EXISTENCE AND NONEXISTENCE OF SUBSOLUTIONS FOR AUGMENTED HESSIAN EQUATIONS

. In this paper, we consider the augmented Hessian equations S 1 k k [ D 2 u + σ ( x ) I ] = f ( u ) in R n or R n + . We ﬁrst give the necessary and suﬃcient condition of the existence of classical subsolutions to the equations in R n for σ ( x ) = α , which is an extended Keller-Osserman condition. Then we obtain the nonexistence of positive viscosity subsolutions of the equations in R n or R n + for f ( u ) = u p with p > 1.


Introduction. The augmented Hessian equation
is a class of fully nonlinear elliptic equation which has attracted many interests and has many applications, where D 2 u is the second derivative of u, Ω is an open set in R n , A : Ω × R × R n → S n×n , S n×n denotes the set of n × n real symmetric matrices, B : Ω × R × R n → R is a positive function and S k is the k−Hessian operator defined by S k [W ] = S k (λ) with λ being the eigenvalues λ 1 , λ 2 , · · · , λ n of the n × n symmetric matrix W and S k (λ) being the k−th elementary symmetric function given by S k (λ) = 1≤i1<···<i k ≤n λ i1 · · · λ i k , 1 ≤ k ≤ n.
If A ≡ 0, (1) becomes the standard Hessian equation. In this case, when k = n it is the standard Monge-Ampère equation det(D 2 u) = B(x, u, Du). For these standard Hessian equations and Monge-Ampère equations, they are very well known and investigated extensively, see [2,3,7,27,28], etc. If A = 0, in this case, for k = n, (1) corresponds to a class of Monge-Ampère type equations which has been studied in [15,18,23,25] and the references therein. For 1 < k < n, if A depends only on x, Li [22] established the existence and uniqueness results and Guan [14] investigated the Dirichlet problem on Riemannian manifolds under some very general structure conditions. If A depends on x and Du or on x, u and Du, the second order derivative 580 LIMEI DAI estimates and the existence of the solutions to the Dirichlet problem in bounded domains are given in [17,19].
In this paper we consider the subsolutions for the augmented Hessian equations (2) in the whole space R n or in the half space R n + := {x = (x , x n ) ∈ R n |x n > 0}, where σ(x) is a positive continuous function in R n , I is the identity matrix and f is a positive continuous function on R. To work in the realm of elliptic equations, we let Γ k = {λ ∈ R n |S j (λ) > 0, j = 1, 2, · · · , k}. Γ k is a symmetric cone, that is, any permutation of λ is in Γ k if λ ∈ Γ k . We always assume that λ(D 2 u + σ(x)I) ∈ Γ k for x ∈ R n . A function u ∈ C 2 (R n ) is called a classical subsolution of (2), if u satisfies We first study the necessary and sufficient condition for the existence of subsolutions to the augmented Hessian equations where 1 ≤ k < n, α ≥ 0 is a constant. Osserman [26] proved that the equation has a subsolution if and only if where we omit the lower limit to admit any positive constant. For f (u) = u p , p > 1, Keller [21] proved that (4) has no positive solution. Then the growth condition (5) is the well known Keller-Osserman condition. Ji and Bao [16] proved that if f is nonnegative, nondecreasing and continuous on R, the Hessian equation Covei [8] studied the Keller-Osserman condition on the boundary blow up solution of Hessian equations. The Keller-Osserman condition on the existence of entire solutions is similar to that on the existence of boundary blow up solutions, we can also refer to [11]. For the Keller-Osserman condition on k−Yamabe type equations, we can see [1]. Analogous results for fully nonlinear equations have been obtained by Dolcetta, Leoni and Vitolo [4,5], Cutrì and Leoni [10], Felmer and Quaas [12], Lu and Zhu [24] and the references therein. Let the positive constant α 0 satisfy where C m n = n! m!(n−m)! , 0 ≤ m ≤ n.
