© Wuhan University 2024

0 Introduction

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

$S_k(D^{\mathrm{2}}(-\phi ))=\frac{|\mathit{x}{|}^{\lambda -\mathrm{2}}}{(\mathrm{1}+|\mathit{x}{|}^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}}{\phi }^q,\mathit{x}\in \mathrm{\Omega }$(1)

where $S_k(D^{\mathrm{2}}(-\phi ))$ is the k-Hessian operator, $q>k>\mathrm{1},\lambda >\mathrm{0},n>\mathrm{2}k,k\in \mathbb{N}$, and $\mathrm{\Omega }$ is a suitable bounded domain in ${\mathbb{R}}^n$. The operators $\left\{S_k:k=\mathrm{1},\cdots ,n\right\}$ are a family of operators including Laplace operator when $k=\mathrm{1}$ and Monge-Ampère operator when $k=n$. The $k$-Hessian equation admits several significant applications in fluid mechanic, geometric problem and other applied subjects. For example, the $k$-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 $k=\mathrm{1}$, equation (1) reduces to the classical Matukuma equation^[4]

$\mathrm{\Delta }\phi +\frac{|\mathit{x}{|}^{\lambda -\mathrm{2}}}{(\mathrm{1}+|\mathit{x}{|}^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}}{\phi }^q=\mathrm{0},\mathit{x}\in {\mathbb{R}}^{\mathrm{n}}$(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 ${\mathbb{R}}^{\mathrm{3}}$, and displayed that there exist three different types of solutions: M-solutions (singular at $r=\mathrm{0}$), E-solutions (regular at $r=\mathrm{0}$) and F-solutions (whose existence begins away from $r=\mathrm{0}$). By applying an iterative method^[11], Wang et al^[12] generalized asymptotic expansions of M-solutions of (2) from $n=\mathrm{3}$ to $n>\mathrm{3}$. This iterative method also could be used for the Hénon equation $-{\mathrm{\Delta }}_p\phi =|\mathit{x}{|}^{\sigma }{\phi }^q$ with $p=\mathrm{2}$, 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 $p=\mathrm{2}$ to $\mathrm{1}<p<N$. When $\lambda =\mathrm{2}$, equation (2) reduces to

$\mathrm{\Delta }\phi +\frac{\mathrm{1}}{(\mathrm{1}+|\mathit{x}{|}^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}}{\phi }^q=\mathrm{0},\mathit{x}\in {\mathbb{R}}^n$(3)

which was presented by astrophysicist Matukuma^[17] for the description of certain stellar globular clusters in a steady state, where $q>\mathrm{1},n=\mathrm{3}$, and $\phi >\mathrm{0}$ is the gravitational potential. Li^[18] gave a nearly complete description of the structure of positive radial solutions to (3) when $\mathrm{1}<q<\mathrm{5}$ 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 $n\ge \mathrm{3}$ and $\mathrm{1}<q<(n+\mathrm{2})/(n-\mathrm{2})$. We refer to Refs. [20-22] about the Matukuma equation.

When $k>\mathrm{1}$, Sánchez and Vergara^[23] considered the problem

$\{\begin{array}{l}{S}_k(D^{\mathrm{2}}\phi )=\lambda |\mathit{x}{|}^{\sigma }{\left(\mathrm{1}-\phi \right)}^q,\mathit{x}\in B\\ \phi <\mathrm{0},\mathit{x}\in B\\ \phi =\mathrm{0},\mathit{x}\in \partial B\end{array}$(4)

where $B$ is the unit ball in ${\mathbb{R}}^n,n>\mathrm{2}k,k\in \mathbb{N},\lambda >\mathrm{0},q>k$, and $\sigma \ge \mathrm{0}$. 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

