Existence and regularity estimates for quasilinear equations with measure data : the case 1 < p ≤ (3n − 2)∕(2n − 1)

Nguyen, Quoc-Hung; Nguyen Cong, Phuc

doi:10.2140/apde.2022.15.1879

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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

DOI: 10.2140/apde.2022.15.1879

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 $- div (| \nabla u |^{p - 2} \nabla u) = δ | \nabla u |^{q} + μ$ in a bounded domain $Ω \subset ℝ^{n}$ potentially with nonsmooth boundary. Here either $δ = 0$ or $δ = 1$ , $μ$ is a finite signed Radon measure in $Ω$ , and $q \geq 1$ . Our main concern is to extend earlier results to the strongly singular case $1 < p \leq (3 n - 2) ∕ (2 n - 1)$ . 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

Vol. 15, No. 8, 2022