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

R. A. Mashiyev, Dicle University, Diyarbakir, Turkey
B. Cekic, Dicle University, Diyarbakir, Turkey
O. M. Buhrii, Ivan Franko National University of Lviv, Lviv, Ukraine

E. J. Qualitative Theory of Diff. Equ., No. 65. (2010), pp. 1-13.

Abstract: We discuss the problem
\begin{equation*}
\left\{
\begin{array}{ll}
-\func{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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