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