Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

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 ℝN. 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)<N⋅m(x)N-m(x) for any x∈Ω̅. In the first case we show th...

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Main Author: Teodora-Liliana Dinu
Format: Article
Language:English
Published: Wiley 2006-01-01
Series:Journal of Function Spaces and Applications
Online Access:http://dx.doi.org/10.1155/2006/515496
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author Teodora-Liliana Dinu
author_facet Teodora-Liliana Dinu
author_sort Teodora-Liliana Dinu
collection DOAJ
description 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 ℝN. 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)<N⋅m(x)N-m(x) for any x∈Ω̅. 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 ℤ2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.
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spelling doaj-art-7e46695bec8e440cbc19281d3da08b3c2025-02-03T01:25:59ZengWileyJournal of Function Spaces and Applications0972-68022006-01-014322524210.1155/2006/515496Nonlinear eigenvalue problems in Sobolev spaces with variable exponentTeodora-Liliana Dinu0Department of Mathematics, “Fraţii Buzeşti” College, Bd. Ştirbei–Vodă No. 5, 200409 Craiova, RomaniaWe 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 ℝN. 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)<N⋅m(x)N-m(x) for any x∈Ω̅. 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 ℤ2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.http://dx.doi.org/10.1155/2006/515496
spellingShingle Teodora-Liliana Dinu
Nonlinear eigenvalue problems in Sobolev spaces with variable exponent
Journal of Function Spaces and Applications
title Nonlinear eigenvalue problems in Sobolev spaces with variable exponent
title_full Nonlinear eigenvalue problems in Sobolev spaces with variable exponent
title_fullStr Nonlinear eigenvalue problems in Sobolev spaces with variable exponent
title_full_unstemmed Nonlinear eigenvalue problems in Sobolev spaces with variable exponent
title_short Nonlinear eigenvalue problems in Sobolev spaces with variable exponent
title_sort nonlinear eigenvalue problems in sobolev spaces with variable exponent
url http://dx.doi.org/10.1155/2006/515496
work_keys_str_mv AT teodoralilianadinu nonlineareigenvalueproblemsinsobolevspaceswithvariableexponent