Boundary blow-up solutions for the Monge-Ampère equation with an invariant gradient type term

Abstract

We prove existence and non-existence of the boundary blow-up solutions for the Monge–Ampère equation of the form detD2u+g(u)ϕ(∇u,D2u)=b(x)f(u) in a smooth, bounded, strictly convex domain Ω⊂Rn(n≥2), where b∈C∞(Ω) is positive in Ω, the pair of functions f and g satisfies conditions of Keller–Osserman type on [0,∞), and ϕ:Rn×Rn2→R is a quadratic term which is invariant for both the gradient ∇u and the Hessian D2u in the sense discussed by Kazdan and Kramer (1978).