Boundary blow-up and degenerate equations

Let Ω⊂R2 be a bounded domain of class C2+α, 0<α<1. We show that if u is the solution of Δu=4 exp(2u) which tends to +∞ as (x,y)→∂Ω, then the hyperbolic radius v=exp(−u) is also of class C2+α up to the boundary. The proof relies on new Schauder estimates for degenerate elliptic equations of Fuchsian type.