Regularity of Weak Solutions for Singular Elliptic Problems Driven by m-Laplace Operator

TL;DR

This paper establishes optimal regularity results in Sobolev spaces for solutions of a singular elliptic PDE involving the m-Laplace operator, considering the behavior of the coefficient function near the boundary.

Contribution

It provides the first sharp regularity bounds for weak solutions of singular m-Laplace problems with boundary-dependent coefficients.

Findings

Solutions belong to specific Sobolev spaces depending on p and q.

The regularity range for solutions is proven to be optimal.

Results extend understanding of singular elliptic PDE regularity.

Abstract

We obtain optimal regularity in the Sobolev space $W_0^{1,\tau}(\Omega)$ for the unique solution of $$ -\Delta_m u=K(x)u^{-p} \mbox{in} \Omega, \quad u=0\mbox{on}\partial \Omega. $$ Here $\Omega\subset{\mathbb R}^N$ is a smooth and bounded domain, $m>1$, $p\geq 0$ and $K\in C(\Omega)$ is a positive function that behaves like ${\rm dist}(x,\partial\Omega)^{-q}$ for some $q\geq 0$ with $p+q< 2-\frac{1-p}{m}$. We obtain that the unique weak solution to the above problem belongs to $W_0^{1,\tau}(\Omega)$ for $$ m\leq \tau<\frac{m+p-1}{p+q-1}\quad \mbox{if}p+q>1, $$ and $$ m\leq \tau<\infty\quad \mbox{if}p+q=1. $$ The above range of $\tau$ is optimal.