exists a weak solution va of (7.1) such that 0 < uA < va in St and Ilea - va 11 Ho (n) = r.

As a consequence, in this case, problem (7.1) has infinitely many solutions.

Proof Fix0 < r < ro and let p > 0 be such that o < r - p < r + p < ro. Define

B:={uEA:0 oo.

(7.24)

Because the measure of the set {vn+E0 < ua < va+E0} tends to zero as n -> oo, we also have Ovn©wn,Edx =

J{uA>v,,+ect} {u >vA+et'}

+ Jua> rf

VvnV (ua - vn - Eq)dx

VvnV(ua - va - 0)dx (7.25)

Vvn0(v,\ - vn)dx + 0(1)

J VvVwdx + 0(1).


172

Using now (7.24) and (7.25) in (7.23) and then letting n --- oo we obtain (VvAV 10

j

- fa(va)we)dx - fa(va)O)dx > -E J (VvAVwE n

f (h(VA) - fa(ua))wedx

=E J V(ua - va)VwEdx+

sa

sfa

>1

J{ua>va+eO}

+

{uA>va

f

V (uA - VA)V (ua - va - eo)dx (VA

uA -) (ua - vA - E(5)dx

V(va - ua)V dx + f

ua>va+eq5}

(v, ` - ua a)gdx. uA>va+eq5}

Using again the fact that the Lebesgue's measure of the set {u,\ > va +ej} tends to zero as E - 0, the last estimate yields

i

(Vv\V - fa(va))dx > o(1)

as E->0.

Letting e -+ 0 and then reversing the sign of 0 we derive that va is a weak solution of (7.1).

El

Lemma 7.5.3 The sequence (vn)n>1 converges strongly to VA in Ho (52).

Proof Arguing as in the proof of Proposition 7.4.2, as n -+ oo we have 2

2

IIvnhII.1(0) = IIvn - val1H o(n) + IIVAIIHo(s1) + 0(1),

Ilvnllp+11 = IIvn - vAIl pp+1 + IIva

(7.26)

(7.27)

IIp+1 + 0(1),

r vn-adx = J va-adx + 0(1),

(7.28)

Ivn - val1-`tdx = o(1).

(7.29)

n

st

and

Let us take w = VA in (7.22). We have

JIV(vn - v),)I2dx + J v,,iavadx < J va-adx +. vn(vn - va)dx + o(1) J in S2

< Java-adx+.\IIvn-vaIIP+1+0(1). By Lebesgue's theorem, the previous estimate produces IIvn - vallH1(-) -< AIIv, - vaIIp+i + o(1)

as n

oo.

(7.30)


173

Letting now w = 2vn in (7.22) we have IIvn112

(r?) - AlIvnllp+1 - J vn-ad2 > o(1)

as n -r oo.

(7.31)

Because vA is a solution of (7.1) we also have IIvxllp+i

- Jn v

ads = 0.

(7.32)

From (7.26) through (7.28) and (7.31), (7.32) we find Ilvn - VAIIHo(c) -> ) I]vn - vAllp+i +o(1)

as n -> oo.

(7.33)

as n -> co.

(7.34)

Combining (7.30) and (7.33) it follows that llvn - VAIIHa(O) = AIlvn - vallp+i + o(1)

The last equality together with (7.26) through (7.29) provides

J,$v,, - va) = JA(vn) - J,\ (v,$ + 0(1)

as n -> oo.

(7.35)

Recall that vn - va in HO '(Q) as n -9 oo. Hence, by the definition of B we deduce

fIIvn-ual1Hi(n) llva - u,\IIH(f) va in Ho (S2) as n -F oo, it follows that Ilea - ua ll H,l (c2) = r. This yields va 0 ua. Notice that ua < va in Q. Furthermore, applying Corollary 1.3.8 as we did in the proof of Lemma 7.3.2 we derive that

va - u,\ > 0 in any subdomain w CC Q. Hence, ua < va in ft This completes the proof.


174

7.5.2

Second case

Similar to Proposition 7.5.1 we have the following.

Proposition 7.5.4 Let 0 < A < A*. Then there exists a solution va of problem (7.1) such that 0 < ua < va in Q.

Proof Consider the complete metric space

P=

E c([o, 11, fl)

77(o) = uA, Ja(rl(1)) < JA(ua) II71(1) - uAIIH;(Q) > r1

endowed with the distance max IIn'(t) -17(t) II

tE [0,1]

for all 77,77' E P.

Set

cP := inf max J,,(77(t)). nEP tEJ0,1]

We first prove that P is nonempty. When this claim is established, we may proceed to determine the second solution va of (7.1). This will be done by making use of Ekeland's variational principle, but in a quite different manner to that in the previous case.

To show that P is nonempty, let S be the best Sobolev constant for the embedding Ho (S2) y Lp+1(SZ)-that is,

S = inf{IIouII2 : u E H0(Il),

IIuIIp+1 =1}.

(7.38)

Then, S is independent of Si and depends only on N. Furthermore, the infimum in (7.38) is never achieved in bounded domains. If Si = R ' then the infimum is achieved by CN (1 + Ix12)(N-2)/2 ,

for some normalization constant CN > 0. Also we have S

where

A=

f

B

Up+1dx, N

Bf

Let us fix y c i and 0 E Co (SZ) such that and define

(7.39)

= A2/(p-F1) '

IVUI2dx.

= 1 in a neighborhood of y,

CNE(N-2)/2 Ue(x)

Lemma 7.5.5 We have

(e2 + Ix -

(7.40)

N

yI2)(N-2)/245{x}.


175

B+O(EN-2) and IIUEII1+I = A+0(EN); (ii) for all v E HO '(Q) and p > 0, the following estimate holds: (l)

=IIvIIp+1+pp+1IIUeIIp+1+pp

IIv+pUellp+l

fn U vpdx (7.41)

+ ppp

Upvdx + o(,- (N-2)/2).

J

Proof We give here a complete proof of (i). Remark first that VUe = CNE(N-2)/2

IX l (E2 + Ix - y12) (N-2)/2 - ((E2 + yl )N2)

Because,O = 1 in a neighborhood of y we obtain CNEN-2

II2 =

(IIoUe -

(N -2) 2 f

(E2

N

=CN(N-2)2

f

J

I

+

Ix-yl2y 12)N dx + O(Ely-2)

2 121NdX +O(EN-2)

x (1+1 1

= B + O(EN-2).

We also have II Ue IIP+1

= C2Nl N (N-2)EN

I

,52N/(N-2)

(x) dx x (EY2+Ix-y12)N

- 1 d2 NJRN (E2 + Ix - yi2)N dx + f x (E202N/(N-2) + Ix - y12)N I

- C2N/(N-2) EN

=

Cr2N/(N-2) eN N

= C2N/(N-2) N

f

1

RN

1 dx + O (Ely) (E2 + IX - y12)N

1

N (1 + Izi2)N

dz + O(EN)

= A+O(EN).

Estimate (7.41) follows with similar arguments.

Lemma 7.5.6 There exist so > 0 and po > 1 such that for all 0 < E < Co we have

for all p > pi),

JJ(ua + pUe) <JA(ua) SN/2

JA(UA + pUe) <JA(ua) + N.)(N-2)/2

for all 0 < p < po.


176

51

Proof A straightforward computation yields

f F JA(ua+pU6) =2 J 1VU I2dx+p ( Du.,VUEdx+ n

1

a f(uA + pU)1dx -

2

fIVUeI2dx (7.42)

l,lua +

Because uis a solution of (7.1) we have

Jn

Du,\VUEdx = j ua OUedx +A / uXUdx.

(7.43)

n

1

Next, from (7.41) through (7.43) we obtain

J,\(ua + pUe) =Ja(ua) +

PB

- p+

1

pP+1A

- app fn Upu,dx

(7.44)

Pp0(E(N-2)/2),

+ DE +

for some 0 < ,0 0, in view of (7.60) we have A(Vk,e) < JA,(vk) < A (7JE(tk,e)) + k This yields

JA(77e(tk,E))

- JA(vk,e) >

1

E

-

k

for all k > 1.

for allk>1.

(7.61)

On the other hand, 7IE

(t k,E ) = 'IX (t k,e ) + E (tk,E) - 7Ik(tk,e) Ck(tk,E)

= Vk,E +

Ee(tk) - 7Ik(tk)

(7.62)

Ck(tk) +E(S(tk,e) - rlk(tk,E) Ck(tk,E)

_ b(tk) COO -?1k(tk) Ck(tk)

)


181

Notice that C(tk:e) -> 1;(tk) and C(tk,e) -F c(tk) as E -> 0. Thus, from (7.62) we obtain Wk) - ?7k(tk) + O(E) as E -> 0. ?7e(tk,e) = Vk,e + E

Ck(tk)

Using this relation in (7.61) and passing to the limit with E - 0, we deduce

\

tr

J

max{1,

(VvkV(C(tk) - Vk) - MVk)(b(tk) - Vk))dx >

k

VkIIH1(o)}

0

The previous estimate contradicts (7.59). This concludes the proof. By Lemma 7.5.7 and (7.54) we also have JA(vk) --+ cp

ask -* oo.

(7.63)

Furthermore, letting w = 2ua in (7.56) we deduce CP + 0(1) =

I

1 1

> \2

a J IvkI1-°dx -

p+llllvkllHn(SZ) - (1 1

cx

p+1

I

IVklp+1dX

p+ll f

Ivkll-adx

1

k(p+ 1) (1 + IIVkIIH(;(n)) Therefore, (Vk)k>1 is bounded in Ho (S2). Hence, there exists vA E Ho (Q) such

that, up to a subsequence, we have vk - va in Ho (S2) and vk - va almost everywhere in 9 as k -* oo. At this point, we can proceed as in the proof of Proposition 7.5.1 to derive that (vk)k>1 converges weakly to va in Ha (S2) and va is a weak solution of (7.1). Let us set Ok := vk - VA, k > 1. Then, as k --> oo we have (7.64)

IAIIH,(n) = AIIfikIIp}1 + 0(1), 2II0kIIH, (S)

p+ 1

IIOkIIp}1 - CP - Ja(va) +0(1).

(7.65)

By (7.53) and for a suitable eo we obtain

l

2

S'N12

A

p+1

11411p+1
0 in Ho (S1) as k -> oo. Supposing, to the contrary, it follows that, up to a subsequence, (IIQikIIH,,I(o))k>1 is bounded away from zero. Furthermore, from (7.38) we have II0kIIH.(SL)

SIIIkkIIp+1


182

Therefore, by (7.64) we deduce S 1kl{p+1

as k --oo.

+o(1)

(-i)

(7.67)

Combining (7.64), (7.66), and (7.67), it follows that SN/2 NA(N-2)/2

/1

- CO > 12 -

\

1

p+1 1

Afl4kitp+l + o(1)

SN/2

N-(N2)/2 + o(1)

as k --> 00,

which is a contradiction. Therefore, vk -> vA strongly in HO '(11) as k --> oo. By (7.63) this yields .JA(VA) = Cr > JA(uA).

uA in Q. Moreover, by Corollary 1.3.8 we have uA < vA in Q. This 0 completes the proof of Proposition 7.5.4 and Theorem 7.1.2. Hence, vA

.

7.6 C1 regularity of solution The C' regularity of solution is based upon standard procedure for elliptic equations related to this issue. However, a restriction on the exponent of the singular term u- is required. Before we start, let us point out that the C' regularity is a particular feature of positive weak solutions to (7.1) in the sense of Definition 7.1.1. More precisely, if a weak solution u E Ho (S2) is nonnegative and vanishes at xo E SZ, then u is not differentiable at xo. Indeed, supposing to the contrary it follows that Vu(xo) = 0

and for each e > 0 we can find r > 0 such that Br(xo) cc Sl and u(x) = u(x) - u(xo) < Elx - xoI

for all x E Br(xo).

Let S E CC(Br(xo)) be such that 0 < ¢ < 1 in Br(xo), AO < cr-1_01 in B,.(xo) for some constant c > 0.

= I on B,./2(xo), and

Then, by (7.2) we have

f

uzq5dx =

Note that

f

uLodx > (xo)

J

sfA(u)gdx.

(7.68)

f u-abdx (7.69)

B ,-/2 (xo)

e:--

f

Ix - xoi--dx r/2(xo)

> E-"clrN -a

Ct regularity of solution

183

where cl > 0 is independent of T. On the other hand,

J

uzaq dx
cifc2, which is a contradiction if e is small enough. Hence, u is not differentiable in x0_ The main result in this section is the following.

Theorem 7.6.1 Assume 0 < a < 1/N and 1 0, 0 < y < 1, and C > 0 that are independent of p such that any solution u of (7.1) satisfies u E C2(9) n and (7.71) IIUIICi,,(sz) < C(1 + }}

Proof Assume first that u E C2(Q) n C(St) is a classical solution of (7.1). To obtain u E Ho (9), it suffices to prove that for all 0 E Ho (5l) there holds L,

u a¢dx < oo.

Let v E C'(fi) be the unique solution of problem (7.20). With similar arguments to those in Proposition 7.4.4 and by virtue of Corollary 1.3.8 we have

u(x) > v(x) > cdist(x, 9Q)

in Q,

(7.72)

for some positive constant c > 0 that does not depend on p. Because 0 < a < 1/N, by Holder's inequality we obtain

f

(N+2)/(2N)

(

u-aq dx < (J u,-2Na/(N+2)dx)

110112.

\n

1.

