Abstract and Applied Analysis

Volume 2014 (2014), Article ID 435125, 7 pages

http://dx.doi.org/10.1155/2014/435125

Research Article

Infinitely Many Homoclinic Solutions for Nonperiodic Fourth Order Differential Equations with General Potentials

Liu Yang

Department of Mathematics and Computing Sciences, Hengyang Normal University, Hengyang, Hunan 421008, China

Received 24 March 2014; Revised 30 May 2014; Accepted 30 May 2014; Published 16 June 2014

Academic Editor: Juntao Sun

Copyright © 2014 Liu Yang. 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.

We investigate a class of nonperiodic fourth order differential equations with general potentials. By using variational methods and genus properties in critical point theory, we obtain that such equations possess infinitely homoclinic solutions.

1. Introduction

In this paper, we consider the following a class of fourth order differential equations:

where is a constant, , and .

Recently, a lot of attention has been focused on the study of homoclinic and heteroclinic solutions for this problem; see [1–8]. This may be due to its concrete applications, such as physics and mechanics; see [9, 10]. More precisely, Tersian and Chaparova [5] studied periodic case. They obtained nontrivial homoclinic solutions by using mountain pass theorem. For nonperiodic case, Li [7] studied the existence of nontrivial homoclinic solutions for this class of equations. Sun and Wu [8] studied multiple homoclinic solutions for the following nonperiodic fourth order equations with a perturbation:

By using the mountain pass theorem and Ekeland variational principle, two homoclinic solutions for these equations are obtained under the assumption that and are superlinear or asymptotically linear as , where is the following condition:

there exists a positive constant such that as and .

The assumption is too strict to be satisfied by many general functions ; for example, . In addition, although there is perturbation, the right of (2) is superlinear or asymptotically linear as . In the present paper we study the infinitely many homoclinic solutions for (1) under more general assumption than and sublinear condition on .

Before stating our results we introduce some notations. Throughout this paper, we denote by the norm, . is the Banach space of essentially bounded functions equipped with the norm

If we take a subsequence of a sequence we will denote it again by .

Now we state our main result.

Theorem 1. Assume that the following conditions hold:

  there exists a positive constant such that and ;  there exists a constant and positive function such that

  there exist and such that

where is the primitive .

Then, problem (1) possesses at least one nontrivial homoclinic solution.

In addition, if is odd symmetry in , that is,

 , ,

then one gets the existence of infinitely many nontrivial homoclinic solutions.

Theorem 2. Under the assumptions of , problem (1) possesses infinitely many nontrivial homoclinic solutions.

Example 3. If , clearly,

Thus , , and are satisfied with .

The paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we give the proof of our main results.

2. Preliminaries

In order to prove our main results, we first give some properties of space on which the variational setting for problem (1) is defined.

Lemma 4 (see [5]). Assume that hold. Then there exists a constant such that

where is the norm of Sobolev space .

Letting

then is a Hilbert space with the inner product

and the corresponding norm . Note that

for all , with the embedding being continuous. Hence, for any , there is such that

Now we begin describing the variational formulation of problem (1). Consider the functional defined by

Lemma 5. Under the conditions of , and its derivative is given by the following;

for all . In addition, any critical point of on is a classical solution of problem (1).

Proof. We firstly show that . From , one has

By the Hölder inequality and (14), we have

where is constant in (11). Hence, defined by (12) is well defined on .

Next we prove that . To this end, we rewrite as follows:

It is easy to check that , and we have , for all . On the other hand, we will show that and

for any given . For any given , let us define

It is obvious that is linear. Now we show that is bounded. In fact, for any , by the Hölder inequality and , we can obtain that

Moreover, for , using the Mean Value Theorem, we get

for some . On the other hand, , for any , there exists such that

Therefore, on account of the Sobolev compact theorem ( is compactly embedded in ) and Hölder inequality, we have

which, together with (19), implies that (17) holds. It remains to show that is continuous. Suppose that in , then we have

uniformly with respect to , which implies that is continuous. Therefore, we obtain and its derivative is given by

for all . In addition, from [8], we can know that any critical point of on is a classical solution of problem (1).

Next, we give some useful Lemmas which can be seen in [11].

Definition 6. is said to satisfy the condition if any sequence , for which is bounded and as and possesses a convergent subsequence in .

Lemma 7. Let be a real Banach space and let satisfy the (PS) condition. If is bounded from below, then is a critical value of .

To obtain the existence of infinitely many homoclinic solutions for problem (1) under the assumptions of Theorem 2, we will employ the “genus” properties in critical point theory; see [11].

Let be a Banach space, , and . We set

Definition 8. For , we say the genus of is (denoted by ) if there is an odd map and is the smallest integer with this property.

Lemma 9. Let be an even functional on and satisfy the (PS) condition. For any , set

(i)If and , then is a critical value of ;(ii)if there exists such that

and , then .

Remark 10. From Remark 7.3 in [11], we know that if and , then has infinitely many distinct critical points in .

3. Proof of the Main Results

To prove our main results, we first give the following Lemma.

Lemma 11. If and hold, then defined by (12) satisfies (PS) condition.

Proof. By (12) and (14) and Hölder inequality, one has

Since , (28) implies that as . Consequently, is bounded from below.

Now, we show that satisfies the condition. Assume that is a sequence such that is bounded and as . Then by (28), there exists a constant such that

So passing to a subsequence if necessary, it can be assumed that in . Since , for any , there exists such that

Since the embedding of is compact, in implies

Then we have

Hence, by (30), (31), and the fact that is arbitrary, one can get

as . It follows from (13) that

In view of the definition of weak convergence, we have

Therefore, we can obtain that in . Hence, satisfies condition.

Now we are in the position to complete the proof of Theorems 1 and 2.

Proof of Theorem 1. It is obvious that , and by Lemmas 5 and 11 we know that is a functional on satisfying the condition. In view of (28), we have is bounded below on . Hence, by Lemma 7, is a critical value of ; that is, there exists a critical point such that .

In addition, by , there exists an open set with , , such that

Let and ; then we have

where . Since , one has , for small enough. Hence, , ; therefore is a nontrivial homoclinic solution for problem (1).

Proof of Theorem 2. Now, by , we have is even and . In order to apply Lemma 9, we prove that there exists such that

for any . For any , we take disjoint open sets such that . For , let with , and

Then, for any , there exist , such that

Then,

Since all norms of a finite dimensional norm space are equivalent, so there exists a constant such that

For all and sufficient small , we have

In this case (36) is applicable, since is continuous on and so , , can be true for sufficiently small . Therefore, it follows from (44) that there exist and such that

Let

Then, it follows from (45) that

which together with the fact that is even functional on , yields that

where and have been previously introduced in Section 2. On the other hand, it follows from (40) and (42) that there exists an odd homeomorphism . By some properties of the genus (see of Propositions 7.5 and 7.7 in [11]), we infer

so the proof of (38) follows. Set

where is defined in Lemma 9. It follows from (50) and the fact that is bounded from below in that we have , which implies that, for any , is a real negative number. By Lemma 9 and Remark 10, has infinitely many nontrivial critical points, and consequently, problem (1) possesses infinitely many nontrivial homoclinic solutions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of Hunan Province (12JJ9001) and Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117).

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