Existence of ground state solutions for a Choquard double phase problem

Abstract

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form −Lp,qa(u)+|u|p−2u+a(x)|u|q−2u=∫RNF(y,u)|x−y|μdyf(x,u)inRN,where Lp,qa is the double phase operator given by Lp,qa(u)≔div(|∇u|p−2∇u+a(x)|∇u|q−2∇u),u∈W1,H(RN),0<μ<N, 1<p<N, p<q<p+αpN, 0≤a(⋅)∈C0,α(RN) with α∈(0,1] and f:RN×R→R is a continuous function that satisfies a subcritical growth. Based on the Hardy–Littlewood–Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.