Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs

Abstract

In this paper, let G=(ZN,E,μ,ω) be a weighted lattice graph, where μ is a uniformly bounded measure and ω is a positive symmetric weight. We are concerned with the following biharmonic Choquard equation with a nonlocal nonlinearity on ZN, for any α∈(0,N),Δ2u−Δu+hu=(∑y∈ZN,y≠xF(u(y))d(x,y)N−α)f(u), where h denotes a function on ZN, f:R→R is a continuous function, F(u)=∫0uf(s)ds represents the primitive function of f, and d(x,y) stands for the distance between x and y. In particular, under some suitable conditions on h, we prove that if the nonlinearity f satisfies distinct growth conditions, then there exists a mountain-pass solution and a ground state solution respectively. As far as we know, there are no existing results for the biharmonic Choquard equation with the nonlinearity f on weighted lattice graphs.