On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent
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- by Mihai Mihăilescu and Vicenţiu Rădulescu PDF
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Abstract:
We consider the nonlinear eigenvalue problem \[ -\textrm {div}\left (|\nabla u|^{p(x)-2}\nabla u\right )=\lambda |u|^{q(x)-2}u\] 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.References
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Additional Information
- Mihai Mihăilescu
- Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
- MR Author ID: 694712
- Email: mmihailes@yahoo.com
- Vicenţiu Rădulescu
- Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
- MR Author ID: 143765
- ORCID: 0000-0003-4615-5537
- Email: vicentiu.radulescu@math.cnrs.fr
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2929-2937
- MSC (2000): Primary 35J70; Secondary 35D05, 35J60, 58E05, 74M05, 76A05
- DOI: https://doi.org/10.1090/S0002-9939-07-08815-6
- MathSciNet review: 2317971