Existence of Positive Solutions for Kirchhoff Type Problems with Critical Exponent (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.
Keywords: Freedricksz transition; variational problem; liquid crystals; Landau-de Gennes functional