A uniqueness result for some singular semilinear elliptic equations

Given $\mathrm{\Omega }$ a bounded open subset of ${\mathbb{R}}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-\mathrm{\Delta }u=\frac{f}{u^{\beta }}$ in $H_{loc}^1(\mathrm{\Omega })$, under zero Dirichlet boundary conditions. For $\beta >0$ and $f\in L^1(\mathrm{\Omega })$, we prove that the solution is unique.