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Existence and regularity estimates for quasilinear equations with measure data: the case $1\lt p\leq (3n-2)/(2n-1)$

### Quoc-Hung Nguyen and Nguyen Cong Phuc

Vol. 15 (2022), No. 8, 1879–1895
##### 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 $-\mathrm{div}\left(|\nabla u{|}^{p-2}\nabla u\right)=\delta |\nabla u{|}^{q}+\mu$ 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 $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.

##### Keywords
quasilinear equation, Riccati-type equation, measure data, good-$\lambda$ inequality, Muckenhoupt–Wheeden-type inequality, weighted norm inequality, capacity
##### Mathematical Subject Classification 2010
Primary: 35J60, 35J61, 35J62
Secondary: 35J75, 42B37
##### Milestones
Received: 28 November 2019
Revised: 15 January 2021
Accepted: 25 March 2021
Published: 10 February 2023
##### Authors
 Quoc-Hung Nguyen ShanghaiTech University Shanghai China Nguyen Cong Phuc Department of Mathematics Louisiana State University Baton Rouge, LA United States