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

摘要: We consider the Monge–Ampere equation det D2u = b(x)f(u) > 0 in Ω, subject to singular boundary condition u ∞ on ∂Ω. assume that \(b\in C^\infty(\overline{\Omega})\) is positive Ω and non-negative Under suitable conditions f, we establish existence of strictly convex solutions if a smooth convex, bounded domain \({\mathbb R}^N\) with N ≥ 2. give asymptotic estimates behaviour such near ∂Ω uniqueness result when variation f at regular index q greater than (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ 0). Using theory, treat both cases: b \(b\equiv 0\)