An existence result for a class of (p(x),q(x))-Laplacian system via sub-supersolution method

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= ۰ \quad\text{on } \partial \Omega\end{gather*}where p(x),q(x) \in C^۱(\mathbb{R}^N) are radial symmetric functions such that \sup|\nabla p(x)| < \infty, \sup|\nabla q(x)|<\infty and ۱ < \inf p(x) \leq \sup p(x) <\infty,۱ < \inf q(x) \leq \sup q(x) < \infty, and where -\Delta_{p(x)} u = -\mathop{\rm div}|\nabla u|^{p(x)-۲}\nabla u,-\Delta_{q(x)} v =-\mathop{\rm div}|\nabla v|^{q(x)-۲}\nabla v respectively are called p(x)-Laplacian and q(x)-Laplacian, \Omega = B(۰ , R) = \{x | |x| < R\} is a bounded radial symmetric domain, where R > ۰ is a sufficiently large constant. We discuss the existence of positive solution via sub-supersolutions without assuming sign conditions on f(۰) and g(۰).

Keywords:

Positive radial solutions , (p(x) , q(x))-Laplacian system , Sub-super solutions