Infinitely Many Solutions for a Robin Boundary Value Problem

Aixia Qian1 and Chong Li2 1 School of Mathematic Sciences, Qufu Normal University, Qufu Shandong 273165, China 2 Institute of Mathematics, AMSS, Academia Sinica, Beijing 100080, China Correspondence should be addressed to Aixia Qian, qaixia@amss.ac.cn and Chong Li, lichong@amss.ac.cn Received 29 August 2009; Accepted 7 November 2009 Academic Editor: Wenming Zou Copyright q 2010 A. Qian and C. 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. 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.

Because of f 2 , 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 f 1 , f 4 and Obviously, f 2 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 θ 1 in f 3 .f 3 is from 3, 4 .
We need the following condition C , see 3, 5, 6 .
Definition 1.1.Assume that X is a Banach space, we say that J ∈ C 1 X, R satisfies Cerami condition C , if for all c ∈ R: 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 X ∞ j 1 X j , where X j are finite dimensional subspace of X.
Proposition 1.2.Assume that J ∈ C 1 X, R satisfies condition (C), and J −u J u .For each k ∈ N, there exist ρ k > r k > 0 such that Then J has a sequence of critical points u n , such that J u n → ∞ as n → ∞.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 f 2 and f 3 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 .

Preliminaries
Let the Sobolev space X H 1 Ω .Denote to be the norm of u in X, and |u| q the norm of u in L q Ω .Consider the functional J : X → R: The critical point of J is just the weak solution of problem 1.1 .

International Journal of Differential Equations
Since we do not assume condition AR , we have to prove that the functional J satisfies condition C instead of condition PS .Lemma 2.1.Under (f 1 )-(f 3 ), J satisfies condition (C).
Proof.For all c ∈ R, we assume that {u n } ⊂ X is bounded and Going, if necessary, to a subsequence, we can assume that u n u in X, then

2.5
that is,

2.6
Since the Sobolev imbedding W 1,2 Ω → L γ Ω 1 ≤ γ < 2 * is compact, we have the right-hand side of 2.6 converges to 0. While ∂Ω b x u n − u 2 dS ≥ 0, we have u n − u 2 → 0. It follows that u n → u in X and J u 0, that is, condition i of Definition 1.1 holds.Next, we prove condition ii of Definition 1.1, if not, there exist c ∈ R and e. in Ω.

2.9
If v 0, define a sequence {t n } ⊂ R as in 4 J t n u n max t∈ 0,1 J tu n .

2.10
If for some n ∈ N, there is a number of t n satisfying 2.10 , we choose one of them.For all m > 0, let Then for n large enough, by 2.9 , 2.11 , and ∂Ω b x v 2 n ≥ 0, we have

2.12
That is, lim n → ∞ J t n u n ∞.Since J 0 0 and J u n → c, then 0 < t n < 1.Thus

2.13
We see that

International Journal of Differential Equations
From the aforementioned, we infer that

2.16
That is, 2 exists, and by v n v in X the weakly convergent sequence is bounded , we get where C is the constant of Sobolev Trace imbedding from H 1 Ω → L 2 ∂Ω , see 10 .We have

2.20
By using Fatou lemma, since the Lebesgue measure

2.23
Together with 2.19 and 2.21 , 2.23 , it is a contradiction.This proves that J satisfies condition C .

Proof of Theorem 1.3
We will apply the Fountain theorem of Proposition 1.2 to the functional in 2.2 .Let It shows that J ∈ C 1 X, R by f 1 and satisfies condition C by Lemma 2.1.
i After integrating, we obtain from f 1 that there exist c 1 > 0 such that Let us define β k sup u∈Z k ∩S 1 |u| q .By 2, Lemma 3.8 , we get β k → 0 as k → ∞.Since |u| 2 ≤ C Ω |u| q , let c c 1 1/2 C Ω , and r k cqβ q k 1/2−q , then by 3.2 , for u ∈ Z k with u r k , we have

International Journal of Differential Equations
Notice that β k → 0 and q > 2, we infer that we can deduce that Ω |∇u| 2 dx ∂Ω α x u 2 dS is the equivalent norm of u 2 in X.Since dim Y k < ∞ and all norms are equivalent in the finite-dimensional space, there exists C k > 0, for all u ∈ Y k , we get

3.8
Therefore, we get that for ρ k large enough ρ k > r k , a k max u∈Y k , u ρ k J u ≤ 0. 3.9 By Fountain theorem of Proposition 1.2, J has a sequence of critical points u n ∈ X, such that J u n → ∞ as n → ∞, that is, 1.1 has infinitely many solutions.

whereF 2 International
Ω is a bounded domain in R n with smooth boundary ∂Ω and 0≤ b ∈ L ∞ ∂Ω .Denote t dt, F f x, s s − 2F x, s , 1.2 and let λ 1 ≤ λ 2 ≤ • • • ≤ λ j < • •• be the eigenvalues of −Δ with the Robin boundary conditions.We assume that the following hold: Journal of Differential Equations

Theorem 1 . 3 .
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.Under assumptions (f 1 )-(f 4 ), 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 3 . 1 . 9 Remark 3 . 2 .
By Theorem 1.3, the following equation:− Δu 2u log 1 |u| , in Ω, ∂u ∂n b x u 0, on ∂Ω,3.10 has infinitely many solutions, while the results cannot be obtained by 1, 2, 8, In the next paper, we wish to consider the sign-changing solutions for problem 1.1 .