Existence and estimates of solutions to a singular Dirichlet problem for the Monge–Ampère equation

Given a strictly convex, smooth, and bounded domain Ω in Rn we establish the existence of a negative convex solution in C∞(Ω) ∩ C(Ω) with zero boundary value to the singular Monge–Ampère equation det(D2u) = p(x)g(−u). An associated Dirichlet problem will be employed to provide a necessary and sufficient condition for the solvability of the singular boundary value problem. Estimates of solutions will also be given and regularity of solutions will be deduced from the estimates. © 2007 Elsevier Inc. All rights reserved.