Existence of weak solutions for nonlinear elliptic systems involving the (p(x), q(x))-Laplacian

TL;DR

This paper establishes the existence of weak solutions for a class of nonlinear elliptic systems involving variable exponent p(x)-Laplacian operators, using monotone operator theory, applicable to bounded domains and the entire space.

Contribution

It extends the existence results for nonlinear elliptic systems with variable exponents involving the p(x)-Laplacian, including unbounded domains, using monotone operator methods.

Findings

Existence of weak solutions in bounded domains.

Extension to solutions in al^N.

Application of monotone operator theory.

Abstract

In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system {lll} -\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u - b(x)|u|^{\alpha(x)}|v|^{\beta(x)} v + f(x) in \Omega, \Delta_{q(x)}v = c(x) |v|^{q(x)-2}v - d(x)|v|^{\beta(x)}|u|^{\alpha(x)} u + g(x) in \Omega, u = v = 0 \quad on \partial\Omega, where $\Omega$ is an open bounded domains of $\mathbb{R}^N$ with a smooth boundary $\partial\Omega$ and $\Delta_{p(x)}$ denotes the $p(x)$-Laplacian.The existence of weak solutions is proved using the theory of monotone operators. Similar result will be proved when $\Omega=\mathbb{R}^N$.