Theorem 1.1. Let 1 ≤ k < n and f (t) be a continuous and nondecreasing function on R and f (t) ≥ α 0 . Then there exists a subsolution u ∈ C 2 (R n ) of (3) if and only if (6) holds.
In the proof of necessity of Theorem 1.1, due to the impact of αI, different from the standard Hessian equations, we cannot have a good estimate as [16]. By giving the condition f ≥ α 0 , through Lemma 2.2 and the convexity inequality, we overcome the impact of αI. We need not only ϕ (r) > 0, but also Remark 2. One could observe that, if u is a subsolution of (3) for a given α, then u is an entire subsolution for any bigger α. In fact, remaining in the cone Γ k of admissible functions, S k is elliptic and therefore, if α 1 > α, then . So assuming f (t) ≥ α 0 : if the Keller-Osserman type condition (6) holds, there exist entire subsolutions for all α > 0; vice versa, if there exists an entire subsolution with α small enough as in condition (7), then the Keller-Osserman type condition (6) must hold.
Then for α ≥ 0 there exists a subsolution of (3) if and only if p ≤ 1.
Then we consider the viscosity subsolutions (the definition of viscosity subsolutions will be given in section 4) of the equation in the whole space R n or in the half space R n + , where σ(x) ∈ C(R n ) is a positive function, σ(x) ≤ σ 0 for some positive constant σ 0 and p is a positive constant. Jin, Li and Xu [20] investigated the viscosity subsolutions for the equation k (D 2 u) = u p in the whole space R n or in the half space R n + . In this paper, we extend the results in [20] to the equation (8). Let g(x, t) be defined on ∂R n + × [0, ∞) such that g(x, t) > 0 for any x ∈ ∂R n + and t > 0.
has no positive continuous viscosity subsolution.
This paper is arranged as follows. In section 2, we give some basic results of radial functions. Theorem 1.1 will be proved in section 3 and Theorems 1.2 and 1.3 will be proved in section 4.
2. Some basic results of radial functions. In this section, some results for the radial functions will be given.
Next we prove an existence result for (11).
is continuous and nondecreasing on R. Then for any constant a, there exists a positive number R such that (11) with the initial value Proof. As the proof of Lemma 2.3 in [16]. We define a functional F (·, ·) on where l and h are positive constants small enough. Then (11) can be rewritten as By Lemma 2.2, F > 0 for r > 0.
Define an Euler's break line on [0, l] as where 0 = r 0 < r 1 < · · · < r m = l. Then ψ ∈ C 2 [0, l]. We claim that (r, ψ) ∈ R. Indeed, for any (r, ψ) ∈ R, we have Then for the break line ψ, we have Thus if h is fixed, we can choose l sufficiently small such that Next, we prove that the Euler's break line ψ is an ε−approximation solution of (11). For this, we only need to prove that for any ε > 0, we can choose points {r i } i=1,··· ,m such that the break line satisfies Indeed, we see from the second line in (17) that So for any ε > 0, there existsr ∈ (0, l) such that for 0 ≤ r <r, we have Then On the other hand, for r ≤ r ≤ l, we have Since r n is Lipschitz continuous on [r, l], for the above ε, there exists 0 < δ(ε) < ε such that for r , r ∈ [r, l] and |r − r | < δ(ε), , and r (n−k) − r (n−k) < ε k k nC k−1 n−1 r 2(n−k) l n f k (a + h) .
Let r 1 =r and max 2≤i≤m |r i−1 − r i | < min{r, δ(ε)}, then we have (18). The rest of the proof is similar to the proof of Lemma 2.3 in [16].
We only need to prove that u(x) ≤ v(x) in B R . Suppose on the contrary that u > v at some point. Then there exists some positive constant c such that u − c touches v from below at some interior point x ∈ B R , i.e., u(x) − c − v(x) = 0, and u − c − v ≤ 0 in B R . Due to the fact that v(x) → +∞ as |x| → R, we can choose some |x| < R < R such that u − c − v ≤ 0 in B R and sup ∂B R (u − c − v) < 0.
