Existence of solutions for $p(x)$-Laplacian equations
We discuss the problem \begin{equation*} \left\{ \begin{array}{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right) =\lambda (a\left( x\right) \left\vert u\right\vert ^{q(x)-2}u+b(x)\left\vert u\right\vert ^{h(x)-2}u)\text{,} & \text{for }x\in \Omega , \\ u=0\te...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2010-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=525 |
| Summary: | We discuss the problem
\begin{equation*}
\left\{
\begin{array}{ll}
-\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right)
=\lambda (a\left( x\right) \left\vert u\right\vert ^{q(x)-2}u+b(x)\left\vert u\right\vert ^{h(x)-2}u)\text{,} & \text{for }x\in \Omega , \\ u=0\text{,} & \text{for }x\in \partial \Omega .
\end{array}
\right.
\end{equation*}
where $\Omega $ is a bounded domain with smooth boundary in $\mathbb{R}^{N}$ $\left( N\geq 2\right)$ and $p$ is Lipschitz continuous, $q$ and $h$ are continuous functions on $\overline{\Omega }$ such that $1<q(x)<p(x)<h(x)<p^{\ast }(x)$ and $p(x)<N$. We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem. |
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| ISSN: | 1417-3875 1417-3875 |