A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained. 


Introduction
Let D 2 u be a square matrix of second-order partial derivatives for the scalar-valued function u, which is called the Hessian matrix in mathematics. In general, the Hessian matrix can describe the local curvature of the function u with multiple variables. Suppose λ 1 , λ 2 , . . . , λ N are the eigenvalues of the Hessian matrix D 2 u, define the k-Hessian operator S k (λ(D 2 u)) as follows: where λ(D 2 u) = (λ 1 , λ 2 , . . . , λ N ) is the vector of eigenvalues of D 2 u, that is S k (λ(D 2 u)) is the sum of the kth principal minors of the Hessian matrix, and here we define Γ k := λ ∈ R N : S l λ D 2 u > 0, 1 l k .
On the other hand, the k-Hessian operator can also be expressed as the following form of divergence: where S ij k = ∂S k (D 2 u)/∂u ij , and for more details, see Trudinger et al. [50] and Gavitone [7].
For k > 1, S k (λ(D 2 u)) is a second-order fully nonlinear differential operator. When k = N , the k-Hessian operator reduces to the Monge-Ampère operator det D 2 u, and if k = 1, then S k (λ(D 2 u)) turns into a Laplace operator, which implies that the k-Hessian operator constructs a discrete collection of partial differential operators including the Monge-Ampère operator det D 2 u and the Laplace operator ∆u as special cases.
In this paper, we consider the existence and nonexistence of blow-up solutions for the following k-Hessian equation with a nonlinear operator: where f : C((0, +∞) × R → [0, +∞)) is nondecreasing, and the nonlinear operator B ∈ X satisfies the following property: Many existing work for the k-Hessian equation is devoted to constructing mathematical theory rather than modeling or exploring new applications. Here we will briefly mention some potential implications in physics and applied mathematics. Escudero [3] described some phenomena of non-equilibrium phase transitions and statistical physics by using the k-Hessian equation. In addition, the k-Hessian equation also gives a class of important fully nonlinear elliptic operators related to geometric optics [51], study of quasilinear parabolic problems [40], optimal transportation, and isometric embedding [19].
The k-Hessian equation of form (1) has attracted much research interest, and there are many rich literatures concerning the k-Hessian equation. If B(x) = x k−1 , then the k-Hessian equation (1) reduces to the standard Hessian equation S k (λ(D 2 u)) = f and the standard Monge-Ampère equation when k = n. Caffarelli et al. [1] studied the existence and a priori estimate of the smooth solutions for the k-Hessian equation and the work was then extended to more general equations in [25,49]. If B(x) = const = 0, then the k-Hessian equation becomes the Hessian equation Covei [2] considered the existence of positive radial solutions for the above Hessian equation, provided that f = p(|x|)h(u). If B(x) = |x| p−2 , p 2, the k-Hessian equation (1) turns into the form which is a p-Poisson Hessian equation, and few work was reported. Thus, since the k-Hessian equation (1) involves a nonlinear operator B, it includes many interesting and important equations such as the Hessian equation, the Monge-Ampère equation, the p-Poisson Hessian equation as special cases. To the best of our knowledge, no result has been reported on the existence and nonexistence of blow-up solutions for the k-Hessian equation (1), and this is the first paper using the iterative method to study the k-Hessian equation involving a nonlinear operator. The rest of the paper is organized as follows. In Section 2, we firstly study the property of the nonlinear operator B and then transform the k-Hessian partial differential equation (1) to an ordinary differential equation. In Section 3, we establish the results of the nonexistence of blow-up solutions for the k-Hessian equation (1) by some estimations. In Section 4, a necessary and sufficient condition on existence of blow-up solutions for equation (1) is established and some further results are obtained.

Preliminary results on radial solutions
Before we give a detailed description of our main results, we firstly establish a property for the inverse operator of the operator sB(s). Lemma 1. Let L(s) = sB(s), B ∈ X , then there exists a nonnegative increasing inverse mapping L −1 (s), and for any 0 < b < 1, Proof. Firstly, we assert that B is an increasing operator if B ∈ X . In fact, for any B ∈ X and s, t ∈ [0, +∞), without loss of the generality, let 0 s < t. If s = 0, then, obviously, B(s) B(t) holds. If s = 0, let c 0 = s/t, then 0 < c 0 < 1. It follows from the property of B that http://www.journals.vu.lt/nonlinear-analysis which implies that B is an increasing operator. Thus we have L (s) = (sB(s)) > 0 for any s > 0, which implies that L is a bijection on (0, ∞) and has a nonnegative increasing inverse mapping L −1 (s).

Remark 2. The operator set X includes a large class of operators and the standard type of operators is
In fact, take α = min{α 1 , . . . , α n } > 0, then for any 0 < c < 1, one has B(cs) c α B(s).
Lemma 2. (See [18].) Assume y(r) ∈ C 2 [0, R) is radially symmetric and y( 0) = 0. Then the function u(|x|) = y(r) with r = |x| < R is C 2 (B R ), and Now we consider the k-Hessian equation (1). By using Lemma 1, equation (1) can be transformed into the following form: Make the radial transformation u(|x|) = y(r), then from Lemma 2 and (2), u(|x|) = y(r) is a radial solution of equation (1) if and only if y(r) is a solution of the ODE Take any central value y(0) = d > 0 for equation (1). Since which implies that y(r) ∈ C 2 [0, R). Thus it follows from (4) that the following lemma holds.

