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International Journal of Differential Equations
Volume 2010 (2010), Article ID 548702, 9 pages
http://dx.doi.org/10.1155/2010/548702
Review Article

Infinitely Many Solutions for a Robin Boundary Value Problem

1School of Mathematic Sciences, Qufu Normal University, Qufu Shandong 273165, China
2Institute of Mathematics, AMSS, Academia Sinica, Beijing 100080, China

Received 29 August 2009; Accepted 7 November 2009

Academic Editor: Wenming Zou

Copyright © 2010 Aixia Qian and Chong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

By combining the embedding arguments and the variational methods, we obtain infinitely many solutions for a class of superlinear elliptic problems with the Robin boundary value under weaker conditions.

1. Introduction

In this paper, we consider the following equation:

where is a bounded domain in with smooth boundary and . Denote

and let be the eigenvalues of with the Robin boundary conditions. We assume that the following hold:

(), such that

where , . If , let ;

(), uniformly for .()there exist s.t.

(), .

Because of , (1.1) is usually called a superlinear problem. In [1, 2], the author obtained infinitely many solutions of (1.1) with Dirichlet boundary value condition under , and

, such that

Obviously, can be deduced form (AR). Under (AR), the (PS) sequence can be deduced bounded. However, it is easy to see that the example [3]

does not satisfy (AR), while it satisfies the aforementioned conditions (take in ). is from [3, 4].

We need the following condition (C), see [3, 5, 6].

Definition 1.1. Assume that is a Banach space, we say that satisfies Cerami condition (C), if for all :(i)any bounded sequence satisfying , possesses a convergent subsequence;(ii)there exist s.t. for any with , .

In the work in [2, 7], the Fountain theorem was obtained under the condition (PS). Though condition (C) is weaker than (PS), the well-known deformation theorem is still true under condition (C) (see [5]). There is the following Fountain theorem under condition (C).

Assume , where are finite dimensional subspace of . For each , let

Denote .

Proposition 1.2. Assume that satisfies condition (C), and . For each , there exist such that(i), ,(ii).Then has a sequence of critical points , such that as .

As a particular linking theorem, Fountain theorem is a version of the symmetric Mountain-Pass theorem. Using the aforementioned theorem, the author in [6] proved multiple solutions for the problem (1.1) with Neumann boundary value condition; the author in [3] proved multiple solutions for the problem (1.1) with Dirichlet boundary value condition. In the present paper, we also use the theorem to give infinitely many solutions for problem (1.1). The main results are follows.

Theorem 1.3. Under assumptions ()–(), problem (1.1) has infinitely many solutions.

Remark 1.4. In the work in [1, 2], they got infinitely many solutions for problem (1.1) with Dirichlet boundary value condition under condition (AR).

Remark 1.5. In the work in [8], they showed the existence of one nontrivial solution for problem (1.1), while we get its infinitely many solutions under weaker conditions than [8].

Remark 1.6. In the work in [9], they also obtained infinitely many solutions for problem (1.1) with Dirichlet boundary value condition under stronger conditions than the aforementioned and above. Furthermore, function (1.6) does not satisfy all conditions in [9]. Therefore, Theorem 1.3 applied to Dirichlet boundary value problem improves those results in [1, 2, 8, 9].

2. Preliminaries

Let the Sobolev space . Denote

to be the norm of in , and the norm of in . Consider the functional :

Then by , is and

The critical point of is just the weak solution of problem (1.1).

Since we do not assume condition (AR), we have to prove that the functional satisfies condition (C) instead of condition (PS).

Lemma 2.1. Under ()–(), satisfies condition (C).

Proof. For all , we assume that is bounded and Going, if necessary, to a subsequence, we can assume that in , then that is,
Since the Sobolev imbedding is compact, we have the right-hand side of (2.6) converges to 0. While , we have . It follows that in and , that is, condition (i) of Definition 1.1 holds.
Next, we prove condition (ii) of Definition 1.1, if not, there exist and satisfying, as then we have
Denote , then , that is, is bounded in , thus for some , we get
If , define a sequence as in [4] If for some , there is a number of satisfying (2.10), we choose one of them. For all , let , it follows by a.e. that Then for large enough, by (2.9), (2.11), and , we have That is, . Since and , then . Thus We see that From the aforementioned, we infer that which contradicts (2.8).
If , by (2.7) That is, Since exists, and by in (the weakly convergent sequence is bounded), we get where is the constant of Sobolev Trace imbedding from , see [10]. We have For , we get . Then by By using Fatou lemma, since the Lebesgue measure , On the other hand, by there exists , such that for . Moreover, Now, there is s.t. Together with (2.19) and (2.21), (2.23), it is a contradiction.
This proves that satisfies condition (C).

3. Proof of Theorem 1.3

We will apply the Fountain theorem of Proposition 1.2 to the functional in (2.2). Let

then . It shows that by and satisfies condition (C) by Lemma 2.1.

(i)After integrating, we obtain from that there exist such that

Let us define . By [2, Lemma ], we get as . Since , let and , then by (3.2), for with we have

Notice that and , we infer that

(ii)While

we can deduce that is the equivalent norm of in . Since and all norms are equivalent in the finite-dimensional space, there exists , for all , we get

Next by , there is such that for . Take := , , then for all , we obtain

It follows from (3.6), (3.7), for all that

Therefore, we get that for large enough (),

By Fountain theorem of Proposition 1.2, has a sequence of critical points , such that as , that is, (1.1) has infinitely many solutions.

Remark 3.1. By Theorem 1.3, the following equation: has infinitely many solutions, while the results cannot be obtained by [1, 2, 8, 9]

Remark 3.2. In the next paper, we wish to consider the sign-changing solutions for problem (1.1).

Acknowledgments

We thank the referee for useful comments. C. Li is supported by NSFC (10601058, 10471098, 10571096). This work was supported by the Chinese National Science Foundation (10726003), the National Science Foundation of Shandong (Q2008A03), and the Foundation of Qufu Normal University.

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