If q1a < p, we replace 2* with q1 in (7.73) and (7.74). We obtain Ilullgz 2*ao-

q1/p

After a finite number of such iterations we find a positive integer m, which is independent of p, and q,,,, > 1 such that p and Iluh- 0 such that A(l + b) < A1. By virtue of Theorem 5.6.1, there exists w E C2(Q) fl C1,1-a(SZ) such that

-dw = w-a + A(l + b)w w>0 w=0

in Q, in fl, on 8S1.

Hence,

-©w > w-' + Awp + A((' + b) - llw 1lo0 1) w >w_a+Awp

in0 asp\1.

Therefore, w is a supersolution of (7.1). Notice that v defined in (7.20) is a subsolution of (7.1). As in the proof of Proposition 7.4.4, we find v < w in fl. Thus, by Theorem 7.2.2, problem (7.1) has at least one solution. Therefore, lim infpNl Ap > A. Because A < Ai was arbitrary, we obtain lim infp\, 1 Ap > Al. Now let A > A1. Then we can find pp > 1 such that

t-«+Atp>Jilt forallt>Oand1 1 such that A > claim that problem (7.1) has no solutions for 1 < p < po. Indeed, if u would be a solution of (7.1), multiplying in (7.1) by the first eigenvalue cl of (-A) we have f (fx(u) - A,u)W 1dx = 0.

Using (7.76) and the fact that 0 in fl, the previous equality leads us to a

'

contradiction. Hence, AP < A for all I Al was arbitrary, we deduce lim supp\l A* < A1. This completes the proof.

Let 0 < A < Ai and 0 < a < 1/N be fixed. By Theorem 7.7.1, for all I < p < (N + 2)/(N - 2), problem (7.1) has at least two solutions. Let up be

the solution of (7.1) obtained by Theorem 7.4.4. Recall that JA(up) < - 0, which means that 1 1 1

and

a J 1Lp-'dx - p+1 IluPllp+1

Hf upj 1IHI(o) =

2UP -adx+AlluPljp+l

0

(7.77)


186

Combining these relations, by Holder's inequality we deduce

(2

p+i! f JuPll «dx

p+1/IIuPIIHo(O) < (1 la C1llupll2-«

>1 such that Pk \ 1 and 1lupkliHo(s1) -> oo as k -+ oo. Set Mk := IjupkiIH, (ft) and let Wk := upk/Mk, k > 1. Because (wk)k>1 is bounded in Ho(S2), there exists 0 < wo E Ho (9) such that, up to a subsequence, wk - wo weakly in Ho (&) and wk -} wo'strongly in L2(1) as k --> oo. We first claim that wo # 0. From (7.77) we have _

M2 =

I u1-«dx + A Pk

uPk+1dx. In,

Pk

Dividing by Mk in the previous equality and using the fact that Mk -3 co as k -1 oo, we have

1=o(1)+A

upk+1

in

Pk2 dx

Mk

= AIlwoII 22 + A 1

(7.80)

P' Mk

pk dx +

o(1) ask-' oo.

Let us evaluate the last integral in (7.80). By the mean value theorem and estimate (7.79), we have

L.

.upk+1 - u2 Pk Pk dx

Mk2

1 still denoted by (vp)p>1 such that IIv Ik < M < oo as p \ 1. Note that by Corollary 1.3.8 there exists cp > 0 such that up, vp > cp dist (x, 8S1)

in Q.


189

Because up,vp E C2(1l) fl C1,7(i ), it follows that up2

-2

2

yP up

UP

-

2

VP

E C2 (n) fl Ci 1

VP

Then, 2

0
0

in fl,

u=0

on 8S2

(7.88)

in a smooth bounded domain 52 C RN (N > 1), p E L2(Q) is nonnegative, and f is a positive decreasing function such that fo f (t)dt < oo. It was shown that (7.88) has a unique weak solution u E H01(92). The approach given here follows an idea of Haitao [101], but the sharp esti-

mates in proving the existence of the second solution originate in the works of Brezis and Nirenberg [33], Tarantello [187], and Badiale and Tarantello [10]. For a detailed proof of the estimate (7.41), the interested reader may consult Brezis and Nirenberg [33], (34].

This chapter completes the study of bifurcation and asymptotic analysis for the model problem

1-Du=u-°`+Aup

u>0

in9, in 92,

(7.89)

u=0 on 852, where 0 < p:< (N + 2)/(N - 2), 0 < a < 1, A > 0. The results obtained so far concerning (7.89) can be summarized as follows:

If 0 < p < 1, then problem (7.89) has a unique solution ua E C2(52) fl Cl,l-«(52) (see Theorem 4.3.2). If p = 1, then problem (7.89) has solutions if and only if A < A I. Moreover, for all 0 < A < Al there exists a unique solution ua E C2(Q) fl C1,1-«(S2) of (7.89) that satisfies limb, fa, uA = oo uniformly on compact subsets of SZ (see Theorem 5.6.1). If 1 3, then there exists A* > 0 such that (i) for all 0 < A < A*, problem (7.89) has at least two weak solutions ua and va such that uA < va in 0; (ii) for A = A*, problem (7.89) has at least one weak solution ua; (iii) for all A > A*, problem (7.89) has no weak solution (see Theorem 7.1.2).

Furthermore, if 0 < a < 1/N and 1 < p < (N+2)/(N - 2), then any weak solution of (7.89) is actually a C2(S2) n C1'7(52) (0 < ry < 1) solution and (iv) limp\,, A* = A1(-0) (see Theorem 7.7.1); (v) problem (7.89) has a minimal solution;

(vi) any other solution of (7.89) blows up in the L°° norm as p

(see

Theorem 7.7.3). As remarked by Meadows [138], the C1''Y regularity is a particular feature of the fact that the weak solutions of (7.89) are positive inside the domain.

8

STABILITY OF THE SOLUTION OF A SINGULAR PROBLEM All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. James C. Maxwell (1831-1879)

The issue of the stability of a solution was raised in Chapters 5 and 6 for elliptic equations involving smooth nonlinearities. In this chapter we provide more results related to this matter, being mainly interested in the case of singular nonlinearities. The analysis we develop here is carried out in a general framework that includes the classical singular elliptic problems already considered so far and for which we have discussed the existence, bifurcation, and regularity. This fact will be illustrated by some examples presented in the last part of this chapter. 8.1 Stability of the solution in a general singular setting We are mainly interested in the stability of solutions corresponding to the problem

1-Du = f (x, u)

in Il,

u>0 u=0

in Q,

on81,

where f C RN (N > 1) is a smooth bounded domain of class C3'7, 0 < ,y < 1. We assume that the nonlinearity f : fI x (0, oo) -* R satisfies the following: (f 1) There exists an integer m > 0 such that f,

a3f &f ate atj-lark

for all1<j<m+l and1 0 in a neighborhood of &Q.

Proof As a result of the regularity of the domain, we can find S > 0 such that d(x) E C2'7(f& ), where Q6 :_ {x E Q : d(x) < b}. Let us fix -1 0

forallt>0,

pi(t) = tp+i IP(t)=0

for all 0 < t < e < p,

forallt>

where s > 0 will be defined later in this proof. Consider now cp+(x) := e+O(d(x))

x E Ti

and

vt(x) := cp'(x)v(x) x Then

E SZ.

vi- verifies

vt = 0 av± an

_

av

an

on aQ, on 0Q,

and

-Av+ + 2Vv+ - V (O o d)(x) = a} (x)v± + Ab(x)v±

in Q,

where

a=':(x) := a(x) + ((,O'(d(x))2 ± "(d(x))) IVd(x)12 ± 0'(d(x))Md(x)

in U.

Setting 4)(x) := -2V(i o d)(x), x E Ti, we obtain (8.4). Now, to have fag > 0 in a neighborhood of an, it suffices to take c > 0 such that for all x E Il with d(x) < e there holds ±a(x)d(x)'-v

+ (p + 1) (1

- p+ 1 d(x)p+l) IVd(x) I2 + p + 1 d(x)Ad(x) > C.

Because' Vd(x) I is bounded away from zero in the neighborhood of 8SZ and the mapping x --* d(x)i-aIa(x)j is bounded in 11 according to assumption (H), we El easily can find e > 0 with the previous property. This concludes the proof.

Stability of the solution of a singular problem

194

Remark 8.1.2 Lemma 8.1.1 implies that the spectrum of (8.3) coincides with the spectrum of (8.4), provided that the eigenfunctions are required to be in C2(52) n Co (S2). Moreover, we have:

(i) v > 0 in SZ if and only if vt > 0 in Q. (ii) v = 0 in 1l if and only if v+ = 0 in Q. (iii) an < 0 on 852 if and only if an < 0 on 852. (iv) an = 0 on 852 if and only if an = 0 on 812. The following result states that the strong maximum principle holds for the operators of type L+ + a(x).

Proposition 8.1.3 Let a > 0 satisfy (H) and let f be a differential operator defined as

Lv :_ -Av + (D(x) Vv - for all v E C2(12), where -D E C1(SZ) (in particular, C may be any of two operators 1. (8.5)). Assume that v E C2(1) n C1"19(1) satisfies

in Il, in 0,

f Lv + a(x)v > 0

v>0 and let xo E U be such that v(xo) = 0. (i) If x0 E Ii then v 0. (ii) If xo E 852 and v > 0 in SZ then 8 0)

defined in

< 0.

Proof Because the coefficients of the linear operator L+a(x) are locally bounded in 52, the proof of (i) follows directly by the strong maximum principle as stated in Theorem 1.3.5.

(ii) Assume by contradiction that v > 0 in 52 and v(xo) = 8v(xo)/8n = 0. Because v E C1'p(52), there exists c > 0 such that

v(x) = Iv(x) - v(xo)I < clx - xol'+A

for all x E Q.

(8.6)

Note also that 52 satisfies the interior sphere condition at x0. Hence, there exists an open ball B C 52 with a center at yo and a radius r > 0 such that B n 852 = {xo}. Define w(x) := (r - Ix - y0l)1+0/2 for all x E B. Then,

Cw + a(x)w = - Aw +,t(x) Vw + a(x)w

_2+Q N-1 2

Ix-yol(r-

x - yoI)"/2

- P( 2 -r a) (r - Ix - yol)-1+0/2 2

2 +,6 (r - Ix - Y01) ,I/ 2 Vx) - (x - y)o + a(x)w 2 Ix - yol .

in B

Stability of the solution in a general singular setting

195

Because d(x)1-pa(x) is bounded, it is readily seen that there exists ro close to r such that Lw + a(x)w < 0 in the annulus

A := {xEc:ro < Ix - xci < r}. Let e > 0 and set vE := v - ew. Then CvE + a(x)vE > 0 in A. By the standard maximum principle it follows that vE attains its minimum at a point x1 E A.

Thus, if Ixl - yol = r, then

v(x1) > 0, and if Ixi - yoI = ro, then

v6(xl) = v(xl) -ew(xl). We can chooses > 0 small enough such that 0. Hence, in both cases we have vE > 0 in A. This property holds true in particular on the rectilinear segment S C A that joins yo with xo and for which we have r - Ix - yoI = Ix - xol. Therefore, v(x) > eIx - X0111,912 for all x E S n A. Because c > 0, the previous relation contradicts (8.6). This concludes the proof. For w E Co" (Ti), consider the linear problem

Av = a(x)w v=0

in 52,

on 852.

The following result provides a continuous dependence on data for solutions of problem (8.8) even when a is a singular potential. We omit the proof, which is lengthy and beyond the purposes of this chapter. Proposition 8.1.4 Let a satisfy (H). Then, for all w E 0-00,1 (-Q), there exists a unique solution v E C2(52) n C''(52), 0 0 that is independent of w such that (8.9) llvllc1.o() < CllwllCO,l( ).

Let GQ be Green's operator of (8.8)-that is, for all w E C0'1(f), Qa(w) is the unique solution v E C2(Sl) fl C(Q) of the problem (8.8). Thus, Proposition 8.1.4 states that the Green operator !g,, : Co'1(52) - C1',3(S2) is continuous. Proposition 8.1.5 Assume that a and b satisfy the assumption (H) and b > 0 in f2. Then; there exists ko E R such that for all k > ko and all w E Co'1(52), the problem

-Av - a(x)v + kb(x)v = b(x)w

v=0

in n, (8.10)

on 852

has a unique solution v E C2(52) n C1"3 (S2) that fulfills

llvllcl,s(r) 0 that does not depend on w. Moreover, if w > 0 in 52 and w is not identically zero, then 8v/8n < 0 on 852.

196


Proof We first choose ko which ensures the uniqueness. To this aim we rewrite problem (8.10) in the form

f G-v- - a-(x)v- + kb(x)v- = b(x){o w

in 52, (8.12)

V- = 0

on 852,

where G-,a-,v-, and cp are defined in Lemma 8.1.1 and v- = cp-v. Because a- > 0 in a neighborhood 525 of 852 and b > 0 in S2, we can define ko := sup xEn\IZ&

a (x) b(x)

This yields kb(x) - a- (x) > 0 in l for all k > ko. Furthermore, by Proposition 8.1.3 we derive the uniqueness of the solution to problem (8.12) and hence the uniqueness of (8.10). Moreover, 8v-/8n < 0 on 8S2 provided that w > 0 is not identically zero. Using Remark 8.1.2, this yields 8v/8n < 0 on 852. Let us prove now the existence for all k > ko. For this purpose let Cp,P(u)

9a,

be the Green's operators associated with problems (8.8) and Ov = b(x)w v=0

in S2,

on 852,

respectively. Then problem (8.10) may be formulated as fl(v) = Gb(w),

(8.13)

where

l(v) := v - qa(v) + kGb(v) v E Cl'p(S2). Because the embedding i : Cp'Q(?) -, Ca"(S2)

(8.14)

is compact, we derive that N := N o i is a compact perturbation of.the identity in CC'A(1l). Furthermore, N is injective for all k > ko. By standard Riesz theory on compact operators, we note that 7-l is a linear homeomorphism. Thus, if w E Co"(52), then problem (8.13) has a unique solution v E Cl"a(1). Moreover, according to Proposition 8.1.4 we also have C1 IIcb(w)IIcI.fl(-a)

c'211w11cO.1(),

where C1, C2 > 0 are independent of w. This finishes the proof.

