On the boundary blow-up problem for real (n−1) Monge–Ampère equation

Abstract

In this paper, we establish a necessary and sufficient condition for the solvability of the real $(n-1)$ Monge–Ampère equation $\stackrel{1/n}{det}(\Delta uI-D^{2}u)=g(x,u)$ in bounded domains with infinite Dirichlet boundary condition. The $(n-1)$ Monge–Ampère operator is derived from geometry and has recently received much attention. Our result embraces the case $g(x,u)=h\left(x\right)f\left(u\right)$ where $h\in C^{\infty }\left(\bar{\Omega }\right)$ is positive and $f$ satisfies the Keller–Osserman type condition. We describe the asymptotic behavior of the solution by constructing suitable sub-solutions and super-solutions, and obtain a uniqueness result in star-shaped domains by using a scaling technique.