The Bernstein problem for affine maximal type hypersurfaces under decaying convexity.

We study a fourth order partial differential equation (*) u^ijD_ijw = 0, w ≡ [det D²u]^-θ, θ ≠ 0 of affine maximal type, which has attracted much attention in recent years. These models include the affine maximal equation for θ = N+1/N+2 (see, for example, [Invent. Math. 140 (2000), pp. 399-422]) and the Abreu equation for θ = 1 (see, for example, [Int. J. Math. 9 (1998), pp. 641-651] or [Calc. Var. Partial Differential Equations 43 (2012), pp. 25-44]. In this paper, we will prove a Bernstein theorem of (*) for all dimension N ≥ 2 and θ ∈ (0, 1/2 − √N−2/2√N) ∪ (1/2 + √N−2/2√N,+ ∞) under decaying convexity. Our result covers the affine maximal equation (θ = N+1/N+2) for dimension N ≤ 3 or the Abreu equation (θ = 1) for all dimension N\geq 1, which largely improves both cases by Du in [Nonlinear Anal. 187 (2019), pp. 170-179] (N=2 for θ = N+1/N+2 and N ≤ 9 for θ = 1). [ABSTRACT FROM AUTHOR]