Boundary blow-up and degenerate equations

Let $Ω\subset\mathbb R^2$ be a bounded domain of class $C^{2+α}$, $0<α<1$. We show that if $u$ is the solution of $Δu = 4\exp(2u)$ which tends to $+\infty$ as $(x,y)\to\partialΩ$, then the hyperbolic radius $v=\exp(-u)$ is also of class $C^{2+α}$ up to the boundary. The proof relies on new Schauder estimates for degenerate elliptic equations of Fuchsian type.