Electron. J. Diff. Equ., Vol. 2011 (2011), No. 105, pp. 1-8.

Existence of solutions for p-Kirchhoff type problems with critical exponent

Ahmed Hamydy, Mohammed Massar, Najib Tsouli

Abstract:
We study the existence of solutions for the p-Kirchhoff type problem involving the critical Sobolev exponent,
$$\displaylines{ -\Big[g\Big(\int_\Omega|\nabla u|^pdx\Big)\Big]\Delta_pu =\lambda f(x,u)+|u|^{p^\star-2}u\quad\hbox{in }\Omega,\cr u=0\quad\hbox{on }\partial\Omega, }$$
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.

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Ahmed Hamydy
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: a.hamydy@yahoo.fr
Mohammed Massar
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: massarmed@hotmail.com
Najib Tsouli
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: tsouli@hotmail.com

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