(untitled) (Journal of Partial Differential Equations, 2020)
In this paper,we consider the following Kirchhoff type problemwith critical exponent -(a+b∫_Ω|∇u|^2dx)Δu=λu^q+u^5, in Ω, u=0, on ∂Ω, where Ω⊂R^3 is a bounded smooth domain, 0< q < 1 and the parameters a,b,λ > 0. We show that there exists a positive constant T_4(a) depending only on a, such that for each a > 0 and 0 < λ < T_4(a), the above problem has at least one positive solution. The method we used here is based on the Nehari manifold, Ekeland's variational principle and the concentration compactness principle.