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

Abstract

This paper investigates the positive radial solutions of a nonlinear -Hessian system. where is a nonlinear operator and , , , are continuous functions. With the help of Keller–Osserman type conditions and monotone iterative technique, the positive radial solutions of the above problem are given in cases of finite, infinite, and semifinite. Our results complement the work in by Wang, Yang, Zhang, and Baleanu (Radial solutions of a nonlinear -Hessian system involving a nonlinear operator, Commun. Nonlinear Sci. Numer. Simul. 91(2020), 105396).

1. 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 field 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 , and Osserman [6] studied the existence of solutions for the nonlinear differential inequality .

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 systemwhere , , , 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 -Hessian operator as follows:where is the Hessian matrix of and the is the vector, which consists of eigenvalues of . It is easy to observe that is a family of operators including many well-known operators. For example, for , is Laplace operator, which is studied widely in [15–19], for , is Monge-Ampère operator, which is studied extensively in [20–26].

The -Hessian equations play an important role in differential geometry [27, 28]. They 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 -convex solution to the boundary blow up problem for the following -Hessian equation:where and are smooth positive functions and is a smooth, bounded, strictly, convex domain of with .

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 -Hessian system:where , is a nonlinear operator in the set

What interests us in this paper is whether the similar results in [1] hold under the Keller–Osserman type conditions (4) of the -Hessian system (8). Motivated by the idea, we reconsider the -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 -Hessian system (8) under the cases of finite, infinite, and semifinite. For details of the monotone iterative method, see [1, 7, 8, 33–36]. In addition, our results complement the work in [1] and extend works of many authors in [7, 8, 15–17, 37, 38].

2. Preliminaries

For the convenience of subsequent proofs, we list a definition, notations, assumptions, and related lemmas.

We first recall the classification of solutions.

Definition 1 (see [8]). A solution of system (8) is called an entire bounded solution if condition (12) holds; it is called an entire blow up solution if condition (13) holds; it is called a semifinite entire blow up solution if condition (14) or (15) holds.
Finite case: both components , are bounded, namelyInfinite case: both components , are blow up, namelySemifinite case: one of the components is bounded, whereas the other is blow up, namelyorNext, we introduce the notations as follows: and are suitably chosen,We see thatwhich mean that has the inverse function on , has the inverse function on , and has the inverse function on .
We assume that and satisfy the following assumptions:
are increasing for each variable and for all ;
For fixed parameters , there exist such thatwhere , , , , , ; ; ; ; ; ; ; ; .

Lemma 1 (see [39]). If , letting , then
(1): has a nonnegative increasing inverse mapping ;
(2): when , one has(3): when , one has

Lemma 2 (see [40]). We assume that is radially symmetric and , then the function with belongs to , andwhere and .

Lemma 3 (see [1]). is a radial solution of the k-Hessian system (4) if and only if is a solution of the following ordinary differential system:

3. 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 , 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 and on byUsing the same arguments as in [1], we get the sequences and that are increasing for andNext, integrating the above inequality from 0 to , we getConsequently,It follows from the above inequality and the fact that is a bijection with the inverse function strictly increasing on thatSince , , , and are increasing, by means of Lemma 1, , (35), (39), we getSimilar to the above, by Lemma 3, , (40), (41), we obtainNext, we haveMultiplying the last inequality in (44) by and the last inequality in (45) by , we arrive atBy and Lemma 1, integrating (46) and (47) from 0 to , we getBy (48) and (49), we getFrom the above two inequalities, we easily deduce thatIntegrating (52) and (53) from 0 to, we arrive atNow, the above two inequalities can be written asandFinally, by the fact that and are strictly increasing on and separately, we getThen, we prove that the sequences and are bounded on for arbitrary . Indeed, sinceIt follows thatwhere and are positive constants. Moreover, from (50) and (51), we can deduce that and are bounded on for arbitrary . Thus, the sequences and are bounded and equicontinuous on for arbitrary . By Arzela–Ascoli theorem, there exist subsequences of and converging uniformly to and on . Since and are increasing on , then and converge uniformly to and on . By the arbitrariness of , we deduce that is an entire positive radial solution of system (32). Thus, by Lemma 3, we get that is an entire positive radial solution of system (8). Repeating the proof in [1], we get . Thus, by Lemma 3, we deduce that system (8) has an entire positive radial solution . This completes the proof.

