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Title: On a class of nonlinear problems involving a $p(x)$-Laplace type operator (English)
Author: Mihăilescu, Mihai
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 58
Issue: 1
Year: 2008
Pages: 155-172
Summary lang: English
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Category: math
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Summary: We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)<q(x)< \frac{N\cdot m(x)}{(N-m(x))}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda $ 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 ${\mathbb{Z}} _2$-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods. (English)
Keyword: $p(x)$-Laplace operator
Keyword: generalized Lebesgue-Sobolev space
Keyword: critical point
Keyword: weak solution
Keyword: electrorheological fluid
MSC: 35D05
MSC: 35J60
MSC: 35J70
MSC: 47J30
MSC: 58E05
MSC: 68T40
MSC: 76A02
MSC: 76A05
idZBL: Zbl 1165.35336
idMR: MR2402532
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Date available: 2009-09-24T11:54:15Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/128252
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