Weak Solutions for Singular Quasilinear Elliptic Systems

TL;DR

This paper establishes existence, uniqueness, and regularity results for weak solutions to a class of singular quasilinear elliptic systems with specific boundary conditions.

Contribution

It provides new conditions for existence and uniqueness of weak solutions and analyzes their regularity in Sobolev spaces for a class of singular elliptic systems.

Findings

Existence and uniqueness of weak solutions under certain conditions.

Determination of the optimal regularity range in Sobolev spaces.

Analysis of solution regularity and boundary behavior.

Abstract

We investigate the quasilinear elliptic system $-\Delta_{m} u&=u^{-p}v^{-q}$, $u>0 \quad\mbox{ in } \Omega$, $-\Delta_{m} v&=u^{r}v^{-s}$, $v>0 \quad\mbox{ in }\Omega$, $u=v=0 \quad\mbox{ on } \partial{\Omega}$, where $\Omega \subset{\mathbb R}^{N}(N\geq 1)$ is a bounded and smooth domain, $1<m<\infty, p, q, r, s>0$. Under certain conditions imposed on the exponents we obtain the existence and uniqueness of a weak solution $(u, v)$ with $u, v \in W_{0}^{1, m}(\Omega)\cap C(\Omega)$. We also investigate the $W_{0}^{1, \tau}(\Omega)$ regularity of solution and determine the optimal range of $\tau \geq m$ for such regularity.