Solutions to discrete nonlinear Kirchhoff–Choquard equations

In this paper, we study the discrete Kirchhoff–Choquard equation \(\begin{aligned} -\left( a+b \int _{{\mathbb {Z}}^3}|\nabla u|^{2} d \mu \right) \Delta u+V(x) u=\left( R_{\alpha } *F(u)\right) f(u),\quad x\in {\mathbb {Z}}^3, \end{aligned}\) $$\begin{array}{c}\hfill -\left(a,+,b,{\int }_{{\mathbb{Z}}^3},{|\mathrm{\nabla }u|}^2,d,\mu \right)\mathrm{\Delta }u+V(x)u=\left(R_{\alpha },,\ast ,F,(u)\right)f(u),x\in {\mathbb{Z}}^3,\end{array}$$ where \(a,\,b>0\) $a,b>0$ , \(\alpha \in (0,3)\) $\alpha \in (0,3)$ are constants and \(R_{\alpha }\) $R_{\alpha }$ 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.