The critical problem of Kirchhoff type elliptic equations in dimension four

We study the following Kirchhoff type elliptic problem,equation(P){−(a+b∫Ω|∇u|2dx)Δu=λuq+μu3,u>0in Ω,u=0on ∂Ω, where Ω⊂R4Ω⊂R4 is a bounded domain with smooth boundary ∂Ω  . Moreover, we assume a,λ,μ>0a,λ,μ>0, b≥0b≥0 and 1≤q<31≤q<3. In this paper, we prove the existence of solutions of (P). Our tools are the variational method and the concentration compactness argument for PS sequences.