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Euclidean Complete Hypersurfaces of a Monge-Ampere Equation

We study the Monge-Ampere equation with some power nonlinear term. A solution u is called to be Euclidean complete if it is an entire solution defined over the whole R^n or its graph is a large hypersurface satisfying the large condition on boundary \partial\Omega in case \Omega\not=R^n. In this paper, we will give various sharp conditions on p and \Omega classifying the Euclidean complete solution. Our results clarify and extend largely the existence theorem of Cirstea-Trombetti (Calc. Var., 31, 2008, 167-186) for bounded convex domain and p>n.


Introduction
The Monge-Ampère Equation with a semilinear term f has been studied extensively, see for examples [2,5,11,15,16,[21][22][23][24][25] about the existence and regularity of Dirichelt problems and the references therein. A problem concerning the solutions of (1.1) satisfying the boundary blow-up condition u = ∞ on ∂Ω has been proposed and studied by Cheng-Yau [6,7] for exponential nonlinear term f (u) = e u due to their applications in geometry. It was later been considered by Lazer-McKenna [17] for nonlinear term f (u) = u p of power law. Some other results can be found in [3,12,18,19,26]. Throughout this paper, a solution u of (1.1) is called to be Euclidean complete if it is an entire graph defined over the whole Euclidean space R n or it is a large graph satisfying the large condition  This result was later generalized to Hessian equations by Huang in [14], and was generalized to Monge-Ampère equations with weight by Zhang, Du-Zhang in [9,26,28]. See also [10,20,27] for some other extensions. The first part of this paper is devoted to prove the counterpart of Theorem 1.1 as following. Theorem 1.2. Supposing that f is an unbounded function satisfying (1.3) lim sup u→∞ f (u) u n < ∞, then for any bounded smooth convex domain Ω ⊂ R n , there is no convex large solution to (1.1).
Unlike the case of bounded domain, when one considers the entire solution defined on whole Ω = R n , the situation changes dramatically. In fact, we have the following sharp non-existence result for p > n. Theorem 1.3. Supposing that f ∈ C((0, ∞)) is a positive monotone nondecreasing function on u and satisfies (1.4) f (u) ≥ Au p , ∀u > 0 for some p > n and A > 0,, then there is no positive entire convex solution of (1.1) on R n .
With the help of the technics developed in [8], we can also prove the following sharp existence result. Finally, we turn to the case of unbounded domain Ω R n and prove the following result for p > n. Theorem 1.5. Supposing that f ∈ C((0, ∞)) is a positive monotone nondecreasing function on u and satisfies (1.4) for some p > n, n ≥ 2 and A > 0, then there is no positive entire convex solution of (1.1) for ideal domain Ω.
As illustrated in Theorem 1.3-1.5, it's natural to conjecture that for p < n and unbounded domain Ω R n , there exist some positive convex solutions which are Euclidean complete. However, focusing on the case of n = 2 and p ∈ (0, 1/2), we have shown the following surprising non-existence result. Theorem 1.6. Considering (0.1) for n = 2, p ∈ (0, 1/2) and unbounded domain Ω R 2 , there is no positive convex solution which is Euclidean complete.
The contents of this paper are organized as follows: We will prove Theorem 1.2 in Sect. 2, and prove Theorem 1.3 in Sect. 3. Using the technics developed in [8], we show Theorem 1.4 in Sect. 4-5. Finally, the proofs of Theorem 1.5 and 1.6 were presented in Sect. 6 and Sect. 7 respectively.

Non-existence of large solution on bounded domain
Considering the Dirichlet problem let's first recall a lemma in [1].
As a counterpart of Theorem 1.1, we have the following nonexistence result.
Theorem 2.1. Supposing that f is an unbounded function satisfying (2.3) lim sup u→∞ f (u) u n < ∞, then for any bounded domain Ω ⊂ R n , there is no convex large solution to (0.1).
Proof. By assumption (2.3), there exists an increasing sequence u k → ∞ such that We claim that holds for any B < (C 2 A 1/n ) −1/γ . In fact, setting for some large k 0 ∈ N, it follows from (2.6) and Lemma 2.1 that Sending k → ∞ in (2.7) and using (2.4) for B < (C 2 A 1/n ) −1/γ , we conclude that u is infinite everywhere in Σ B . The claim holds true. By an bootstrapping argument we reach the conclusion that u is infinite everywhere in the whole domain Ω. The proof of the theorem was done.
3. Non-existence of entire solution on R n In this section, we will show the non-existence of entire solution as following.
Theorem 3.1. Supposing that f ∈ C((0, ∞)) is a positive monotone nondecreasing function on u and satisfies for some p > n and A > 0, then there is no positive entire convex solution of (1.1) on R n .
Before proving the theorem, we need the following existence result by Lazer-McKenna [17]. Letting Ω be a bounded smooth convex domain in R n and f (u) = u p for some p > n, there exists an unique large convex solution u ∈ C ∞ (Ω) of (1.1) satisfying Proof of Theorem 3.1 By (3.1), each positive convex solution u of is a sup-solution of (1.1). By Lemma 3.1, we denote u R to be the unique convex large solution of (1.1) for f (u) = u p and Ω = B R and rescale u R by it follows from Lemma 3.1 again that and thus a sup-solution to (1.1), using the relation As a result, Fixing y and sending R tends to infinity, it's inferred from (3.5) that u is identical to zero for p > n. The proof was done.

