The Scientific World Journal

Volume 2014 (2014), Article ID 276372, 6 pages

http://dx.doi.org/10.1155/2014/276372

Research Article

Infinitely Many Homoclinic Solutions for Second Order Nonlinear Difference Equations with -Laplacian

Guowei Sun¹ and Ali Mai1,2

¹Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, China

²School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong 510006, China

Received 12 March 2014; Accepted 18 April 2014; Published 14 May 2014

Academic Editor: Maoan Han

Copyright © 2014 Guowei Sun and Ali Mai. 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 employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discrete -Laplacian equations with a coercive weight function and superlinear nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and without any periodicity assumptions, we prove the existence and multiplicity results of the equations.

1. Introduction

Difference equations represent the discrete counterpart of ordinary differential equations and are usually studied in connection with numerical analysis. As it is well known, the critical point theory is used to deal with the existence of solutions of difference equations. For example, in 2003, Guo and Yu [1] introduced a variational structure associated with second order difference equations; they employ Rabinowitz's saddle point theorem (see [2]) to obtain the existence of -periodic solutions for the -periodic system:

The forward difference operator is defined by . They assume that is bounded and is coercive with respect to or satisfies a subquadratic Ambrosetti-Rabinowitz condition and a related coercivity condition. In particular, when for all , they prove the existence of nontrivial -periodic solutions of (1). In [3], they assume that satisfies a superquadratic Ambrosetti-Rabinowitz condition and satisfies a superlinear condition near and prove the existence of two nontrivial -periodic solutions of (1) by using the similar methods. A survey of those results is given in [4]. In 2004, Zhou et al. [5] consider the case, where the nonlinearity is neither superlinear nor sublinear and generalize the results of [3]. In these papers, the critical point theory is applied to find the periodic solutions of difference equations. The main idea of these papers is to construct a suitable variational structure, so that the critical points of the variational functional correspond to the periodic solutions of the difference equations. Naturally, the critical point theory is also applied to find homoclinic solutions of difference equations; see [6–11] and the reference therein.

In this paper, we consider the following second order nonlinear difference equations with -Laplacian:

where for all , . is a positive and coercive weight function and is a continuous function on . The forward difference operator is defined by

As usual, and denote the set of all integers and real numbers, respectively.

Assume further that ; then is a solution of (2), which is called the trivial solution. As usual, we say that a solution of (2) is homoclinic (to 0): if

In addition, we are interested in the existence of nontrivial homoclinic solution for (2), that is, solutions that are not equal to identically. In this paper, we also obtain infinitely many homoclinic solutions of (2) for case, where is odd in .

Moreover, we may regard (2) as being a discrete analogue of the following second order differential equation:

The study of homoclinic solutions for (2) in case has been motivated in part by searching standing waves for the nonlinear discrete Schrödinger equation:

namely, solutions of the form . Periodic assumptions on (6) can be found in [6, 7]. Without any periodic assumptions, the existence and multiplicity of standing wave solutions of (6) are obtained in [8, 9]. We are going to extend the approach of [8] to nonlinear discrete -Laplacian equations.

Throughout this paper, we always suppose that the following conditions hold:

function satisfies for all and

and there exist , , such that

uniformly for . uniformly for , where is the primitive function of ; that is,

is strictly increasing on and .

In many studies of -Laplacian equations, the following classical Ambrosetti-Rabinowitz superlinear condition ([12, 13]) is assumed:

It is easy to see that (10) implies , for some constants and .

In this paper, instead of (10), we assume the -superlinear condition . It is easy to see that (10) implies . For example, the -superlinear function,

does not satisfy (10). However, it satisfies the condition . A crucial role that (10) plays is to ensure the boundedness of Palais-Smale sequences. This is very crucial in applying the critical point theory.

The rest of the paper is organized as follows. In Section 2, we establish the variational framework associated with (2) and then present the main results of this paper. Section 3 is devoted to prove some useful lemmas, and in Section 4 we prove the main result.

2. Preliminaries

We will establish the corresponding variational framework associated with (2).

Consider the real sequence spaces

Then the following embedding between spaces holds:

Define the space

Then is a Hilbert space equipped with the norm

is the usual absolute value in .

Now we consider the variational functional defined on by

Then , for all ,

Thus, is a critical point of on only if is homoclinic solutions of (2). We have reduced the problem of finding homoclinic solutions of (2) to that of seeking critical points of the functional on . This means that functional is just the variational framework of (2).

The following lemma plays an important role in this paper; it was established in [11].

Lemma 1. If satisfies the condition , for any , then the embedding map from into is compact.

The main result is as follows.

Theorem 2. Suppose conditions are satisfied. Then we have the following conclusions.

(1)Equation (2) has a nontrivial ground state homoclinic solution, that is, homoclinic solutions corresponding to the least positive critical value of the variational functional.(2)If is odd in for each , (2) has infinitely many pairs of homoclinic solutions in .

