Concentrating Ground State for Linearly Coupled Schrödinger Systems Involving Critical Exponent Cases

We study the following linearly coupled Schrödinger system:{−Δu+(μV1(x)+a)u=f(x)|u|p−2u+β(x)v,x∈RN,−Δv+(μV2(x)+b)v=g(x)|v|2⁎−2v+β(x)u,x∈RN, where N≥3,2−λ1(Ω1),b>−λ1(Ω2) and λ1(Ωi) is the first eigenvalue of −Δ in H01(Ωi). Under some suitable assumptions on β(x) which relate to the potentials V1,V2 and the parameters a,b, the existence of positive ground states is obtained by variational method. Some interesting phenomena are that we relax the upper control condition of the coupling function β(x) and we do not need the weight function f(x) to be integrable or bounded in the subcritical case 2