Chipot m (ed ) handbook of differential equations vol 5

Semilinear Elliptic Systems: Existence, Multiplicity, Symmetry of Solutions 1 T.. Semilinear elliptic systems: existence, multiplicity, symmetry of solutions 31.. We shall discuss mainly

Trang 3

This page intentionally left blank

Trang 4

Institute of Mathematics, University of Zürich, Zürich, Switzerland

Amsterdam•Boston•Heidelberg•London•New York•OxfordParis•San Diego•San Francisco•Singapore•Sydney•Tokyo

Trang 5

North-Holland is an imprint of Elsevier

Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands

Linacre House, Jordan Hill, Oxford OX2 8DP, UK

First edition 2008

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 written permission of the publisher

Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0)1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and

selecting Obtaining permission to use Elsevier material

Notice

No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter

of products liability, negligence or otherwise, or from any use or operation of any methods, products, tions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made

instruc-Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

ISBN: 978-0-444-53217-6

Set ISBN: 0-444-51743-x

For information on all North-Holland publications

visit our website at books.elsevier.com

Printed and bound in Hungary

08 09 10 11 12 10 9 8 7 6 5 4 3 2 1

Trang 6

This handbook is the fifth volume in the series devoted to stationary partial differentialequations As the preceding volumes, it is a collection of self-contained, state-of-the-artsurveys written by well-known experts in the field

The topics covered by this volume include in particular semilinear and superlinear tic systems, the fibering method for nonlinear variational problems, some nonlinear eigen-value problems, the studies of the stationary Boltzmann equation and the Gierer–Meinhardtsystem I hope that these surveys will be useful for both beginners and experts and help tothe diffusion of these recent deep results in mathematical science

ellip-I would like to thank all the contributors for their elegant articles ellip-I also thank LaurenSchultz Yuhasz and Mara Vos-Sarmiento at Elsevier for the excellent editing work of thisvolume

M Chipot

v

Trang 7

Trang 8

Ruf, B., Dipartimento de Matematica, Università degli Studi di Milano, Via Saldini 50,

I-20133 Milano, Italy (Ch 3)

Suzuki, T., Division of Mathematical Science, Department of Systems Innovation,

Gradu-ate School of Science, Osaka University, Japan (Ch 4)

Takahashi, F., Department of Mathematics, Faculty of Science, Osaka City University,

Japan (Ch 4)

Ukai, S., Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences,

City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong (Ch 5)

Wei, J., Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong

Kong (Ch 6)

Yang, T., Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences,

City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong (Ch 5)

vii

Trang 9

Trang 10

1 Semilinear Elliptic Systems: Existence, Multiplicity, Symmetry of Solutions 1

T Suzuki and F Takahashi

S Ukai and T Yang

6 Existence and Stability of Spikes for the Gierer–Meinhardt System 487

J Wei

ix

Trang 11

Trang 12

Contents of Volume I

1 Solutions of Quasilinear Second-Order Elliptic Boundary Value Problems via

C Bandle and W Reichel

2 Stationary Navier–Stokes Problem in a Two-Dimensional Exterior Domain 71

G.P Galdi

W.-M Ni

4 On Some Basic Aspects of the Relationship between the Calculus of Variations

Trang 13

Trang 14

Contents of Volume II

T Bartsch, Z.-Q Wang and M Willem

2 Nonconvex Problems of the Calculus of Variations and Differential Inclusions 57

G Rozenblum and M Melgaard

S Solimini

xiii

Trang 15

Trang 16

Contents of Volume III

1 Elliptic Equations with Anisotropic Nonlinearity and Nonstandard Growth

S Antontsev and S Shmarev

A Braides

M del Pino and M Musso

J Hernández and F.J Mancebo

Trang 17

Trang 18

Contents of Volume IV

6 Maximum Principles for Elliptic Partial Differential Equations 355

P Pucci and J Serrin

7 Singular Phenomena in Nonlinear Elliptic Problems: From Blow-Up Boundary

V.D R˘adulescu

xvii

Trang 19

Trang 20

CHAPTER 1

Semilinear Elliptic Systems:

