Boundary blow-up solutions for the Monge-Ampère equation with an invariant gradient type term

We prove existence and non-existence of the boundary blow-up solutions for the Monge-Amp & egrave;re equation of the form det D 2 u + g ( u ) 0 (del u, D 2 u ) = b ( x ) f ( u ) in a smooth, bounded, strictly convex domain S2 c R n ( n >= 2) , where b is an element of C infinity ( S2 ) is positive in S2 , the pair of functions f and g satisfies conditions of Keller-Osserman type on [0 , infinity) , and 0 R n x R n 2 -+ R is a quadratic term which is invariant for both the gradient del u and the Hessian D 2 u in the sense discussed by Kazdan and Kramer (1978).