A Bernstein theorem for affine maximal type hypersurfaces under decaying convexity

Abstract

In this paper, we will study a fourth order fully nonlinear PDE uijDijw=0,w≡[detD2u]−θ,θ>0 of affine maximal type. By a formula of quasi-norm of third derivatives derived from pure PDE tricks, we will prove that any entire convex solution must be a quadratic function, provided θ∈(0,(12−N−12(N+8))+)⋃(12+N−12(N+8),+∞) and the minimal convex constant λ(R) decays slower than R−α(θ,N) for some positive constant α(θ,N) given explicitly below.