D

Theorem 8.1.6 Assume that a and b satisfy the assumption (H) and b > 0 in Q. Then the spectrum of the linear eigenvalue problem (8.3) with v E C2(52) fl C0I'3 (0) has the following properties:

Stability of the solution in a general singular setting

197

(i) It consists of at most a countable set of eigenvalues which are isolated. (ii) The first eigenvalue is simple and the associated eigenfunction v1 satisfies v1 > 0 in n and 8v1/8n < 0 on 812. (iii) It does not change when the eigenfunctions are required to be in C2(Q) rl Ha (Sl).

Proof (i) Let ko be as in Proposition 8.1.5. Then, for all k > ko, problem (8.10) defines a Green's operator G : C00'1(12) -4 C011'3(- Q), which, in view of (8.11), is bounded. Furthermore, if i is the compact embedding defined in (8.14), then

G=goi

: CO, -6 (S') -4 C"(?) is compact. Thus, (i) follows by the spectral characterization of the compact operators and the fact that the eigenvalues of (8.3) and the eigenvalues it of 9 are related by

µ(A + k) = 1.

(8.15)

(ii) By Proposition 8.1.5, the compact operator G maps the positive cone of Co'a(S2) into its interior-that is, into the set of those functions u E C01'1(i) that fulfill u > 0 in 12 and 8u/8n < 0 on 81. Hence, by the Krein-Rutman theorem (see Theorem A.2.3 in Appendix A), it follows that the first eigenvalue µl of a is simple and positive. Taking into account the relation (8.15), we derive that A, = 1/µl - k is also simple. Moreover, the first eigenfunction vl is positive in 9 and satisfies 8v1/8n < 0 on 852. (iii) Let X be the completion of Ca(a) with respect to the norm IIvIIt :=

J5

b(x)v2dx.

It turns out that X is a Hilbert space with respect to the inner product (v, w) =

Jsi b(x)vwdx

v, w E X.

X is compact. Taking into account the standard compact embedding theorems into LP spaces with weights, we need only to prove that j is continuous. Indeed, by the assumption (H) it follows that d(x)2b(x) is bounded in Q. Hence, there exists c > 0 such that b(x) < d2(x) in Q. Furthermore, we claim that the embedding j : Ho (1Z)

By the Hardy-Sobolev inequality there exists co > 0, which is independent of v such that f51

'

dx < co fn

IVvj2dx.

(8.16)

Hence,

IIvIIb 0 independent of v and w. This means that Green's operator 9 : X -* H01(9) of problem (8.10) is bounded. Then 9 :_ 9 0 j :

Hp (Il) -+ Hp (Ii) is compact. From now on, we proceed as in the proof of (i) and use the fact that the eigenvalues of (8.3) and those of G are related by (8.15). 0 This finishes the proof of Theorem 8.1.6. Theorem 8.1.6 does not give a precise answer on the sign of the first eigenvalue of (8.2). In other words, we cannot decide whether the solution u of (8.1) is stable. In this sense, the following result provides some natural and simple conditions

to derive that the first eigenvalue of (8.2) is strictly positive. It is a very useful result when applying the implicit function theorem as we did in Theorem 5.2.1.

Proposition 8.1.7 Suppose that (8.1) has a solution u E C2(Sl) n C'(52) such that (i) an < 0 on 91l; (ii) f (x, u) - uft(x, u) > 0 in 0. Then the first eigenvalue Ai of (8.2) is positive-that is, u is a stable solution of problem (8.1).

A min-max characterization of the first eigenvalue for the linearized problem

199

8.2 A min-max characterization of the first eigenvalue for the linearized problem We start this section with an equivalent condition for the sign of the first eigenvalue of problem (8.3).

Lemma 8.2.1 The first eigenvalue A of the spectral problem (8.3) is positive if and only if the operator -A - a(x) satisfies the strong maximum principle-that is, for all v E C2(11) n C' (11) not identically zero such that

Av - a(x)v > 0

v>0

in ll, on an,

(8.22)

then v > 0 in S2 and 8v(x)/8n > 0 for all x E aQ with v(x) = 0.

Proof If -A - a(x) satisfies the strong maximum principle, then we clearly have Al > 0. To prove the converse, we transform (8.22) according to Lemma 8.1.1 in

,C-v- - a-(x)v- > 0

in 1, on an,

v->0

(8.23)

and there exists w CC S2 such that a- (x) < 0 in S2 \ w. Let v1 be the first eigenvalue of (8.3) corresponding to Al. Notice that v1 > 0 in Q. Set

k := sup

2la (x)l

xE1\w \Jb(x)vl(x)

and for e > 0 define

> Q,

(8.24)

w:=v-+e+ekvl.

Thus, by (8.24) we have

L-w - a-(x)w = (iv - - a-(x)v-) + ek(G-vi - a-(x)v1) - ea-(x) > E(kA1b(x)vi -a-(x) ) > 0

in Q.

Using the continuity of w, there exists S = 8(e) > 0 such that w > 0 in 128. Moreover, by the strong maximum principle (see Theorem 1.3.5) applied to w in St \ 128 we also have to > 0 in Q \ Q. Therefore, to > 0 in St and letting e -i 0, we obtain v- > 0 in Q. Thus, we have obtained

5 G-v- - a- (x)v- > 0

in St,

v->0

in Q.

Sl

Because v- is not identically zero, by Proposition 8.1.3 we find v- > 0 in S2 and

8v-(x)/8n > 0 for all x E an with v-(x) = 0. By virtue of Remark 8.1.2, the last conclusion also holds for v. This completes the proof.


200

A min-max characterization of the first eigenvalue to (8.3) is stated in the following proposition.

Proposition 8.2.2 Assume that a, b satisfy (H) and b > 0 in Q. Then, the first eigenvalue Al of (8.3) is stated by

Al =

sup

inf

-Av - a(x)v

(8.25)

b(x)v

vEC2(n)nco(ri) 2: En v>O in S1

Proof Denote by m the quantity in the right-hand side of (8.25) and let vl be the corresponding eigenfunction of the first eigenvalue Al to (8.3). Then vi E Cz(0)flCol (52),v>0in 92,and

_ Al

-Avl - a(x)vl b(x)vl

zESO

This means that Al < m. To prove that m < A1, we argue by contradiction and we assume that there exists e > 0 and w E C2(c) fl Co (St), w > 0 in S2, such

that i nf

-©w - a(x)w b(x)w

> A l + E.

( 8 . 26 )

Let us consider the spectral problem

Av - a(x)v - (Al - e)b(x)v = Ab(x)v V=0

in S2,

on 852.

(8.27)

Clearly, E > 0 is the first eigenvalue of (8.27). Furthermore, by Lemma 8.2.1 the

operator -Av - a(x) - (Al - E)b(x) satisfies the strong maximum principle. To raise a contradiction, let r := min{tl, tz}, where t1 := sup{t > 0 : w - tvl > O in S2}, t2 :=SUP {t > 0 :

8(w - tvl)

< 0 on on }

.

1JJ

Define now W := w - rv1. Clearly, W > 0 in 52. Moreover, if r = tl then W = 0

at some point in St and if r = t2 then W = 8W/an = 0 at some point on aSt. On the other hand, by (8.26) we have

-AW - a(x)W - (Al - s)b(x)W > sb(x)(w + W) > 0

in 52.

By Lemma 8.2.1 we find that W > 0 in S2 and aW/8n < 0 on aft, which is clearly a contradiction. Hence, Al = m and the proof is now complete.

Differentiability of some singular nonlinear problems

201

8.3 Differentiability of some singular nonlinear problems In this section we are concerned with the differentiability of the semilinear elliptic problem (8.1) around positive solutions u E C2(S2)f1C1(Sl) that satisfy au/an < 0 on aQ. To this aim, let 9 : CQ (St) - Co (S2) be Green's operator of AV =W

v=0

in 0, on ad,

defined as v = G(w). By virtue of Proposition 8.1.4, 9 is bounded and it can be extended as a bounded operator to C01,8 (Ti). Furthermore, u E C2(d) fl C1(12) is a solution of (8.1) if and only if 0 0 in Si and au/8n < 0 on 812. We are concerned here with the Frechet differentiability of the operator T : Co (d) - Co (Q).

Theorem 8.3.1 Under the assumptions (f 1) and (f2), the operator T is of class Cm on the positive cone C, and for all v E C the linear operator T'(v) : Co (12) - Co (Q) is given by

T'(v)(w) = w v)w). If m > I and I < j < m then the j linear operator

(8.29)

T('} (v) : [Co(d)]' -+ CO, M)

is given by

T(i)(v)(wi,w2,...,w?) _ -G(`f( - v) (w1,w2,...,w5)). ati

(8.30)

Proof Remark first that T = I - T, where I : Co (SZ) -+ CC (12) is the identity v)). Notice that I is linear and bounded; hence, I is of class C. Furthermore, its first derivative is I and its higher order derivatives are zero. Thus, it suffices to show that for all 1 < j < m the jth derivative of T

operator and T(v) = c(f is given by

T(j)(v)(w1,w2,...,wj)=JC( 2 and the mth derivative of T is continuous.

,

v)(w1,w2,...,wj)

(8.31)


202

We prove (8.31) by induction on j. Consider first the case j = 1 and let a E C2(SZ) be such that a > 1 in Sl and a(x) = d(x)'-1 in a neighborhood of 89. Then, a

and d(x)2-alVa(x)I are uniformly bounded in Q.

(8.32)

a(x) Let v E C and W E Col(U) with IIwIIco(cz) sufficiently small. Then there exist two

positive constants cl, c2 that are independent of w such that 0 < cld(x) < v + Ow < c2d(x)

in S2,

(8.33)

for all 0 < 9 < 1. Also define

W(x)

:= a (x) (f (x'v + w) - AX, v) - ft (X, v)w}.

By the mean value theorem, relation (8.33), and the hypotheses (f 1) and (f 2) we have

IW(x)1=

a Ift(x,v+91(x)w)-ft(x,v)IIwI

(0 < 91(x) < 1) (8.34)

2 Iftt(x, v + 02(x)w)Iw2 < CiIIwIICO(n) < a(x)

(0 < 02(x) < 1)

and

Wxk(x)I
0 independent of wi, w2, ... , wj+1. This estimate implies that T(j+1)(v) exists and is given by (8.31). This conipletes the induction argument. Finally, the same reasoning that led us to the previous estimate yields rT(m) (v + wj+1) - T(m) (v), (w1, w2, .... wj ) I

< CIIwi11q(si)

Co(w)

Ilwj+1llc,l(-n)T(Ilwj+1IICi(n)),

(8.37)

which implies that T(m) is continuous. This finishes the proof. Assume next that the nonlinearity f depends on a parameter A E A, where A C R is an interval, and the derivatives 8P f /aA , 1 Co(S2). defined as

T(u, A) := u - c(f (', u, A)), satisfies

(i) for all 1 < j < m and 1 

Co (? )

is given by Oi+PT(v, A)

atiaAP

(WI, w2, ... ,

f aj+Pf (._ v_ A)

(WI, w2, ... , wj}),

for all w1i w2, ... , wj E Co (S2).

8.4 Examples The results obtained in this chapter concerning the stability and the differentiability in singular elliptic problems are illustrated by the following two examples.

Example 8.1 Let us first consider the problem

-Du = Aa(x)u'

u>0 u=0

in S2,

in Q,

(8.38)

ono)

in a smooth bounded domain SZ with -1 < a < 1, A > 0 and

(i) a E C1(9) is positive and there exists /3 E (-1 - a,1 - a) such that a(x) < cd(x)p in SZ for some c > 0; (ii) the mapping f (x, t) = a(x)ta satisfies (f 2).

If 0 < a < 1 then (8.38) arises in population dynamics when dealing with stationary solutions of the usual logistic equation with nonlinear diffusion. For

-1 < a < 0 and a E C(?i), a > 0 in Q, problem (8.38) was considered in Chapter 4, where we proved the existence and uniqueness of a solution u), E C2(1l) fl C1, (St), (0 < 7 < 1) for all A > 0. Similar results hold in the case 0 < a < 1. Concerning the stability and the differentiability of (8.38) we have the following.

Proposition 8.4.1 For all_A > 0, problem (8.38) has a unique solution ua E C2(9) fl C1 "(S2), 0 < ry < min{1, I + a + 0}, such that (i) a < 0 on c3S2; (ii) ua is stable; (iii) the mapping (0, co) D A'--4 ua E C'"-Y(52) is of class C.

The existence follows in a similar way as in the proof of Theorem 4.3.2. The conclusion in statements (ii) and (iii) follows by Proposition 8.1.7 and Theorem 8.3.1, respectively.

