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