Existence and regularity of weak solutions for singular elliptic equations

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 $\mathrm{\Omega }$ is a regular bounded domain of ${\mathbb{R}}^N$, $\alpha \in \mathbb{R}$, $p\in C(\mathrm{\Omega })$ which behaves as $d(x{)}^{-\beta }$ as $x\to \mathrm{\partial }\mathrm{\Omega }$ with $d$ the distance function up to the boundary and $0\le \beta <2$. 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 $\mathrm{\partial }\mathrm{\Omega }$. Consequently, we can prove that the solution belongs to $W_0^{1,q\mathrm{\S }}(\mathrm{\Omega })$ for $1<q<{\bar{q}}_{\alpha ,\beta }\text{eqdef}\frac{1+\alpha }{\alpha +\beta -1}$ optimal if $\alpha +\beta >1$.