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.