Existence and regularity of weak solutions for singular elliptic equations

TL;DR

This paper studies the existence, uniqueness, and regularity of weak solutions for a class of singular elliptic equations with boundary behavior, providing precise gradient estimates and optimal Sobolev space membership conditions.

Contribution

It establishes new existence and regularity results for singular elliptic problems with boundary-dependent coefficients, including optimal gradient estimates and Sobolev space inclusion criteria.

Findings

Existence and uniqueness of weak solutions under specified conditions.

Precise boundary gradient estimates for solutions.

Solutions belong to optimal Sobolev spaces $W^{1,q}_0( abla)$ for certain $q$.

Abstract

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0 \text{on} \Omega, \end{array} \right . \end{equation*} where $\Omega$ is a regular bounded domain of $\mathbb R^{N}$, $\alpha\in\mathbb R$, $p\in C(\Omega)$ which behaves as $d(x)^{-\beta}$ as $x\to\partial\Omega$ with $d$ the distance function up to the boundary and $0\leq \beta <2$. We discuss below the existence, the uniqueness and the stability of the weak solution $u$ of the problem (P). We also prove accurate estimates on the gradient of the solution near the boundary $\partial \Omega$. Consequently, we can prove that the solution belongs to $W^{1,q{\S}}_0(\Omega)$ for $1<q<\bar{q}_{\alpha,\beta}\eqdef\frac{1+\alpha}{\alpha+\beta-1}$ optimal if $\alpha+\beta>1$.