Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 24, pp. 1-12.

Title: Continuous spectrum of a fourth order nonhomogeneous
       elliptic equation with variable exponent

Authors: Abdesslem  Ayoujil (Univ. Mohamed I, Oujda, Morocco)
         Abdel Rachid El Amrouss (Univ. Mohamed I, Oujda, Morocco)

Abstract:
 In this article, we consider the  nonlinear eigenvalue problem
 $$\displaylines{
 \Delta(|\Delta u|^{p(x)-2}\Delta u )=\lambda
 |u|^{q(x)-2}u\quad \hbox{in }\Omega, \cr
 u=\Delta u = 0\quad \hbox{on }\partial\Omega,
 }$$
 where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth
 boundary and $ p, q: \overline{\Omega} \to (1,+\infty)$ are
 continuous functions. Considering different situations concerning
 the growth rates involved in the above quoted problem, we prove the
 existence of a continuous family of eigenvalues. The proofs of the
 main results are based on the mountain pass lemma and Ekeland’s
 variational principle.

Submitted October 6, 2010. Published February 09, 2011.
Math Subject Classifications: 35G30, 35K61, 46E35.
Key Words: Fourth order elliptic equation; eigenvalue;
           Navier condition;  variable exponent; Sobolev space;
           mountain pass theorem; Ekeland’s variational principle.