Solutions for Discrete Periodic Schrödinger Equations with Spectrum 0

In this paper we study the discrete nonlinear equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u_{n}+\varepsilon_{n}u_{n}-\omega u_{n}=\sigma \chi_{n}g_{n}(u_{n})u_{n},$$\end{document} where σ=±1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta u_{n}=u_{n+1}+u_{n-1}-2u_{n}$$\end{document} is the discrete Laplacian in one spatial dimension. The sequences εn and χn are assumed to be N-periodic in n, i.e. εn+N=εn and χn+N=χn. We prove the existence of solutions in l2 for this equation with ω a lower edge of a finite spectral gap and the nonlinearities satisfying very general superlinear assumptions.