A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Ampere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain Ω, we prove the existence for solutions to the problem in the space C∞(Ω) ∩ C(Ω). We also obtain the local C1/2-estimate up to the ∂Ω and the estimate for the lower bound of the solutions.