Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 105, pp. 1-8.

Title: Existence of solutions for p-Kirchhoff
       type problems with critical exponent
       
Authors: Ahmed Hamydy (Univ. Mohamed I, Oujda, Morocco)
         Mohammed Massar (Univ. Mohamed I, Oujda, Morocco)
	 Najib Tsouli (Univ. Mohamed I, Oujda, Morocco)

Abstract:
 We study the  existence of solutions for the
 p-Kirchhoff type problem involving the critical Sobolev
 exponent,
 $$\displaylines{
 -\Big[g\Big(\int_\Omega|\nabla u|^pdx\Big)\Big]\Delta_pu
 =\lambda f(x,u)+|u|^{p^\star-2}u\quad\text{in }\Omega,\cr
 u=0\quad\text{on }\partial\Omega,
 }$$
 where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$,
 $1<p<N$, $p^\star=Np/(N-p)$ is  the  critical  Sobolev
 exponent, $\lambda$ is a positive parameter, f and g are
 continuous functions. The main results of this paper establish, via
 the variational method. The concentration-compactness principle
 allows to prove that the Palais-Smale condition is satisfied below a
 certain level.
 
Submitted July 26, 2011. Published August 16, 2011.
Math Subject Classifications: 35A15, 35B33, 35J62.
Key Words: p-Kirchhoff; critical exponent; parameter; Lions principle.