A singular Dirichlet problem for the Monge-Ampère type equation

Abstract

We consider the singular Dirichlet problem for the Monge-Ampère type equation \({\rm det}\ D^2 u=b(x)g(-u)(1+|\nabla u|^2)^{q/2}, \ u<0, \ x \in \Omega, \ u|_{\partial \Omega}=0\), 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 \(\lim\limits_{s\rightarrow 0^+}g(s)=\infty\), 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 b and g. Our approach is based on the Karamata regular variation theory and the construction of suitable sub-and super-solutions.