If a(x) changes sign in SZ, the analysis of (8.38) becomes more delicate. A detailed study in this sense was carried out by Bandle, Pozio, and Tesei [17].


205

Because 0 < a < 1, the nonlinearity f is not locally Lipschitz near the origin, which may give rise to dead core solutions. Existence of this kind of solutions was obtained in [17] by the sub- and supersolution method. Also, the uniqueness is provided under some additional hypotheses.

Example 8.2 Our second example concerns the problem 5 -©u = a(x)u-°` + Au1'

u=0

in 52, on 852,

(8.39)

withO 0; (ii) the mapping f (x, t) = + AtP satisfies (f 2). If 0 < p < 1, in view of Theorem 4.3.2 we have the same conclusion as in Proposition 8.4.1. If p = 1, taking into account the bifurcation results in Theorem a(x)t_0

5.6.1 we obtain the following.

Proposition 8.4.2 Assume p = 1. Then, for all 0 < A < Al(-0), problem (14.71) has a unique solution ua E C2(52) n C1°"r(? ), 0 < -y < min{1,1 + a + Of such that (i) as < 0 on 852;

(ii) ua is stable; (iii) the mapping (0, A1) 9 A H ua E C1'"(52) is of class CO° and uA -+ 00 uniformly on compact subsets of 52.

8.5 Comments and historical notes In this chapter we pointed out new features of singular elliptic problems, being concerned with stability and differentiability of such types of problems. First we studied the spectrum of some linearized singular elliptic problems in a general setting involving singular weights. In this sense we considered the general problem (8.3), where the (possible singular) potential b is positive in Q. If the exponent a in the assumptions (f2) and (H) satisfies a > -1/N, then the standard regularity theory for elliptic equations applies to obtain eigenfunctions in the class Co(52). However, we preferred to state our problems in an integral form by means of Green's operator. Thus, Ca (S2) turns out to be a suitable space in our analysis for all exponents -1 < a < 1. The min-max characterization of the first eigenvalue in Theorem 8.2.2 was first introduced in Donsker and Varadhan [68] and then extended in Berestycki, Nirenberg, and Varadhan (24]. Next we have studied the Frechet differentiability of the associated integral problem with respect to u and parameters. The analysis presented in this chapter follows the general line in Hernandez and Mancebo [102] or Hernandez, Mancebo, and Vega [103], [104] for general

206


elliptic operators. We also mention here the work of Bertsch and Rostamian [25], where similar singular eigenvalue problems in divergence form have been studied. By means of Hardy-Sobolev-type inequalities, it is established in [25] that the eigenfunctions belong to the class C2(1) f1 Ha (1).

9

THE INFLUENCE OF A NONLINEAR CONVECTION TERM IN SINGULAR ELLIPTIC PROBLEMS To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.

Henri Poincare (1854-1912), Science et Hypothese, 1902

9.1

Introduction

In the previous chapters we discussed the existence, the bifurcation, and the stability of classical solutions for elliptic problems involving smooth or singular nonlinearities. Our aim in this chapter is to study the following class of singular elliptic problems Au = p(d(x))g(u) + AI VuIa + E.tf (x, u)

in S2,

u>0

in 52,

U=0

on 852,

(9.1)

in a smooth bounded domain 52 C II!N (N > 1). Here, d(x) = dist(x, 8Q), A E R,

µ>0,and0 p(d(x))g(u) u>0

u=0

in S1,

in ),

(9.3)

on 852

has no classical solutions.

Proof Without losing the generality, it suffices to consider A > 0. We argue by contradiction and assume that there exists ua E C2(Q) n C(52) a solution of (9.3). Then, from (gl), we can find c > 0 small enough such that u := cco verifies

-Au +.IDi12 < p(d(x))g(u)

in Q.

Indeed, it suffices to consider c > 0 such that Al 1 +

Ac2IVVI I2 < p(d(x))g(cI PIIII)

in 52.

A general nonexistence result

209

This is obviously possible if we take into account the hypotheses on g and p(d(x)). Furthermore, we claim that (9.4) ua > u in St. -

Assuming the contrary, it follows that minj(ua - u) is achieved in R. At this point, say x0, we have i1(x0) > ua(xo), V(ua-u)(xo) = 0, and /..(u,\-u)(xo) > 0. Because g is decreasing, we obtain 0 > - A(ua - u)(xo)

_ ( - Aua + \I Vua l2\1 (xo) - (- Au + )tFViF2)\ (xo) ?p(d(xo))(g(ua(xo))/- g(u(xo))) >0,

which is a contradiction. Hence, ua > u in ft.

Let us perform the change of variable va = 1 - e- 'in (9.3). Then, 0 < VA

1 in Cl and

-Ova = all - va) (.IVual2 - ouA)

in Cl.

Furthermore, from (9.3) we have

- A(1 - v,\)p(d(x))g (_ln(1_VA))

-OVA >

va > 0 va = 0

l

in S2, in $Z,

on BQ.

To avoid the singularities in (9.5), let us consider the approximated problem

ln(1- v)) -Ov = all - v)p(d(x))g e -

v>0 v=0

in S2, in S2,

onc9Q,

with 0 < a < 1. Clearly, va is a supersolution of (9.6). Moreover, by (9.4) and the fact that limtNo(1 - e-)t)/t = A > 0, there exists c` > 0 such that vj,, > ccpr in Cl. On the other hand, there exists 0 < m < c" small enough such that mcpl is a subsolution of (9.6) and obviously mcpl < va in Cl. Then, problem (9.6) has a solution vE E C2(St) such that rnco < vE < vA Multiplying by cc Ai

f

in Cl.

in (9.6) and then integrating over Cl we find

AJ (1 - v£)Wip(d(x))g

(s - ln(1- ye)) dx.

(9.7)

The influence of a nonlinear convection term in singular elliptic problems

210

Using (9.7) we obtain

ai

(1 - va)cPip{d(x))91 -

colvadx > .1 si

ln(1

v,\)

dx

Jsa

> CJ pip(d(x))dx, na

where C > 0 and 08 = {x E 1

d(x) < 8}, for some b > 0 sufficiently small. Because cp1(x) behaves like d(x) in SZ6 and fo tp(t)dt = oo, by (9.8) we find a contradiction. It follows that problem (9.3) has no classical solutions and the proof is now complete. 0 :

A direct consequence of Theorem 9.2.1 is the following corollary.

Corollary 9.2.2 Assume that f' tp(t)dt = oo and conditions (gl), 0 < a < 2 are fulfilled. Then for all p > 0 and ) E R, problem (9.1) has no classical solutions.

9.3 A singular elliptic problem in one dimension We are concerned in this section with the following singular elliptic problem in one dimension -y"(t) = q(t)g(y(t)) 0 < t < 1, 0 < t < 1, y(t) > 0 (9.9) Y(O) = y(1) = 0, where g verifies (gl) and q : (0, 1) --+ (0, oo) is a Holder continuous function that

may be singular at 0 or 1. In the particular case q = 1 and g(t) = t-«, a > 1, we obtained in Section 4.1 some qualitative properties of the solution to (9.9). The main novelty here is the presence of the (possible singular) potential q. The existence results for problem (9.9) are of particular interest in the study of (9.1), because any solution of (9.9) may provide a supersolution in higher dimensions for problem (9.1). The main ingredient here is Theorem 1.1.2. To this aim we assume that q verifies 1

t(1 - t)q(t)dt < oo.

(9.10)

Theorem 9.3.1 Assume that (gl) and (9.10) hold. Then problem (9.9) has a unique solution y E C2(0, 1) fl C[0,1].

Proof Let us first remark that the mapping : [0, oo) -+ [0, ao),

I(t) = fat {s)

(9.11)

is bijective. Thus, by (9.10) we can choose M > 0 and e > 0 small enough such

that

A singular elliptic problem in one dimension

M ds > 2 1 1 t(1 - t)q(t)dt. 9(3)

211

(9.12)

o

We fix an integer ko > 1 with the property koe > 1, and for all k > ko consider T the problem -y"(t) = q(t)Gk(y(t)) 0 0 (which depends on A) and c > 0 such that v,), := MH(ccpl) is a supersolution of (9.1).

Proof Let us first consider c > 0 such that cWj < min{b, d(x)}

in ft

(9.35)

By the strong maximum principle we have &W1 /On < 0 on i9Q. Hence, there exist

w CC Q and 6 > 0 such that IVVI I > 6

in f2 \ w.

(9.36)

Moreover, because lim {c 2p(cpl)g(H(cco1))IV9pjI2 - 3f (x,H(cco1))} = oo,

d(z)\o

we can assume that c2p(ccPl)g(H(c'pl))IV lI2 > 3f (x, H(ccoi))

in St \ w.

(9.37)

Let M > 1 be such that Me 262 > 3.

(9.38)

Using the fact that H'(0+) > 0 and 0 < a < 1, we can choose M > 1 with M(cb) 2 cl

H'(ccPi) ? 3A(McH'(cW,)IvWiI)a

in 0 \w,

where cl is the constant from (9.33). By (9.33), (9.35), (9.36), and (9.38) we derive

Existence results in the sublinear case

217

Mc2p(cpi)g(H(cpi))IVcpi12 > 3A(McH'(ecpi)1Vcpi1)a

in fI \ w.

(9.39)

Because g is decreasing and H'(cpi) > 0 in -0, there exists M > 0 such that McAicpiH'(ccpi) > 3p(d(x))g(H(ccpi))

in w.

(9.40)

In the same manner, using (f2) and the fact that 0 in 0, we can choose M > 1 large enough such that in w

McA1w1H'(ccpi) > 3)(MH'(ccpi)1Vtpil)a

(9.41)

and

Mc)1 piH'(cwl) > 3f (x, MH(coi))

in w.

(9.42)

For M satisfying (9.38) through (9.42), we claim that

U,\(x) := MH(cpi(x))

for all x E 9

(9.43)

is a supersolution of (9.1). We have

-qua = Mc2p(cSoi),9(H(cpi))IV iI2 + McAicPiH'(cpi)

in Sl.

(9.44)

We first show that in SZ \ w there holds

f(x,ua).

Mc2p(ccoi)g(H(ccai))I©WiI2

(9.45)

Indeed, by (9.35), (9.36), and (9.38) we have

Mc2p(cWi)g(H(epi ))IVWi 12 > p(d(x))g(H(ccpi))

> (d(x))9(MH( -1

c

= p(d(x))g(ii)

in fl \ w.

Pi ))

(9.46)

The assumption (f 1) and (9.37) produce Mc2p(cpi)g(H(ccpi))l Vtpi12 > Mf (x, H(ccP1))

2 f (x, MH(ccpl)) = f(x,z9,)

(9.47)

in S2\w.

From (9.36) and (9.39) we obtain 3

c2p(coi)g(H(cpi))lVWi12> A(McH'(ccpi)lVcpil)a = AIVuala

in fl \ w.

Now, estimate (9.45) follows by (9.46), (9.47), and (9.48).

(9.48)

218


Next we prove that f (x, U ,

McA1VjH'(c0

ua=S=0

in S2,

inn, on 852.

Because A( E L1(S2) (note that ( E C2(S2)), by Theorem 1.3.17 we obtain < ua in Q. The conclusion in this case follows now by the sub- and supersolution

method for the ordered pair CASE A < 0, We fix v > 0 and let u E C2()) fl C(II) be a solution of (9.1) for A = v. Then u, is a supersolution of (9.1) for all A < 0. Set r n:=

inf

(p(d(x))g(t)

(x,t)ESZx (0,00)

+ f(x,t)).

(9.53)

Because g(0+) = oo and the mapping (0, oo) E) t 1--4 minxE? f (x, t) is positive and nondecreasing, we deduce that m is a positive. Consider the problem Av = m + AIDvI°

v=0

in S2,

on 911.

(9.54)


219

Clearly, zero is a subsolution of (9.54). Because A < 0, the solution w of the problem

-Aw=m W=0

in St,

on 99

is asupersolution of (9.54). Hence, (9.54) has at least one solution v E C2 (Q) n

C(fl). We claim that v > 0 in Q. Indeed, if not, we deduce that min.,, E-0 V is achieved at some point xo E Q. Then Vv(xo) = 0 and

-Av(xo) = m + AIVv(xo)[6 = m > 0,

which is a contradiction. Therefore, v > 0 in Q. It is easy to see that v is a subsolution of (9.1) and -Av < m < -Au in Sl, which yields v < u,, in 1 Again by the sub- and supersolution method we conclude that (9.1) has at least one classical solution ua E C2 (Q) n C(?!). This completes the proof.

In the case 1 < a < 2, the complete description is given in the following result.

Theorem 9.4.3 Assume that 1 < a < 2, u = 1, and conditions (f 1), (f 2), (gl), and (9.28) are fulfilled. Then there exists )1* > 0 such that (9.1) has at least one classical solution for all A < )1*, and no solutions exist if A > A*.

Proof We proceed in the same manner as in the proof of Theorem 9.4.1. The only difference is that (9.39) and (9.41) are no longer valid for any A > 0. The main difficulty when dealing with estimates like (9.39) is that H'(ccpi) may blow up at the boundary. However, combining the assumption 1 < a < 2 with (9.34), we can choose A > 0 small enough such that (9.39) and (9.41) hold. This implies that problem (9.1) has a classical solution provided A > 0 is sufficiently small. Set

A := {A > 0: problem (9.1) has at least one classical solution}.

