Abstract and Applied Analysis

Volume 2013 (2013), Article ID 597193, 8 pages

http://dx.doi.org/10.1155/2013/597193

Research Article

Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

Chun Li, Zeng-Qi Ou, and Chun-Lei Tang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received 13 August 2013; Accepted 28 October 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Chun Li et al. 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.

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

1. Introduction and Main Results

Consider the fourth-order Navier boundary value problem

where is a bounded smooth domain, , and . is the Laplace operator and is the biharmonic operator.

Let be the eigenvalues of in . The eigenvalue problem

has infinitely many eigenvalues .

We will always assume . Let be the Hilbert space .   is equipped with the inner product

and the norm

A weak solution of problem (1) is any such that

for any .

Let be the functional defined by

where . And, one has

for any , so that a critical point of the functional in corresponds to a weak solution of problem (1).

In recent years, fourth-order problems have been studied by many authors. In [1], Lazer and McKenna have pointed out that problem (1) furnishes a model to study travelling waves in suspension bridges if , where and . Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied.

In [2, 3], Micheletti and Pistoia proved that problem

admits two or three solutions by variational method. In [4], Zhang obtained the existence of weak solutions for problem (8) when is sublinear at . In [5], Zhang and Li showed that problem (8) has at least two nontrivial solutions by means of Morse theory and local linking. When is asymptotically linear at infinity, the existence of three nontrivial solutions has been obtained in [6] by using Morse theory. In [7], by using the mountain pass theorem, An and Liu gave the existence result for nontrivial solutions for a class of asymptotically linear fourth-order elliptic equations. In [8], Zhou and Wu got the existence of four sign-changing solutions or infinitely many sign-changing solutions for (8) by using the sign-changing critical point theorems. In [9], Yang and Zhang showed new results on invariant sets of the gradient flows of the corresponding variational functionals and proved the existence of positive, negative, and sign-changing solutions for some fourth-order semilinear elliptic boundary value problems. In [10], by using the variational method, Liu and Huang obtained an existence result of sign-changing solutions as well as positive and negative solutions for a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity.

In this paper, we will study the existence of nontrivial solutions of problem (1). Our main results are the following theorems.

Theorem 1. Assume that is even in and the following conditions hold:

()

 uniformly in ;()there exist two constants and such that

 uniformly for all ;()there exists a constant such that

 uniformly for all .

Then problem (1) has infinitely many nontrivial solutions.

Theorem 2. Assume that satisfies () and

()there exist three positive constants , , and such that

(), if ;(), if , where .

If is even in , problem (1) has infinitely many nontrivial solutions.

Remark 3. For Schrödinger equation, the corresponding condition () is due to Ding and Luan [11]. The condition () is weaker than the usual Ambrosetti-Rabinowitz-type condition (see [11, 12]).

Theorem 4. Assume that satisfies (), (), (), and

()

 uniformly in .

If is an eigenvalue of (with Navier boundary condition), assume also the condition that

()there exists such that

(i),  for all ; or(ii),  for all .

Then problem (1) has at least one nontrivial solution.

Theorem 5. Suppose that satisfies (), (), and (). If is an eigenvalue of (with Navier boundary condition), assuming also (), then problem (1) has at least one nontrivial solution.

Here, we have

where , and is the space spanned by the eigenvectors corresponding to negative (positive) eigenvalues of . It is easy to know that under the conditions of our theorems.

It is well known that is continuously embedded in for every . If , the embedding is compact. It follows from (), (), and () that

Hence, there is a positive constant such that

for , , where denotes the norm of .

2. Proof of Main Results

In this paper, we will use the Fountain Theorem of Bartsch ([13, Theorem 2.5], [14, Theorem 3.6]) to prove our Theorems 1 and 2. And, we will prove Theorems 4 and 5 by using a version of Local Linking theorem [12, Theorem 2.2] which extends theorems given by Li and Willem [15], Li and Szulkin [16].

In [13, 14], Bartsch established the Fountain Theorem under the condition. Since the Deformation Theorem is still valid under the Cerami condition, the Fountain Theorem is true under the Cerami condition. So, we have the following Fountain Theorem.

Let be a reflexive and separable Banach space. It is well known that there exist such that

(1) where for and for .(2), .

Let ; then . We define

Theorem A (Fountain theorem). Assume that satisfies the Cerami condition (), . If for almost every , there exist such that

()

()

 then has an unbounded sequence of critical values.

