Existence of Positive Solutions for Kirchhoff Type Problems with Critical Exponent

In this paper, we consider the following Kirchhoff type problem with critical exponent {(u = 0,)-(a+b integral(Omega)vertical bar del u vertical bar(2)dx) Delta u = lambda u(q) + u(5) , in Omega, on partial derivative Omega, where Omega subset of R-3 is a bounded smooth domain, 0 < q < 1 and the parameters a,b,A.> 0. We show that there exists a positive constant T-4 (a) depending only on a, such that for each a>0 and 0 < lambda < 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.