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

Shakeri, Saleh

doi:10.30495/maca.2021.1937515.1024

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

https://doi.org/10.30495/maca.2021.1937515.1024

Volume 3, Issue 4
Autumn 2021

Pages 1-8

Mathematical Analysis and its Contemporary Applications

Document Type : Original Article

Author

Saleh Shakeri

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 $g(0)$.