Schauder's fix point scheme and entire hypersurface for p < n
Letting u = u(r) be a radial symmetric solution to (0.1), there holds that Noting that for p < n, is a nonnegative convex entire solution of (1.1) for f (u) = u p with one degenerate singular point x = 0, it can not be expect to prove an A-priori positive lower bound and regularity for entire solution of (1.1) as that in [3] for large solution. However, one may also ask the following question of existence of positive strictly convex entire solution of (1.1).
Question. Considering (0.1) for p < n and Ω = R n , whether it admits a smooth positive convex entire solution?
We will give a confirm answer in the following theorem.
to be a radial symmetric positive entire solution is given by Proof. The lemma can be easily verified by substituting u into (4.1) at r = 0.
At first step, we will introduce an iteration scheme to prove a local existence result for (4.1). Letting a ≡ (a 0 , a 1 , · · · , · · · , a 2κ ) ∈ R 2κ , ∀κ ∈ N to be a vector satisfying (4.3) and σ, δ ∈ (0, 1), we define . For any function ϕ ∈ Γ a,δ,σ,κ , we define ξ ≡ T ϕ to be the solution determined by Next, we will show that T is a continuous and compact mapping from Γ a,δ,σ,κ to Γ a,δ,σ,κ . Thus, it's inferred from the following variant of Schauders' fix point lemma that T admits a fix point in the closure of Γ a,δ,σ,κ .
Therefore, supposing that let's introduce a new notation for conjugate indices which is determined by f i , i = 0, 1, · · · , l for each l ≤ k. We need the following lemma.
Moreover, by taking higher derivatives on (4.4) and applying the Cauchy's intermediate value theorem repeatedly, it's not hard to see that the following proposition holds true.
Remark. Without special indication, the functions on left hand side of (4.16) take values on r, while the functions on right hand side of take values on 0 < · · · < r 3 < r 2 < r 1 < r, which come from the Cauchy's intermediate value theorem.
As a corollary, we have the following local existence result of positive entire solution of (4.1).

Long time existence and validity of Theorem 4.1
Letting u be the solution derived in Corollary 4.2, we will show that the solution exist for all r > 0, provided p < n.  n, p, R, a 0 , δ, σ, κ).
Lemma 5.2. Let u be a solution of (4.1) derived in Corollary 4.2. If p < n, there exists a positive constant C 5 = C 5 (n, p, R, a 0 , δ, σ, κ) such that holds for any R > 0.
Complete the proof of Theorem 4.1. Lemma 5.1 and 5.2 ensure local solutions u of (4.2) derived from Corollary 4.2 would not be blowup in finite r, and hence give the derived entire positive entire solutions of (1.1). Moreover, these solutions are affine independent.

