Regularity of Weak Solutions for Singular Elliptic Problems Driven by m-Laplace Operator

We obtain optimal regularity in the Sobolev space $W_0^{1,\tau }(\mathrm{\Omega })$ for the unique solution of $$-{\mathrm{\Delta }}_{m}u=K(x)u^{-p}in\mathrm{\Omega },u=0on\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ Here $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth and bounded domain, $m>1$, $p\ge 0$ and $K\in C(\mathrm{\Omega })$ is a positive function that behaves like $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{-q}$ for some $q\geq 0$with$p+q< 2-\frac{1-p}{m}$. We obtain that the unique weak solution to the above problem belongs to $W_0^{1,\tau }(\mathrm{\Omega })$ for $$m\le \tau <\frac{m+p-1}{p+q-1}ifp+q>1,$$ and $$m\le \tau <\mathrm{\infty }ifp+q=1.$$ The above range of $\tau$ is optimal.