Positive ground state solutions for a nonlinearly coupled Schrödinger system with critical exponents in R4R4

Consider the following nonlinear Schrödinger equations:{−△u+λ1u=μ1u3+3αu2v+βuv2+γv3,x∈Ω,−△v+λ2v=μ2v3+3γuv2+βu2v+αu3,x∈Ω,u≥0,v≥0in Ω;u=v=0on ∂Ω, where Ω⊂R4Ω⊂R4 is a smooth bounded domain, −λ(Ω)<λ1−λ(Ω)<λ1, λ2<0λ2<0, μ1,μ2,α,γ>0μ1,μ2,α,γ>0, β≥0β≥0, and λ(Ω)λ(Ω) is the first eigenvalue of −△ with the Dirichlet boundary condition. Note that all the cubic nonlinearities, including the coupling terms, are of critical growth. The existence of positive ground state solutions for the system is established.