Positive ground states for coupled nonlinear Choquard equations involving Hardy–Littlewood–Sobolev critical exponent

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In this paper, we consider the following coupled nonlinear Schrödinger equations with Choquard type nonlinearities: −Δu+λ1u=μ1(1|x|μ∗u2μ∗)u2μ∗−1+β(1|x|μ∗v2μ∗)u2μ∗−1,x∈Ω,−Δv+λ2v=μ2(1|x|μ∗v2μ∗)v2μ∗−1+β(1|x|μ∗u2μ∗)v2μ∗−1,x∈Ω,u,v≥0inΩ,u=v=0on∂Ω,where Ω⊂RN is a smooth bounded domain, 2μ∗≔2N−μN−2 is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, −λ1(Ω)<λ1,λ2<0,λ1(Ω) is the first eigenvalue of (−Δ,H01(Ω)), μ1,μ2>0 and β≠0 is a coupling constant. We show that the critical nonlocal system has a positive ground state solution for negative β and positive large β via variational methods. Moreover, we study the limit behavior of the positive ground state solution (uβ,vβ) as β→−∞ and some different phenomenon arises comparing with the local Schrödinger system.