Boundary behavior in the Loewner–Nirenberg problem

Let ⊂ Rn be a bounded domain of class C2+ , 0< <1. We show that if n 3 and u is the maximal solution of equation u = n(n − 2)u(n+2)/(n−2) in , then the hyperbolic radius v =u−2/(n−2) is of class C2+ up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE. © 2004 Elsevier Inc. All rights reserved.