On a degenerate singular elliptic problem

TL;DR

This paper establishes existence, uniqueness, and regularity of solutions for a class of degenerate singular elliptic boundary value problems involving weighted p-Laplacian operators.

Contribution

It provides new mathematical results on the well-posedness and regularity of solutions for a broad class of degenerate singular elliptic equations with weights in the Muckenhoupt class.

Findings

Existence of solutions under given conditions.

Uniqueness of solutions for the problem.

Regularity results for solutions.

Abstract

In this article we provide existence, uniqueness and regularity results of a degenerate singular elliptic boundary value problem whose prototype is given by \begin{gather*} \begin{cases} -\operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u^\delta}\,\,\text{ in }\,\,\Omega, u>0\text{ in }\Omega,\\ u = 0 \text{ on } \partial\Omega, \end{cases} \end{gather*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 2$, $w$ belong to the Muckenhoupt class $A_p$ for some $1<p<\infty$, $f$ is a nonnegative function belong to some Lebesgue space and $\delta>0$.