Classical solutions of singular Monge–Ampère equations in a ball

Our concern is on existence, uniqueness and regularity of convex, negative, radially symmetric classical solutions to det D2u = ψ(x,−u) in B, u =0 on∂B, where (D2u) is the Hessian of u, B ⊂ RN, N 1, is the unit ball with boundary ∂B, ψ :B × (0,∞)→[0,∞) is continuous and ψ(x, t) = ψ(|x|, t), where |x| is the euclidean norm of x. The main interest is in the case ψ is singular at |x| = 1 and/or u = 0, although several nonsingular cases are covered by the main result. Our approach to show existence, exploits fixed point arguments and the shooting method. Uniqueness and regularity are achieved through suitable estimates.  2004 Elsevier Inc. All rights reserved.