Existence of solutions for $p(x)$-Laplacian equations

• Main Authors: Rabil Ayazoglu (Mashiyev), B. Cekic, O. M. Buhrii
• Format: Article
• Language: English
• Published: University of Szeged 2010-11-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=525

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.