Entire solutions to the Monge–Ampère equation

We consider the Monge–Ampère equation det(D2u)=Ψ(x,u,Du) in Rn, n⩾3, where Ψ is a positive function in C2(Rn×R×Rn). We prove the existence of convex solutions, provided there exist a subsolution of the form and a superharmonic bounded positive function φ satisfying: .