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

TL;DR

This paper proves the existence and describes the asymptotic behavior of solutions for a nonlinear Choquard equation on groups of polynomial growth, using variational methods on Cayley graphs.

Contribution

It establishes the existence and asymptotic properties of ground state solutions for the nonlinear Choquard equation on Cayley graphs of polynomial growth, extending previous results to discrete groups.

Findings

Existence of ground state solutions under certain conditions.

Asymptotic behavior of solutions characterized.

Application of Nehari manifold method to discrete groups.

Abstract

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 with the 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 fractional Laplacian, which has same asymptotics as the Riesz potential. Under some assumptions on $a(x)$, we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.