A SINGULAR DIRICHLET PROBLEM FOR THE MONGE-AMPÈRE TYPE EQUATION

We consider the singular Dirichlet problem for the Monge-Amp & egrave;re type equation det D(2)u = b (x) g(-u)(1+|del u|(2))(q/2), u < 0, x is an element of Omega, u|(partial derivative Omega )= 0, where Omega is a strictly convex and bounded smooth domain in & Ropf;(n), q is an element of [0, n +1), g is an element of C-infinity (0, infinity) is positive and strictly decreasing in (0, infinity) with lim(s -> 0+ )g(s) = infinity, and b is an element of C-infinity (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.