Positive least energy solutions for a coupled Schrِdinger 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: { − Δ u + λ 1 u = μ 1 u 5 + β u 2 v 3 , x ∈ Ω , − Δ v + λ 2 v = μ 2 v 5 + β v 2 u 3 , x ∈ Ω , u > 0 , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω , where Ω ⊂ R 3 is a smooth bounded domain,  λ 1 , λ 2 < 0 , μ 1 , μ 2 > 0 and β > 0 . Under certain conditions on λ 1 , λ 2 and β, we show that this problem has at least one positive least energy solution.