On Kirchhoff type equations with critical Sobolev exponent

We study the following Brezis-Nirenberg problem of Kirchhoff type{−(a+b∫Ω|∇u|2dx)Δu=λ|u|q−2u+δ|u|2u,inΩ,u=0,on∂Ω, where Ω⊂R4 is a bounded domain with the smooth boundary ∂Ω, 2≤q<4 and a, b, λ, δ are positive parameters. We obtain some new existence and nonexistence results, depending on the values of the above parameters, which improve some known results.