Refined second boundary behavior of the unique strictly convex solution to a singular Monge-Ampère equation

: In this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation where Ω is a bounded, smooth and strictly convex domain in R N ( N ≥ 2), b ∈ C ∞ ( Ω ) is positive and may be singular (including critical singular) or vanish on the boundary, g ∈ C 1 ((0, ∞), (0, ∞)) is decreasing on (0, ∞) with lim t →0 + g ( t ) = ∞ and g is normalized regularly varying at zero with index − γ ( γ > 1). Our results reveal the refined influence of the highest and the lowest values of the ( N − 1)-th curvature on the second boundary behavior of the unique strictly convex solution to the problem.


Introduction and main results
This presentation is to establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation det(D u) = b(x)g(−u), u < in Ω and u = on ∂Ω, (1.1) where Ω is a bounded, smooth and strictly convex domain in R N (N ≥ ), and denotes the Hessian of u and D u is the so called Monge-Ampère operator. The nonlinearity g satis es (g ) g ∈ C (( , ∞), ( , ∞)) is decreasing on ( , ∞) and lim t→ + g(t) = ∞; (g ) there exist a constant γ > and some function f ∈ C ( , a ) ∩ C[ , a ) for a su ciently small constant a > such that −tg ′ (t) g(t) := γ + f (t) with lim t→ + f (t) = , i.e., where the function f satis es (S ) f ≡ on ( , a ] (or (S ) f (t) ≠ , ∀ t ∈ ( , a] for some a ≤ a ).
The weight b satis es (b ) b ∈ C ∞ (Ω) is positive in Ω and one of the following two conditions (b ) there exist k ∈ Λ, B ∈ R and µ ∈ R + such that where c ∈ R + , y ∈ C( , t ] and lim t→ + y(t) = . The set Λ in (b ) was rst introduced by Cîrstea and Rădulescu [6]- [8] for non-decreasing functions and by Mohammed [33] for non-increasing functions to study the exact boundary behavior and uniqueness of boundary blow-up elliptic problems. When b satis es (b ) with D k > , we see by Lemma 3.1 (iii)-(iv) that b may be singular on the boundary with the index ( − D k )(N + ) D k > −(N + ).
The condition (b ) implies that b is critical singular with the index −(N + ). Problem (1.1) has a wide range of applications in Riemannian geometry and optical physics and one important geometric application is to structure a Riemannian metric in Ω that is invariant under projective transformations. When g(t) = t −(N+ ) , t > and b ≡ in Ω, Nirenberg [38], Loewner and Nirenberg [31] for N = , Cheng and Yau [5] for N ≥ studied the existence and uniqueness of solutions to problem (1.1). In particular, Cheng and Yau [5] showed that if Ω is convex and bounded but not necessarily strictly convex then problem (1.1) possesses a unique solution u ∈ C ∞ (Ω) ∩ C(Ω) which is negative in Ω. When g(t) = t −γ (t > ) with γ > and b ∈ C ∞ (Ω) with b(x) > for all x ∈ Ω, Lazer and McKenna [27] proved the existence and uniqueness of solutions to problem (1.1). Moreover, they also obtained the following global estimate When b satis es (b ) and g : ( , ∞) → ( , ∞) is a non-increasing, smooth function, Mohammed [34] showed that problem (1.1) has a strictly convex solution u ∈ C ∞ (Ω Later, Yang and Chang [47] extended the above results (i )-(i ) to the following cases: (i ) if b(x) ≤ C(d(x)) −(N+ ) (− ln d(x)) −q near ∂Ω for some q > N and C > , then problem (1.2) has a strictly convex solution; (i ) if b(x) ≥ C(d(x)) −(N+ ) (− ln d(x)) −N near ∂Ω for some C > , then problem (1.2) has no strictly convex solution. Let P ∈ C ( , ∞) satisfy P ′ (t) < and lim t→ + P(t) = ∞ and de ne P(t) = t P(s)ds. Recently, under the hypothesis of (b ), Zhang and Du [49] obtain the following results: (i ) if b(x) ≤ P(d(x)) near ∂Ω and (P(s)) /N ds < ∞, then problem (1.2) has a strictly convex solution; (i ) if b(x) ≥ P(d(x)) and (P(s)) /N ds = ∞, then problem (1.2) has no strictly convex solution. The above facts imply that problem (1.2) has a strictly convex solution if b satis es (b ) and In [28], Li and Ma studied the existence and the rst boundary