Euclidean complete solution on unbounded domain Ω R n
In this section, we assume that Ω R n is an unbounded convex domain, which satisfies the following hypothesis of "ideal domain": (H3) There exist a nonempty bounded Lipschitz convex domain Ω 0 ⊂ R n and sequences a k ∈ Ω, λ k → ∞ such that holds for some a ∞ ∈ ∞ k=1 Ω k . We swill call Ω to be an "ideal domain" and call a ∞ to be an "ideal center" of Ω. It's clearly that any unbounded convex set Ω ⊂ R n containing an infinite cone C ≡ λx ∈ R n x ∈ C 0 , λ ∈ R + , which is defined for a nonempty domain C 0 ⊂ R n , are all ideal domains. We will prove the following nonexistence result for ideal domain. Theorem 6.1. Supposing that f ∈ C((0, ∞)) is a positive monotone nondecreasing function on u and satisfies (3.1) for some p > n, n ≥ 2 and A > 0, then there is no positive entire convex solution of (1.1) for ideal domain Ω.
In the definition of ideal domain, Ω 0 is only bounded Lipschitz convex domain. Therefore, Lemma 3.1 can not be applied directly to Ω 0 . Fortunately, we have the following variant version of Lemma 3.1.
Letting Ω be a bounded Lipschitz convex domain in R n and f (u) = u p for some p > n, there exists a large convex solution u ∈ C ∞ (Ω) of (1.1) satisfying (3.2) for some positive constant C 3 = C 3 (n, p, Ω), where α = n+1 p−n . Proof. Assuming Ω is a bounded Lipschitz convex domain, there exists a sequence of smooth convex domains Ω j ⊂ Ω, j ∈ N which is monotone increasing in j and satisfies ∞ j=1 Ω j = Ω.
Applying Lemma 3.1 to Ω j , there exists a sequence of positive constants C 3, j such that (3.2) holds for solution u j of (1.1) on Ω j . Noting that by maximum principle (1.1), we have u j is monotone decreasing in j and C 3, j can also be chosen to be monotone decreasing. As a result, passing to the limit j → ∞ in (3.2), we obtain that the limiting solution for any x ∈ Ω and some positive constant C 3,∞ . To show the left inequality of (3.2), one needs only to choose a monotone shrinking bounded smooth convex domains Ω j ⊃ Ω satisfying ∞ j=1 Ω j = Ω.
Then, applying the monotone increasing of u j and C −1 3, j , we arrive at for any x ∈ Ω and some positive constant C 3,∞ . Choosing we conclude that u satisfies (3.2) for C 3 . Now, we can complete the proof of Theorem 6.1. Proof. As in proof of Theorem 3.1, each positive convex solution u of is a sup-solution of (1.1) due to (3.1). Letting Ω 0 ⊂ Ω, a k ∈ Ω, λ k → ∞, k ∈ N as given in definition of ideal domain, we set By Lemma 6.1, we denote u 0 to be a convex large solution of (1.1) for f (u) = u p on Ω 0 , which satisfies that Rescaling u 0 by we have it is also a sup-solution to (1.1). Using the relation and [U i j ] being the cofactor-matrix of tD 2 w k + (1 − t)D 2 u, it inferred from maximum principle of uniformly elliptic equation that u ≤ w k in Ω k . As a result, , ∀y ∈ Ω k . Taking y to be an ideal center a ∞ of Ω and sending k to be large, it's inferred from (6.5) that u(a ∞ ) must be identical to zero since p > n, which contradicts with the positivity of u. The proof of the theorem was done.

Wiping barrier and large solution on unbounded domain
In spired of Theorems 3.1, 4.1 and 6.1, it's naturally to ask the following question.
Question: For n ≥ 2 and p < n, given any unbounded convex domain Ω R n , whether there is some Euclidean complete solution of (0.1) on Ω?
It would be surprising that we have the following non-existence result of large solution of (0.1). Our proofs of Theorem 7.1 are relied on a construction of zero barrier solution on some wiping domain defined by where r 0 ∈ (0, ∞) and β < 0. We need to use the following existence result in proof of Theorem 7.1.
Proof. To verify the compactness of T , we use (7.5) to calculate provided q is chosen to close 1 and then δ is chosen small, where has been used. Integrating over ϕ yields that Consequently, T is a continuous and compact mapping from Γ β,q,δ to Γ β,q,δ . The proof was done.
As a corollary, we obtain the following local solvability of (7.2).
Corollary 7.1. Under the assumptions of Lemma 7.1, the mapping T admits a fix point ζ in Γ β,q,δ which is a local smooth solution of (7.2) satisfying (7.8).
Finally, let's complete the proof of Proposition 7.1 with the help of the following long time existence and asymptotic results. Proof. It's clear that the solution of (7.2) preserves positivity and monotonicity. To show that the solution exists for all ϕ, one needs only using (7.2) to deduce that So, ζ is A-priori bounded from above and thus exists for all ϕ.
The second proposition gives the asymptotic behavior of ζ at infinity. We divided the proof into two lemmas.
Lemma 7.2 is a direct consequence of proof of Proposition 7.2.
Solving this first order O.D.E. as in proof of Lemma 7.3, it yields the desired inequality (7.6) with the help of (7.8).
At the end of this section, let's complete the proof of Proposition 7.1 and Theorem 7.