Boundary behavior in the Loewner-Nirenberg problem

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded domain of class $C^{2+\alpha }$, $0<\alpha <1$. We show that if $n\ge 3$ and $u_{\mathrm{\Omega }}$ is the maximal solution of equation $\mathrm{\Delta }u=n(n-2)u^{(n+2)\mathrm{/}(n-2)}$ in $\mathrm{\Omega }$, then the hyperbolic radius $v_{\mathrm{\Omega }}=u_{\mathrm{\Omega }}^{-2\mathrm{/}(n-2)}$ is of class $C^{2+\alpha }$ up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.