Document Type : Original Article

Author

Islamic Azad University-Ayatollah Amoli, Branch, Amol, 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 $g(0)$.

Keywords