behavior of the strictly convex solutions to problem (1.1) by using regularity theory and sub-supersolution method. In particular, when b ∈ C (Ω) is positive in Ω and satis es (b ) there exist k ∈ Λ and positive constants b and b such that g satis es (g ) and they showed that the unique strictly convex solution u to problem (1.1) satis es where ψ is uniquely determined by denotes the (N − )-th curvature atx and κ (x), · · ·, κ N− (x) denote the principal curvatures of ∂Ω atx. In [51], Zhang showed that if b satis es (b ) and (b ), g satis es (g ) and (g ) and ND k + ( + N)Cg > + N, then the unique strictly convex solution u to problem (1.1) satis es where ϕg is uniquely determined by Especially, if (b ) is replaced by the following condition (b ) there existL ∈ L with (1.3) and positive constants b and b such that Zhang [51] showed that the unique strictly convex solution u to problem (1.1) satis es Then, Sun and Feng [43] and Li and Ma [29] generalized the above boundary behavior results to the case of the following Hessian equation for i = , · · ·, N and λ , · · ·, λ N are the eigenvalues of D u. Furthermore S (λ) ≡ for λ ∈ R N . Espically, Li and Ma [29] also studied the existence and uniqueness of viscosity solution to the problem. For related insights on the existence, regularity and asymptotic behavior of solutions to the Monge-Ampère equations, please refer to [4], [11], [19], [21]- [25], [30], [35]- [37], [44]- [45] and the references therein. When the Monge-Ampère operator (det(D u)) is replaced by the Laplace operator (∆), many papers have been dedicated to resolving existence, uniqueness and asymptotic behavior issues for solutions, please refer to [1]- [2], [10], [12]- [17], [26], [39], [46], [48], [50] and the references therein. In this paper, by making a complete and detailed analysis to some indexes in various cases, we establish the exact second boundary behavior of the unique strictly convex solution to problem (1.1), which is quite di erent from the rst behavior of this solution. For all we know, in literature there aren't articles on the second boundary behavior of the strictly convex solution to problem (1.1).
To our aims, we de ne the following subclasses of Λ and L as follows: where β is a positive constant and the relation betweenL and y is given in (b ). Our results are summarized as follows andm± (given in Theorems 1.1-1.3) are de ned by (1.5).
Corollary 1.1. In Theorem 1.1, if Ω is a ball with radius R and center x , then (I) When (S ) holds (or (S ) and (g )-(g ) hold with θ = in (g )), we have (i) If k ∈ Λ , then the unique strictly convex solution u to problem (1.1) satis es and C is given by (1.9).
, we further suppose that (g ) with θ > and the following hold If k ∈ Λ , then the unique strictly convex solution u to problem (1.1) satis es where ψ is uniquely determined by (1.4), ξ± are given by (1.7) and where ξ R is given by (1.13), r = |x − x | and where ψ is uniquely determined by (1.4), if X+ ≥ and X− ≥ ,   Since lim t→ + (− ln t) β y(t) = E , for any ε > we can choose a small enough constantt < min{t , } such that A simple calculation shows that Moreover, we have It follows by the de nition ofL in (b ) that there exists a positive constantC such that (1.20) forL ∈ L β with (1.17), then the conclusion of Theorem 1.3 still holds.
Remark 1.5.L ∈ L is normalized slowly varying at zero and lim t→ + tL ′ (t) The rest of the paper is organized as follows. In Section 2, we give some bases of Karamata regular variation theory. In Section 3, we show some auxiliary lemmas. The proofs of Theorems 1.1-1.3 are given in Section 4.
De nition 2.1. A positive continuous function g de ned on ( , a ], for some a > , is called regularly varying at zero with index p, denoted by g ∈ RVZp, if for each ξ > and some p ∈ R, In particular, when p = , g is called slowly varying at zero. Clearly, if g ∈ RVZp, then L(t) := g(t)/t p is slowly varying at zero.

