Solutions for Discrete Periodic Schrodinger Equations with Spectrum 0

In this paper we study the discrete nonlinear equation -Delta u(n) + epsilon(n)u(n) - omega u(n) = sigma chi(n)g(n)(u(n))u(n), where sigma = +/-1, Delta u(n) = u(n+1) + u(n-1) - 2u(n) is the discrete Laplacian in one spatial dimension. The sequences epsilon(n) and chi(n) are assumed to be N-periodic in n, i.e. epsilon(n+N) = epsilon(n) and chi(n+N) = chi(n). We prove the existence of solutions in l(2) for this equation with. a lower edge of a finite spectral gap and the nonlinearities satisfying very general superlinear assumptions.