The ground state solutions of nonlinear Schr\"{o}dinger equations with Hardy weights on lattice graphs

Abstract

In this paper, we study the nonlinear Schr\"{o}dinger equation $$ -\Delta u+(V(x)- \frac{\rho}{(|x|^2+1)})u=f(x,u) $$ on the lattice graph $\mathbb{Z}^N$ with $N\geq 3$, where $V$ is a bounded periodic potential and $0$ lies in a spectral gap of the Schr\"{o}dinger operator $-\Delta+V$. Under some assumptions on the nonlinearity $f$, we prove the existence and asymptotic behavior of ground state solutions with small $\rho\geq 0$ by the generalized linking theorem.