Existence, Multiplicity, Symmetry of Solutions

Djairo G de Figueiredo

IMECC-UNICAMP, C.P 6065, Campinas, S Paulo 13081-970, Brazil

E-mail: djairo@ime.unicamp.br

Contents

1 Introduction 3

2 Gradient systems 5

3 Hamiltonian systems 7

4 Nonlinearities of arbitrary growth 14

5 Multiplicity of solutions for elliptic systems 16

5.1 Some abstract critical point theory 19

5.2 A second class of Hamiltonian systems 21

6 Nonvariational elliptic systems 21

6.1 The blow-up method 24

6.2 A more complete analysis of the blow-up process 28

7 Liouville theorems 31

7.1 Liouville for systems defined in the whole of R N 32

7.2 Liouville theorems for systems defined in half-spaces 38

7.3 A Liouville theorem for a full system 39

7.4 Remarks on the proof of Theorem 7.14 40

7.5 Final remarks on Liouville theorem for systems 41

8 Decay at infinite 42

8.1 Remarks on the proof of the above theorems 43

9 Symmetry properties of the solutions 43

10 Some references to other questions 45

References 45

HANDBOOK OF DIFFERENTIAL EQUATIONS

Stationary Partial Differential Equations, volume 5

Edited by M Chipot

1

Trang 21

Trang 22

Semilinear elliptic systems: existence, multiplicity, symmetry of solutions 3

1 Introduction

Semilinear elliptic systems of the type

−u = f (x, u, v), −v = g(x, u, v) in Ω, (1.1)and the more general one, where the nonlinearities depend also on the gradients, namely

−u = f (x, u, v, ∇u, ∇v), −v = g(x, u, v, ∇u, ∇v) in Ω, (1.2)have been object of intensive research recently In this work we shall discuss some aspects

of this research For the sake of the interested reader we give in Section 10 some references

to other topics that are not treated here

On the above equations u and v are real-valued functions u, v : Ω → R, where Ω is

some domain in R N , N 3, and Ω its closure There is also an extensive literature on the

case of N= 2, but in the present notes we omit the study of this interesting case; only on

Section 4 we make some remarks about this case

We shall discuss mainly the following questions pertaining to the above systems:

• Existence of solutions for the Dirichlet problem for the above systems, when Ω is

some bounded domain in R N

• Systems with nonlinearities of arbitrary growth

• Multiplicity of solutions for problems exhibiting some symmetry

• Behavior of solutions at ∞ in the case that Ω is the whole of R N

• Monotonicity and symmetry of positive solutions

Although we concentrate in the case of the Laplacian differential operator =

op-The nonlinearity of the problems appears only in the real-valued functions f, g : Ω×

R × R → R For that matter, problems involving the p-Laplacian are not studied here.

Some references are given in Section 10

Here we treat only the Dirichlet problem for the above systems Other boundary valueproblems like the Neumann and some nonlinear boundary conditions have been also dis-cussed elsewhere, see Section 10

Some systems of the type (1.1) can be treated by Variational Methods In Sections 2and 3 we study two special classes of such systems, the Gradient systems and the Hamil-

tonian systems We say that the system (1.1) above is of the Gradient type if there exists a function F : Ω × R × R → R of class C1such that

∂F

∂v = g,

and it is said to be of the Hamiltonian type if there exists a function H : Ω× R × R → R

of class C1such that

∂H

∂u = g.

Trang 23

which is defined in the Cartesian product H01(Ω) × H1

0(Ω) provided again that

H (x, u, v) |u| p + |v| q , ∀x ∈ Ω, u, v ∈ R

with p, q 2N

N−2, if the dimension N 3 However, as we shall see, the restriction on the

powers of u and v as above it is too restrictive, in the case of Hamiltonian systems We shall allow different values of p, q, as observed first in [26] and [84].

