Ground State Solutions to Some Indefinite Nonlinear Schrödinger Equations on Lattice Graphs

Abstract

In this paper, we consider the Schrödinger type equation −Δu + V (x)u = f(x, u) on the lattice graph ℤ^N with indefinite variational functional, where Δ is the discrete Laplacian. Specifically, we assume that V (x) and f(x, u) are periodic in x, f satisfies some growth condition and 0 lies in a finite spectral gap of (−Δ + V). We obtain ground state solutions by using the method of generalized Nehari manifold which has been introduced by Pankov.