Boundary blow-up and degenerate equations

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