In addition, we have in B R , By linearizing the operator S k and using the maximum principle of linear elliptic equations [6,13], we know that in B R , This is a contradiction. Then we complete the proof of Lemma 3.1. Proof. Sufficiency. Assume that (11) has a solution ϕ, then clearly v(x) = ϕ(r) = ϕ(|x|) is the desired solution of (3). Necessity. Conversely, suppose that (11) has no such function ϕ(r) existing globally. From Lemma 2.4 and Lemma 2.3, (11) has a C 2 solution ϕ on some interval with ϕ(0) = a, ϕ (0) = 0 which cannot be a global solution. Then there is a maximal interval [0, R) in which the solution exists. From Lemma 2.2, we have ϕ (r) > 0 for r ∈ (0, R), then ϕ(r) → ∞ as r → R. In fact, if ϕ is bounded above, then by (15) ϕ is bounded above as well, and this contradicts the maximality of R if R < +∞. So from Lemma 3.1, any subsolution of (3) satisfies u(x) ≤ ϕ(|x|) for |x| < R. In particular, we have u(0) ≤ ϕ(0) = a. However since a is arbitrary, letting a = u(0)/2, we know that it is a contradiction. Lemma 3.2 is proved. Proof. Sufficiency. Suppose that there is no such solution of (11). Just as the proof of Lemma 3.2, (11) has a C 2 solution ϕ(r) on the maximal interval [0, R) satisfying ϕ (0) = 0, ϕ(0) = 0 and ϕ(r) → ∞ as r → R. Hence ϕ satisfies (10). Since by Lemma 2.2, we have ϕ (r) > 0, ϕ (r) + α > 0, then we know that that is,
Proof of Theorem 1.1. By Lemma 3.2 and Lemma 3.3, we know that Theorem 1.1 is true.
Proof of Corollary 1. For 0 < p ≤ 1, For p > 1, Therefore by Theorem 1.1, we can prove that the corollary is true.
Definition 4.1. We say a nonnegative function u ∈ C(R n ) is a viscosity subsolution of (8) in R n , if for any x 0 ∈ R n there exists an > 0 such that for any φ ∈ C 2 (B (x 0 )) satisfying φ(x 0 ) = u(x 0 ),

LIMEI DAI
Similarly, we say a nonnegative function u ∈ C(R n ) is a viscosity supersolution of (8) in R n , if for any x 0 ∈ R n there exists an > 0 such that for any φ ∈ C 2 (B (x 0 )) satisfying φ(x 0 ) = u(x 0 ), φ ≤ u and λ(D 2 φ + σ(x)I) ∈ Γ k in B (x 0 ), there holds S 1 k k [D 2 φ(x 0 ) + σ(x 0 )I] ≤ φ p (x 0 ). We say a nonnegative function u ∈ C(R n ) is a viscosity solution of (8) in R n , if it is both a viscosity subsolution and a viscosity supersolution of (8).
Proof of Theorem 1.2. Let j = 1, 2, . . . be the positive integers. Suppose that B j is the ball with radius j and center at the origin. Then ∞ j=1 B j = R n .
In the following we will construct a sequence of positive functions {v j } with v j ∈ C 2 (B j ) such that λ(D 2 v j + σ(x)I) ∈ Γ k in B j , v j (x) → +∞ uniformly as d(x, ∂B j ) → 0 (26) and v j (x) → 0 as j → ∞ for any fixed x ∈ R n .
Assume that such functions v j exist and u ∈ C(R n ) is a positive viscosity subsolution of (8) in R n , by Lemma 4.2, we can get that u(x) ≤ v j (x), x ∈ B j for any j.
Now we construct such functions v j . Let where β is a positive constant to be determined. Let r = |x|, then v(x) = ϕ(r) = (1 − r 2 ) −β . By straightforward calculation, we know that