Regularity for $H$-systems of bounded distortion

We consider weak solutions $u \in W^{1,n}(U,\mathbf{R}^{n+1})$ of the $H$-system $- \operatorname{div} ( |\mathrm{D} u|^{n-2} \mathrm{D} u ) = (H \circ u) \, \mathbf{J} u$, where $n \ge 2$, $U$ is the open unit ball in $\mathbf{R}^n$, and $H$ is bounded. We show that $u$ is locally bounded if its di