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

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $-{\mathrm{\Delta }}_{p}u=\mu$ 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)$isthe$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$q>0$,whereboth$\mu$and$\sigma$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators $\mathrm{d}\mathrm{i}\mathrm{v}(\mathcal{A}(x,\mathrm{\nabla }\cdot ))$ in place of ${\mathrm{\Delta }}_p$.