On homoclinic solutions for a second order difference equation with p-Laplacian

In this paper, we obtain conditions under which the difference equation $-\Delta \left( a(k)\phi _{p}(\Delta u(k-1))\right) +b(k)\phi_{p}(u(k))=\lambda f(k,u(k)),\quad k\in \mathbb{Z}$, has infinitely many homoclinic solutions. A variant of the fountain theorem is utilized in the proof of our theorem. It improves the results in [L.Kong, homoclinic solutions for a second order difference equation with $p-$Laplacian, \textit{Appl. Math. Comput}., \textbf{247} (2014), 1113--1121], where the set of conditions imposed on nonlinearity is inconsistent.

In this paper, similary to [3], our goal is to apply the variational method and a variant of the fountain theorem to find a sequence of homoclinic solutions for the problem (1). Our theorem improves the results in [3], where the set of conditions imposed on nonlinearity is inconsistent. We not only show that one of the assumptions is in fact superfluous, but also that others can be relaxed. The problem (1) has been studied recently in several papers. Infinitely many solutions were obtained in [8] by employing Nehari manifold methods, in [6] by use of the Ricceri's theorem (see [1], [5]), and in [7] directly applying the variational method.
We assume that potential b(k) and the nonlinearity f (k, t) satisfies the following conditions: (H 1 ) f (k, −t) = −f (k, t) for all k ∈ Z and t ∈ R; (H 2 ) there exist d > 0 and q > p such that |F (k, t)| ≤ d (|t| p + |t q |) for all k ∈ Z and t ∈ R; |t| p−1 = 0 uniformly for all k ∈ Z; Kong [3] gave conditions for existence of a sequence of solutions of the problem (1). In additions to hypotheses (B), (H 1 ), (H 5 ), he offered also the following conditions: Obviously, (H ′ 4 ) is stronger than (H 4 ), and (H ′ 2 ) is stronger than both (H 2 ) and (H 3 ). In [3], as an example of function, which satisfied conditions ( with µ > 1 and ν ≥ 1. But this does not satisfy the condition (H ′ 4 ). Moreover, the conditions ( . As f is continuous we have for T > T 1 and k ∈ Z and so |F (·, T )| / ∈ l 1 , contrary to (H ′ 3 ). It is easy to verify that the function (2) does satisfy conditions (H 1 ) − (H 5 ). Note also that it does not satisfy the standard Ambrosetti-Rabinowitz condition.

Preliminaries
We repeat the relevant for us material from [3]. We begin by defining some Banach spaces. For all 1 ≤ p < +∞, we denote ℓ p the set of all functions u : Z → R such that Moreover, we denote ℓ ∞ the set of all functions u : Z → R such that We set and Clearly we have Moreover, (X, · ) is a reflexive and separable Banach space and the embedding X ֒→ l p is compact (see Lemma 2.2 in [3]).
Lemma 1 If S is a compact subset of l p , then, for every δ > 0, there exists h > 0 such that (b) Ψ ∈ C 1 (l p ) and Ψ ∈ C 1 (X); (c) J λ ∈ C 1 (X) and every critical point u ∈ X of J λ is a homoclinic solution of problem (1).
This version of the lemma can be proved essentially by the same way as Propositions 5,6 and 7 in [2], where a(k) ≡ 1 on Z and the norm on X is slightly different. See also Lemma 2.3 in [3].

Main results
Now we are ready to state our result.
Our main tool is the following version of the fountain theorem with Cerami's condition (see [4]). We say that I, a C 1 -functional defined on a Banach space X, satisfies the Cerami condition if any sequence {u n } ⊂ X such that {I(u n )} is bounded and (1 + u n ) I ′ (u n ) → 0 has a convergent subsequence; such a sequence is then called a Cerami sequence. Now, let X be a reflexive and separable Banach space. It is well known that there exists e i ∈ X and e * i ∈ X * such that Theorem 4 Assume that I ∈ C 1 (X, R) satisfies the Cerami condition and I(−u) = I(u). If for almost every n ∈ N, there exist ρ n > r n > 0 such that (i) a n = inf u∈Zn, u =rn I(u) → +∞ as n → ∞; (ii) b n = max u∈Yn, u =ρ n I(u) ≤ 0, then I has a sequence of critical points {u n } such that I(u n ) → +∞.
In the remainder of this paper, let X be defined by (3), and Y n and Z n be given in (5). To prove Theorem 1, we will also need the following. Then, lim n→∞ β q,n = 0.
For q > p this is Lemma 3.2 in [3]. As proof shows, the instance q = p is also true.
Since F (k, 0) = 0, we obtain F (k, t) ≥ 0 and f (k, t)t ≥ 0 for all k ∈ Z and t ≥ 0. Arguing similarly for the case t ≤ 0, we complete the proof.