To prove the multiplicity results, we need the following lemma.

Lemma 3 (see [14]). Let . If is an infinite-dimensional Hilbert space, is even and bounded below and satisfies the Palais-Smale condition. Then has infinitely many pairs of critical points.

3. Some Useful Lemmas

We define the Nehari manifold:

To prove the main results, we need some lemmas.

Lemma 4. Suppose conditions are satisfied. Then, for each , there exists a unique such that .

Proof. Let . By , we have

From , for all and , we have

Let be a weakly compact subset and ; we claim that

Indeed, let . It suffices to show that

as . Passing to a subsequence if necessary, and for every , as .

Note that, from and , it is easy to get that

Since and , by and (23), we have

Therefore, (21) holds.

Let . Then

from (19)–(21); then there exists a unique , such that whenever , whenever , and . So .

Lemma 5. Suppose conditions are satisfied. Then satisfies the Palais-Smale condition on .

Proof. Let be a sequence such that for some and as .

Firstly, we prove that is bounded. In fact, if not, we may assume by contradiction that as . Let . Then there exists a subsequence, still denoted by the same notation, such that   in   as .

Suppose . For every , from Lemma 4, we have

This is a contradiction if . Therefore, .

According to (21), we have

a contradiction again. Thus, is bounded.

Finally, we show that there exists a convergent subsequence of . Actually, there exists a subsequence, still denoted by the same notation, such that . By Lemma 1, for any , then

Note that

By the weak convergence, the first term on the right hand side of (29) approaches as .

By and , it is easy to show that, for any , there exists , such that

By Hölder's inequality, we have

Combining (28) and the boundedness of , the above inequality implies

It follows from (29) that in  . This implies that satisfies the Palais-Smale condition.

4. Proof of Main Results

Proof of Theorem 2. (1) Now we need five steps to finish this proof.

Step  1. We claim that is homeomorphic to the unit sphere in .

By (19) and (21), for small and for large. So is a unique maximum of and is the unique point on the ray () which intersects . That is, is the unique maximum of on the ray. Therefore, by Lemma 4, we may define the mapping by setting

Next we show that the mapping is continuous. Indeed, suppose . Since , for each , we may assume for all . Write . Then is bounded. If not, as .

Note that, by , for all ,

Therefore, for all , we have

Combining with and Lemma 4, we have

this is a contradiction. Therefore, after passing to a subsequence if needed, since is closed and . Hence by the uniqueness of of Lemma 4. Therefore, is continuous.

Then we define a mapping by setting , then is a homeomorphism between and , and the inverse of is given by .

Step  2. Now we define the functional and by

Then we have

and . Moreover,

In fact, let and . By Lemma 4 and the mean value theorem, we obtain

where is small enough and . Similarly,

where . From the above, the function is continuous, combining these two inequalities that

Hence the Gâteaux derivative of   is bounded linear in and continuous in . It follows that is a class of (see [15], Proposition 1.3) and (38) holds. Note only that, since , , so (39) is clear.

Step  3. is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .

Let be a Palais-Smale sequence for and let . Since for every we have an orthogonal splitting, , we have

Then

Therefore

By (35), for , , so there exists a constant such that . And since , . Together with Lemma 5, . Hence is a Palais-Smale sequence for if and only if is a Palais-Smale sequence for .

Step  4. By (45), if and only if . So is a critical point of if and only if is a nontrivial critical point of . Moreover, the corresponding values of and coincide and .

Step  5. satisfies the Palais-Smale condition.

Let be a Palais-Smale sequence for ; then is a Palais-Smale sequence for by Step 3, where . From Lemma 5, after passing to a subsequence and , so satisfies the Palais-Smale condition.

Let be a minimizing sequence for . By Ekeland's variational principle we may assume as , so is a Palais-Smale sequence for . By the Palais-Smale condition, after passing to a subsequence if needed. Hence is a minimizer for and therefore a critical point of  ; then is a critical point of and also is a minimizer for . Therefore, is a ground state solution of (2).

(2) If is odd in for each , then is even, so is . Since and satisfies the Palais-Smale condition, has infinitely many pairs of critical points by Lemma 3. It follows that (2) has infinitely many pairs of homoclinic solutions in .

This completes Theorem 2.

Finally, we exhibit examples to demonstrate the applicability of Theorem 2.

Example 6. Consider the second order difference equation:

where such that . Let

for all . Then it is clear that all conditions of Theorem 2 are satisfied. By Theorem 2, (46) has infinitely many pairs of homoclinic solutions.

Example 7. Consider the -Laplacian difference equation:

where such that . Let

for all and . It is easy to verify that satisfies all conditions in Theorem 2. Therefore, (48) has infinitely many pairs of homoclinic solutions.

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 anonymous referee for his/her valuable suggestions. This work is supported by Program for the National Natural Science Foundation of China (no. 11371313) and Biomathematics Laboratory of Yuncheng University (SWSX201302, SWSX201305).

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