Interior estimates for solutions of a fourth order nonlinear partial differential equation

Let x:M→An+1 be a locally strongly convex hypersurface, given by the graph of a convex function xn+1=f(x1,…,xn)xn+1=f(x1,…,xn) defined in a convex domain Ω⊂RnΩ⊂Rn. M is called a α-extremal hypersurface, if f is a solution of∑Uij∂2∂xi∂xj((det(∂2f∂xi∂xj))−n+1−αn+2)=0,α≠n+1. The purpose of this paper is to establish an interior estimates for solutions of fourth order nonlinear PDE Δρ=α‖∇ρ‖G2ρ and prove that Euclidean complete α  -extremal hypersurface must be an elliptic paraboloid for |α|⩾K(n)|α|⩾K(n), where K(n)K(n) is a positive constant depending only on the dimension n.