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 $\Omega$ 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 $\alpha+\beta>1$.