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

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 $\Delta_{p}$.