BMO solutions to quasilinear equations of $p$-Laplace type

TL;DR

This paper establishes conditions for the existence and characterization of BMO solutions to certain quasilinear $p$-Laplace equations involving Radon measures, extending to more general operators.

Contribution

It provides necessary and sufficient conditions for BMO solutions to quasilinear equations with Radon measure data, generalizing previous results to broader operators.

Findings

Characterization of BMO solutions for $- ext{div}( extbf{A}(x, abla u))= ext{measure}$

Conditions for solutions to equations with measure data and nonlinear terms

Extension to more general quasilinear operators

Abstract

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $-\Delta_{p} u = \mu$ in $\mathbb{R}^n$, $u\ge 0$, where $\mu$ is a locally finite Radon measure, and $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($p>1$). We also characterize BMO solutions to equations $-\Delta_{p} u = \sigma u^{q} + \mu$ in $\mathbb{R}^n$, $u\ge 0$, with $q>0$, where both $\mu$ and $\sigma$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators ${\rm div}(\mathcal{A}(x, \nabla \cdot))$ in place of $\Delta_{p}$.