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 $\mathrm{div}(\mathcal{A}(x,\nabla \cdot ))$ in place of ${\Delta }_p$.

To appear in Annales de l'Institut Fourier