Existence of solutions for fourth-order PDEs with variable exponents

El Amrouss, Abdelrachid ; Moradi, Fouzia ; Moussaoui, Mimoun

Electronic journal of differential equations, Tome 2009 (2009)

Résumé

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.

Détail

Classification : 35G30, 35K61, 46E35
Keywords: fourth-order pdes, variable exponent, palais Smale condition, mountain pass theorem, Fountain theorem