On the Monge-Ampere equation with boundary blow-up:existence,uniqueness and asymptotics.

We consider the Monge-Ampere equation det D(2)u = b(x) f (u) > 0 in Omega, subject to the singular boundary condition u = infinity on partial derivative Omega. We assume that b epsilon C infinity(Omega) is positive in Omega and non- negative on partial derivative Omega. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Omega is a smooth strictly convex, bounded domain in R-N with N >= 2. We give asymptotic estimates of the behaviour of such solutions near partial derivative Omega and a uniqueness result when the variation of f at integral infinity regular of index q greater than N ( that is, lim(u ->infinity) f (lambda u) / f (u) = lambda(q), for every lambda > 0). Using regular variation theory, we treat both cases: b > 0 on partial derivative Omega and b equivalent to 0 on partial derivative Omega.