Nonexistence of blow-up solutions
Now let us list a growth assumption, which is to be used in the rest of this paper. Under the above growth condition, we have the following result of nonexistence of blow-up solution for the k-Hessian equation (1). Theorem 1. Suppose (A) holds, and the fast decay condition is satisfied: then the k-Hessian equation (1) has no positive entire blow-up solutions.
Proof. We shall establish the result of nonexistence by using the method of contradiction. Suppose that the k-Hessian equation (1) has a positive entire blow-up solution, say u = y(r). Then by (5), there exists a sufficiently large r > 0 such that Since y is radially symmetric, by using the standard integrating procedure for (3), we have Consequently, for any r > r > 0, (6) yields Thus it follows from (7), (A), and Remark 1 that which contradicts with the assumption that y is the positive entire blow-up solution of equation (1) Proof. Sufficient condition. We firstly prove that the k-Hessian equation (1) has a positive radial solution. For this purpose, according to Lemma 3 and (3), we consider the equivalent integral equation In the following, we shall construct a positive increasing sequence {y (m) (r)} m 1 by using (9), which is bounded above on Obviously, {y (m) (r)} m 1 is an increasing sequence of nonnegative and increasing functions, and y (m) (r) d for all r 0 and m ∈ N . Thus it is sufficient to prove that the sequences {y (m) (r)} m 1 are bounded from above on bounded subsets for obtaining our desired conclusion.
Fix R > 0, since f is increasing with respect to the second variable y on [0, ∞), then it follows from (A) and Remarks 1 and 3 that Hence from the monotonicity of {y (m) (R)} m 1 we have We assert that L(R) is finite. In fact, if not, by (11), (8), and 0 < β < 1 + α, one has which is a contradiction. So L(R) is finite, and the mapping L : (0, ∞) → (0, ∞) is increasing since y (m) (r) is an increasing function in (0, ∞). Thus for any r ∈ [0, R] and m 1, we have which implies that the sequence {(y (m) (r))} m 1 is bounded from above on bounded sets. Letting y(r) := lim m→∞ y (m) (r) ∀r 0, and taking the limit for (10), we get that y is a positive solution of (1). Now to complete the proof, we only need to prove that y blows up. Indeed, by (9) and (8), we have which implies that y is a blow-up solution of the k-Hessian equation (1). Noticing that d ∈ (0, ∞) is chosen arbitrarily, the k-Hessian equation (1) has infinitely many positive entire blow-up solutions.
Necessary condition. For the proof of necessity, we still adopt the method of contradiction. Suppose y is any positive entire blow-up solution of the k-Hessian equation (1) and (8) is not true, that is which implies that y(r) is increasing on [0, ∞). By (A), (9), and Remark 1, one has Noticing that lim r→+∞ y(r) = +∞, we have which is a contradiction. The necessity of Theorem 2 is proved.

Further results
In this section, we discuss some alternative conditions of (A) and give some further results.

Now we show that assumption (B) implies assumption (A).
Proof. Notice that f (s, u)/u β is decreasing on the variable u ∈ (0, +∞), for any 0 < c 1 and u ∈ (0, +∞), we have which implies that f (s, cu) c β f (s, u). The proof is completed. Thus we have the following corollaries: Obviously, (12) implies that f (s, u)/u β is decreasing on the variable u ∈ (0, +∞), and thus Corollaries 1 and 2 can be rewritten as:  (D) f ∈ C([0, +∞), (0, +∞)) is nondecreasing and p(t) 0 for t ∈ [0, +∞), and there exists a positive constant β < 1 + α such that By checking the proof of Theorems 1 and 2, the conclusions can be rewritten as follows: Theorem 3. Suppose (D) holds, and the following fast decay condition holds: then the k-Hessian equation (1) has no positive entire blow-up solution.
Theorem 4. Assume that (D) holds, and the weigh function p satisfies the following slow decay condition: Then the k-Hessian equation (1) has infinitely many positive entire blow-up radial solutions if and only if the slow decay condition (13) is satisfied.
In addition, noticing that always holds, if the weigh p ≡ const = 0, then we only have the result of existence of blow-up positive solutions, that is Corollary 5. Assume that (D) holds, then the k-Hessian equation (1) has infinitely many positive entire blow-up radial solutions.

Numerical example
In this section, we present an example to illustrate our main results. Example 1. Consider the existence and nonexistence of radial blow-up solutions for the following 3-Hessian equation: where p(s) > 0, s ∈ (0, +∞). Then According to Theorem 1, the k-Hessian equation (14) has no positive entire blow-up solutions.

Conclusion
The k-Hessian operator is an important fully nonlinear and non-uniform elliptic partial differential operator, which includes the Monge-Ampère operator and the Laplace operator as special cases, thus the study of the k-Hessian equation is interesting and challenging. In this paper, we introduce a new nonlinear operator B with suitable growth condition, and the k-Hessian equation is extended to more generalized cases. In addition, we also put forward some suitable growth conditions for nonlinearity for obtaining the results of nonexistence and existence of blow-up radial solutions for the k-Hessian equation. The conditions are easy to be checked, and the corresponding results improve and generalize some recent results such as those in [1,2,18,25,49].