From the previous arguments. A is nonempty. Let A* := sup A > 0. We first claim that if A E A, then (0, A) C A. To this aim, let Al E A and 0 < A2 < A1. If ua, is a solution of (9.1) with A = A1i then ual is a supersolution of (9.1) with A = A2, whereas c defined in (9.29) is a subsolution. Using Theorem 1.3.17 once more, we derive that (< ux> in 1 so that problem (9.1) has at least one classical solution for A = )12. This proves the claim. Because Al E A was arbitrary, we conclude that (0, A*) C_ A.

Next we prove that A* < oo. This claim will be achieved in a more general framework. We have the following proposition. Proposition 9.4.4 Let F : [0, oo) --+ [0, oo) be a C1 convex function such that

(Fl) (0)Ff0 v=0

(9.55)

on &Q

has no classical solutions for a > Q.

Proof Let v E C2(0) n C(Sl) be a solution of (9.55) and fix 0 E Cu-(St). Multiplying by 0 in (9.55) and then integrating over SZ we obtain v fn Odx

4F(IVu[))dx = 1. (vuVO - F(JVuJ) ) dx.

0 (ivui1

(9.56)

Consider now the convex conjugate F* of F defined as

F*(t) = sup{at - F(a)}

for all t E R.

aER

Now, from (9.56) we deduce

v J 4'dx < fn OF* (i.Y1) dx

for all 0 E Co (St).

(9.57)

holds

for any 0 E W1.OO(1l). Hence, if we By density, the previous estimate such that 0 > 0 in S2 and fa OF* ([04I /gi)dx < oo, then, construct 0 E by (9.57), it follows that or is finite. Without loss of generality, we may assume that St is the unit ball in RN. Let t

fi(t)-Jo F(s)+M

forallt>0.

Using the assumption (F2) we can choose M > 0 large enough such that t limt-,, ' (t) < 1. Then 4P : [0, oo) -> [0, e) is bijective and let r : [0, e) [0, co) be the inverse of 4. Define now 0 : [0, 1] - [0, oo) by 1

fi(t) =

F(r(t)) + M 10

0 .\IVuala + m

in Il,

where m > 0 is the infimum in (9.53). Because 1 < a < 2, it follows that va := Al/(a-1)u,\ verifies

-©v,\ > IVVAIa +

mAl/(a-1)

in Q.

Hence, vA is a supersolution of

-AV = IQvIa +

m)11/(a-1)

v=0

in R

on ac.

(9.58)

Because zero is a subsolution, it follows that' problem (9.58) has a classical solution that, in view of the maximum principle, is positive. According to Proposition 9.4.4 with F(t) = ta, we obtain mAlAa-1) < a--that is, A < This means that A* = inf A < < oo. Hence, )i* is finite. The existence of a solution in the case A _< 0 can be achieved exactly in the same way as in Theorem 9.4.1. This finishes the proof of Theorem 9.4.3. (Q/m)a-1

The case a = 2 is a special one, because by the change of variable (often called the Gelfand transform) v = eau - 1, we obtain a new singular problem without a gradient term. If f (x, u) depends on u, this change of variable does not preserve

either the sublinearity conditions (f 1) and (f2) on f , or the monotonicity of g. In turn, if f (x, u) does not depend on u, this approach can be successfully used. This may help us to understand better the dependence between A and it in problem (9.1). Let m := lim g(t) E [0, oo).

too


222

Theorem 9.4.5 Assume that a = 2, A > 0, it > 0 and p m 1, f - 1. (i) Problem (9.1) has a solution if and only if A(m + µ) < \,. (ii) Assume it > 0 is fixed and let A* = Aj/(m+p). Then (9.1) has a unique solution ua for every 0 < A < A*, and the sequence (ua)o 0. With the change of variable v = eau - 1, problem (9.1) becomes

-Av =

in 52,

v>0

in Q,

v=0

on BSZ,

(9.59)

where

(t) = A(t + 1)g ( ln(t + 1) I + A (t + 1)

for all t > 0.

Obviously, 0a is not monotone, but we still have that the mapping (0, co) t 1--1 fa(t)/t is decreasing and

E)


223

U

(0,0)

A=

A

A,

m+N

FIGURE 9.2. The bifurcation diagram in Theorem 9.4.5 (ii). li

t

fi

t (t) = A(m + p) 00 m

and

li

o

fi(t) fi = 00,

for all A > 0. We first remark that fia satisfies the hypotheses of Theorem 1.2.5 provided that A(m + p) < A1. Hence, problem (9.59) has at least one solution. On the other hand, because g > m on (0, oo), we obtain

fia(t) > A(m + tc)(t + 1)

for all A, t e (0, oo).

(9.60)

This implies that (9.59) has no classical solutions if .(m+p) > al. Indeed, if ua,,, would be a solution of (9.59) with A(m+,u) > A1, then ua+, is a supersolution of the problem Au = A(u + 1) in c,

U=0

on all,

(9.61)

where A = A(m + µ). Clearly zero is a subsolution of (9.61), so there exists a classical solution u of (9.61) such that u < u,\, in Q. By the maximum principle and elliptic regularity, it follows that u is positive in 12 and u E C2(ci). To raise a contradiction, we multiply by c.p in (9.61) and then integrate over Q. We find

- I w1Oudx = AJ ucpldx + AJ coidx. J S2

in

S2

This implies Al ff uW,dx = A fn u A1. The proof of the first part in Theorem 9.4.5 is therefore complete. (ii) Follows exactly in the same manner as in Theorem 5.6.1. This concludes the O proof.


224

In what follows we discuss the case a = 1. Note that the method used in Theorem 9.4.1 does not apply here for large values of A.

Assume that Il = BR for some R > 0, where BR denotes the open ball centered at the origin, with a radius R > 0. In this case, and with u = 1, problem (9.1) reads

-Au = p(R - Ixl)g(u) + AIVul + f (x, u) u>0

lxi < R, lxj < R,

u=0

(9.62)

jxl=R.

Theorem 9.4.6 Assume that 11 = BR for some R > 0, a = 1, p = 1, and conditions (f 1), (f2), and (gl) hold. Then problem (9.62) has at least one solution for all A E R.

Proof The proof in the case A < 0 is the same as in Theorem 9.4.1. In what follows we assume that A > 0. By virtue of Theorem 9.4.1, there exists u E C2(SZ) fl C(Sl) such that

-Au = p(R - lxj)9(u)

lxi < R,

u>0

ixl 0. To provide a supersolution of (9.62) we consider the problem

-Au = p(R - lxl)g(u) + AIVuI + 1 u>0 U=0

lxi < R, lxi < R,

(9.63)

1xl = R.

We need the following auxiliary result.

Lemma 9.4.7 Problem (9.63) has at least one classical solution.

Proof We are looking for radially symmetric solutions u of (9.63)-that is,

u(x) = u(r)

for all 0 < r = jxj < R.

In this case, problem (9.63) reads

-u" - N - 1 u'(r) = p(R - r)g(u(r)) + Alu'(r)j + 1

r

u > 0

t

0 < r < R,

u(R) = 0.

The first equality in (9.64) implies

-(rN-'u (r))' > 0

0 < r < R,

for all 0 < r < R.

(9.64)


225

This yields u'(r) < 0 for all 0 < r < R. Again by (9.64) we find

- (U,, + N - 1 u(r) + Au'(r)) = p(R - r)g(u(r)) + 1 The previous relation may be written as -(earrN-lu

0 < r < R.

0 < r < R,

(r))' =

(9-65)

where z)(r,t) = p(R - r)g(t) + 1

(r, t) E 10, R) x (0, oo).

From (9.65) we have

u(r) = u(0) -

jr e-)tt-N+1

f t eAs8N-10(S, u(s))dsdt

0 < r < R.

(9.66)

0

On the other hand, in view of Theorem 9.4.1 and using the fact that g is decreasing, there exists a unique solution w e C2(BR) fl C(BR) of the problem

Aw = p(R - I a[)g(w) + 1

IxI < R,

w>0

IxI 0 in (0, R). Let A = w(0) and define the sequence (vk)k>o by vo = w and

vk(r) = A - j e-att-N+f eA$sN-1 /ls, vk-1(s))dsdt,

(9.69)

a

for all 0 < r < R and k > 1. Note that vk is decreasing in [0, R) for all k > 0. From (9.68) and (9.69) it is easy to see that v1 > vo in [0, R). A further induction argument yields vk > vk_ 1 in [0, R) for all k > 1. Hence,

w=vo 1 we have

-A(Mu) = Mp(R - IzI)g(u) + ) I V(Mu)I + M

p(R - IxI)g(Mu) + \IV(Mu)I + M in BR.

(9.70)

Because f is sublinear, we can choose M > 1 such that

M > f (x, MIIulI.)

in BR.

Then, ua := Mu satisfies -AU. x> p(R - IxI)g(ua) + AIVuaI + f(x,ua)

in BR.

It follows that ua is a supersolution of (9.62). Because g is decreasing, we easily deduce u < ua in BR. Hence, problem (9.1) has at least one solution. The proof of Theorem 9.4.6 is now complete. If p is bounded in the neighborhood of the origin, the previous procedure can be applied for general bounded domains. Let fl be a smooth bounded domain

and R > 0 be such that fl C BR. According to Theorem 9.4.6, there exists V E C2(BR) fl C(BR) such that

-Av = Mg(v) + ) I Vvl + f(x, v)

IxI < R,

v>0 v=0

IxI 0 w=0

in fl, in ft, on 19Q

is a subsolution of (9.1). It is easy to see that to < v in ) and, by Theorem 1.2.3, we deduce that problem (9.1) has at least one classical solution.

9.5 Existence results in the linear case In this section we study problem (9.1) in which we drop the sublinearity assumptions (f 1) and (f2) on f , but we require, in turn, that f is linear. More precisely, we assume that f (x, t) = t for all (x, t) E ft x [0, co) and consider the problem

Existence results in the linear case

227

-Au = p(d(x))g(u) + AI Vula + pu

in 52,

u > 0

in 52,

u=0

on 852,

(9.71)

where A > 0, p > 0 and p, g verify (g1) and (9.28). We shall be concerned in this section with the case 0 < a < 1. Note that the existence result in Theorem 1.2.5 does not apply here because the mapping '(x,t) = p(d(x))g(t) + pt

(x, t) E SZ x (0,00)

is not defined on 852 x (0, oo).

Theorem 9.5.1 Assume that 0 < a < 1 and conditions (gl), (9.28) are fulfilled. Then for all A > 0, problem (9.71) has solutions if and only if p < Al. Proof Fix p E (0, Al) and A > 0. By Theorem 9.4.1 there exists u E C2(Q) n C(Q) such that in 9, -Au = p(d(x))g(u) +.1IVul'

u>0 u=0

in S2,

on8Sl.

Obviously, uap := u is a subsolution of (9.71). Because p < A1i there exists v E C2(S2) such that

1-Ov = pv + 2

in 92,

v>0 v=0

in 0,

(9.72)

on 852.

Using the fact that 0 < a < 1, we can choose M > 0 large enough such that (9.73) M > pljull,, and M > A(MIVvl)a in Q. From (9.72) and (9.73) we note that w := My satisfies -Aw > AIvwIa + p(u + w) in 9. We claim that uxu := u + w is a supersolution of (9.71). Indeed, we have

-Duaµ > p(d(x))g(u) + AjVuIa + ANVwia + piiaµ

in 52.

(9.74)

Using the assumption 0 < a < 1, we can easily deduce

t1 + t2 > (tl + t2)a Hence,

for all tl, t2 >0-

IVuia + IVwla > (IVul+ IVwl)a > IV( u + w)!a

in 52.

(9.75)

Combining (9.74) with (9.75) we obtain

in 9. is an ordered pair of sub- and supersolutions of (9.71), and

-Du), ,, > p(d(x))g(ua,) + AI VzAu is + puau Hence,

thus problem (9.71) has a classical solution uaµ provided A > 0 and 0 A1, using the same method as in the proof of Theorem 9.4.5, we deduce that problem (9.71) has no classical solutions. This finishes the proof.

9.6 Boundary estimates of the solution In this section we are concerned with the asymptotic analysis of solutions of problem (9.1) in the special case p(t) = t-a, g(t) = t-p, where a, 6 > 0. Hence, we study the boundary behavior of classical solutions to

-Au = d(x)-au-0 + AIDula + f (x, u)

in 0,

u>0

in S2,

U=0

on 852,

(9.76)

where 0 < a < 2, A > 0, and f satisfies (f 1) and (f2). Recall that if fo tp(t)dt < oo and A belongs to a certain range, then Theorems 9.4.1 and 9.4.3 assert that (9.76) has at least one classical solution ua satisfying ua < MH(cpi) in fl, for some M, c > 0. Here, H is the solution of

-H"(t) = t-aH-a(t) H,H' > 0

for all 0 < t 0 small enough such that v := mH(cpi) satisfies

-Av
0

for all 0 < t tH'(t) for all 0 < t < b. Relations (9.83) and (9.84) yield (H,(t))«-1

-H"(t) < Hence,

for all 0 < t 0 and 81 E (0, b) such that

H'(t) < cl(-

for all 0 < t < 81.

int)1/(2-«)

(9.86)

Fix t E (0, 61]. Integrating in (9.86) over [E, t], 0 < e < t, we have

H(t) - H(e) < clt(- In

t)1/(2-a)

+

2

cl

t

a J (- In

s)(a-1)/(2-a)ds.

