The Existence and Convergence of Solutions for the Nonlinear Choquard Equations on Groups of Polynomial Growth

Abstract

In this paper, we study the nonlinear Choquard equation

$\Delta^{2} u-\Delta u+(1+\lambda a(x)) u=\left(R_{\alpha} *|u|^{p}\right)|u|^{p-2} u$

on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension N≥1, where $\alpha \in(0, N)$, $p>\frac{N+\alpha}{N}$, λ is a positive parameter and R_α stands for the Green’s function of the discrete fractional Laplacian, which has no singularity at the origin but has same asymptotics as the Riesz potential at infinity. 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.

Keywords

Nonlinear Choquard equation / discrete Green’s function / ground state solutions / Cayley graphs.