Ground state solutions to some Indefinite Nonlinear Schrödinger equations on lattice graphs

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