Ground state solution for strongly indefinite Choquard system

Abstract

In this paper, we consider the following Choquard system in RN−Δu+V(x)u=2qp+q(Iα∗|u|p)|v|q−2v,−Δv+V(x)v=2pp+q(Iα∗|v|q)|u|p−2u,u(x)→0andv(x)→0as|x|→∞, where α∈(0,N) and N+αN<p,q<2∗α, in which 2∗α denotes N+αN−2 if N≥3 and 2∗α≔∞ if N=1,2. Iα is a Riesz potential. V is continuous, 1-periodic in x and 0 belongs to a spectral gap of −Δ+V. This system is subcritical in the sense of the Hardy–Littlewood–Sobolev inequality. Based on the generalized Nehari manifold, we obtain the existence of ground state solutions. Our results can be looked on as a generalization to results by Szulkin and Weth (2009).