Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 153, pp. 1-13.

Title: Existence of solutions for fourth-order PDEs with variable exponents
       
Authors: Abdelrachid El Amrouss (Univ. Mohamed I, Oujda, Morocco)
         Fouzia Moradi     (Univ. Mohamed I, Oujda, Morocco)
	 Mimoun Moussaoui  (Univ. Mohamed I, Oujda, Morocco)

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