*Accepted Paper*

**Inserted:** 20 apr 2021

**Last Updated:** 20 apr 2021

**Journal:** Analysis and PDEs

**Year:** 2021

**Abstract:**

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form
$-div (

\nabla u

^{p-2} \nabla u)= \delta\,

\nabla u

^q +\mu$ in a bounded main $\Omega\subset\mathbb{R}^n$ potentially with non-smooth boundary. Here either $\delta=0$ or $\delta=1$, $\mu$ is
a finite signed Radon measure in $\Omega$, and $q$ is of linear or super-linear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. 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.