Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities

Ali Mai,Zhan Zhou

doi:10.1155/2013/317139

Abstract

We consider the periodic discrete nonlinear Schrödinger equations with the temporal frequency belonging to a spectral gap. By using the generalized Nehari manifold approach developed by Szulkin and Weth, we prove the existence of ground state solutions of the equations. We obtain infinitely many geometrically distinct solutions of the equations when specially the nonlinearity is odd. The classical Ambrosetti-Rabinowitz superlinear condition is improved.

Highlights

The following discrete nonlinear Schrodinger equation (DLNS): iψn = −Δψn + εnψn − σχnfn, n ∈ Z, (1) where σ = ±1 andΔψn = ψn+1 + ψn−1 − 2ψn (2)is the discrete Laplacian operator, appears in many physical problems, like polarons, energy transfer in biological materials, nonlinear optics, and so forth
The following are the basic hypotheses to establish the main results of this paper: (V1) ω ∈ (α, β), (f1) fn ∈ C(R, R) and fn+T(u) = fn(u), and there exist a > 0 and p ∈ (2, ∞) such that
We introduce some notations

Summary

Introduction

The existence of solitons for the periodic DNLS equations with superlinear nonlinearity [7,8,9,10] and with saturable nonlinearity [11,12,13]. If ω is a lower edge of a finite spectral gap, the existence of solitons was obtained by using variant generalized weak linking theorem in [10]. If ω lies in a finite spectral gap, the existence of solitons was proved by using periodic approximations in combination with the linking theorem in [8] and the generalized Nehari manifold approach in [9], respectively. We employ the generalized Nehari manifold approach instead of periodic approximation technique to obtain the existence of a kind of special solitons of (7), which called ground state solutions, that is, nontrivial solutions with least possible energy in l2.

Preliminaries and Main Results

Proofs of Main Results