(9.87)

E

Note that t (-Ins)(a-1)/(2-a)ds

< 00

s)(a-1)/(2-a)dS

Jp (- In

and

t\O

10

t(- In t) 1/(2-a)

= 0. (9.88)

Hence, taking e -* 0 in (9.87), there exist c2 > 0 and 62 E (0, 81) such that

H(t)
0 such that

forall0 0.

III

ds < oo,

(9.94)

232

1 'he influence of a nonlinear convection term in singular elliptic problems

A nonexistence result We establish here the following nonexistence result related to problem (9.92). 9.7.1

Theorem 9.7.1 Assume that f' p(t)g(t)dt = oo and let 4 : St x [0, oo) -> [0, oo) be a continuous function such thatP ;6 0. Then the problem

-Au + p(d(x))g(u) < 4)(x, u) u>0

u=0

in f2, in S2,

(9.95)

on on

has no classical solutions.

Proof Assume that problem (9.95) has a classical solution u and fix C > maxi D(x, u). Let v E C2 (cl) be the unique solution of -AV = C v>0 v=0

in Q,

in il,

(9.96)

on BSl.

Moreover, there exist cl, c2 > 0 such that

cld(x) < v < c2d(x)

for all x E Q.

(9.97)

By the weak maximum principle, it follows that u < v in 0. Next we consider the perturbed problem -Au + p(d(x) + e)g(u + c) = C u>0 U=0

in Q, in Sl,

(9.98)

on an.

Obviously, u and v are, respectively, sub- and supersolution of (9.98). By standard arguments, there exists uE E C2(?) a solution of (9.98) such that u < uE < v in Q. Integrating in (9.98) we obtain - in Au,dx + J p(d(x) + e)g(uE + e)dx = CjQj,

in

which yields

-1

u dv(x) + I p(d(x) + e)g(u + e)dx < M,

(9.99)

where M is a positive constant. Taking into account the fact that Sue/8n < 0 on 81Z, from (9.99) we find

fP(d(x)+e)9(u. + e)dx < M.

The case of a negative singular potential

233

Because g is decreasing and uE < v in S2, it follows that

j p(d(x) + e)g(v + -)dx < M, for any compact subset rv CC Q. Passing to the limit with e -> 0 in the previous estimate we obtain f p(d(x))g(v)dx < M, for all w CC Q. Therefore

II p(d(x))g(v)dx < M.

(9.100)

On the other hand, by (9.97) and fo p(t)g(t)ds = oo, it follows that M > f p(d(x))g(v)dx > r p(d(x))g(c2d(x))dx = oo, Jtz t

which is a contradiction. Hence, problem (9.95) has no classical solutions. This completes the proof.

Corollary 9.7.2 Assume that fo p(t)g(t)dt = oo. Then, for all ,a < 0, problem (9.92) has no classical solutions. 9.7.2

Existence result

Theorem 9.7.3 Assume that fa p(t)g(t)dt < oo and conditions (f 1), (f 2), and (gl) are satisfied.

(i) If ti = -1 and 0 < a < 2, then there exists A* > 0 such that problem (9.92) has at least one classical solution if A > A*, and no solutions exist if 0 < A < A*. (ii) If IL = 1 and 0 < a < 1, then there exists A* > 0 such that problem (9.92) has at least one classical solution for all A > A*, and no solutions exist if

00 U=0

in 0,

(9.101)

on 49Q

has at least one classical solution U. Using the regularity of f it follows that Ua E C2(ii) and there exist CI, C2 > 0 depending on A such that

cid(x) < UA(x) < c2d(x)

in Q.

(9.102)

Fix A > 0 and observe that UA, is a supersolution of (9.92). The main point is to find a subsolution uA of (9.92) such that ua < U., in Q. For this purpose, let 41(t) = p(t)g(t), t > 0, and define

234


W : [0, 00) -+ [0, 00),

4(t) =

ft (j8 (r)d'r f

1/2

ds.

j

.

As in the proof of Theorem 4.5.1, we obtain that ' is a bijective map. Let h : [0, oo) -+ [0, oo) be the inverse of T. Then h satisfies

in (0, oo), 2 f.h(t}

h'(t) h"(t} _ '(h(t))

in (0, oo), in (0, co),

(s)ds

(9.103)

h(0) = h'(0) = 0.

Thus, h E C2(0, 00) n C1 [0, 00).

The key result for this part of the proof is the following lemma.

Lemma 9.7.4 There exist two positive constants c > 0 and M > 0 such that ua := Mh(ccp1) is a subsolution of (9.92) provided A > 0 is large enough.

Proof Because h E C'[0, oo) and h(0) = 0, we can take c > 0 small enough such that h(ccpl) < d(x)

in Q.

(9.104)

By the strong maximum principle, there exist S > 0 and w CC fl such that

IV it>Sin 1\w. Let M = max{1, 2(cS)-2}.

(9.105)

Because

(Mch'(ccpl)IVwi[)a} = -oo,

lim I -p(d(x))9(h(cco1)) + d(x) \0

we can assume that in 0 \ w there holds -p(d(x))g(h(ccpl)) + McAlcplh'(ccpl) + (Mch'(ccpl)I owl I)a < 0.

(9.106)

We are now able to show that ua := Mh(ccpl) is a subsolution of (9.92), provided A > 0 is sufficiently large. Indeed, we have

-4ua =

-Mc2p(h(cp1))g(h(cp1))IVcp1I2 + Mc.lcp1h'(ccpl).

By the monotonicity of g and (9.104) we obtain

-oua +p(d(x))9(ua) + Iouala 0

in ft,

u=0

on M.

(9.115)

By Theorem 1.3.17 we have uo < UA in ft. Furthermore, from (9.102) we have cuo < o 1 in S2 for some positive constant 0 < c < 1. Note that cuo is still a subsolution of (9.115) whereas ipl is a supersolution of (9.115). Hence, problem (9.115) has a solution u E C2(ft). Multiplying by S°1 in (9.115) and then integrating over SZ we have

j

- J cp]4udx = nt

12 J A

uc1dx.

That is,

Al f uW,dx = -

in

Jo uA(ldx = Al2 J nuW,dx.

The previous equality yields fnucpldx = 0, but this is clearly a contradiction, because both u and W, are positive in Q. It follows that (9.92) has no classical solutions for 0 < A < Ao.

Step 3: Dependence on A > 0. Set

A:= {A > 0: problem (9.92) has at least one classical solution}. From the previous arguments we deduce that A is nonempty and A* := inf A is positive. We show that if A E A, then (A, oo) C A. To this aim, let Al E A and A2 > A1. If ua, is a solution of (9.92) with A = A1i then ua, is a subsolution of (9.92) with A = A2 whereas Ua2 defined by (9.101) for A = A2 is a supersolution. Furthermore, we have DUa2 + A2 f (x, Ua2) < 0 < Aua, + A2 f (x, uA,)

Ua2,u,\l>0 Ua2 = ua, = O

in Q,

in n, on Oft,

Ua2 E Ll (SZ).

Again by Theorem 1.3.17 we find ua, < Ua2 in Q. Therefore, problem (9.92) with A = A2 has at least one classical solution. Because A E A was arbitrary, we conclude that (A*, oo) C A. This completes the proof of (i). (ii) Step 1: Existence of a solution for large A. According to Lemma 9.7.4, there exists A* > 0 such that (9.92) has a subso-

lution ua for A > A* and p = -1. Then ua is also a subsolution in case p = 1,

The case of a negative singular potential

237

provided that A > A*. Let us now construct a supersolution. By Theorem 1.2.5, for all A > A* there exists v, E C2(St) a solution of

Av=Af(x,v)+1

in ),

v>0 v=0

ins),

onl3l.

Because 0 < a < 1, we can choose M = M(A) > 1 large enough such that

M > MaIVvaIa

in Q

Then, using (f 1) we obtain

-0(MvA) = AM! (x, v,\) + M > Af (x, Mva) + IV(Mva)I'

in Q.

Hence, uA := MVA E C2(S2) is a supersolution of (9.92) for all A > A*. On the other hand, because i 1i + X f (x, ua) _< 0 < Dua + A f (x, ua) in 11, by Theorem 1.3.17 we obtain uA < 9A, and finally problem (9.92) has at least one solution for all A > A*.

Step 2: Nonexistence for small A > 0. We first extend Theorem 1.3.17 in the following way.

Lemma 9.7.5 Let 0 < a < 1 and I : Sz x (0, oo)

]l8 be a Holder continuous

*(x, t)/t is decreasing for all function such that the mapping (0, oo) t x E 1. Assume that there exist v, w E C2 (I) fl C(f) such that 1

)

(a) Ow+41(x,w)+IVwla 0,

which yields

0 < (-wOv + vzw) (xo).

(9.116)

Because w(xo) < v(xo), from assumption (a), the property of ', and (9.116), it follows that 0 < (wIVvIa -vjVwIa)(xo). From Vr (xo) = 0 we also have w(xo)IVvI(xo) = v(xo)IVwI(xo).

238


Using the previous two relations we find à

0 < f (v J in - v] IOwla(xo) = va I wl-a - vl-a) I0wla(xo), LL W

which contradicts w(xo) < v(xo). This concludes the proof of our lemma. Assume by contradiction that there exists a sequence of solutions (un,),,>1 of problem (9.92) associated with a sequence of parameters 1 C (0, oo) such that A . -* 0 as n -+ oo. A simple computation shows that w(x) := A(R2 - 1x12) is positive and satisfies the inequality

Aw + f (x, w) + lvwla < 0

in S2,

where A, R > 0 are large constants. In particular, it follows from Lemma 9.7.5 that 0 < u, < in whenever A < 1. Let x,, E ) be a maximum point of um. Then Vu,,(x,,) = 0 and 0. Letting d,, = d(x ), M. = u.(xn), it follows from (9.92) that p(dn)g(Mn) < A2f (xn, Mn) : A,,, max f (x, w(x)) < CA, xEO

which yields a contradiction as n --+ oo. Hence, problem (9.92) has no classical solutions when a > 0 is small. The third step concerning the dependence on A follows in exactly the same way as in the case p = -1. This finishes the proof of Theorem 9.7.3.

9.8

Ground-state solutions of singular elliptic problems with gradient term

We continue here the study of singular elliptic problems in unbounded domains started in Section 4.7. We shall be concerned in the sequel with the corresponding problem to (9.1) in the case K = RN, N > 3. More precisely, we consider

-Au = p(x)(g(u) + f (u) + Ivu'la) u>0

I u(x) - 0

in RN,

in RN,

(9.117)

as lxl -+ oo,

where p : RN -+ (0, oo) is a Holder continuous function and 0 < a < 1. We also assume that f fulfills (f 1) and (f2), and g E C'(0, oo) satisfies g > 0, g' < 0 in (0,oo) and (gl). As we have argued in Section 4.7, solutions of such problems are called ground-state solutions, especially for the prescribed condition at infinity. For f =_ 0 and a = 0, we saw in Section 4.7 that a necessary condition have a solution is 00 I 1

tzp(t)dt < oo,

(9.118)

Ground-state solutions of singular elliptic problems with gradient term

239

where 1b(r) = minjxj=r p(x), r > r0. As in Section 4.7 we require that p satisfies 00

J

(9.119)

to(t)dt < oo, 1

where O(r) = maxjz1=r p(x), r > 0. In this framework we have the following theorem.

Theorem 9.8.1 Assume that (f 1), (f2), (gl), and (9.119) are fulfilled. Then, problem (9.117) has at least one solution.

Proof The approach is similar to that in Theorem 4.7.1. The construction of the monotone sequence with a pointwise limit that is the solution of (9.117) relies on Theorem 9.4.1. More precisely, let B,, := {x E RN : Ixl < n}. According to Theorem 9.4.1, for all n > 1 there exists un E C2(B,,,) fl C(B,,) such that

in B,,, in B,,, on 8B,,.

Dun = p(x)(9(un) + f(un) + Ivunla)

un>0 un=0 Set 1(t) := g(t) + f (t), t > 0. Then we have

Dun+1+p(x)(F(un+l)+IVun+lla) :5 0 < Dun+p(x)(W(un)+IVu,Ia)

un=0 1. Extending un by zero outside of Bn, this means that

in RN. The main point is to find an upper bound for the sequence (u,,),,>1.

Lemma 9.8.2 There exists v E C2(RN) such that

-Ov > p(x)(g(v) + f (v) + Ivvla)

v>0

in RN,

inRN,

v(x)-+0

(9.120)

as 1XI -->00.

Proof Let C be defined by (4.56) and fix k > 2 such that kl-a > 2map ca(r). In view of Lemma 2.2.4 we can de(x)

:= k

J

xl

Then 6 satisfies

c(t)dt

for all x E RN.

(9.121)


240

-A = kO(Ixl)

in RN,

>0

in-RN ,

C(x)-0

as IxI-400.

To proceed further, we implicitly define the mapping w : RN -* (0, oo) by w(x)

J

dt

_

g(t)+1 =fi(x)

forallxERN.

It is easy to see that w E C2(RN) and w(x) - 0 as IxI -+ oc. Moreover, we have IVwl =

1) = k 0

for all 0 < t < a,

1

( 10 . 11)

'F(0) = 0.

By the strong maximum principle, there exist w cc 9 and b > 0 such that

IV iI>S in S1\w

and cpl>S

in w.

(10.12)

Singular Gierer-Meinhardt systems

246

Fix M > 1 large enough such that M(cb)2 > C

and McA18W'(cIIw1II... ) >

minzE ((cal))

(10.13)

We have -ATV =

k(W j)) I°cpil2 +

in 52.

