Existence and regularity estimates for quasilinear equations with measure data: the case $1<p\le \frac{3n-2}{2n-1}$

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-{\rm div} (|\nabla u|^{p-2} \nabla u)= \delta, |\nabla u|^q +\mu$inaboundedmain$\Om\subset\RR^n$ potentially with non-smooth boundary. Here either $\delta =0$ or $\delta =1$, $\mu$ is a finite signed Radon measure in $\mathrm{\Omega }$, and $q$ is of linear or super-linear growth, i.e., $q\ge 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\le \frac{3n-2}{2n-1}$. In particular, in the case $\delta =1$ which corresponds to a Riccati type equation, we settle the question of solvability that has been raised for some time in the literature.