Ground state solutions for periodic discrete Schrodinger equations with local super-quadratic conditions

In this paper, we consider the following discrete nonlinear Schrodinger equation{-delta u(n) + is an element of(n)u(n) - omega u(n) = f(n)(u(n)), n is an element of Z,lim(|n|->infinity) u(n) = 0,with periodic potentials and omega belongs to a spectral gap of the operator L = -delta + is an element of defined by Lu-n = -delta u(n) + is an element of(n)u(n). In order to investigate the existence of ground state solutions or infinitely many geometrically distinct solutions, it is commonly assumed that lim(|s|->infinity) integral(s)(0) f(n)(t)dt/s(2) = infinity uniformly for all n is an element of Z in the existing literature. The purpose of this paper is to prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition lim(|s|->infinity) integral(s)(0) f(n)(t)dt/s(2) = infinity for n is an element of G just for some set G subset of Z. Our result sharply extends and improves some existing ones in the literature.