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

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

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