By (10.12) and (10.13) we obtain C -AU > McAi

g(v)

g(M`1'(c(Pi))

in 52.

Furthermore, u satisfies

-Du + au > f (md(x))

in 52,

9(M'1'(ccoi))

u>0 u=0

in 9, on an.

(10.14)

To avoid the singularities in (10.14) near the boundary, we consider the approximated problem

f (md(x)) - A w + aw - g(Ml'(ca1)) +

w=0

in 52 , (10.15)

on an.

Clearly, U7:= u is a supersolution of (10.15) whereas w = 0 is a subsolution. By the sub- and supersolution method and the strong maximum principle, problem (10.15) has a unique solution wE E C2(52) such that 0 < wf < u in Q. To raise

A nonexistence result

247

a contradiction, we multiply by cci in (10.15) and then we integrate over Q. We obtain f (md(x)) dx. (a + A,) wEcPldx = W1 ftt in g(M'1'(ccci))+e Because wE f (Pi g(M'(co ))+ 6 dx cP

for all w CC Q.

Let C := (a + )q) fn ucojdx. Passing to the limit with E -+ 0 in the previous inequality we obtain f (md(x))

J

dx < C < oo

for all w CC Q.

Hence,

f Vi g(MIF(c i)) dx < C < 00. Now let go :_ {x E Il : d(x) < a}. The previous estimate combined with (10.8) produces p

j

f (md(x))

dx < oo,

d(x).9(M`F(d(x)))

but this clearly contradicts (10.3). Hence, system (S) has no positive classical solutions. This finishes the proof.

If k(t) = ts, s > 0, condition (10.3) can be written more explicitly by describing the asymptotic behavior of T. We have the following corollary.

Corollary 10.2.4 Assume that k(t) = t8, s > 0, and for all 0 < m < 1 < M one of the following conditions hold:

(i) s > 1 and fo tf (mt)/g(Mt2/(1+s))dt = oo. (ii) s = 1 and fo '°{a,l/2} t f (mt)/g(Mt - In t)dt = oo. (iii) 0 < s < 1 and fo t f (mt)/g(Mt)dt = oo. Then, system (S) has no positive classical solutions.

Proof The main idea is to describe the asymptotic behavior of ' near the origin. Notice that in our setting the mapping [ : (0, a) --- [0, 1) satisfies

-W"(t) = W-s(t) 'F'(t), '(t) > 0 I

for all 0 < t < a, for all 0 < t < a,

(10.16)

'2 (0) = 0.

The asymptotic behavior of T was studied in a more general framework concerning problem (9.77) in Chapter 9. Using the arguments in the proof of Theorem 9,6.1, there exist cl, c2 > 0 such that


248

c1t2/(1+s)

< qi(t) < c2t2/(1+s)

cit - In t < 9(t)

c2t - In t

clt 0 be such that one of the following conditions hold:

(i) s > 1 and 2q > (s + 1)(p + 2). (ii) s = 1 andq > p + 2. (iii) 0 < s < 1 andq > p + 2. Then, system (10.1) has no positive classical solutions.

Proof The proofs of (i) and (iii) are simple exercices of calculus. For (ii), by Corollary 10.2.4 we have that (10.1) has no classical solutions provided s = 1 and 1/2

fo

tl+p-q(-lnt)-q/2dt = oo.

(10.17)

On the other hand, for a, b E R we have fo /2 to (- In t)bdt < oo if and only if

a > -1 or a = -1 and b < -1. Now condition (10.17) reads q > p + 2. This concludes the proof.

10.3 Existence results In this section we provide existence results for classical solutions to (S) under the additional hypothesis /3 < a. The existence is obtained without assuming any growth condition on p near the boundary, because we are able to establish general bounds for the regularized system associated with (S). In particular, we

obtain that problem (10.1) has solutions provided that r - p = s - q > 0 and

q>p-1.

For all t1, t2 > 0 define

A(t1it2) := f(t1) h(tl) In this section we suppose that A fulfills (A1)

- g(t2) k(t2)

A(t1,t2) < 0 for all t1 > t2 > 0.

We also assume that k E C1(0, oo) is a nonnegative and nondecreasing function such that '") = oo for all c > 0, where K(t) = f0t k(T)dT. (A2) limt-,,. h(t+c)

Here are some examples of nonlinearities that fulfill (Al) and (A2):

Existence results

249

(i) f (t) = tp; g(t) = tQ; h(t) = t1 ; k(t) = t3; t > 0; p, q, r, s > 0; r - p =

s-q>0;andp-q 1 that depends only on n such that sup K(ve) < C sup h(u, + e) < C sup h(v, + 11511. + 1). sa

5

N

Using assumption (A2), we deduce that (v,)E>o is uniformly bounded-that is, 11v, 11,, < m for some m > 0 independent of E. In view of (10.19), this yields uE = vE + wE < m + I]( I],,.

in Q,

and the proof of Lemma 10.3.2 is now complete.

Lemma 10.3.3 For all 0 < C2(a) of system (S)E.

< 1 there exists a solution (uE, v,) E C2(?) X

Existence results

251

Proof We use topological degree arguments. Consider the set

U:=

{(v)EC2()xC2():

Hull.,Ilvll-<M+11 u,v > 0 in fl,ulast= vlast= 0

where M > 0 is the constant in (10.18). Define 4it(u,V) = ((p t (u, v), (1>t (u, v)),

U _4 U, by

4)'(U, v) = u

- t(-0 +a)-'

9(v + E)

+P

'1(u,v)=v-t(-d+p)-L fh(u+E) l k(v + e) Using Lemma 10.3.2, we have bt(u, v) # (0,0) on BU, for all 0 < t < 1. Therefore, by the invariance at homotopy of the topological degree we have deg (4?1, U.

(0, 0)) =deg (%, U, (0, 0)) = 1.

Hence, there exists (u, v) E U such that 1 i (u, v) = (0, 0). This means that system (S)E has at least one classical solution. The proof is now complete. Let us come back to the proof of Theorem 10.3.1. Let (u6, v6) E C2(SI) x C2(SI) be a solution of (S)E. Then,

d(uE - ) - a(u6 - 0 < 0

uE-(=0

in Q, on BSI,

where ( is the unique solution of (10.4). Hence t; < uE in 0. By (10.19) it follows

that Let

w6 0 and Eli" < 0 on (0, oo), and set iJi(5(x)), x E Do. Then

AO(x) _

'/1(5(x))M5(x)

+ 0"(5(x))IV5(x) I2

y10'(E(x)) + O"(5(x))

N < NR O'(5(x)) + 0"(6(x))

in Do.

Let us now choose O(t) = Cvl-t, t > 0, where C > 0. Therefore, LO(x) < 4 5-3/2(x) [2(N -R1)5(x)

- i]

< _C6-312(X) 5-3/2(x) < 0

in Do.

We choose C > 0 large enough such that

Qt (x) < -h(M + 1)

in Do

(10.29)

and

i(x) > K(M) > supK(vE)

on aQo \ asp.

(10.30)

50

Furthermore, by relations (10.28), (10.29), and (10.30), we obtain

0(q(x) - K(vE)) > 0 O(x) - K(vE) > 0

in Do,

on Mo.

This implies O(x) > K(ve) in po-that is, 0 < vE(x) < K-1(O(x))

in Do.

Passing to the limit with s --40 in the last inequality, we have 0 < v < K-1(O(x) ) in Do. Hence,

0 < lim v(x) < lim K(qS(x)) = 0. X-XO

X-.Xo

Because xo E ffl was arbitrarily chosen, it follows that v e C(1i). Using the fact that ue = wE + vE < c + vE in 1, in the same manner we conclude u c C(U). This finishes the proof of Theorem 10.3.1. 0

Existence results

253

The next result concerns the following singular system:

-Du + an = -AV +)9V =

uvqP

+ p(x)

uP+C

in 1Z,

in D'

v4+°

u=v=0

(10.31)

on 090,

where u > 0 is a nonnegative real number.

Theorem 10.3.4 Assume that p, q > 0 satisfy p - q < 1. Then the following properties are valid: (i) System (10.31) has solutions for all v > 0. (ii) For any solution (u, v) of (10.31), there exist c1, c2 > 0 such that in Q.

cid(x) < U, V < c2d(x)

(10.32)

Moreover, the following properties hold:

(iil) If -1 iF+° vQ+Q

in 52.

Because p - q < 1, we deduce that vv := c satisfies p+a

-4v-+Oii> cP+v`pl

in SL ,

v4

for some c > 0 small enough. Therefore, by virtue of Lemma 10.2.1, we obtain

v>c(pl in Q. Let us now prove the second inequality in (10.32). To this aim, set w := u-v. With the same idea as in the proof of Lemma 10.3.2, we find

-Ow + aw < p(x)

in the set {x E St : w(x) > 0},

which yields

w 0, u'(1) < 0, and v'(1) < 0 for any solution (u, v) of system (10.36). The main result of this section is the following.

Theorem 10.4.1 Assume that 0 < q 0. Then system (10.36) has a unique solution (u, v) E C2(Q) X C2(S2).

Proof The existence part follows from Theorem 10.3.4. We prove here only the uniqueness. Suppose that there exist two distinct solutions (u1, vi), (u2, v2) E C2[0,1] X CZ[0,1] of system (10.36). First we claim that we cannot have u2 > u1 or v2 > v1 in [0,1]. Indeed, let us assume that u2 > u1 in (0,1]. Then, V2'.'

-

uply 16V2 +

q+Q

u+Q P+°

= 0 = of - five +L,

v2

v1

in (0,1)

and, by Lemma 10.2.1, we further obtain v2 > v1 in 10, 1]. On the other hand, uP

uP

v2

v2

ui - au1 + e + p(x) < 0 = u2 - au2 + 4 } p(x)

in (0, 1).

(10.37)

Because p < 1, the mapping

1'(x, t) = -at +

tv

+ p(x) V2(x)q

(

- T '

( 0

,

satisfies the hypotheses in Lemma 1.3.17. Hence, u2 < u1 in [0, 1]-that is ul U2. This also implies v1 _= v2i which is a contradiction. Replacing ul with u2 and


256

v1 with v2, we also deduce that the situation ul > u2 or vl > v2 in (0,1) is not possible.

Set U := U2 - u1 and V := v2 - vl. From the previous arguments, both U and V change sign in (0, 1). Moreover, we have the following proposition.

Proposition 10.4.2 The functions U and V vanish only finitely many times in the interval [0, 1].

Proof We write system (10.36) as

5 W"(x) + A(x)W(x) = 0 W(0) = W(1) = o,

in (0,1),

{10.38}

where W = (U,V) and A(x) _ (Atj(x))1 0 such that 0,

uz Vi cl < < c2i (i = 1, 2) in (0,1). - min{x,1 - x} min{x,1 - x}

Then, by the mean value theorem we have xIA12(x)I

ql(xq) max{vi-1(x), v2-1(x)} 0 small enough such that IA12(x)V(x)I > IA11(x)U(x)I

for a1l x E (xo,xo+r7).

(10.40)

Combining now (10.39) with (10.40), we deduce that U" has a constant sign on the interval (xo, xo + r1). This clearly contradicts the fact that U vanishes infinitely many times in (xo, xo + 71).

Similarly, if we assume that V has infinitely many zeros in [a, b], we arrive at a contradiction using the fact that A21(x) # 0, for all x E [a, b]. This finishes the proof of our lemma. 0

Lemma 10.4.4 Let W E C2[0, a], a > 0, be such that

W"(x) + A(x)W(x) = 0 1 W(0) = W'(0) = 0. Assume that A = (A23)1 0 in a neighborhood of x = 0. We may also assume that U > 0 on [0, a1], [a2, a3], ... , [a2n, a2n+1], and U < 0 on [al, a2], [a3, a4], ..., where a2,,+1 = 1 or a2n = 1 depending on the sign of U on the last interval on which U does not vanish. Let us analyze the following distinct situations. CASE 1: V > 0 in a neighborhood of x = 0. If V does not change the sign in (0, al), then we arrive at a contradiction by Lemma 10.4.5 (i). Furthermore, if V changes sign more than once in (0, a1), then we again find a contradiction, by Lemma 10.4.5 (iv). Therefore V changes sign exactly once in (0, a1)-that is,

V(a,) 0 and, more generally, V(ak) < 0 if k is odd and V(ak) > 0 for all even values of k.

If the last interval is [a2ni 1], then V (a2,,) > 0. Using Lemma 10.4.5 (i), it follows that V vanishes on (a2n, 1), but this will contradict case Lemma 10.4.5 (iv).

The case in which the last interval where U < 0 is [a2.}1, 1], follows in the same manner. This finishes the proof of Theorem 10.4.1. As a consequence of Theorem 10.3.1, solution (u, v) of system (10.31) can be approximated by the solutions of (S)e. Furthermore, the shooting method combined with the Broyden method (to avoid the derivatives) are suitable to approximate the solution of (10.31) numerically. We have considered a = 1, d = 0.5,

p = q = 1, s = 10-2, and p(x) = p1(x) = sin(rrx). In the following figures, we have plotted solution (u,v) of (Se) for a = 0 (Figure 10.1) and Q = 2 (Figure 10.2), respectively.


261

FIGURE 10.1. Solution (u,v) of system (SE) for or = 0.

