Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 3, June 2024
Page(s) 242 - 256
DOI https://doi.org/10.1051/wujns/2024293242
Published online 03 July 2024

© Wuhan University 2024

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

The purpose of this paper is to inquire about asymptotic behavior of radially positive solutions to the -Hessian equation with a Matukuma-type source:

(1)

where is the k-Hessian operator, , and is a suitable bounded domain in . The operators are a family of operators including Laplace operator when and Monge-Ampère operator when . The -Hessian equation admits several significant applications in fluid mechanic, geometric problem and other applied subjects. For example, the -Hessian equation is closely related to non-equilibrium phase transitions and statistical physics[1], the problem of prescribing the Gauss curvature of a hypersurface[2] and to the Monge-Ampère equation, which is of interest in complex geometry[3].

When , equation (1) reduces to the classical Matukuma equation[4]

(2)

The existence and nonexistence of positive solutions to (2) could be referred to Refs. [5-9]. Batt et al[10] established a comprehensive theory of radially positive solutions to (2) in , and displayed that there exist three different types of solutions: M-solutions (singular at ), E-solutions (regular at ) and F-solutions (whose existence begins away from ). By applying an iterative method[11], Wang et al[12] generalized asymptotic expansions of M-solutions of (2) from to . This iterative method also could be used for the Hénon equation with , where the accurate asymptotic expansions of M-solutions was systematically derived in Ref. [13]. It is worth noting that the results obtained in Ref. [13] are more precise than those in Refs. [14,15]. Recently, Wang and Zhang[16] extended the work of Ref. [13] from to . When , equation (2) reduces to

(3)

which was presented by astrophysicist Matukuma[17] for the description of certain stellar globular clusters in a steady state, where , and is the gravitational potential. Li[18] gave a nearly complete description of the structure of positive radial solutions to (3) when and proved a symmetry result for general nonlinear elliptic equations. Yanagida[19] established the uniqueness of positive radial entire solution with finite total mass and obtained its explicit structure for and . We refer to Refs. [20-22] about the Matukuma equation.

When , Sánchez and Vergara[23] considered the problem

(4)

where is the unit ball in , and . The existence, multiplicity and uniqueness of radially symmetric bounded solutions to (4) were investigated by a dynamical systems approach. Lately, Miyamoto et al[24] extended the problem (4) into

where denotes the unit ball in , and . Combining dynamical-systems tools, the intersection number between a singular and a regular solution and the super/subsolution method, the existence and multiplicity of solutions for the above problem were obtained. The problems with -Hessian operator have attracted lots of attention, see e.g., Refs. [25-32].

It is known from Refs. [10,12] that the equation (1) with admits three different types of radially positive solutions: the F-, E- and M-solutions. Furthermore, the E- and F-solutions are regular, and the M-solutions are singular. However, when , it turns out that the equation (1) only has the E- and M-solutions, see Section 1.1. From the above literatures, the study of M-solutions to (1) is quite scarce. Hence, we shall pay our attention to the existence and asymptotic behavior of the singular solution (i.e., the M-solutions). To this end, let

When , i.e., , we firstly give some a priori estimates in Theorem 1. Similar to Refs. [10,13], we find the M-solution admits a splitting form: , where is the singular term and is the regular one. To derive more accurate asymptotic expansions of and , we introduce a new parameter , and choose such that in Theorem 1, 2 and Theorem 3. Furthermore, we separate the range into three subcases: (i) ; (ii) ; (iii) in Section 2. It is worth noting that we require more precise ranges of and for the subcase (i), which is the most complicated and difficult case in these three subcases. Combining a priori estimates with an iterative method of Refs. [10,11], we could obtain the precise asymptotic expansions of and near the origin.

The case with Laplace operator (i.e., ) and weight term has been examined in Ref. [12]; the case with -Laplace operator and weight term has been discussed in Ref. [16]. These provided us the significant references to solve problems for the case with -Hessian operator (i.e., ) and weight term . The schemes we used in current paper are as follows.

