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In this paper, we study the nonlinear Choquard equation \begin{eqnarray*}\Delta^{2}u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}\ast|u|^{p})|u|^{p-2}u\end{eqnarray*} on a Cayley graph of a discrete group of polynomial growth withthe homogeneous dimension $N\geq 2$, where$\alpha\in(0,N),\,p>\frac{N+\alpha}{N},\,\lambda$ is a positive parameter and$R_\alpha$ stands for the Green's function of the discrete fractionalLaplacian, which has same asymptotics as the Riesz potential. Under someassumptions on $a(x)$, we establish the existence and asymptotic behavior ofground state solutions for the nonlinear Choquard equation by the method ofNehari manifold.

The existence and convergence of solutions for the nonlinear Choquard equations on groups of polynomial growth