EXISTENCE AND REGULARITY ESTIMATES FOR QUASILINEAR EQUATIONS WITH MEASURE DATA: THE CASE 1<p≤(3n−2)/(2n−1)

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 $-\text{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)=\delta |\mathrm{\nabla }u{|}^q+\mu$ in a bounded domain $\mathrm{\Omega }\subset {ℝ}^n$ potentially with nonsmooth boundary. Here either $\delta =0$ or $\delta =1$, $\mu$ is a finite signed Radon measure in $\mathrm{\Omega }$, and $q\ge 1$. Our main concern is to extend earlier results to the strongly singular case . 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.