In Section 4 we discuss some Hamiltonian systems when one of the nonlinearities mayhave arbitrary growth following [44]

In Section 5 we present results on the multiplicity of solutions for Hamiltonian systems(1.1) exhibiting some sort of symmetry, we follow [9] Also in Section 5, following [34], wepresent a third class of systems which can also be treated by variational methods, namely

−u = H u (x, u, v) in Ω, −v = −H v (x, u, v) in Ω,

u(x) = v(x) = 0 on ∂Ω. (1.5)

In this form some supercritical systems can be treated, see [34]

In Section 6 we discuss classes of nonvariational systems, which are then treated byTopological Methods The difficulty here is obtaining a priori bounds for the solutions.There are several methods to tackle this question We will comment some of them, includ-ing the use of Moving Planes and Hardy type inequalities However the most successfulone in our framework seems to be the blow-up method Here we follow [42] This methodleads naturally to Liouville-type theorems, that is, theorems asserting that certain systems

have no nontrivial solution in the whole space R N or in a half-space R+N.

In Section 7, we present some results on Liouville theorems for systems

In Section 8, systems defined in the whole of R Nare considered again and the behavior

of their solutions is presented

In Section 9 we discuss symmetry and monotonicity of solutions

And finally in Section 10 we give references to other topics that are not treated here

Trang 24

2 Gradient systems

The theory of gradient systems is sort of similar to that of scalar equations

−u = f (x, u) in Ω. (2.1)

This theory has also been considered by several authors in the framework of p-Laplacians,

p u= div|∇u| p−2∇u, p > 1.

We consider the system of equations

−u = F u (x, u, v), −v = F v (x, u, v) (2.2)

subjected to Dirichlet boundary conditions, that is u = v = 0 on ∂Ω In the context of

the Variational Method, here we look for weak solutions, namely solutions in the Sobolev

space H01(Ω) So the variational method consists in looking for such solutions of (2.2) as

critical points of the functional

methods some sort of compactness is required, like a Palais–Smale condition (for short,(PS) condition)

DEFINITION Let X be a Banach space, and Φ : X → R a C1 functional We say that

Φ satisfies the (PS) condition if, all sequences (x n ) such that (Φ(x n )) is bounded and

Φ (x )→ 0 contain a convergent subsequence

Trang 25

6 D.G de Figueiredo

In this section we treat only subcritical problems, which means that the powers in the

nonlinearity F are strictly less than 2∗ Such a restriction is done viewing some (PS) dition to be obtained later So we require, due to Sobolev imbeddings, that

con-(F2) F (x, u, v) C

1+ |u| r + |v| s

, where 0 < r < 2∗and 0 < s < 2∗.

Here, in analogy with the scalar case, a variety of problems have been studied We singleout three noncritical cases, although many other combinations are of interest:

(I) r, s < 2 (“sublinear-like”),

(II) r, s > 2 (“superlinear-like”),

(III) r = s = 2 (“resonant type”).

Systems (2.1) satisfying one of the above conditions, as well as other problems, havebeen discussed in [17,1] Let us mention three of those results, in order to show the sort oftechniques used in this area

THEOREM 2.1 (The coercive case) Assume (F1) and (F2) with r and s as in (I) Then

Φ achieves a global minimum at some point (u0, v0) ∈ E, which is then a weak solution

of (2.1).

This result is an easy consequence of the following theorem on the minimization ofcoercive weakly lower semi-continuous functionals, see [32,48] for instance

A THEOREM FROM TOPOLOGY Let X be a compact topological space Let Φ : X→

R ∪ +∞ be a lower semi-continuous function Then (i) Φ is bounded below, and (ii) the

infimum of Φ is achieved, i.e., there exists x0∈ X such that inf x ∈X Φ(x) = Φ(x0).

Next, if we assume

(F3) F (x, 0, 0) = F u (x, 0, 0) = F v (x, 0, 0) = 0, ∀x ∈ Ω,

then clearly u = v = 0 is a solution of (2.4) And the next result gives conditions for the

existence of nontrivial solutions

THEOREM2.2 (The coercive case, nontrivial solutions) Assume (F1), (F3) and (F2) with

r and s as in (I) Then Φ achieves a global minimum at a point (u0, v0) = (0, 0), provided

that there are positive constants R and Θ < 1, and a continuous function K : Ω×R×R →

R+such that

(F4) F

x, t1u, t1v

t Θ K(x, u, v),

for x ∈ Ω, |u|, |v| R and small t > 0.

