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

 

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent


Authors: Mihai Mihailescu and Vicentiu Radulescu
Journal: Proc. Amer. Math. Soc. 135 (2007), 2929-2937
MSC (2000): Primary 35J70; Secondary 35D05, 35J60, 58E05, 74M05, 76A05
Published electronically: May 9, 2007
MathSciNet review: 2317971
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the nonlinear eigenvalue problem

$\displaystyle -{\rm div}\left(\vert\nabla u\vert^{p(x)-2}\nabla u\right)=\lambda \vert u\vert^{q(x)-2}u$

in $ \Omega$, $ u=0$ on $ \partial\Omega$, where $ \Omega$ is a bounded open set in $ \mathbb{R}^N$ with smooth boundary and $ p$, $ q$ are continuous functions on $ \overline\Omega$ such that $ 1<\inf_\Omega q< \inf_\Omega p<\sup_\Omega q$, $ \sup_\Omega p<N$, and $ q(x)<Np(x)/\left(N-p(x)\right)$ for all $ x\in\overline\Omega$. The main result of this paper establishes 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

Mihai Mihailescu
Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Email: mmihailes@yahoo.com

Vicentiu Radulescu
Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Email: vicentiu.radulescu@math.cnrs.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08815-6
PII: S 0002-9939(07)08815-6
Received by editor(s): February 4, 2006
Received by editor(s) in revised form: June 9, 2006
Published electronically: May 9, 2007
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.