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

Abstract

We obtain optimal regularity in the Sobolev space $W_0^{1,\ddot{I}„}(\hat{I}\text{c◯})$ for the unique solution of $$ -Δ_m u=K(x)u^{-p} \mbox{in} Ω, \quad u=0\mbox{on}\partial Ω. $$ Here $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ is a smooth and bounded domain, $m>1$, $p\ge 0$ and $K\in C(\hat{I}\text{c◯})$ is a positive function that behaves like $\mathrm{dist}(x,\partial \hat{I}\text{c◯}{)}^{-q}$ for some $q\ge 0$ with $p+q<2-\frac{1-p}{m}$. We obtain that the unique weak solution to the above problem belongs to $W_0^{1,\ddot{I}„}(\hat{I}\text{c◯})$ 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 $\ddot{I}„$ is optimal.

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