REMARK As in Theorem 2.1, Φ achieves its infimum All we have to do in order to prove Theorem 2.2 is to show that there is a point (u , v ) ∈ E where Φ(u , v ) < 0 Let ϕ be a

Trang 26

first eigenfunction of the Laplacian, subject to Dirichlet data The function ϕ1can be taken

> 0 in Ω So we can use u1, v1= t1

ϕ1, with t positive and small in order to construct (u1, v1).

Now let us go to the “superlinear cases.” Viewing the need of a Palais–Smale condition

we assume, in addition, a condition of the Ambrosetti–Rabinowitz type, namely

THEOREM2.3 Assume (F1), (F3), (F5) and (F2) with r and s as in (II) Assume also that

there are positive constants C and ε, and numbers r, s > 2 such that

(F6) F (x, u, v) C

|u| r + |v| s

,

for |u|, |v| ε, x ∈ Ω Then Φ has a nontrivial critical point.

The proof goes by an application of the Mountain-Pass Theorem [3,87]

THEMOUNTAIN-PASSTHEOREM Let X be a Banach space, and Φ : X → R of class C1

and satisfying the PS condition Suppose that Φ(0) = 0, and

(i) there exists ρ > 0 and α > 0 such that Φ(u) α for all u ∈ X with u = ρ,

(ii) there exists an u1∈ X such that u1 > ρ and Φ(u1) < α.

Then Φ has a critical point u0= 0, which is at the level c given by

c:= inf

γ ∈Γ u ∈γ [0,1]max Φ(u),

where Γ := {γ ∈ C([0, 1], X), with γ (0) = 0, γ (1) = u1}

The analysis of the resonant case requires the study of some eigenvalue problem for

systems, and this can be done even for systems involving p-Laplacians, see [17].

3 Hamiltonian systems

In this section we study elliptic systems of the form

−u = H (x, u, v), −v = H (x, u, v) in Ω, (3.1)

Trang 27

where H : Ω × R × R → R is a C1-function and Ω ⊂ R N , N 3, is a smooth bounded

domain.We shall later consider also the case when Ω = R N, and in this latter case, thesystem takes the form

−u + u = H v (x, u, v), −v + v = H u (x, u, v), (3.2)and existence and multiplicity of solutions will be discussed in Section 5

In the bounded case, we look for solutions of (3.1) subject to Dirichlet boundary

condi-tions, u = v = 0 on ∂Ω This has been object of intensive research starting with the work

via a Topological argument, using a theorem of Krasnoselski˘ı on Fixed Point Index forcompact mappings in cones in Banach spaces, see Krasnoselski˘ı Theorem in Section 6.The model of a superlinear system as in (3.3) is

−u = |v| p−1v, −v = |u| q−1u in Ω. (3.4)

By analogy with the scalar case one would guess that the subcritical case occurs when

1 p, q < N+2

N−2 However, if p= 1, system (3.4) is equivalent to the biharmonic equation

2u = |u| q−1u, and the Dirichlet problem for the system becomes the Navier problem

for the biharmonic, that is u = u = 0, on ∂Ω Since the biharmonic is a fourth order

operator the critical exponent is (N + 4)/(N − 4), which is greater than (N + 2)/(N − 2).

So this raises the suspicion (!) that for systems the notion of criticality should take, verycarefully, into consideration the fact that the system is coupled It appeared in [26] and

independently in [84] the notion of the Critical Hyperbola, which replaces the notion of

the critical exponent of the scalar case:

We call a system (3.4) to be subcritical if the powers p, q are the coordinates of a point below the critical hyperbola And similarly, system (3.3) is subcritical when f (v) grows like v p as v → +∞, and g(u) grows like u q as u→ +∞ A similar definition will be

given later for system (3.1)

Trang 28

Fig 1. p+11 + 1

q+1 = 1 − 2

N.

