关键词: Monge-Ampère equation, a singular boundary value problem, the unique convex solution, global asymptotic behavior

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 $\Omega$ is a strictly convex and bounded smooth domain in $\mathbb R^n$, $q\in [0, n+1)$, $g\in C^\infty(0,\infty)$ is positive and strictly decreasing in $(0, \infty)$ with $\lim\limits_{s \rightarrow 0^+}g(s)=\infty$, and $b \in C^{\infty}(\Omega)$ is positive in $\Omega$. 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.

Key words: Monge-Ampère equation, a singular boundary value problem, the unique convex solution, global asymptotic behavior