Boundary blow-up solutions to the Monge-Ampère equation : Sharp conditions and asymptotic behavior

HereM[u] = det (uxixj ) is the Monge-Ampère operator, and Ω is a smooth, bounded, strictly convex domain in RN (N ≥ 2). Under K(x) satisfying appropriate conditions, we rst prove that the boundary blow-up MongeAmpère problem has a strictly convex solution if and only if f satis es Keller-Osserman type condition. Then the asymptotic behavior of strictly convex solutions to the boundary blow-up Monge-Ampère problem is considered under weaker condition with respect to previous references. Finally, if f does not satisfy KellerOsserman type condition, we show the existence of strictly convex solutions under di erent conditions on K(x). The proof combines standard techniques based upon the sub-supersolutionmethodwith non-standard arguments, such as the Karamata regular variation theory.

Mohammed [33] proved that if K(x) satis es (K) and is such that the Dirichlet problem has a strictly convex solution, then (1.1) has a strictly convex solution if f satis es (f ) and the  and [38]) type condition Here In [34], the authors showed that, in the case that η > −∞, (1.3) alone does not guarantee the existence of a strictly convex solution to (1.1). One needs additionally Obviously, (1.4) is equivalent to lim r→η + Ψ(r) = ∞. From Theorem 1.1 and Theorem 1.2 of [34] we know, if K(x) satis es (K) and K ∈ L ∞ (Ω), the Keller-Osserman type condition is necessary and su cient (combing with (1.4) if η ∈ R ) for the existence of strictly convex solution, but if K(x) satis es (K) and is such that (1.2) has a strictly convex solution, it is only su cient. We would like to prove the necessity in this paper.
So the rst main result of this paper is the following. At the same time, in [34], the authors did not consider the boundary asymptotic behavior of the strictly convex solution. The study of boundary asymptotic behavior of blow-up solutions is also a hot topic, see [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], and the references therein. Recently, in [35], Zhang studied the boundary behavior of the strictly convex solution to (1.1) with K(x) ∈ C(Ω). Very recently, in [36], Zhang studied the boundary behavior of the strictly convex solution to (1.1) with f (u) including gradient terms and K(x) is in general case or borderline case.
In [35] and [36], there is an important condition on f , i.e. We can see C f has the same meaning of I∞. By Theorem 1.1 the existence of strictly convex solution to problem (1.2) is the key point for the existence of strictly convex solution to problem (1.1). If K(x) is bounded on Ω, according to Theorem 1 of [52], (1.2) always has a strictly convex solution. If K(x) is unbounded near ∂Ω, problem (1.2) is not always having solution. The existence of solution depends on the increasing speed of K(x) when x approaches ∂Ω. In [34], the authors gave a su cient condition for the existence of strictly convex solutions. For ease of composition, we rst introduce some notations.
For a positive function p(t) in C ( , ∞) satisfying p (t) < and lim t→ + p(t) = +∞, to distinguish its behavior near t = we set P(τ) = τ p(t)dt. We say such a function p(t) is of class P nite if In Theorem 1.5 of [34], the author proved that, if K(x) satis es (K), then (1.2) has no strictly convex solution if there exists a function p(t) of class P∞ such that K(x) ≥ p(d(x)) near ∂Ω, and has a strictly convex solution if there exists a function p(t) of class P nite such that K(x) ≤ p(d(x)) near ∂Ω.
If p(t) is of class P nite , we may modify p(t) for large t and assume that p(t) = c e −t for some positive constant c and all large t, say t ≥ M . With p(t) modi ed as above, if we de nẽ Set The second main result of this paper is the following. Here For more articles about boundary blow-up solutions in a ball, please see [53][54][55][56][57].

Remark 1.4. We can determine that the condition imposed on b(x) in [36] is equivalent to
We can see that (1.6) is a weaker condition than (1.12).
Then we have Meanwhile, by Theorem 1.5 of [34] we know (1.6) is sharper than (1.12) for the existence of strictly convex solutions of (1.2) and the existence of strictly convex solutions of (1.1).
If K(x) is such that (1.2) has no strictly convex solution, then (1.1) may have or have no strictly convex solution, depending on the behavior of f . In [34], the authors only examined some such cases for the radially symmetric situation. In this paper, we'll consider the general case. But we have to impose some su cient condition such that (1.2) has no strictly convex solution. It is (1.14) We have The rest of the paper is organized in the following way. In Section 2 we will collect some known results to be used in the subsequent sections. Section 3 is devoted to the proofs of the Theorem 1.1 and Theorem 1.2. In Section 4 we prove that Theorem 1.5 holds. In Appendix we will introduce the theory of regular variation for the proof of Corollary 1.3.

Some preliminary results
In this section, we collect some results for the convenience of later use and reference.

Lemma 2.1. (Lemma 2.1 of [23])
Let Ω be a bounded domain in R N , N ≥ , and let u k ∈ C (Ω) ∩ C(Ω) for k = , . Let f (x, u) be de ned for x ∈ Ω and u in some interval containing the ranges of u and u and assume

Remark 2.2.
From the proof in [23], it is easily seen that the condition "f (x, u) is strictly increasing in u for all x ∈ Ω" in Lemma 2.1 can be relaxed to "f (x, u) is nondecreasing in u for all x ∈ Ω" provided that one of the inequalities in (ii) and (iii) is replaced by a strict inequality. This observation will be used later in the paper.

