New Results on the Radial Solutions to a Class of Nonlinear k-Hessian System

functions.With the help ofKeller–Osserman type conditions andmonotone iterative technique, the positive radial solutions of the above problem are given in cases of nite, innite, and seminite. Our results complement the work in by Wang, Yang, Zhang, and Baleanu (Radial solutions of a nonlinear k-Hessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simul. 91(2020), 105396).


Introduction
It is well known that Laplace equation has a wide range of applications in mathematics and physics, for instance [1], it can be used to describe the relationship between the curvature of liquid surface and the surface pressure of liquid [2] as well as in solving electrostatic eld problems [3,4]. In a mathematical sense, the existence of solutions of Laplace equation has attracted increasing attention, and numerous excellent results have been obtained.
In 1957, under the condition Keller [5] studied the existence of solutions for the nonlinear equation Δz φ(z), and Osserman [6] studied the existence of solutions for the nonlinear di erential inequality Δz ≥ φ(z).
In 2011, under the Keller-Osserman condition Peterson and Wood [7] presented the existence of entire positive blow up radial solutions of semilinear elliptic system In 2019, under the Keller-Osserman type conditions Covei [8] studied the existence and asymptotic behavior of positive radial solutions for the semilinear elliptic system where b, h, φ, ψ are continuous functions. Besides, under the Keller-Osserman condition and its extension, many excellent results have been obtained, such as the existence and uniqueness of the nonnegative viscosity solutions [9], the existence and uniqueness of blow up solutions [10], the existence and uniqueness of entire blow up solutions [11], the existence of entire classical weak solutions of the differential inequality [12], the existence and uniqueness of solutions with strong isolated singularity [13], and the existence of solutions for the boundary blow up problem in one dimensional case [14]. we define the k-Hessian operator (k � 1, 2, . . . , N) as follows: where D 2 z � (z 2 z(x)/zx i zx j ) N×N , (i, j � 1, 2, . . . , N) is the Hessian matrix of z ∈ C 2 (R N ) and the λ(D 2 z) � (λ 1 , . . . , λ N ) ∈ R N is the vector, which consists of eigenvalues of D 2 z. It is easy to observe that S k (λ(D 2 z)) is a family of operators including many well-known operators. For example, for k � 1, S 1 (λ(D 2 z)) is Laplace operator, which is studied widely in [15][16][17][18][19] Monge-Ampère operator, which is studied extensively in [20][21][22][23][24][25][26]. e k-Hessian equations play an important role in differential geometry [27,28]. ey can describe Weingarten curvature or reflector shape [29] and some phenomena of non-equilibrium phase transitions and statistical physics [30,31]. In 2019, Zhang and Feng [32] considered the existence and asymptotic behavior of k-convex solution to the boundary blow up problem for the following k-Hessian equation: where b and φ are smooth positive functions and Ω is a smooth, bounded, strictly, convex domain of R N with N ≥ 2.
In 2020, by means of monotone iterative technique, Wang, Yang, Zhang, and Baleanu [1] established the existence of the entire positive bounded radial solutions and entire positive blow up radial solutions for the following k-Hessian system: What interests us in this paper is whether the similar results in [1] hold under the Keller-Osserman type conditions (4) of the k-Hessian system (8). Motivated by the idea, we reconsider the k-Hessian system (8) under the Keller-Osserman type conditions: As a continuation of previous work, in this paper, by employing monotone iterative method, we establish some new existence results on positive radial solutions of the k-Hessian system (8) under the cases of finite, infinite, and semifinite. For details of the monotone iterative method, see [1,7,8,[33][34][35][36]. In addition, our results complement the work in [1] and extend works of many authors in [7, 8, 15-17, 37, 38].

Preliminaries
For the convenience of subsequent proofs, we list a definition, notations, assumptions, and related lemmas.
We first recall the classification of solutions.

Entire Positive Bounded Radial Solution
In this section, we investigate the entire positive bounded radial solution of system (8), and the main results are as follows.
Theorem 1. We assume that (N1),(N2) hold, then system (8) has an entire positive bounded radial solution Proof. Obviously, the solutions of system (33) are equivalent to the solutions of the following system: We define the sequences y (m) 1 (r) m≥0 and y (m) 2 (r) m≥0 on [0, ∞) by Using the same arguments as in [1], we get the sequences y (m) 1 (r) m≥0 and y (m) 2 (r) m≥0 that are increasing for ∀m ≥ 0 and

Journal of Mathematics
Next, integrating the above inequality from 0 to r, we get Consequently, It follows from the above inequality and the fact that H is a bijection with the inverse function H − 1 strictly increasing on [0, H(∞)) that Since φ, ψ, y (m) 1 (r) m≥0 , and y (m) 2 (r) m≥0 are increasing, by means of Lemma 1, (N2), (35), (39), we get Similar to the above, by Lemma 3, (N2), (40), (41), we obtain 6 Journal of Mathematics Next, we have Multiplying the last inequality in (44) by (y (m) 1 (r)) ′ and the last inequality in (45) by (y (m) 2 (r)) ′ , we arrive at Journal of Mathematics By (N1) and Lemma 1, integrating (46) and (47) from 0 to r, we get ds.
From the above two inequalities, we easily deduce that (53) Integrating (52) and (53) from 0 tor, we arrive at ds.