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

Abstract

We consider the Monge–Ampere equation det D2u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) 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 \({\mathbb R}^N\) 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, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ∂Ω and \(b\equiv 0\) on ∂Ω.