Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent

This paper deals with the following system linearly coupled by nonlinear elliptic equations{−Δu+λ1u=|u|2⁎−2u+βv,x∈Ω,−Δv+λ2v=|v|2⁎−2v+βu,x∈Ω,u=v=0on∂Ω. Here Ω is a smooth bounded domain in RN(N≥3), λ1,λ2>−λ1(Ω) are constants, λ1(Ω) is the first eigenvalue of (−Δ,H01(Ω)), 2⁎=2NN−2 is the Sobolev critical exponent and β∈R is a coupling parameter. By variational method, we prove that this system has a positive ground state solution for some β>0. Via a perturbation argument, we show that this system also admits a positive higher energy solution when |β| is small. Moreover, the asymptotic behaviors of the positive ground state and higher energy solutions as β→0 are analyzed.