Existence and regularity estimates for quasilinear equations with measure data: the case $1<p\leq \frac{3n2}{2n1}$
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 ${\rm div} (\nabla u^{p2} \nabla u)= \delta\, \nabla u^q +\mu$ in a bounded main $\Om\subset\RR^n$ potentially with nonsmooth boundary. Here either $\delta=0$ or $\delta=1$, $\mu$ is a finite signed Radon measure in $\Omega$, and $q$ is of linear or superlinear growth, i.e., $q\geq 1$. Our main concern is to extend earlier results to the strongly singular case $1<p\leq \frac{3n2}{2n1}$. 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 DOI:
 10.48550/arXiv.2003.03725
 arXiv:
 arXiv:2003.03725
 Bibcode:
 2020arXiv200303725N
 Keywords:

 Mathematics  Analysis of PDEs;
 35J60;
 35J61;
 35J62
 EPrint:
 18 pages