Existence of solutions for p-Kirchhoff type problems with critical exponent - Ahmed Hamydy; Mohammed Massar; Najib Tsouli (Electronic Journal of Differential Equations, 2011)
We study the existence of solutions for the p-Kirchhoff type problem involving the critical Sobolev exponent, $$displaylines{ -Big[gBig(int_Omega|abla u|^pdxBig)Big]Delta_pu =lambda f(x,u)+|u|^{p^star-2}uquadextin Omega,cr u=0quadexton partialOmega, }$$ where $Omega$ is a bounded smooth domain of $mathbbR^N$, $1<p<N$, $p^star=Np/(N-p)$ 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.