First, motivated by Ref. [12], we replace Laplace operator (i.e., ) with -Hessian operator (i.e., ), which leads to some computational challenges as follows. Since , we find that the exponent of is . A straightforward ordinary differential equation (ODE) analysis implies that where . Based on the Taylor expansion for , the asymptotic expansion of could be obtained. However, since we have that the exponent of is , and then deduce

(5)

Not only do we need to use Taylor expansion for , but we also need to use Taylor expansion for . The repeated application of Taylor expansion makes the calculation more complex. Inspired by Ref. [16], we replace weight term with . Since , we have the exponent of is , and then derive

(6)

In a similar manner with (5), we also need to use Taylor expansion for corresponding to in (5), however, in this paper we require to use Taylor expansion for once more. Second, we shall state that the precise ranges for and are necessary in Theorem 1. Wang and Zhang[16] obtained that is in the form of

where , , and is a positive integer.

When , they split the term into singular term (i.e., ) and regular term (i.e., ). In this paper, we obtain for ,

where and are some constants depending upon and . The presence of the term is due to the repeated use of Taylor expansion. In a similar manner with , we shall split the term into singular and regular terms when . But this range is no longer sufficient to divide the term into singular and regular terms. To solve this problem, we require the following precise range of :

On the other hand, when , since the fact that the size of the exponents and could not be determined, we introduce a precise range on , i.e., .

1 Preliminaries

1.1 Classification of Positive Solutions

In this subsection, we will separate radially positive solutions of a more general problem including (1) into two distinct types. Firstly, we state the definitions of the -Hessian operator and maximal solution.

Definition 1   Let and be the eigenvalues of the Hessian matrix . Then the -Hessian operator is given by the formula where is the -th elementary symmetric polynomial in the eigenvalues , see Ref. [33].

Note that the -Hessian operators are fully nonlinear for . Furthermore, they are not elliptic in general, unless they are restricted to the class

(7)

Observe that belongs to the class of subharmonic functions. Moreover, it follows from the maximum principle[33] that the functions in are negative in .

To investigate positive solutions of (1), under the change of variable , it is not hard to obtain by the -homogeneity of the -Hessian operator [23,34].

Definition 2   A function is called a supersolution (resp. subsolution) of (1) if

Observe that the trivial function is always a subsolution.

Definition 3   We say that a function is a maximal solution of (1) if is a solution of (1) and, for every subsolution of (1), we have .

Remark 1   Introduction of functional space is to ensure that the -Hessian operators are elliptic. Then the maximum principle and the super/subsolutions method could be applied to investigate existence of the solutions to (1).

Let be a positive function in such that is bounded as . Suppose that is a maximal solution of the problem

(8)

where . Now we introduce the space of functions defined on and as in (7), for problem (8):

Note that the functions in are non-negative on . If for every , then any function in is positive and strictly decreasing on . Let and

(9)

It follows that and . Hence the limit exists.

We claim that . If not, there exists some such that in . Hence, , which is impossible.

For , we claim that . We argue by contradiction. Suppose that . Then there exists such that and are bounded in . Therefore, could be extended beyond , which is a contradiction. Thus and in . Therefore, the limit exists. In this case, we define and have .

The solutions of (8) could be classified as follows:

(i) if , then we call an M-solution;

(ii) if , then we call an E-solution.

Remark 2   It is known Refs. [10,12] that when , there exists an F-solution. Because the functions are restricted to , they are positive and strictly decreasing on . It follows that . Hence, the -Hessian equation (8) has no F-solution.

In this paper, we set with , the equation (8) reduces to the radial form of (1) , i.e.,

(10)

where .

In the following, we introduce a lemma that will be frequently used to examine the existence and uniqueness of regular term of the M-solutions when . The proof can be established by similar argument as in Ref. [4].

Lemma 1   Assume that and satisfies the following conditions:

(i) ;

(ii) ;

(iii) there exist a number and a function

such that for every and ,

Then the initial value problem:

admits a unique solution on with .

1.2 Transformation to Lotka-Volterra System

In this subsection we discuss the solutions of (8) when . Inspired by Refs. [23,24], we adopt a more general transformation

(11)

where . Set and . We find that is a maximal solution of a non-autonomous Lotka-Volterra system

(12)

where denotes differentiation with respect to , is an invariant set of system (12), i.e., the positive - and - axes are invariant, and .

Furthermore, the inverse of (11) could be characterized by

(13)

In particular, when , we have .

Define . Thus . Then the limiting system of system (12) as could be written as

(14)

2 Singular Solutions

In this section, we mainly study singular solutions in and their corresponding solutions in , where and . To establish more precisely asymptotic expansions of singular solutions (i.e., the M-solutions) near the origin, we need to divide into the following three subcases:

(a) ;

(b) ;

(c) .

