OPTIMAL GLOBAL ASYMPTOTIC BEHAVIOR OF THE SOLUTION TO A SINGULAR MONGE-AMP`ERE EQUATION

. This paper is mainly concerned with the optimal global asymptotic behavior of the unique convex solution to a singular Dirichlet problem for the Monge-Amp`ere equation det D 2 u = b ( x ) g ( − u ) , u < 0 , x ∈ Ω , u | ∂ Ω = 0 , where Ω is a strict convex and bounded smooth domain in R n with n ≥ 2, g ∈ C 1 ((0 , ∞ )) is positive and decreasing in (0 , ∞ ) with lim s → 0 + g ( s ) = ∞ , b ∈ C ∞ (Ω) is positive in Ω, but may vanish or blow up on the boundary properly. Our approach is based on the construction of suitable sub- and super-solutions.

Some basic results are listed as follows.
In this paper, we show optimal global asymptotic behavior of the unique convex solution to problem (3) under the following local structure conditions (g 2 ): there exists C g ≥ 0 such that We also give some results with regard to nonexistence and regularity of such solution.
For convenience, let v = −φ and assume max where φ is given as in the beginning.
Our main results are summarized as follows.
where θ = n+1+σ n+γ , m 1 and M 1 satisfy m n+γ with and uniformly for x ∈ Ω 1 which is an arbitrary compact subset of Ω. Where In particular, when Ω = B R , which is a ball of radius R centered at the origin, is the unique convex solution to problem (3).
For the more general regular Dirichlet problem det Tso [23] first established the functional and showed that any critical point of J n is a convex solution of problem (13) in a suitable convex space.
When σ = γ − 1 in Theorem 1.1, we have the following result.
then problem (3) has a unique classical convex solution u η which satisfies where ξ 1 and ξ 2 are positive constants with ξ 1 ≤ ξ 2 , and For more general g, we have the following results.
When b is in a borderline case near the boundary ∂Ω, we have the following result.
The outline of this paper is as follows. In Section 2, we give some preliminaries. The proofs of Theorems 1.1-1.4 are provided in Section 3.
2. Some preliminaries. In this section, we present some basics of Karamata regular variation theory in order to show the complete characterization of g in (g 1 )-(g 3 ) and the exact behavior near zero of ψ in (5).
Incidentally, Cîrstea and Trombetti [6] first introduced the theory to study boundary behavior of the blow-up boundary solutions of the Monge-Ampère equations.
In particular, when ρ = 0, g is called slowly varying at zero.
where the functions l and y are continuous and for s → 0 + , y(s) → 0 and l(s) → c 0 , with c 0 > 0.
. We call that is normalized slowly varying at zero, and is normalized regularly varying at zero with index ρ (and denoted by g ∈ N RV Z ρ ).
Similarly, we have the following results.
(i 1 ): If g satisfies (g 3 ), then E g ≤ 1; (i 2 ): (g 3 ) holds with E g ∈ (0, 1) if and only if g is normalized regularly varying at infinity with index −γn with γ > 0. In this case γ = E g /(1 − E g ); (i 3 ): (g 3 ) holds with E g = 0 if and only if g is normalized slowly varying at infinity; (i 4 ): if (g 3 ) holds with E g = 1, then g grows faster than any s −p (p > 1) at infinity; then g satisfies (g 3 ) with E g = 1.
For completeness, we give its proof.
Proof. (i 1 ) It follows from (g 1 ) that lim s→∞ Proof. We give a different proof here. Let w(x) = f (v(x)), we have and (i 1 ) It follows that where I is the identity matrix.
Firstly, we see from Remark 1.2 that one of the assumptions (b 1 ), (b 2 ) and (b 3 ) implies that (B 2 ) holds. Moreover, (B 2 ) and (g 1 ) implies that problem (3) has a unique convex solution u ∈ C ∞ (Ω) ∩ C(Ω) (Lemmas 2.6 and 2.7). Thus, our main purpose in this section is to show global asymptotic behavior of such solution to problem (3) in the following.
(i 3 ) It follows from (i 2 ) and Lemma 2.9 that we see that (i 3 ) holds.