For N 3, the “critical hyperbola” plays an important role on the existence of nontrivial

solutions For instance, for the model problem (3.4) with (p, q)∈ R2on and above thiscurve, one finds the typical problems of noncompactness, and nonexistence of solutions,

as it was proved in [97,26,77], using Pohozaev type arguments

If the growths of H u and H v with respect to u and v as u, v→ +∞ were both less that

(N + 2)/(N − 2) one could consider the functional

|ϕ n|2= 1, with the following properties:

(i) λ1is a positive and simple eigenvalue, and ϕ1(x) > 0 for x ∈ Ω,

(ii) λ → +∞,

Trang 29

Now we define the fractional Sobolev spaces as appropriate subsets of L2(Ω):

DEFINITION For s 0, we define

Trang 30

Let now E = E s × E t If z = (u, v) ∈ E, then H (x, u, v) ∈ L1 So the functional below

where η = (φ, ψ) ∈ E So the critical points of the functional Φ given by (3.14) are the

weak solutions (u, v) ∈ E s × E t of the system

REMARK3.1 The following regularity theorem was proved in [35]:

“these weak solutions (u, v) are indeed u ∈ W 1, p+1p

Trang 31

for all x ∈ Ω and |(u, v)| r.

Then, system (3.1) has a strong solution.

ON THE PROOF OF THEOREM 3.1 The proof consists in obtaining a critical point of

the functional (3.14) First we observe that Φ is strongly indefinite This means that the space E, where the functional Φ is defined, decomposes as E = E+⊕ E−, and E± areinfinite dimensional subspaces and the quadratic part

Q(z)=

Ω

A s uA t v, for z = (u, v)

is positive definite in E+and negative definite in E− This fact and (H4) induce a geometry

on the functional Φ that calls for the use of a linking theorem of Benci–Rabinowitz [11] in

a version due to Felmer [54] Conditions (H2), and (H3) are used to prove a Palais–Smalecondition

REMARK 3.2 Condition (H4) in the previous theorem excludes cases when H u and H v

have linear terms Indeed, on the one hand, the superlinearity condition (3.11) gives pq >

1, and, on the other hand, linear terms would imply that (H4) cannot be satisfied with

p = q = 1 Let us now treat this case.

Suppose now that H has a quadratic part, namely 12cu2+1

2bv2+ auv In this case the

system becomes

−u = au + bv + H v , −v = cu + av + H u , (3.17)

where H satisfies part of the assumptions of the previous theorem This situation has been

studied in special cases by Hulshof–van der Vorst [69] and de Figueiredo–Magalhães [36].The result we present below extends the previous ones, and it is due to de Figueiredo–Ramos [38]

We replace conditions (H3) and (H4) of the previous theorem by the following ones

In [38] we consider more general conditions

(H3 ) There exist R > 0 and a positive constant C such that

Trang 32

(H5) there exists r > 0 such that either

THEOREM3.2 Let a, b, c be real constants and p, q as in (3.11) Assume that H satisfies

(H1), (H2), (H3 ), (H4 ) and (H5) Then system (3.17) admits a nonzero strong solution.

ON THE PROOF OFTHEOREM3.2 The proof relies on a Linking Theorem for stronglyindefinite functionals, see [74,75,90] In order to state this result we need two further con-cepts

DEFINITION We say that a functional Φ : E → R defined in a Banach space E has a local

linking at the origin if there is a splitting of the space E = E+⊕ E−, and a r > 0 such that

n We say that a C1functional Φ : E→ R defined in a Banach space

E satisfies a (PS)∗condition if, every sequence (z

n ), z n ∈ E n, such that

Φ(z n ) const, and Φ (z

n )η ε n η E , ∀η ∈ E n , ε n → 0,

contains a convergent subsequence

Now we state the theorem used to prove Theorem 3.2

THEOREM 3 (Li–Liu–Willem) Let Φ : E → R be a C1 functional defined in a Banach space E which satisfies a (PS)∗ condition and has a local linking at the origin Assume

that Φ maps bounded sets into bounded sets Suppose further that the following holds

Φ(z) → −∞, as z E → ∞, z ∈ E+

n ⊕ E−.

