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

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.