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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ p(x)$-Laplacian with indefinite weight


Author: Khaled Kefi
Journal: Proc. Amer. Math. Soc. 139 (2011), 4351-4360
MSC (2000): Primary 35D05, 35J60, 35J70, 58E05, 76A02
Published electronically: April 20, 2011
MathSciNet review: 2823080
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Abstract: We consider the eigenvalue problem $ -\mathop{\rm div}\big(\vert\nabla u\vert^{p(x)-2}\nabla u\big)=$ $ \lambda V(x)\vert u\vert^{q(x)-2}u$, in $ \Omega$, $ u=0$ on $ \partial\Omega$, where $ \Omega$ is a smooth bounded domain in $ \mathbb{R}^{N}$, $ \lambda>0$, $ p,q$ are continuous functions on $ \overline{\Omega}$ and $ V$ is a given function in a generalized Lebesgue space $ L^{s(x)}(\Omega)$ such that $ V>0$ in an open set $ \Omega_{0}\subset \Omega$, where $ \vert\Omega_{0}\vert >0$. We prove under appropriate conditions on the functions $ p,q$ and $ s$ that any $ \lambda>0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.


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Additional Information

Khaled Kefi
Affiliation: Institut Supérieur du Transport et de la Logistique de Sousse, 12 rue abdallah Ibn Zoubër, 4029-Sousse, Tunisia
Email: khaled_kefi@yahoo.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10850-5
PII: S 0002-9939(2011)10850-5
Keywords: $p(x)$-Laplace operator, Ekeland’s variational principle, generalized Sobolev spaces, Mountain Pass Theorem, weak solution.
Received by editor(s): August 12, 2010
Received by editor(s) in revised form: October 11, 2010
Published electronically: April 20, 2011
Communicated by: Varghese Mathai
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.