Centroaffine Bernstein problems

C.P. Wang [Geom. Dedicata 51 (1994) 63–74] studied the Euler–Lagrange equation for the centroaffine area functional of hypersurfaces. In terms of a local representation of the hypersurface as a graph, this equation is a complicated, strongly non-linear fourth order PDE. We consider classes of solutions satisfying these equations together with completeness conditions. We also formulate appropriate centroaffine Bernstein problems and give partial solutions.