Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

Abstract

In this paper, we study the nonlinear Schrödinger equation $-\Delta u+V(x)u=f(x,u)$on the lattice graph ${\mathbb{Z}}^N$. Using the Nehari method, we prove that when $f$ satisfies some growth conditions and the potential function $V$ is periodic or bounded, the above equation admits a ground state solution. Moreover, we extend our results from ${\mathbb{Z}}^N$ to quasi-transitive graphs.

19 pages