Existence of ground state solutions for a Choquard double phase problem

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of 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 given 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 continuous function that satisfies subcritical growth. Based on Hardy–Littlewood–Sobolev inequality, Nehari manifold variational tools, prove existence ground state solutions such problems under different assumptions data.