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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 div(|u|p2u) = δ |u|q + μ in a bounded domain Ω n potentially with nonsmooth boundary. Here either δ = 0 or δ = 1, μ is a finite signed Radon measure in Ω, and q 1. Our main concern is to extend earlier results to the strongly singular case 1 < p (3n 2)(2n 1). In particular, in the case δ = 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