Positive ground state solutions and multiple nontrivial solutions for coupled critical elliptic systems

In this study, we consider the following coupled elliptic system with a Sobolev critical exponent:equation(PP){−Δu1+λ1u1=ν1u1p1−1+μ1u12⁎−1+βu12⁎2−1u22⁎2,x∈Ω,−Δu2+λ2u2=ν2u2p2−1+μ2u22⁎−1+βu12⁎2u22⁎2−1,x∈Ω,u1,u2≥0inΩ,u1=u2=0on∂Ω, where Ω⊂RNΩ⊂RN is a bounded smooth domain, N≥5N≥5, 20μj>0 for j=1,2j=1,2, and λ1(Ω)λ1(Ω) is the first eigenvalue of −Δ with the Dirichlet boundary condition. We demonstrate the existence of a positive ground state solution for problem (PP) when the coupling parameter β≥−μ1μ2. Under some other conditions, we show the nonexistence of positive solutions for (PP) when N≥3N≥3. We also construct multiple nontrivial solutions and sign-changing solutions for the following system:{−Δu1+λ1u1=μ1u13+βu22u1,x∈Ω,−Δu2+λ2u2=μ2u23+βu12u2,x∈Ω,u1=u2=0on∂Ω, where Ω⊂RNΩ⊂RN is a bounded smooth domain and N≤4N≤4.