$p(x)$-Laplacian with indefinite weight
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- by Khaled Kefi
- Proc. Amer. Math. Soc. 139 (2011), 4351-4360
- DOI: https://doi.org/10.1090/S0002-9939-2011-10850-5
- Published electronically: April 20, 2011
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Abstract:
We consider the eigenvalue problem $-\textrm {div}\big (|\nabla u|^{p(x)-2}\nabla u\big )=$ $\lambda V(x)|u|^{q(x)-2}u$, 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 $|\Omega _{0}| >0$. 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.References
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Bibliographic 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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 4351-4360
- MSC (2000): Primary 35D05, 35J60, 35J70, 58E05, 76A02
- DOI: https://doi.org/10.1090/S0002-9939-2011-10850-5
- MathSciNet review: 2823080