For the reader's convenience, we state the following Local Linking theorem [12, Theorem 2.2]. Let be a real Banach space with and such that . For every multi-index , let . We define that . A sequence is admissible if for every there is such that . We say that satisfies the condition if every sequence such that is admissible and satisfies

contains a subsequence which converges to a critical point of , where .

Theorem B (see [12, Theorem 2.2]). Suppose that satisfies the following assumptions:

() and has a local linking at ; that is, for some ,

() satisfies condition;() maps bounded sets into bounded sets;()for every as , on .

Then has at least one nonzero critical point.

Now, we can give the proof of our theorems.

Proof of Theorem 1. At first, we claim that satisfies the Cerami condition (). Consider a sequence such that is bounded and as . Then there exists a constant such that

By a standard argument, we only need to prove that is a bounded sequence in . Otherwise, going if necessary to a subsequence, we can assume that as . From (), there exist two constants such that

So, by (21) and (22), we have

which implies that

for all and some positive constant .

Since

On the one hand, we consider the case

Putting

one has . Let

where . We can obtain from Hölder's inequality, (15), and (24) that

for all , where .

By () and (29), one has

for all . Since , we have

On the other hand, if satisfies,

then one sees . So, we get

It follows from (24) that

By () and (33), we obtain

for all . Note that and imply that . So, it follows from (34) and the above expression that

Hence, we conclude from (31) and (36) that

Similarly for , we get

It follows from the equivalence of the norms on the finite dimensional space that there exists such that

Putting , one has

It follows from (24), (39), and Hölder's inequality that

and consequently

Hence, by (37), (38), and (42), one sees

as , which is a contradiction. So, we obtain that is bounded in . By a standard argument, we get that satisfies the condition .

Let with for any . Set

Since , all the norms are equivalent. For , there exists a constant such that

From condition (), there exists such that

For , it follows from (45) and (46) that

which implies that

So, () of Theorem A is satisfied for every large enough.

Here, we obtain from () that there exists a positive constant such that

uniformly for all . Let us define

For large enough, one has . By (49), on , we have

Choosing , we obtain, if and ,

Since, by Lemma 3.8 of [14], as , () is proved. Hence, the proof is completed by using Fountain theorem.

Proof of Theorem 2. Firstly, we claim that satisfies the Cerami condition (). Consider a sequence such that is bounded from above and as . By a standard argument, we only need to prove that is a bounded sequence in . For otherwise, we can assume that as .

From assumption (), there exist two positive constants and , such that

So, one has

for all and some positive constant .

Let ; then and for all . By (54), we have

as . So, for , it follows from Hölder's inequality and the above expression that

as . It follows from (39) that

Hence, we get

From (), (53), and (56), there exists a positive constant such that

as ; therefore, , which is a contradiction. Hence, is bounded.

In a way similar to the proof of Theorem 1, we can obtain that satisfies () of Theorem A.

It follows from () that there is such that

Then, by () and (60), for , one has

which implies that

for .

Therefore, there exist two positive constants and such that

where . As the proof of Theorem 1, we can get (). Therefore, Theorem 2 holds.

Proof of Theorem 4. The proof of this theorem is divided in several steps.

Step  1. We claim that has a local linking at zero with respect to .

By (), for any , there exists such that

We obtain from the above expression and (49) that

where . Hence, we can get from (15) and (65) that

for all .

Here, we consider only the case where is an eigenvalue of and case (ii) of () holds. The case (i) is similar.

Let , and , where . Choose a Hilbertian basis for and define

Now, by (66), for each , one has

Letting and by , we have

for small enough.

Let be such that . Put

Then, for all and , by (39), one sees

On one hand, from above expression, for any , we have

Hence, by condition (ii) of (), we get

On the other hand, for any , one has

Hence, for all and with , we can obtain from (65) that

which implies that

Letting in above expression, then for all and with , we have

which implies that

for small enough. Hence, has a local linking at zero with respect to for small enough.

Step  2. In a way similar to the proof of Theorem 1, we can get that satisfies the condition.

Step  3. Now, we claim that for each , one has

Since and , all the norms are equivalent. For , there exists a constant such that

From condition (), there exists a constant such that

For , it follows from (80) and (81) that

which implies that

Hence, all the assumptions of Theorem B are verified. Then, the proof of Theorem 4 is completed.

Proof of Theorem 5. In a way similar to the proof of Theorems 2 and 4, we can obtain that satisfies (), (), (), and () of Theorem B. Therefore, Theorem 5 holds.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for the valuable suggestions. This paper is supported by the National Natural Science Foundation of China (no. 11071198) and the Fundamental Research Funds for the Central Universities (no. XDJK2010C055).

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