Geodesic
Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent
Electronic journal of differential equations, Tome 2011 (2011)
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.
Classification : 35G30, 35K61, 46E35
Keywords: fourth order elliptic equation, eigenvalue, Navier condition, variable exponent, Sobolev space, mountain pass theorem, Ekeland's variational principle 
@article{EJDE_2011__2011__a11,
     author = {Ayoujil,  Abdesslem and El Amrouss,  Abdel Rachid},
     title = {Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent},
     journal = {Electronic journal of differential equations},
     year = {2011},
     volume = {2011},
     zbl = {1227.35229},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a11/}
}
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Ayoujil,  Abdesslem; El Amrouss,  Abdel Rachid. Continuous spectrum of a fourth order nonhomogeneous elliptic equation with variable exponent. Electronic journal of differential equations, Tome 2011 (2011). http://geodesic.mathdoc.fr/item/EJDE_2011__2011__a11/