Electron. J. Diff. Equ., Vol. 2011 (2011), No. 24, pp. 1-12.

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

Abdesslem Ayoujil, Abdel Rachid El Amrouss

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 9, 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.

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Abdesslem Ayoujil
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: abayoujil@yahoo.fr
Abdel Rachid El Amrouss
University Mohamed I, Faculty of sciences
Department of Mathematics, Oujda, Morocco
email: elamrouss@fso.ump.ma

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