Theorem 2. We assume , , , and hold, then system (8) has an entire positive bounded radial solution such that

Proof. On the basis of , , by a proof similar to one of Theorem 1, it is easy to prove that (8) has an entire positive radial solution . Moreover, it follows from (56), (57), and thatSince and are strictly increasing on and separately, we getLetting in the above two inequalities, we obtainLetting in (35), we getThen, it follows from , , , and Lemma 1 thatAs in the preceding lines, we can proveThis completes the proof.

4. Entire Positive Blow up Radial Solution

In this section, we investigate entire positive blow up radial solution of system (8), and the main result is as follows:

Theorem 3. We assume that , , , and hold, then the nonlinear -Hessian system (8) has an entire positive blow up radial solution .

Proof. It follows from the conditions , that a similar proof of Theorem 1 ensures that system (8) has an entire positive radial solution . Moreover, it follows from (56) and (57) thatSince and are strictly increasing on and separately, we getWhen holds, we see that . By the condition , letting in the above two inequalities, we obtainBy the condition , letting in the above two inequalities, we obtainThen, it follows from , (72), and (73) thatConsequently,Therefore, system (33) has an entire positive blow up radial solution . By Lemma 3, system (8) has an entire positive blow up radial solution . The proof is completed.

5. Semifinite Entire Positive Blow up Radial Solution

In this section, we investigate semifinite entire positive blow up radial solution of system (8), and the main results are as follows:

Theorem 4. We assume that , , , and hold, then the nonlinear -Hessian system (8) has one semifinite entire positive blow up radial solution .

Proof. In view of , , the same arguments as in Theorem 1, we can know that system (8) has an entire positive radial solution . By , (79), and (81), we getwhich imply thatMoreover, by , (69), and (73), we obtainwhich imply thatTherefore, system (33) has a positive semifinite entire blow up radial solution . By Lemma 3, system (8) has a positive semifinite entire blow up radial solution . The proof is finished.

Theorem 5. We assume that , , , and hold, then the nonlinear -Hessian system (8) has a positive semifinite entire blow up radial solution .

Proof. Same as Theorem 4, we can know that system (8) has an entire positive radial solution . By , (81), and (83), we getwhich imply thatMoreover, by , (68), and (72), we obtainwhich imply thatTherefore, system (33) has a positive semifinite entire blow up radial solution . By Lemma 3, system (8) has a positive semifinite entire blow up radial solution . The proof is finished.

6. Example

Example 1. we considerLetting , , then . Here, , , , , then and are increasing for each variable which satisfies . Obviously, when , we have , , , , , ,which mean that is satisfied. By Theorem 1, the 4-Hessian system (97) has an entire positive radial solution .

Example 2. we considerLetting , , then . Here , , , , then and are increasing for each variable which satisfies . Obviously, when , we have , , , , , , which mean that is satisfied. After a simple calculation, one haswhich mean that is satisfied. Then, we havewhich mean that is satisfied. By Theorem 3, the 4-Hessian system (101) has an entire positive blow up radial solution .

7. Conclusion

The -Hessian equations, a deep extension of the Laplace equation, have played an important role in mathematics and other applied sciences. In this paper, we studied the radial solutions of a class of -Hessian system involving a nonlinear operator. The Keller–Osserman type conditions are also crucial throughout the proof process. By employing the monotone iterative method, we establish some new existence results on positive radial solutions of the -Hessian system (8) in cases of finite, infinite, and semifinite. Our results complement the work in [1] and extend the existing results [7, 8, 15–17, 38], which is a meaningful contribution to the topic of nonlinear elliptic system.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors equally contributed this manuscript and approved the final version.

Acknowledgments

The authors thank the referees for their useful comments on our work that led to its improvement. The work was supported by NSF of Shanxi Province,China (No.20210302123339), the Graduate Education and Teaching Innovation Project of Shanxi, China (No.2021YJJG142), and the Graduate Innovation Program of Shanxi, China (No.2021Y488).