Positive least energy solutions for a coupled Schrödinger system with critical exponent

In this paper, we consider the following coupled Schrödinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis–Nirenberg problem:{−Δu+λ1u=μ1u5+βu2v3,x∈Ω,−Δv+λ2v=μ2v5+βv2u3,x∈Ω,u>0,v>0,x∈Ω,u=v=0,x∈∂Ω, where Ω⊂R3Ω⊂R3 is a smooth bounded domain,  λ1,λ2<0λ1,λ2<0, μ1,μ2>0μ1,μ2>0 and β>0β>0. Under certain conditions on λ1,λ2λ1,λ2 and β, we show that this problem has at least one positive least energy solution.