A singular Dirichlet problem for the Monge-Ampère type equation

We consider the singular Dirichlet problem for the Monge-Ampère type equation detD2u=b(x)g(−u)(1+|∇u|2)q/2,u<0,x∈Ω,u|∂Ω=0, where Ω is a strictly convex and bounded smooth domain in ℝn, q∈ [0, n+1), g∈ C∞(0, ∞) is positive and strictly decreasing in (0, ∞) with lims→0+g(s)=∞, and b∈ C∞(Ω) is positive in Ω. We obtain the existence, nonexistence and global asymptotic behavior of the convex solution to such a problem for more general band g. Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.