Topological Methods in Nonlinear Analysis

Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain

Fu Yongqiang

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In this paper we study the following $p(x)$-Laplacian problem: $$ \begin{alignat}{2} -{\rm div}(a(x)|\nabla u|^{p(x)-2}\nabla u)+b(x)|u|^{p(x)-2}u&=f(x,u) &\quad& x\in \Omega,\\ u&=0 &\quad&\text{on }\partial\Omega, \end{alignat} $$ where $1< p_{1}\le p(x)\le p_{2}< n$, $\Omega\subset {\mathbb R}^{n}$ is an exterior domain. Applying Mountain Pass Theorem we obtain the existence of solutions in $W_{0}^{1,p(x)}(\Omega)$ for the $p(x)$-Laplacian problem in the superlinear case.

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Topol. Methods Nonlinear Anal., Volume 30, Number 2 (2007), 235-249.

First available in Project Euclid: 13 May 2016

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Yongqiang, Fu. Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain. Topol. Methods Nonlinear Anal. 30 (2007), no. 2, 235--249.

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