Boundary behavior in the Loewner-Nirenberg problem

Let Ω⊂Rn be a bounded domain of class C2+α, 0<α<1. We show that if n⩾3 and uΩ is the maximal solution of equation Δu=n(n-2)u(n+2)/(n-2) in Ω, then the hyperbolic radius vΩ=uΩ-2/(n-2) is of class C2+α up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.