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

We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discrete p-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.


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 ofperiodic solutions for the -periodic system: The forward difference operator Δ is defined by Δ ( ) = ( + 1) − ( ). 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 ( , 0) = 0 for all ∈ Z, 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 = 0 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][7][8][9][10][11] and the reference therein.
In this paper, we consider the following second order nonlinear difference equations with -Laplacian: where ( ) = | | −2 for all ∈ R, > 1. : Z → R is a positive and coercive weight function and ( , ) : Z × R → R is a continuous function on . The forward difference operator Δ is defined by As usual, Z and R denote the set of all integers and real numbers, respectively. Assume further that ( , 0) = 0; then ( ) ≡ 0 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 2 The Scientific World Journal In addition, we are interested in the existence of nontrivial homoclinic solution for (2), that is, solutions that are not equal to 0 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: −( ( ( ))) + ( ) ( ( )) = ( , ( )) , ∈ R. (5) The study of homoclinic solutions for (2) in case = 2 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.
In this paper, instead of (10), we assume the -superlinear condition ( 3 ). It is easy to see that (10) implies ( 3 ). For example, the -superlinear function, does not satisfy (10). However, it satisfies the condition ( 1 )-( 3 ). 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.

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 R. Now we consider the variational functional defined on by The Scientific World Journal 3 Then ∈ 1 ( , R), for all V ∈ , 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].  (2) If ( , ) is odd in for each ∈ Z, (2) has infinitely many pairs of homoclinic solutions ± ( ) in .
To prove the multiplicity results, we need the following lemma.
Suppose V = 0. For every > 0, from Lemma 4, we have This is a contradiction if ≥ √ . Therefore, V ̸ = 0. 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 > 1, then By the weak convergence, the first term on the right hand side of (29) approaches 0 as → ∞.
By ( 1 ) and ( 2 ), it is easy to show that, for any > 0, there exists > 0, such that By Hölder's inequality, we have Combining (28) It follows from (29) that → in . This implies that satisfies the Palais-Smale condition.

Proof of Main Results
Proof of Theorem 2. (1) Now we need five steps to finish this proof.
Step 1. We claim that N is homeomorphic to the unit sphere in .
Then we define a mapping : → N by setting := | , then is a homeomorphism between and N, and the inverse of is given by −1 ( ) = /‖ ‖.
Let { } be a Palais-Smale sequence for Ψ; then { } is a Palais-Smale sequence for by Step 3, where := ( ) ∈ N. From Lemma 5, → after passing to a subsequence and → −1 ( ), so Ψ satisfies the Palais-Smale condition. Let { } ⊂ be a minimizing sequence for Ψ. By Ekeland's variational principle we may assume Ψ ( ) → 0 as → ∞, so { } is a Palais-Smale sequence for Ψ. By the Palais-Smale condition, → after passing to a subsequence if needed. Hence is a minimizer for Ψ and therefore a critical point of Ψ; 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 ∈ Z, then is even, so is Ψ. Since inf Ψ = inf N > 0 and Ψ satisfies the Palais-Smale condition, Ψ has infinitely many pairs of critical points by Lemma 3. It follows that (2) has infinitely many pairs of homoclinic solutions ± in .
Finally, we exhibit examples to demonstrate the applicability of Theorem 2.
for all ∈ Z. Then it is clear that all conditions of Theorem 2 are satisfied. By Theorem 2, (46) has infinitely many pairs of homoclinic solutions.