The ground state solutions of nonlinear Schrödinger equations with Hardy weights on lattice graphs

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