Then Φ has a nontrivial critical point.

Trang 33

4 Nonlinearities of arbitrary growth

In the previous section we discussed subcritical systems that include the simpler one below

Existence was proved under the condition (3.11)

We should observe that under the hypothesis p, q > 1 we are leaving out a region below the critical hyperbola and still in the first quadrant, that is p, q > 0 One may guess that ex-

istence of solutions should still persist in this case This section is devoted to this question,see [44]

• The case N = 2.

For N = 2 any (p, q) ∈ R+satisfies the inequality (3.11) Actually, a higher growth than

polynomial is admitted: by the inequality of Trudinger–Moser, see [96,83], subcritical

growth for a single equation is given by the condition (see [46])

lim

|t|→∞

g(t)

e αt2 = 0, ∀α > 0.

In [46] it is proved that system (4.1) has a nontrivial solution for nonlinearities f and

g which have this type of subcritical growth and satisfying an Ambrosetti–Rabinowitz

condition Also existence results for certain nonlinearities with critical growth are given in[46] Here, we consider a different type of extension of these known results: we show that

if one nonlinearity, say g, has polynomial growth (of any order), then, to prove existence

of solutions, no growth restriction is required on the other nonlinearity f (other than a

Ambrosetti–Rabinowitz condition)

• The case N = 3.

For N = 3 the critical hyperbola has the asymptotes p∞= 2 and q∞= 2 In particular, if

g(s) = s p with 1 < p < 2, then the cited existence results tell us that there exists a solution (u, v) for the system (4.1) with f (s) = s q , for any q > 1 Also in this case, existence of solutions can be proved requiring no growth restriction on the nonlinearity f (other than a

N−2 1 Note that for an exponent p < 1, the corresponding equation in the system

is sublinear, i.e we have a system with one sublinear and one superlinear equation In this

situation, the approach used in previous sections is no longer applicable However, in thiscase a reduction of the system to a single equation is possible (see [56]), which allows toprove again a result of the same form; moreover this approach also allows to extend to the

Trang 34

whole range the cases N = 2 and N = 3, that is for N = 2: 0 < p < +∞, and for N = 3:

0 < p < 2.

We have the following theorems:

THEOREM4.1 Suppose that

– and for s near 0:

has a nontrivial (strong) solution.

THEOREM4.2 Suppose that

(1) (p, q) satisfy p+11 + 1

q+1> 1− 2

N , and N2−2 p 1.

(2) f ∈ C(R), and there exist constants θ > p+1

p and s0 0 such that

f (s) c |s| q + d, for some constants c, d > 0.

Then the system

Trang 35

The proofs are variational and use fractional Sobolev spaces as in Section 3 For the case

when p > 1 one use the Linking Theorem, Theorem 3 of Section 3 For the case of p 1

we apply the Mountain-Pass Theorem to the functional

which implies that the second term of the functional I is defined if F is continuous, and so

no growth restriction on F is necessary!

5 Multiplicity of solutions for elliptic systems

In this section we discuss the multiplicity of solutions for elliptic systems of the formstudied previously Namely

−u = H

v (x, u, v) in Ω,

−v = H u (x, u, v) in Ω, (5.1)where Ω ⊂ R N , N 3, is a smooth bounded domain and H : Ω × R × R → R is a C1-

function We shall also consider here the case when Ω = R N, and in this case the systemtakes the form

−u + u = H v (x, u, v) in R N,

−v + v = H u (x, u, v) in R N (5.2)

As before we look for solutions of (5.1) subjected to Dirichlet boundary conditions u=

v = 0 on ∂Ω In the case when Ω = R N we assume that some symmetry with respect to x holds; for instance, that the x-dependence of H is radial, or that H is invariant with respect

to certain subgroups of O(N ) acting on R N We shall obtain both radial and nonradialsolutions in the radial symmetric case, thus observing a symmetry breaking effect Theresults presented next are due to Bartsch–de Figueiredo [9] Related results have beenobtained also by Felmer–Wang [57], see also [60,53,51]

Let us start with the case when Ω is bounded In such a case, the following set of

hypotheses is assumed

(H1) H : Ω × R × R → R is C1and H 0.

