Solutions to discrete nonlinear Kirchhoff–Choquard equations

Abstract

In this paper, we study the discrete Kirchhoff–Choquard equation

$$\begin{aligned} -\left( a+b \int _{{\mathbb {Z}}^3}|\nabla u|^{2} d \mu \right) \Delta u+V(x) u=\left( R_{\alpha } *F(u)\right) f(u),\quad x\in {\mathbb {Z}}^3, \end{aligned}$$

where \(a,\,b>0\), \(\alpha \in (0,3)\) are constants and \(R_{\alpha }\) is the Green’s function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some suitable assumptions on V and f, we prove the existence of nontrivial solutions and ground state solutions respectively by variational methods.