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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} -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 $Ω$ is a regular bounded domain of $\mathbb R^{N}$, $α\in\mathbb R$, $p\in C(Ω)$ which behaves as $d(x)^{-β}$ as $x\to\partialΩ$ with $d$ the distance function up to the boundary and $0\leq β<2$. 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 $\partial Ω$. Consequently, we can prove that the solution belongs to $W^{1,q§}_0(Ω)$ for $1<q<\bar{q}_{α,β}\eqdef\frac{1+α}{α+β-1}$ optimal if $α+β>1$.