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

Defang Ma,Zhan Zhou

doi:10.1155/2012/703596

Abstract

A class of difference equations which include discrete nonlinear Schrödinger equations as special cases are considered. New sufficient conditions of the existence and multiplicity results of homoclinic solutions for the difference equations are obtained by making use of the mountain pass theorem and the fountain theorem, respectively. Recent results in the literature are generalized and greatly improved.

Highlights

Assume that m is a positive integer
We assume that f n, 0 0 for n ∈ Zm, un 0 is a solution of 1.1, which is called the trivial solution
We are interested in the existence and multiplicity of the nontrivial homoclinic solutions for 1.1

Summary

Introduction

Assume that m is a positive integer. Consider the following difference equation in infinite m dimensional lattices, Lun vnun − ωun σf n, un , n ∈ Zm, 1.1 where σ ±1, n n1, n2, . . . , nm ∈ Zm, {un} is a real valued sequence, ω ∈ R, L is a Jacobi operator 1 given by. 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 Schrodinger DNLS equation: iψn Δψn − vnψn σf n, ψn 0, n ∈ Zm, 1.4 where Δψn ψ n1 1,n2,...,nm ψ n1,n2 1,...,nm · · · ψ n1,n2,...,nm 1 − 2mψ n1,n2,...,nm 1.5 ψ n1−1,n2,...,nm ψ n1,n2−1,...,nm · · · ψ n1,n2,...,nm−1 is the discrete Laplacian in m spatial dimension. 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. The other aim of this paper is to obtain sufficient conditions for the existence of infinitely many nontrivial homoclinic solutions of 1.1.

Preliminaries and Main Results

Proofs of Main Results