Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

and Applied Analysis 3 DNLS equation is one of the most important inherently discrete models, which models many phenomena in various areas of applications see 2–4 and reference therein . For example, in nonlinear optics, DNLS equation appears as a model of infinite wave guide arrays. In the past decade, the existence and properties of mobile discrete solitons/breathers in DNLS equations have been considered in a number of studies 5–9 . When m 1, vn ≡ 0, and {an}, {bn}, and f n, u are T -periodic in n, the existence of homoclinic solutions for the 1.1 have been studied in 5, 6, 10 for the case where f is with superlinear nonlinearity kerr or cubic , in 9, 11–14 for the case where f is with saturable nonlinearity, respectively. When {an}, {bn}, and f n, u are not periodic in n, the existence of homoclinic solutions for some special case of 1.1 can be found in 7, 8, 15, 16 . Especially, in 17, 18 , the authors obtained sufficient conditions for the existence of at least a pair of nontrivial homoclinic solutions for the special case of 1.1 when {vn} is unbounded by Nehari manifold method. It is worth pointing out that the so-called global AmbrosettiRabinowitz condition of f plays a crucial role in 17, 18 . One aim of this paper is to replace the global Ambrosetti-Rabinowitz condition by a general one. The other aim of this paper is to obtain sufficient conditions for the existence of infinitely many nontrivial homoclinic solutions of 1.1 . We will see that in Section 2, our results greatly improves those in 17, 18 . Our proofs of the main results are based on Mountain Pass Lemma and Fountain theorem. Our main ideas come from the papers 19–22 . This paper is organized as follows: in Section 2, we will first define some basic spaces. Then, we give the main results of this paper, and a comparison with the existing results is stated. Third, we establish the variational framework associated with 1.1 and transfer the problem of the existence and multiplicity of solutions in E defined in Section 2 of 1.1 into that of the existence and multiplicity of critical points of the corresponding functional. We also recall some basic results from critical point theory. Last, in Section 3, we present the proofs of our main results. 2. Preliminaries and Main Results Let l ≡ l Z ⎧ ⎨ ⎩ u {un}n∈Zm : ∀n ∈ Z, un ∈ R, ‖u‖lp ( ∑ n∈Zm |un| )1/p <∞ ⎫ ⎬ ⎭ . 2.1 Then the following embedding between l spaces holds: l ⊂ l, ‖u‖lp ≤ ‖u‖lq , 1 ≤ q ≤ p ≤ ∞. 2.2 Assume the following condition on {vn} holds. V1 the discrete potential V {vn}n∈Zm satisfies