Lemma 2.3. (Proposition 2.1 of [24])
Let u ∈ C (Ω) be such that the matrix (ux i x j ) is invertible for x ∈ Ω, and let g be a C function de ned on an interval containing the range of u. Then

1)
where A T denotes the transpose of the matrix A, B(u) denotes the inverse of the matrix (ux i x j ), and ∇u = (ux , ux , · · · , ux N ) T . where are the principal curvatures of ∂Ω atx.
The following interior estimate for derivatives of smooth solutions of Monge-Ampère equations is a simple variant of Lemma 2.2 in [23], which follows from [58,59].
The existence result below is a variant of Lemma 2.3 in [23], which is a special case of Theorem 7.1 in [52]. Since (zx i x j ) is negative de nite on Ω, its trace is negative, that is ∆z < , and hence one can apply the Hopf boundary lemma to conclude that |∇z| > for x ∈ ∂Ω. It follows that there exist positive constants b and b

Proof of Theorem 1.1 and Theorem 1.2
Proof of Theorem 1.1. Su ciency. It was proved in [34]. Necessity. Assume to the contrary that (1.1) has a strictly convex solution u. We aim to derive a contradiction.
Denote by g(t) the inverse of G(t), i.e., where G(t) is de ned by (1.13). Then Since Ω is bounded in R N , there exists R such that Ω ⊂ B( , R ). Then de ne where K is a positive constant to be determined. Then we obtain, for x ∈ Ω, where B(w) is the inverse matrix of (wx i x j ), λ is the minimal eigenvalue of B(w). Since w is strictly convex, all the eigenvalue of B(w) is positive. We thus obtain provided that c is chosen large enough. Fix x ∈ Ω and by further enlarging c if necessary we may assume that

Since u(x) → ∞ as d(x) → , while v(x) is continuous on Ω, there exists an open connected set D such that
On the other hand, since M[u] = K(x)f (u) in D and v = u on ∂D, and the matrix (vx i x j ) is positive de nite on D (since w (x), y(x) are strictly convex in Ω and g , g > ), we can apply Lemma 3.1 to conclude that v(x) ≤ u(x) in D. This contradiction completes our proof.

Proof of Theorem 1.2.
For small δ > , let For an arbitrary ε ∈ ( , min{ / , k }), let where m , M , k , k are given in Theorem 1.2, I∞, J are given (1.5) and (1.9). From the de nition of J , I∞, ξ ε , ξ ε we see that where δε ∈ ( , min{ , δ / }) is su ciently small such that for x ∈ Ω δε i.e.ūε is a supersolution to (1.1) in D − σ . Similarly, we can prove u ε is a subsolution to (1.1) in D + σ . By (1.10) and Theorem F, (1.2) has a strictly convex solution. It follows from Theorem 1.1 that (1.1) has a strictly convex solution u. Let A be large enough such that By the de nition ofūε and u ε , we know thatūε(x) → ∞ as d(x) → σ and u ε | ∂Ω < u| ∂Ω . By Lemma 2.1 we have Then Then, for x ∈ D − σ ∩ D + σ , the two formulas above hold. Letting σ → , then we obtain and ≤ lim inf .
Proof of Corollary 1.3.
Combing this with Theorem 1.2 we obtain

Proof of Theorem 1.5
For the proof of Theorem 1.5, we rst introduce a lemma which is about radial solutions. Let In the radially symmetric setting, the smoothness requirements for K and f can be greatly relaxed. But for convenience, we still use (K), (K ) and (f ). In the case (K ) can be state as: there exist constants d , d > and a function p(t) of class P∞ such that We modify p(t) as in Section 1 and de ne σ(t) by Proof of Theorem 1.5.
Since We have Then By the de nition of z we have (zx i x j ) is negative de nite. It follows that there exist e , e > such that Combining this with the fact that R∞ ≠ ∞, we can conclude that ∆ is positive on Ω. By (4.6) we have for large c , i.e. w (x) = g(c σ N N+ ( b z(x))) is a subsolution of (1.1).
Let {σn} ∞ be a strictly increasing sequence of positive numbers such that σn → ∞ as n → ∞, and let Ωn = {x ∈ Ω|w (x) < σn}. Since any level surface of w is a level surface of z, for each n ≥ , ∂Ωn is a strictly convex C ∞ -submanifold of R N of dimension N − . By Lemma 2.5 there exists un ∈ C ∞ (Ωn) for n ≥ such that un ∂Ωn = σn = w ∂Ωn .
Then by (2.2) and (4.2) we have we are in a position to apply Lemma 2.4 to conclude that, for any xed integer k ≥ , there exists a constant C = C k,m independent of n such that for all n > m, It follows that the convergence un(x) → u(x) holds in C k loc (Ω) for every k ≥ , and u ∈ C ∞ (Ω). Moreover, for Since each un is strictly convex, u(x) is strictly convex in Ω. Thus u is a strictly convex solution of (1.1). (1) f p for every p ∈ R, c f + c g(c , c ≥ ),f • g(if g(s) → as s → + ) are also slowly varying at in nity.
(3) For ρ ∈ R and s → ∞, ln(L(s)) ln s → and ln(s ρ L(s)) ln s → ρ. Proposition AP.5. If f ∈ RVρ , f ∈ RVρ , then f f ∈ RVρ +ρ and f • f ∈ RVρ ρ . Proposition AP.6. (Asymptotic behavior) If a function L is slowly varying at in nity, then for a ≥ and t → ∞, . The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.