Journal of Function Spaces

Journal of Function Spaces / 2006 / Article
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Research Article | Open Access

Volume 4 |Article ID 515496 | https://doi.org/10.1155/2006/515496

Teodora-Liliana Dinu, "Nonlinear eigenvalue problems in Sobolev spaces with variable exponent", Journal of Function Spaces, vol. 4, Article ID 515496, 18 pages, 2006. https://doi.org/10.1155/2006/515496

Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

Academic Editor: George Isac
Received01 Jul 2005

Abstract

We study the boundary value problem -div((|u|p1(x)-2+|u|p2(x)-2)u)=f(x,u) in Ω, u=0 on Ω, where Ω is a smooth bounded domain in N. We focus on the cases when f±(x,  u)=±(-λ|u|m(x)-2u+|u|q(x)-2u), where m(x)max{p1(x),p2(x)}<q(x)<Nm(x)N-m(x) for any xΩ̅. In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a 2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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