We assume that f n, 0 0 for n ∈ Z m , then u n 0 is a solution of 1.1 , which is called the trivial solution. As usual, we say that a solution u {u n } of 1.1 is homoclinic to 0 if lim |n| → ∞ u n 0, 1.3 where |n| |n 1 | |n 2 | · · · |n m | is the length of multiindex n. In addition, if u n / ≡ 0, then u is called a nontrivial homoclinic solution. We are interested in the existence and multiplicity of the nontrivial homoclinic solutions for 1. 1 . This problem appears when we look for the discrete solitons of the following Discrete Nonlinear Schrödinger DNLS equation: iψ n Δψ n − v n ψ n σf n, ψ n 0, n ∈ Z m , 1.4 where Δψ n ψ n 1 1,n 2 ,...,n m ψ n 1 ,n 2 1,...,n m · · · ψ n 1 ,n 2 ,...,n m 1 − 2mψ n 1 ,n 2 ,...,n m ψ n 1 −1,n 2 ,...,n m ψ n 1 ,n 2 −1,...,n m · · · ψ n 1 ,n 2 ,...,n m −1 DNLS equation is one of the most important inherently discrete models, which models many phenomena in various areas of applications see 2-4 and reference therein . For example, in nonlinear optics, DNLS equation appears as a model of infinite wave guide arrays. In the past decade, the existence and properties of mobile discrete solitons/breathers in DNLS equations have been considered in a number of studies 5-9 .
When m 1, v n ≡ 0, and {a n }, {b n }, and f n, u are T -periodic in n, the existence of homoclinic solutions for the 1.1 have been studied in 5, 6, 10 for the case where f is with superlinear nonlinearity kerr or cubic , in 9, 11-14 for the case where f is with saturable nonlinearity, respectively. When {a n }, {b n }, and f n, u are not periodic in n, the existence of homoclinic solutions for some special case of 1.1 can be found in 7, 8, 15, 16 . Especially, in 17, 18 , the authors obtained sufficient conditions for the existence of at least a pair of nontrivial homoclinic solutions for the special case of 1.1 when {v n } is unbounded by Nehari manifold method. It is worth pointing out that the so-called global Ambrosetti-Rabinowitz condition of f plays a crucial role in 17, 18 . One aim of this paper is to replace the global Ambrosetti-Rabinowitz condition by a general one. The other aim of this paper is to obtain sufficient conditions for the existence of infinitely many nontrivial homoclinic solutions of 1.1 . We will see that in Section 2, our results greatly improves those in 17, 18 . Our proofs of the main results are based on Mountain Pass Lemma and Fountain theorem. Our main ideas come from the papers 19-22 . This paper is organized as follows: in Section 2, we will first define some basic spaces. Then, we give the main results of this paper, and a comparison with the existing results is stated. Third, we establish the variational framework associated with 1.1 and transfer the problem of the existence and multiplicity of solutions in E defined in Section 2 of 1.1 into that of the existence and multiplicity of critical points of the corresponding functional. We also recall some basic results from critical point theory. Last, in Section 3, we present the proofs of our main results.

Preliminaries and Main Results
Let Then the following embedding between l p spaces holds: Assume the following condition on {v n } holds.
Since the operator L is bounded and self-adjoint in the space l 2 Z m with the norm L see 1 , and by the condition V 1 , we know that the potential V is bounded below, without loss of generality, we suppose v n > L for all n ∈ Z m . Then the operator H is an unbounded positive self-adjoint operator in l 2 Z m . Define the space Then E is a Hilbert space equipped with the norm Since V 1 holds, we see that the spectrum σ H is discrete and let λ 1 be the smallest eigenvalue of H, that is Now, we present the following basic hypotheses in order to establish the main results in this paper: 2.9 f 4 f n, u /u is increasing in u > 0 and decreasing in u < 0, for all n ∈ Z m . Under the above hypotheses, our results can be stated as follows.
3 The solutions obtained in case (2) exponentially decay at infinity, that is, 2.10 holds.
We notice that, in 17, 18 , the authors consider the following DNLS equation which is a special case of 1.1 , where H −Δ V . They obtain the following results.
Theorem A. Assume that the DNLS 2.12 satisfies (V 1 ) and (A 1 ) there exist two positive constants γ and γ, such that for any n ∈ Z m , γ ≤ γ n ≤ γ.

2.13
The nonlinearity f is odd and satisfies (A 2 ) there are two positive constants C 1 , C 2 , and 2 < p < ∞ such that Then we have the following conclusions.
2 If σ 1, ω < λ 1 , 2.12 has at least a pair of nontrivial solutions ±u in l 2 Z m .
3 The solutions obtained in case (2)  Now, we will make some preparations for the proofs of our main results. Since the operator L is bounded in l 2 Z m , the following two norms are equivalent in the Hilbert space The following theorem plays an important role in this paper, which gives a discrete version of compact embedding theorem 16-18 .

Lemma 2.5.
If V satisfies the condition V 1 , then for any 2 ≤ p ≤ ∞, the embedding map from E into l p Z m is compact, denote the best embedding constant c p max u l p 1 1/ u .

