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It has been brought to the journal's attention that the above paper is almost identical with the paper “M. Mihailescu, On a class of nonlinear problems involving a p(x)-Laplace type operator, Czechoslovak Math. J., 58 (2008), 155-172.” Therefore, the paper by T.-L. Dinu has been retracted by the Journal of Function Spaces and Applications. This has been approved by the author as well.

Journal of Function Spaces and Applications
Volume 4 (2006), Issue 3, Pages 225-242

Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

Department of Mathematics, “Fraţii Buzeşti” College, Bd. Ştirbei–Vodă No. 5, 200409 Craiova, Romania

Received 1 July 2005

Academic Editor: George Isac

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.


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.