For the case one can apply a dynamical system approach to obtain asymptotic expansions of singular solutions, see Refs. [10,12,16]. Firstly, we give a prior estimates of singular solution for .

Lemma 2   Let . Then the following statements are equivalent:

(i) is an M-solution.

(ii) There exists a constant such that .

(iii) There exists a constant such that .

Moreover, satisfies

(15)

where and are uniquely determined.

Proof   This proof can be established by similar argument as in Refs. [4,10], and is omitted here.

We proceed to prove that the M-solution has a splitting form , where is the singular term in the form of with as , and is the regular term which satisfies

(16)

It follows that

(17)

It is clear that if satisfies the assumptions in Lemma 1, then the problem (16) admits a unique solution. In the following, we will discuss three different subcases , and , and establish the expansions of and near the origin, respectively. In order to do so, we need the following Taylor expansions

which are crucial in the following process of proof.

Theorem 1   Let . Define and choose such that and .Then there exist numbers where depending on and such that

(i) Every M-solution has the form , where

(18)

and is the solution of (16) with the expansion

for uniquely determined constants .

(ii) Conversely, given any , there exists a unique solution of (16) such that is an M-solution, where is given by (18), and satisfies (17) with

Moreover, the solution possesses the following expansion:

Proof   (i) Let be determined by Lemma 2. We compute

Using (15), we have

Thus

where is a constant.

Repeating the above iterative process, we deduce

Hence,

Then

If , i.e., , then there exists such that when ,

Similarly, we repeat above process again and get

and

Therefore,

with some adequate constants depending on and . Integration yields

The singular term and regular term can be acquired. By induction, for the case , we may suppose that where are some constants depending upon and . In a similar manner, we compute

where and are some appropriate constants which depend on and . Hence,

Then

with some adequate constants depending upon and . By integrating, we derive

where

(19)

Thus the singular term and regular term could be obtained precisely. For the case , we can apply similar arguments to obtain its corresponding conclusion.

(ii) To prove that (16) admits a unique solution , it suffices to verify that fulfills the assumptions in Lemma 1. For the case , we find

and

due to the relations (19) between and . The other case can be similarly handled.

For the case , we obtain

and

Remark 3   The ranges of the parameters are not empty under the assumptions in Theorem 1. Since and , we can choose and so that and . By some computations, it is not difficult to find and such that and .

Theorem 2   Let and select such that . Then the following results are valid.

(i) Every M-solution has the form , where

(20)

and solves the initial value problem (16) with the following expansion

as for some uniquely determined constants .

(ii) Conversely, for any given and , there exists a unique solution of (16) such that is an M-solution, where is given by (20), and satisfies (17) with . In addition, the solution has the following expansion:

Proof   (i) Let be determined in Lemma 2. By some calculations, we have

It follows from (15) that

Hence,

(21)

It is clear that each term on the expansion of is still singular. To obtain the regular term of , we need to repeat the above arguments. Using (21), we find

Therefore,

Let Note that is integrable and exists. Therefore,

(ii) For any given and , we denote . Then

and

which satisfies the assumptions of Lemma 1, thus (16) possesses a unique solution . Hence is an M-solution. We compute

and

Theorem 3   Let . Define and choose such that . Then the following statements are true.

(i) Every M-solution has the form , where , and solves the initial value problem (16). Moreover,

for some uniquely determined constants .

(ii) Conversely, given any and , there exists a unique solution of (16) such that is an M-solution with . In this case, satisfies (17) with . In addition, the solution has the following expansion:

Proof   (i) Let be determined by Lemma 2. It follows from (15) that is integrable near the origin. We compute

Thus, we obtain for certain ,

Clearly, satisfies (16) and (17). Suppose that . It is obvious that and , which implies that and are unique. It follows from that

The expansion for could be obtained directly by integrating.

(ii) For any given and , we denote . Then and

Clearly, satisfies the assumptions in Lemma 1. Therefore, the problem (16) has a unique solution . Thus is an M-solution. We compute

and

Remark 4   We discuss the expansions of singular solutions in different intervals with p as the index in this section. For each specific subinterval of p, the singular solution is unique, and its corresponding singular and regular terms are also unique. However, for the entire interval, the singular solution is not unique, and its singular and regular terms are not unique either.

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