Electron. J. Diff. Equ., Vol. 2009(2009), No. 153, pp. 1-13.

Existence of solutions for fourth-order PDEs with variable exponents

Abdelrachid El Amrouss, Fouzia Moradi, Mimoun Moussaoui

Abstract:
In this article, we study the following problem with Navier boundary conditions
$$\displaylines{ \Delta _{p(x)}^2u=\lambda | u| ^{p(x)-2}u+f(x,u)\quad \hbox{in }\Omega , \cr u=\Delta u=0\quad \hbox{on }\partial \Omega . }$$
Where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega $, $N\geq 1$, $\Delta _{p(x)}^2u:=\Delta (|\Delta u| ^{p(x)-2}\Delta u) $, is the $p(x)$-biharmonic operator, $\lambda \leq 0$, $p$ is a continuous function on $\overline{\Omega } $ with $\inf_{x\in \overline{\Omega }} p(x)>1$ and $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a Caratheodory function. Using the Mountain Pass Theorem, we establish the existence of at least one solution of this problem. Especially, the existence of infinite many solutions is obtained.

Submitted October 5, 2009. Published November 27, 2009.
Math Subject Classifications: 35G30, 35K61, 46E35.
Key Words: Fourth-order PDEs; variable exponent; Palais Smale condition; mountain pass theorem; fountain theorem

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Abdelrachid El Amrouss
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: elamrouss@fso.ump.ma, elamrouss@hotmail.com
Fouzia Moradi
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: f.moradi@hotmail.com
Mimoun Moussaoui
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: moussaoui@est.ump.ma

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