The Existence and Convergence of Solutions for the Nonlinear Choquard Equations on Groups of Polynomial Growth (Journal of Partial Differential Equations, 2025)
In this paper, we study the nonlinear Choquard equation $$Δ^2u-Δ u+(1+λ a(x))u=(R_{α}*|u|^p)|u|^{p-2}u$$on a Cayley graph of a discrete group of polynomial growth with the homogeneousdimension $N ≥ 1,$ where $α ∈ (0,N),$ $p>\frac{N+α}{N},$ $λ$ is a positive parameter and $R_α$ standsfor the Green’s function of the discrete fractional Laplacian, which has no singularityat the origin but has same asymptotics as the Riesz potential at infinity. Under someassumptions on $a(x),$ we establish the existence and asymptotic behavior of groundstate 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.