Existence and regularity of weak solutions for singular elliptic equations

Abstract

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -Δu = \dfrac{p(x)}{u^α}\quad \text{in} Ω\\ u = 0\ \text{on} Ω,\ u>0 \text{on} Ω, \end{array} \right . \end{equation*} where $\hat{I}\text{c◯}$ is a regular bounded domain of ${\mathbb{R}}^N$, $\hat{I}\pm \in \mathbb{R}$, $p\in C(\hat{I}\text{c◯})$ which behaves as $d(x{)}^{-{\hat{I}}^2}$ as $x\to \partial \hat{I}\text{c◯}$ with $d$ the distance function up to the boundary and $0\le {\hat{I}}^2<2$. 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 $\partial \hat{I}\text{c◯}$. Consequently, we can prove that the solution belongs to $W_0^{1,q\hat{A}\S }(\hat{I}\text{c◯})$ for $1<q<\bar{q}_{α,β}\eqdef\frac{1+α}{α+β-1}$ optimal if $\hat{I}\pm +{\hat{I}}^2>1$.

13 pages