Proposition 2.3. (Representation Theorem). A function L is slowly varying at zero if and only if it may be written in the form
where the functions l and y are continuous and for t → + , y(t) → and l(t) → c with c > .

De nition 2.4. The function
is called normalized slowly varying at zero and is called normalized regularly varying at zero with index ρ (written as f ∈ NRVZp).
A function h ∈ C ( , a ] for some a > belongs to NRVZp if and only if Proposition 2.5. Let functions L, L be slowly varying at zero, then (i) L p , p ∈ R, L · L and L • L satisfying lim t→ + L(t) = , are also slowly varying at zero; (iii) for any p ∈ R, ln L(t)/ ln t → and ln(t p L(t))/ ln t = p as t → + .

Auxiliary results
In this section, we show some useful results, which are necessary in the proofs of our results.
there exists a small enough positive constant t * ∈ ( , ) such that A straightforward calculation shows that for any t ∈ ( , t * ], So, we see that there exists a large constant C > such that for any t ∈ ( , t * ], (3.1) This together with (3.1) implies that we can choose a positive integer n * > ( − β) − such that Combining Cases 1-2, we obtain (i) holds.
Proof. (i) It follows by (g ) that (i) holds.
(ii)-(iii) By (g ) we see that It follows by Proposition 2.8 (i) that (ii) holds. Furthermore, we have It follows by Proposition 2.8 (ii) that (iii) holds.
Integrating it from t to a and integration by parts, we obtain i.e., where c is a constant. The condition (g ) implies that g ∈ NRVZ−γ with γ > . Moreover, by t → (− ln t) β ∈ NRVZ and (3.9), we see that We conclude by Proposition 2.7 and Proposition 2.6 that (3.12) Combining with (3.6)-(3.10), by the l'Hospital's rule, we can obtain (iii) By Lemma 3.3 (iv), Proposition 2.7 and Proposition 2.6, we have This implies that (υ(t)) − φ(t) → as t → + .

The Second Boundary Behavior
In this section, we prove Theorems 1.1-1.3. We rst introduce some lemmas as follows.

Suppose h(x, t) is de ned for x ∈ Ω and t in some interval containing the ranges of u and v. If the following hold: (i) h is strictly increasing in t for all x ∈ Ω, (ii) the matrix D v is positive de nite in Ω, (iii) det(D v) ≥ h(x, v) and det(D u) ≤ h(x, u), x ∈ Ω, (iv) u ≥ v on ∂Ω, then, we have u ≥ v in Ω.
For any δ > , let Since Ω is C m -smooth for m ≥ , we can always take δ > such that (see Lemmas 14.16 and 14.17 in [18]) Letx ∈ ∂Ω be the projection of the point x ∈ Ω δ to ∂Ω, and κ i (x) (i = , · · ·, N − ) be the principal curvatures of ∂Ω atx, then , .
Fix ε > and let where ξ± and C± are in Theorem 1.1. By the Lagrange's mean value theorem, we obtain that there exist λ± ∈ ( , ) and Since g ∈ NRVZ−γ, we have by Proposition 2.2 that lim d(x)→ g(ξ±ψ(K(d(x)))) g(Θ±(d(x))) = lim d(x)→ g ′ (ξ±ψ K(d(x)))) g ′ (Θ±(d(x))) = . (4.2) Moreover, by the de nitions of ξ± and C±, we can take a su ciently small positive constant still denoted by δ such that Proof. Our proof is divided into two steps and the outline of the proof is given as below.
• In Step 1, for xed ε > and x ∈ Ω δ , we rst give some functions I ± (d(x)), I ± (d(x)) and I ± (d(x)) (which are corresponding to ε > ), and then by detailed calculation we will show that there exists a su ciently small positive constant δε < δ such that and I − (d(x)) + I − (d(x)) + I − (d(x)) < , x ∈ Ω δε . (4.5) • In Step 2, we will de ne two functions u ε , uε in Ω δε and show they are sub-and super-solutions of Eq.
(1.1) in Ω δε by (4.4) and (4.5), respectively. In particular, we will show u ε is strictly convex in Ω δε . Finally, we will establish the second boundary behavior of the unique strictly convex solution to problem (1.1) by using Lemma 4.1.
By the above analysis, we see that how to get (4.4)-(4.5) is the key of the research. Step 1.We rst de ne functions I ± (d(x)), I ± (d(x)) and I ± (d(x)) as follows.
We obtain by De nition 4.4 that D u ε is positive de nite in Ω δε . Let By the same calculation as (4.13), we obtain i.e., uε is a supersolution of Eq. (1.1) in Ω δε . Let u be the unique strictly convex solution to problem (1.1). Then, there exists a large constant M > such that We assert that and Since D u ε is positive de nite in Ω δε and D (−Md(x)) is positive semide nite in Ω δε , we have by the Minkowski inequality that D (u ε (x) − Md(x)) is positive de nite in Ω δε and Similarly, we have D (u(x) − Md(x)) is positive de nite in Ω δε and By Lemma 4.1, we see that (4.14)-(4.15) hold. Hence, for any x ∈ Ω δε Since (γ + N)D k −(N + ) > , we conclude from Lemma 3.1 (ii), Lemma 3.6 (iii), Proposition 2.7 and Proposition 2.5 (ii) that Letting ε → , the proof is nished.
By Proposition 2.2, we see that (4.2) still holds. Moreover, we can adjust δ > such that (4.3) holds here.
Proof. Similar to the proof of Theorem 1.1, the proof is divided into the following two steps.

. Proof of Theorem 1.3
Now, we prove Theorem 1.3. As before, for xed ε > , we de ne where η±, C * ± and M are given in Theorem 1.3. By the Lagrange's mean value theorem, it is clear that there exist λ± ∈ ( , ) and such that for x ∈ Ω δ g(w±(d(x))) = g η±ψ We obtain by Proposition 2.2 that (4.2) still holds. Moreover, we can adjust δ such that (4.3) holds here.
Proof. The proof is still divided into the following two steps.