Entire subsolutions of Monge-Ampère type equations

In this paper, we consider the subsolutions of the Monge-Ampere type equations \begin{document}$ {\det}^{\frac{1}{n}}(D^2u+\alpha I) = f(u) $\end{document} in \begin{document}$ \mathbb{R}^{n} $\end{document} . We obtain the necessary and sufficient condition of the existence of subsolutions.

1. Introduction. In this paper we consider the existence of subsolutions to the Monge-Ampère type equations which can be written in the general form where Ω is a domain in R n , Du and D 2 u are respectively the gradient vector and Hessian matrix of a function u ∈ C 2 (Ω), A is a given n×n symmetric matrix function in Ω × R × R n and B is a positive scalar valued function in Ω × R × R n . We use (x, z, p) to denote points in Ω × R × R n , then A(x, z, p) ∈ R n×n and B(x, z, p) ∈ R, here R n×n is the n(n + 1)/2 dimensional space of real symmetric n × n matrices.
Equations (1) are elliptic with respect to u whenever D 2 u + A(x, u, Du) > 0. At the same time, a solution u ∈ C 2 (Ω) of (1) is called an elliptic solution. We say a function u ∈ C 2 (Ω) is a subsolution of (1), if u satisfies det[D 2 u + A(x, u, Du)] ≥ B(x, u, Du) in Ω.
In recent years, the Monge-Ampère type equations have attracted many interests. For a recent survey and the earlier history, see for instance [18]. When A ≡ 0, (1) reduce to the standard Monge-Ampère equations det(D 2 u) = B(x, u, Du) which were investigated in many works. For example, the existence of smooth solutions to the Dirichlet problem was settled by Caffarelli, Nirenberg and Spruck [3], Krylov [13] and Ivochkina [8]. When A ≡ 0 in (1), if A is linear in, or independent of p, we have very similar a priori estimates as the standard Monge-Ampère equations [6]. But if A is nonlinear in p, the situation is very different. The interior a priori estimates for (1) are given in [20] for dimension two and [5] for all dimensions. Ma, Trudinger and Wang [16] got the interior C 2 estimates of solutions for (1) under an analytical structure condition on the matrix A and the interior regularity of solutions under a generalized target convexity condition. Trudinger and Wang [19] obtained the global C 2 estimates and the global regularity of the second boundary value problem for (1) under weaker conditions. But the results of C 2,α estimates are not many. Liu, Trudinger and Wang [14] established the interior C 2,α estimates of solutions for (1) with A = A(x, p), B = B(x) under appropriate assumptions. Recently, Huang, Jiang and Liu [7] got the boundary C 2,α estimates of solutions with A = A(x, p), B = B(x). For the existence of solutions, Jiang, Trudinger and Yang [10] proved the existence and uniqueness of classical solutions to the Dirichlet problem of (1). In this paper, we study the existence of subsolutions to the Monge-Ampère type equations where α ≥ 0 is a constant and f is a positive function defined in R.
For the nonlinear partial differential equations where F is a real function defined in R n×n , there have been many results about the existence of solutions. If F is the Laplacian operator, i.e. F (D 2 u) = ∆u, (3) has no positive solution if f (u) = u p , p > 1, see Keller [12], Osserman [17], Loewner and Nirenberg [15] and Brezis [1]. In particular, Osserman [17] considered the necessary and sufficient condition under which the equation has a subsolution if f is a positive nondecreasing continuous function. The following growth condition on f at infinity, is well known as Keller-Osserman condition, where we omit the lower limit to admit any positive constant. Capuzzo Dolcetta, Leoni and Vitolo [4] proved that M + 0,1 (D 2 u) = f (u) in R n has an entire viscosity subsolution u ∈ C(R n ) if and only if f satisfies the Keller-Osserman condition (4). Here λ i1 · · · λ i k is the Hessian operator, Jin, Li and Xu [11] have prove that for p > 1 the equation has no positive subsolution. But the method used in [11] cannot prove that p > 1 is optimal. Ji and Bao [9] proved that if f is nonnegative, nondecreasing and continuous on R, the Hessian equation has a positive subsolution u ∈ C 2 (R n ) if and only if For the case f (u) = u p in [9], (6) indicates that (5) has a positive subsolution if and only if 0 < p ≤ 1 and then p > 1 in [11] is optimal. Bao, Ji and Li [2] also established a generalized Keller-Osserman condition on the existence and nonexistence for positive entire subsolutions of k−Yamabe type equations where A u is given by and I is the n × n identity matrix.
In this paper, we shall establish the necessary and sufficient condition on the solvability for the subsolutions of (2). Let α 0 be a positive constant. In the following, we always assume that α 0 > α. The main results of this paper are the following.
In view of the impact of αI, different from the standard Monge-Ampère equation, we cannot directly integrate the equation on both sides in the proof of necessity. In this paper, by means of Lemma 2.2 and the binomial theorem, we overcome the impact of αI, see the necessity in the proof of Lemma 3.3. We need not only ϕ (r) > 0, but also ϕ (r) ≥ (α 0 − α)r, see Lemma 2.2.
From Theorem 1.1, we can get the following corollaries.
This paper is arranged as follows. In section 2, we give some basic results of radial functions. The main results will be proved in section 3.
2. Some basic results of radial functions. In this section, we collect some results for the radial functions. Let r = |x| = x 2 1 + · · · + x 2 n .

LIMEI DAI AND HONGYU LI
, and the eigenvalues of D 2 u + αI are and so The proof of Lemma 2.1 is similar to the proof of Lemma 2.1 in [9]. Here we omit the proof.
Next we prove an existence result for (10).
The rest of the proof is similar to the proof of Lemma 2.3 in [9].
Then according to Lemma 2.3, we know that the solution ϕ(r) of (10) with the initial value ϕ(0) = a, ϕ (0) = 0 is in C 2 [0, R).  Assume that (9) has a solution ϕ(r) ∈ C 2 [0, R) satisfying ϕ (0) = 0, and ϕ(r) → +∞, as r → R. Then if u is a subsolution of (2), we have u( Suppose on the contrary that u > v at some point. Then there exists some positive constant c 0 such that u − c 0 touches v from below at some interior point This implies from the maximum principle that in B R (x 0 ), which is a contradiction. Then we complete the proof of Lemma 3.1.

LIMEI DAI AND HONGYU LI
where C 1 = ( n+1 C0 ) 1 n+1 . Integrating on r from r 1 to r, we know that Since f is nondecreasing, then for τ ≥ 2ϕ(r 1 ), we have Therefore As a result, from (19), Letting r → ∞, we get a contradiction.
Proof of Theorem 1.1. By Lemma 3.2 and 3.3, we know that Theorem 1.1 is true.
Proof of Corollary 1. Let Then f satisfies the conditions in Theorem 1.1.
For k > 0, let Then f satisfies the conditions in Theorem 1.1. In addition, By Theorem 1.1, there exists no nonnegative subsolution of (8). Then Corollary 2 is proved.
Remark 1. If α ≥ 1, the method we used is not applicable to (8).