The Bernstein problem for affine maximal type hypersurfaces under decaying convexity

We study a fourth order partial differential equation \displaystyle u^{ij}D_{ij}w=0, \ \ w\equiv [\det D^2u]^{-\theta }, \ \ \theta \not =0 ( \ast ) of affine maximal type, which has attracted much attention in recent years. These models include the affine maximal equation for \theta =\frac {N+1}{N+2} (see, for example, [Invent. Math. 140 (2000), pp. 399-422]) and the Abreu equation for \theta =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\geq 2 and \displaystyle \theta \in \Bigg (0,\frac {1}{2}-\frac {\sqrt {N-2}}{2\sqrt {N}}\Bigg )\bigcup \Bigg (\frac {1}{2}+\frac {\sqrt {N-2}}{2\sqrt {N}},+\infty \Bigg ) under decaying convexity. Our result covers the affine maximal equation ( \theta =\frac {N+1}{N+2}) for dimension N\leq 3 or the Abreu equation ( \theta =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 \theta =\frac {N+1}{N+2} and N\leq 9 for \theta =1).