  Bernstein problem for affine maximal type equation has been a core problem in
affine geometry. A conjecture proposed firstly by Chern for entire graph and
then extended by Trudinger-Wang to its fully generality asserts that any
Euclidean complete, affine maximal type, locally uniofrmly convex
C^4-hypersurface in R^{N+1} must be an elliptic paraboloid. At the same time,
this conjecture was solved completely by Trudinger-Wang for dimension N=2 and
\theta=3/4, and later extended by Jia-Li to N=2, \theta\in(3/4,1] (see also
[Zhou]). On the past twenty years, much efforts were done toward higher
dimensional issues but not really successful yet, even for the case of
dimension N=3. In this paper, we will construct non-quadratic affine maximal
type hypersurfaces which are Euclidean complete for N>=3,
\theta\in(1/2,(N-1)/N).