0.2

04

08

08

t

FIGURE 10.2. Solution (u, v) of system (SE) for a = 2. 10.5 Comments and historical notes Activator-inhibitor systems account for many important types of pattern formation and morphogenesis that rely on cell differentiation. A central question is how the cells, which carry identical genetic code, become different from each other. Spontaneous pattern formation is also common in inorganic systems. For instance, large sand dunes are formed despite the fact that the wind permanently redistributes the sand; sharply contoured and branching river systems (which are in fact quite similar to the branching patterns of a nerve) are formed as a result to erosion, despite the fact that the rain falls more or less homogeneously over


262 the ground.

The main feature of these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. Pattern formation requires, in addition, a longer ranging confinement of the locally selfenhancing process. Turing [189] suggested in 1952 that, under certain conditions, chemicals can react and diffuse in such a way to produce steady-state heterogeneous spatial patterns of chemical or morphogen concentration. In 1972 Gierer and Meinhardt [94] proposed a mathematical model for pattern formation of spatial tissue structures in morphogenesis, a biological phenomenon discovered by Trembley [188] in 1744. The influential activator-inhibitor mechanism suggested by Gierer and Meinhardt [94], [140] may be written as P

ut = d1Lu - cru + cpu

v4

+ pop

vs Vt=d2L1v-,Qv+c'p'u

au av

0,

au 49V

=0

in 52 x (0, T),

inI x(0,T),

(10.44)

on tQ

in a smooth bounded region n C IRN (N > 1). In system (10.44), the unknown u represents the concentration of a short-range autocatalytic substance (that is, activator), and v is the concentration of its long-range antagonist (that is, inhibitor). Also p and p' stand for the source distributions of the activator and inhibitor, respectively; dl, d2 are diffusion coefficients with dl 1, q = 1, s = 0. In [44] a priori bounds are obtained via sharp estimates of the associated Green's function. The method we used in proving the uniqueness of the solution to system (10.36) is a result of the work by Choi and McKenna [43], where it

was initially obtained for p = q = r = s = 1 and p - 0.

APPENDIX A SPECTRAL THEORY FOR DIFFERENTIAL OPERATORS All intelligent thoughts have already been thought; what is necessary is only to try to think them again. Johann Wolfgang von Goethe (1749-1832)

A.1 Eigenvalues and eigenfunctions for the Laplace operator Recall first the following theorem. Theorem A.1.1 (Poincare) Let S2 C l[tN be a bounded domain (or, more generally, such that one of its projections to the coordinate axes is bounded). Then there exists a constant c > 0 such that for all u e HQ (SZ), IIUIIH1(n) HH(SZ) defined as follows: For all Consider the operator f E L2(Q), (-A)-'f is the unique u e Ho (S2) such that

Du = f

u=0

in SZ,

onr9ll.

(A.2)

Remark that the operator (-©)-1 is continuous. Indeed, by (A.2) we have fudx -1.

We point out that the existence of a countable family of eigenvalues for the Laplace operator under the Dirichlet boundary condition was established in 1894 by Poincare [165]. This pioneering result is the beginning of the spectral theory, which has played a crucial role in the development of theoretical physics and functional analysis. Linear eigenvalue problems with nonconstant potential of the type in 51, Du = AV(x)u

u=0

on 8S1

(A.3)

were studied by Bocher [28], Minakshisundaran and Pleijel [141], Pleijel [162], and Hess and Kato [107]. For instance, Minakshisundaran and Pleijel [141], [162]

were concerned with the case when V is a variable potential such that V E L°'(11), V > 0 in 11, and V > 0 in no C Sl with 19101 > 0.

A.2 Krein-Rutman theorem Definition A.2.1 Let X be a real Banach space. A nonempty set C is said to be a cone if C is a closed convex set and C fl (-C) = {0}. Definition A.2.2 A real Banach space X is called ordered if there exists a cone C and a partial order relation " < " such that x < y if and only if x - y E C. A very useful tool in the spectral theory of differential operators is the following result.

Theorem A.2.3 (Krein-Rutman) Let X be a real Banach space with an order cone C having a nonempty interior. Assume that T : X -' X is a linear compact operator such that T(C) C C. Then the following properties hold true: (i) T has exactly one eigenfunction x E C. The corresponding eigenvalue equals the spectral radius r(T) of T.

Krein-Rutman theorem

267

(ii) For all complex numbers A in the spectrum of T that are different from r(T) we have JAI < r(T). Instead of using the Krein-Rutman theorem, we can establish by means of elementary arguments that the least eigenvalue of the Laplace operator in Ho(Q) is simple. For this purpose, because eigenfunctions corresponding to higher eigenvalues Ak (k > 2) do necessarily change sign in S2, it is enough to show that the solutions of the problem

-Au = Au

in 0,

u>0

(A.4)

in S2,

u=0 on 8d are unique, up to a multiplicative constant. For this purpose, we apply an idea introduced by Benguria, Brezis, and Lieb [21], which relies on the following facts:

(i) Any solution of problem (A.4) is a minimizer of the Dirichlet energy functional

E(v) := if VvI2dx [

on the manifold

M:={vEHH(d):v>0in0and f v2 dx1 JJJ}. 111

(ii) The energy functional E(v) is convex in v2 on the cone

C:= {vE HH(d):v>Oin0}. Assuming that ul and u2 are arbitrary solutions of (A.4), then both ul and u2 minimize E on M. Set v:= (u2+u2)/2 and w := . Thus, by (ii), we deduce that w E M. A straightforward computation yields Vw = 1L,Vu1 +U2Vu2 2,v/v-

in Q.

Hence,

E(w)

= fn 2v lu1Vu1 +u2Vu212dx VU, + u1 ui + u2 u2 Vu2 2

IVw12dx

I

n

2

< f ui+u2 2 fln

_

2

dx

ui + u2 u2 1Vui12 + ui u2 IVu2I2 dx u1 + u2 u2 ) u2 + u2 ui

u 2 + u2 u1

f (IVuil2 + JVu212) dx E(ul) + E(u2) 2

2

Because ul and u2 are minimizers of E, the previous relation shows that w is a

minimizer of E, too. Thus, E(w) = E(ul) = E(u2). Hence, Vul/ul = Vii2/u2 in Q. Therefore, V(ul/u2) = 0 in St, which implies that ul/u2 is constant.

APPENDIX B IMPLICIT FUNCTION THEOREM Mediocrity knows nothing higher

than itself, but talent instantly recognizes genius.

Arthur Conan Doyle (1859-1930)

The germs of the implicit function theorem appeared in the works of Newton, Leibniz, and Lagrange. In its simplest form, this theorem involves an equation of the form F(u, A) = 0 (we can assume without loss of generality that F(0, 0) = 0). where A is a parameter and u is an unknown function. The basic assumption is that F(u, A) is invertible for u and A "close" to their respective origins, but with some loss of derivatives when computing the inverse. In his celebrated proof of the result that any compact Riemannian manifold may be isometrically embedded in some Euclidean space, Nash [147] developed a deep extension of the implicit function theorem. His ideas have been extended by various people to a technique that is now called the Nash-Moser theory. We

recall that John F. Nash Jr. is an outstanding scientist who received the 1994 Nobel Prize in economics for his pioneering analysis of equilibria in the theory of noncooperative games. Throughout these lecture notes we have used the following "standard" version of the implicit function theorem. This property is an important tool in nonlinear

analysis and, roughly speaking, it asserts that the behavior of a nonlinear system is qualitatively determined by its linearized system around the zeros of the nonlinear system.

Theorem B.1 Let X and Y be real Banach spaces and let (uo, Ao) E X x R. Consider a mapping F = F(u, A) : X x IR -+ Y of class C1 such that (i) F(uo,A0) = 0; (ii) the linear mapping F,d(uo, A0) : X -4 Y is bijective. Then there exists a neighborhood U0 of uo and a neighborhood Vo of \0 such that for every .\ E Vo, there is a unique element u(A) E Uo so that F(u(A), A) = 0. Moreover, the mapping V0 3 A'--f u(A) is of class C1.

Proof Consider the mapping 4)(u, A) : X x R --> Y x R defined by 4(u, A) (F(u, A), A). By our hypotheses, 4i is a mapping of class C1. We apply to 4) the inverse function theorem. To conclude the proof, it remains to verify that 4?`(uo, A0) : X x ll8 -> Y x R is bijective. Indeed, we have

Implicit function theorem

270 it (uo

+ tu, AO + tA) = (F(uo + tu, AO + tA), AO + tA) =

(F(uo, Ao) + Fu(uo, Ao) - (tu) + FA(uo, Ao) - (tA) + o(i), Ao + tA).

It follows that AO) _

\

F.(uo, Ao) F.(uo, Ao) I 0

which is a bijective operator, by our hypotheses. Thus, by the inverse function theorem, there exist a neighborhood U of the point (uo, AO) and a neighborhood V of (0, A) such that the equation -11(u, A) = (f, Ao)

has a unique solution, for every (f, A) E V. Now, it is sufficient to take here f = 0 0 and our conclusion follows.

With a similar proof we can establish the following global version of the implicit function theorem.

Theorem B.2 Assume F : X x R --> Y is a function of class Cl such that (i) F(0, 0) = 0; (ii) the linear mapping F (0, 0) : X -+ Y is bijective. Then there exist an open neighborhood I of 0 and a mapping I a A -- u(A) of class Cl such that u(0) = 0 and F(u(A), A) = 0. The following result has been of particular importance in our arguments in the study of bifurcation problems. Theorem B.3 Assume the same hypotheses on F as in Theorem B.2. Then there exists an open maximal interval I containing the origin and there exists a unique mapping I E) A i--> u(A) of class C' such that the following hold: (i) F(u(A), A) = 0 for every A E I. (ii) The linear mapping Fu(u(A), A) is bijective, for any A E I.

(iii) u(0) = 0.

Proof Let ul, u2 be solutions and consider the corresponding open intervals I, and I2 on which these solutions exist, respectively. It follows that ul(0) _ U2(0) = 0 and F(ul (A), A) = 0

for every A E Il,

F(u2(A), A) = 0

for every A E I2.

Moreover, the mappings F, (ul (A), A) and Fu(U2(A), A) are bijective on I,, respectively I2. But, for A sufficiently close to 0, we have ul(A) = u2(A). We wish to show that we have global uniqueness. For this purpose, let

i:={AEI1n12:u1(A)=u2(A)}. Our aim is to show that I = Il n i2. We first observe that 0 E I, so I # 0. A standard argument implies that I is closed in Il n I2. To prove that I = Il n I2, it

Implicit function theorem

271

is sufficient to argue that I is an open set in Il fl I2. The proof of this statement follows by applying Theorem B.1 for A instead of 0. Thus, I = Il fl I2. Next, to justify the existence of a maximal interval I, we consider the C' curves u,,(A) defined on the corresponding open intervals I, such that 0 E In, A) = 0 and F,,(u7,(.\), A) is an isomorphism, for any A E I,,. u.(0) = uo, A standard argument enables us to construct a maximal solution on the set

U,, I,,. This concludes the proof. We refer to a recent book [121] for various applications of the implicit function theorem.

APPENDIX C EKELAND'S VARIATIONAL PRINCIPLE It is not enough that we do our best; sometimes we have to do what is required. Sir Winston Churchill (1874-1965)

C.1

Minimization of weak lower semicontinuous coercive functionals

An important topology in which many arguments can be carried out in reflexive Banach spaces (such as LP(S2) or Wk-p(St) for 1 < p < oo) is the weak topology, in which the unit ball is weakly compact. The weak lower semicontinuity is a major tool in reflexive Banach spaces. We recall that if E is a Banach space, then a functional 4i : E -+ R is said to be weak lower semicontinuous if 4)(u) < lim inf,,- 0 4)(u,,) for any sequence (un,) in E converging weakly to u. A basic sufficient condition for a continuous functional 4i : E - R to be weakly lower semicontinuous is that 4) is convex. One of the deepest consequences is that weak lower semicontinuous coercive functionals attain their infimum on some suitable sets, as stated in the following basic result.

Theorem C.1.1 Let E be a reflexive Banach space, M C E be a weakly closed subset of E, and 4) : M -+ R such that (i) 4) is coercive on M with respect to E-that is

4)(u) - oo

as jhulI -+ oo.

(ii) 4i is weakly lower semi continuous on M.

Then 4i is bounded from below and attains its infimum on M. The proof of Theorem C.1.1 uses elementary arguments based on the definition of the weak lower semicontinuity. We refer to [27] for complete details and further comments.

C.2 - Ekeland's variational principle Ekeland's variational principle [75] was established in 1974 and is the nonlinear version of the Bishop-Phelps- theorem [161], with its main feature of how to

Ekeland's variational principle

274

_

use the norm completeness and a partial ordering to obtain a point where a linear functional achieves its supremum on a closed bounded convex set. A major

consequence of Ekeland's variational principle is that even if it is not always possible to minimize a nonnegative C1 functional 4) on a Banach space; however, there is always a minimizing sequence (u7,),>, such that 4)'(u,,) --+ 0 as n -+ oo.

We first state the original version of Ekeland's variational principle, which is valid in the general framework of complete metric spaces. Theorem C.2.1 (Ekeland's variational principle) Let (M, d) be a complete metric space and assume that 4) : M (-oo, oo], 4) $ oo, is a lower semicontinuous functional that is bounded from below. Then, for every e > 0 and for any zo E M, there exists z e M such that (i) b(z) 4)(zo) - e d(z, zo); (ii) 4)(x) > D(z) - c d(x, z), for any x E M. Proof We may assume without loss of generality that e = 1. Define the following binary relation on M:

y<x

if and only if

4)(y) - 4)(x) + d(x, y) < 0.

Then "

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