2.20
Standard arguments show that the functional J is well-defined C 1 functional on E and 1.1 is easily recognized as the corresponding Euler-Lagrange equation for J. Thus, to find nontrivial solutions of 1.1 , we need only to look for nonzero critical points of J.
For the derivative of J we have the following formula:

2.21
Definition 2.6 see 22, 23 . Let E be a real Banach Space and J ∈ C 1 E, R . For some c ∈ R, we say J satisfies the so-called C c condition if any sequence {u n } ⊂ E such that J u n → c and J u n 1 u n → 0 as n → ∞, has a convergent subsequence. Let B r be the open ball in H with radius r and center 0, and let ∂B r denote its boundary. In order to obtain the existence of critical points of J on E, we cite some basic lemmas from 24 , which will be used in the proof of Theorem 2.1. The first is the following Mountain Pass Lemma. Lemma 2.7. Let E be a real Banach Space, J ∈ C 1 E, R satisfies the C c condition for any c > 0, J 0 0, and J 1 There exist ρ, σ > 0 such that J| ∂B ρ ≥ α. J 2 There exist e ∈ E \ B ρ such that J e ≤ 0. Then J has a critical value c ≥ α.
In order to prove Theorem 2.2, we shall use the following fountain theorem 23, 25, 26 . Let E be a real Banach Space with the norm · and E j∈N X j with dim X j < ∞ for any j ∈ N. Set Y k k j 0 X j and Z k ∞ j k X j . Lemma 2.8. Let J ∈ C 1 E, R be even. If, for each sufficiently large k ∈ N, there exists ρ k > γ k > 0 such that B 1 a k : max u∈Y k , u ρ k J u ≤ 0.

Proofs of Main Results
ii there exists e ∈ E such that J te → −∞ as |t| → ∞.
Proof. i Let λ 1 − ω /2. According to f 1 and f 2 , it is easy to show that, there exists c 1 > 0, such that, f n, u ≤ |u| c 1 |u| p−1 3.1 for all n ∈ Z m and u ∈ R. This, together with the mean value theorem, leads to |F n, u | |F n, u − F n, 0 | 1 0 f n, su uds ≤ 2 |u| 2 c 1 p |u| p .

8 Abstract and Applied Analysis
Noting that p > 2, we obtain the following estimate: ii It follows from f 3 that for any M > 0, there exists δ δ M > 0 such that for all n ∈ R m , |u| ≥ δ, we have F n, u ≥ M|u| 2 . 3.5 Notice that, from f 2 and f 4 , it is easy to get that F n, u > 0, ∀u / 0.

3.6
Let e ∈ E be the eigenvector of H corresponding to the smallest eigenvalue λ 1 , that is to say He λ 1 e. Then, there exists N > 0, such that Taking t large enough, such that |te n | > δ for all n ∈ A * , then, in view of 3.5 -3.7 , we have
The proof is completed. Proof. Let {u k } ⊂ E be a C c sequence of J, that is, To prove the functional J satisfies the C c condition, first, we prove that {u k } is bounded in E. In fact, if not, we may assume by contradiction that u k → ∞ as k → ∞. Set α k : u k / u k . Up to a sequence, we have 3.12

3.13
Noticing that u 2 Hu, u ≥ λ 1 u 2 2 , we divide both sides of 3.13 by u k 2 and get
Case 2 α 0 . We define J t k u k : max

14
Abstract and Applied Analysis Taking ρ k sufficiently large, we have, a k : max u∈Y k , u ρ k J u ≤ 0.

3.37
The proof is completed.
Proof of Theorem 2.2. The proofs for 1 and 3 are similar to that of 1 and 3 in Theorem 2.1, and we omit them. Now we give the proof of 2 . By Lemma 3.3, the functional J satisfies B 1 and B 2 of Lemma 2.8. Lemma 3.2 implies that J satisfies C c condition for any c ∈ R. f is odd implies that J u is even. It follows from Lemma 2.8 that J has a sequence of critical points {u k } ⊂ E, such that J u k → ∞. Hence, 1.1 has infinitely many high-energy solutions in l 2 Z m . This completes the proof.