Sharp conditions for the existence of boundary blow-up solutions to the Monge–Ampère equation

Abstract

In this paper we give sharp conditions on K(x) and f (u) for the existence of strictly convex solutions to the boundary blow-up Monge-Ampère problem M[u](x) = K(x) f (u) for x ∈ Ω, u(x)→+∞ as dist(x, ∂Ω) → 0. Here M[u] = det (uxi x j ) is the Monge-Ampère operator, and Ω is a smooth, bounded, strictly convex domain in RN (N ≥ 2). Further results are obtained for the special case that Ω is a ball. Our approach is largely based on the construction of suitable sub- and super-solutions.