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

In this paper, we establish a necessary and sufficient condition for the solvability of the real (n - 1) Monge-Ampere equation det(1/ n) ( Delta uI - D(2)u)) = g(x, u) in bounded domains with infinite Dirichlet boundary condition. The (n - 1) Monge-Ampere operator is derived from geometry and has recently received much attention. Our result embraces the case g(x, u) = h(x)f(u) where h is an element of C-infinity (Omega) is positive and integral 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.