Pointwise gradient estimates for a class of singular quasilinear equation with measure data

Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data $-\mathrm{div}(A(x,\mathrm{\nabla }u))=\mu$ in a bounded and possibly nonsmooth domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n$. Here $\mathrm{div}(A(x,\mathrm{\nabla }u))$ is modeled after the $p$-Laplacian. Our results extend earlier known results to the singular case in which $\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}$.