Solutions to discrete nonlinear Kirchhoff-Choquard equations

Abstract

In this paper, we study the discrete Kirchhoff-Choquard equation $$-\left(a+b{\int }_{{\mathbb{Z}}^3}|\nabla u{|}^{2}d\mu \right)\Delta u+V(x)u=\left(R_{\alpha }\ast F(u)\right)f(u),x\in {\mathbb{Z}}^3,$$ where $a,b>0$ are constants, $R_{\alpha }$ is the Green's function of the discrete fractional Laplacian with $\alpha \in (0,3)$, which has no singularity but has same asymptotics as the Riesz potential. Under some suitable assumptions on $V$ and $f$, we prove the existence of nontrivial solutions and ground state solutions by variational methods.

18 pages