Document Type : Original Article
Author
Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, P. O. Box 678, Iran
Abstract
This study concerns the existence of positive solution for the following nonlinear boundary value problem
\begin{gather*}
-\Delta_{p(x)} u= a(x)h(u) + f(v) \quad\text{in }\Omega\\
-\Delta_{q(x)} v=b(x)k(v) + g(u) \quad\text{in }\Omega\\
u=v= 0 \quad\text{on } \partial \Omega
\end{gather*}
where $p(x),q(x) \in C^1(\mathbb{R}^N)$ are radial symmetric functions such that $\sup|\nabla p(x)| < \infty,$ $\sup|\nabla q(x)|<\infty$ and $1 < \inf p(x) \leq \sup p(x) <\infty,1 < \inf q(x) \leq \sup q(x) < \infty$, and where $-\Delta_{p(x)} u = -\mathop{\rm div}|\nabla u|^{p(x)-2}\nabla u,-\Delta_{q(x)} v =-\mathop{\rm div}|\nabla v|^{q(x)-2}\nabla v$ respectively are called $p(x)$-Laplacian and $q(x)$-Laplacian, $\Omega = B(0 , R) = \{x | |x| < R\}$ is a bounded radial symmetric domain, where $R > 0$ is a sufficiently large constant. We discuss the existence of positive solution via sub-supersolutions without assuming sign conditions on $f(0)$ and $g(0)$.
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