Gradient estimates for singular $p$-Laplace type equations with measure data

Abstract

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the $p$-Laplace equation $-{\Delta }_{p}u=\mu$ with $p\in (1,2)$. The cases when $p\in (2-\frac{1}{n},2)$ and $p\in (\frac{3n-2}{2n-1},2-\frac{1}{n}]$ were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379–418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when $p\in (1,\frac{3n-2}{2n-1}]$. Interior and global modulus of continuity estimates of the gradients of solutions are also established.