Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs

In this paper, we study the nonlinear Choquard equation

$$-\mathrm{\Delta }u+V(x)u=\left(\sum _{\genfrac{}{}{0ex}{}{y\ne x}{y\in {\mathbb{Z}}^N}}\frac{|u(y){|}^p}{|x-y{|}^{N-\alpha }}\right)|u{|}^{p-2}u$$

on lattice graph ${\mathbb{Z}}^N$. Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function $V$ is periodic or confining. Moreover, when the potential function $V(x)=\lambda a(x)+1$ is confining, we obtain the asymptotic properties of the solution $u_{\lambda }$ which converges to a solution of a corresponding Dirichlet problem as $\lambda \to \mathrm{\infty }$.