Trang 36

(H2) There exist constants p, q > 1 and c1> 0 with

The next condition is the nonquadraticity condition at infinity introduced by Costa–

Magalhães [29] It is related to the so-called Ambrosetti–Rabinowitz condition and it isdevised to get some sort of Palais–Smale condition for the functionals involved In fact,here we obtain a condition which is related to the so-called Cerami condition, see condi-

The next condition provides the symmetry assumed here

(H4) H (x, −u, −v) = H(x, u, v) for all (x, u, v) ∈ Ω × R × R.

Now we are prepared to state the result in the case of Ω bounded For that matter we introduce a nonincreasing sequence of constants δ n , n ∈ N, with δ n→ 0, which will be

appropriately defined later, and which depend only on p, q, α and β.

Trang 37

holds for K k0, system (5.1), subjected to Dirichlet boundary conditions, has K − k0+ 1

pairs of nontrivial solutions.

(in particular, if H is superquadratic) then system (5.1), subjected to Dirichlet boundary

conditions has infinitely many solutions.

As already observed in Section 3, the solutions obtained in Theorem 5.1 are strong

solutions in the sense that u ∈ W 2,p +1/p (Ω) ∩ W 1,p +1/p

1) H : R N × R × R → R is C1, H 0, H (x, u, v) > 0 for |(u, v)| > 0 and H is

radial in the variable x.

H u (x, u, v) c1

|u| p + |v| p(q +1)/p+1 + |u| a

(5.11)and

Trang 38

5) H (x, u, v) = H(x, −u, −v) for all (x, u, v) ∈ R N × R × R.

REMARK5.1 It follows from (H

3) that there are positive constants c and R such that

H (x, u, v) c|u| p+1+ |v| q+1

for(u, v) R. (5.13)Then (5.13) and assumption (H

system (5.2) has infinitely many radial solutions.

As in the previous results, the solutions obtained in Theorem 5.2 are strong solutions in

the sense that they satisfy u ∈ W 2,p +1/p

loc (R N ) and v ∈ W 2,q +1/q

loc (R N ).

The next result exhibits the breaking of symmetry in certain dimensions Let us justmention an interesting result coming from [9], which in the scalar case was proved byBartsch–Willem [10]

THEOREM5.3 Suppose that (H

1)–(H

5) holds If N = 4 or N 6 then system (5.2) has

infinitely many nonradial solutions.

5.1 Some abstract critical point theory

The two previous theorems are proved by Variational Methods, using some abstract criticalpoint theory that we explain next It is then applied to functionals associated to the systems

as in Section 3

We consider a Hilbert space E and a functional Φ ∈ C1(E, R) Given a sequence

F = (X n ) of finite dimensional subspaces X n ⊂ X n+1, with

has a subsequence which converges to a critical point of Φ In the case when X n = E

for all n∈ N this form of the Palais–Smale condition is due to Cerami [23] It is closely

related to the standard Palais–Smale condition and to the (PS)∗condition that we defined

in Section 3 It also yields a deformation lemma In the present form (PS) F

c was introduced

in Bartsch–Clapp [8]

Trang 39

n we can formulate the hypotheses on Φ which are needed for

our first abstract theorem

(Φ1) Φ ∈ C1(E, R) and satisfies (PS) F c forF = (E n ) n∈Nand c > 0.

(Φ2) For some k 2 and some r > 0 one has

b k:= infΦ(z): z ∈ E+, z ⊥ E k−1, z ... multiplicity, symmetry of solutions 19

5< /small >) H (x, u, v) = H(x, −u, −v) for all (x, u, v) ∈ R N × R × R.

REMARK5. 1... byBartsch–Willem [10]

THEOREM5. 3 Suppose that (H

1)? ??(H

5< /small>) holds If N = or N then system (5. 2). .. symmetry of solutions 21

THEOREM 5. 5 (Fountain theorem) Suppose that (Φ1), (Φ

2)? ??(Φ

Định dạng
Số trang	617
Dung lượng	2,41 MB