On the Monge-Ampère equation with boundary blow-up: Existence, uniqueness and asymptotics

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