Entire solutions to the Monge–Ampère equation

We consider the Monge–Ampère equation det ( D 2 u ) = Ψ ( x , u , D u ) in R n , n ⩾ 3 , where Ψ is a positive function in C 2 ( R n × R × R n ) . We prove the existence of convex solutions, provided there exist a subsolution of the form u ̲ = a | x | 2 and a superharmonic bounded positive function φ satisfying: Ψ > ( 2 a + Δ φ n ) n .