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We give necessary and sufficient conditions for the existence of a BMOsolution 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 moregeneral quasilinear operators ${\rm div}(\mathcal{A}(x, \nabla \cdot))$ inplace of $\Delta_{p}$.

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