On a class of nonlinear problems involving a p ( x ) -Laplace type operator

Mihai Mihăilescu

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 1, page 155-172
  • ISSN: 0011-4642

Abstract

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We study the boundary value problem - d i v ( ( | ∇ u | p 1 ( x ) - 2 + | ∇ u | p 2 ( x ) - 2 ) ∇ u ) = f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in ℝ N . Our attention is focused on two cases when f ( x , u ) = ± ( - λ | u | m ( x ) - 2 u + | u | q ( x ) - 2 u ) , where m ( x ) = max { p 1 ( x ) , p 2 ( x ) } for any x ∈ Ω ¯ or m ( x ) < q ( x ) < N · m ( x ) ( N - m ( x ) ) for any x ∈ Ω ¯ . In the former case we show the existence of infinitely many weak solutions for any λ > 0 . In the latter 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 ℤ 2 -symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.

How to cite

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Mihăilescu, Mihai. "On a class of nonlinear problems involving a $p(x)$-Laplace type operator." Czechoslovak Mathematical Journal 58.1 (2008): 155-172. <http://eudml.org/doc/31205>.

@article{Mihăilescu2008,
abstract = {We study the boundary value problem $-\{\mathrm \{d\}iv\}((|\nabla u|^\{p_1(x) -2\}+|\nabla u|^\{p_2(x)-2\})\nabla u)=f(x,u)$ 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.},
author = {Mihăilescu, Mihai},
journal = {Czechoslovak Mathematical Journal},
keywords = {$p(x)$-Laplace operator; generalized Lebesgue-Sobolev space; critical point; weak solution; electrorheological fluid; -Laplace operator; generalized Lebesgue-Sobolev space; critical point; weak solution; electrorheological fluid},
language = {eng},
number = {1},
pages = {155-172},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a class of nonlinear problems involving a $p(x)$-Laplace type operator},
url = {http://eudml.org/doc/31205},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Mihăilescu, Mihai
TI - On a class of nonlinear problems involving a $p(x)$-Laplace type operator
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 1
SP - 155
EP - 172
AB - We study the boundary value problem $-{\mathrm {d}iv}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ 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.
LA - eng
KW - $p(x)$-Laplace operator; generalized Lebesgue-Sobolev space; critical point; weak solution; electrorheological fluid; -Laplace operator; generalized Lebesgue-Sobolev space; critical point; weak solution; electrorheological fluid
UR - http://eudml.org/doc/31205
ER -

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