Ground state solutions to some Indefinite Nonlinear Schr\"{o}dinger equations on lattice graphs

Abstract

In this paper, we consider the Schr\"odinger type equation $-\Delta u+V(x)u=f(x,u)$ on the lattice graph $\mathbb{Z}^{N}$ with indefinite variational functional, where $-\Delta$ 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 spectral gap of $(-\Delta + V)$. We obtain ground state solutions by using the method of generalized Nehari manifold which has been introduced in arXiv:1801.06872.