The Existence and Convergence of Solutions for the Nonlinear Choquard Equations on Groups of Polynomial Growth
Year: 2025
Author: Ruowei Li, Lidan Wang
Journal of Partial Differential Equations, Vol. 38 (2025), Iss. 2 : pp. 226–248
Abstract
In this paper, we study the nonlinear Choquard equation $$\Delta^2u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}*|u|^p)|u|^{p-2}u$$on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension $N ≥ 1,$ where $α ∈ (0,N),$ $p>\frac{N+α}{N},$ $λ$ 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v38.n2.7
Journal of Partial Differential Equations, Vol. 38 (2025), Iss. 2 : pp. 226–248
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Nonlinear Choquard equation discrete Green’s function ground state solutions Cayley graphs.