We obtain optimal regularity in the Sobolev space $W_0^{ 1,\tau}(\Omega)$ for the unique solution of $$ -\Delta_m u=K(x)u^{ -p} \mbox{ in} \Omega, \quad u=0\mbox{ on}\partial \Omega. $$ Here $\Omega\subset{ \mathbb R}^N$ is a smooth and bounded domain, $m>1$, $p\geq 0$ and $K\in C(\Omega)$ is a positive function that behaves like ${ \rm dist}(x,\partial\Omega)^{ -q}$ 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 $W_0^{ 1,\tau}(\Omega)$ for $$ m\leq \tau<\frac{ m+p-1}{ p+q-1}\quad \mbox{ if}p+q>1, $$ and $$ m\leq \tau<\infty\quad \mbox{ if}p+q=1. $$ The above range of $\tau$ is optimal.
Categories: math.AP