$\{\begin{array}{l}{S}_k(D^{\mathrm{2}}\phi )=\mu \frac{|\mathit{x}{|}^{\lambda -\mathrm{2}}}{(\mathrm{1}+|\mathit{x}{|}^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}}{\left(\mathrm{1}-\phi \right)}^q,\mathit{x}\in B\\ \phi <\mathrm{0},\mathit{x}\in B\\ \phi =\mathrm{0},\mathit{x}\in \partial B\end{array}$

where $B$ denotes the unit ball in ${\mathbb{R}}^n,n>\mathrm{2}k,k\in \mathbb{N},\mu >\mathrm{0},q>k$, and $\lambda \ge \mathrm{2}$. 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 $k$-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 $k=\mathrm{1}$ 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 $k>\mathrm{1}$, 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 $p=n-\mathrm{2}+\lambda .$

When $p>[(n-\mathrm{2}k)q]/k$, i.e., $\mathrm{1}<q<[(n-\mathrm{2}+\lambda )k]/(n-\mathrm{2}k)$, we firstly give some a priori estimates in Theorem 1. Similar to Refs. [10,13], we find the M-solution admits a splitting form: $\phi =S+\mathrm{\Theta }$, where $S$ is the singular term and $\mathrm{\Theta }$ is the regular one. To derive more accurate asymptotic expansions of $S$ and $\mathrm{\Theta }$, we introduce a new parameter $\omega :=p-[(n-\mathrm{2}k)q]/k$, and choose $k_{\mathrm{0}}\in \mathbb{N}$ such that $\mathrm{2}{k}_{\mathrm{0}}<(n-\mathrm{2}k)/k\le \mathrm{2}(k_{\mathrm{0}}+\mathrm{1})$ in Theorem 1, 2 and Theorem 3. Furthermore, we separate the range $p>[(n-\mathrm{2}k)q]/k$ into three subcases: (i) $[(n-\mathrm{2}k)q]/k<p<[(n-\mathrm{2}k)(q+\mathrm{1})]/k$; (ii) $p=[(n-\mathrm{2}k)(q+\mathrm{1})]/k$; (iii) $p>[(n-\mathrm{2}k)(q+\mathrm{1})]/k$ in Section 2. It is worth noting that we require more precise ranges of $(n-\mathrm{2}k)/k$ and $\omega$ 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 $S$ and $\mathrm{\Theta }$ near the origin.

The case with Laplace operator (i.e., $k=\mathrm{1}$) and weight term $K(r)=r^{\lambda -\mathrm{2}}/(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}$ has been examined in Ref. [12]; the case with $p$-Laplace operator and weight term $K(r)=r^{\sigma }$ has been discussed in Ref. [16]. These provided us the significant references to solve problems for the case with $k$-Hessian operator (i.e., $k>\mathrm{1}$) and weight term $K(r)=r^{\lambda -\mathrm{2}}/(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}$. The schemes we used in current paper are as follows.

First, motivated by Ref. [12], we replace Laplace operator (i.e., $k=\mathrm{1}$) with $k$-Hessian operator (i.e., $k>\mathrm{1}$), which leads to some computational challenges as follows. Since $\mathrm{\Delta }\phi =r^{\mathrm{1}-n}(r^{n-\mathrm{1}}{\phi }^{\text{'}}{)}^{\text{'}}$, we find that the exponent of ${\phi }^{\text{'}}$ is $\mathrm{1}$. A straightforward ordinary differential equation (ODE) analysis implies that ${\phi }^{\text{'}}=-\frac{\mathrm{1}}{r^{n-\mathrm{1}}}\left[\widehat{c}+{\int }_{\mathrm{0}}^r{s}^{n-\mathrm{1}}K(s){\phi }^q(s)\mathrm{d}s\right],$ where $K(r)=r^{\lambda -\mathrm{2}}/(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}$. Based on the Taylor expansion for $K(r)$, the asymptotic expansion of ${\phi }^{\text{'}}$ could be obtained. However, since $S_k(D^{\mathrm{2}}(-\phi ))=c_{n,k}{r}^{\mathrm{1}-n}(r^{n-k}(-{\phi }^{\text{'}}{)}^k{)}^{\text{'}},c_{n,k}=\left(\begin{array}{c}n\\ k\end{array}\right)/n,$ we have that the exponent of ${\phi }^{\text{'}}$ is $k$, and then deduce

${\phi }^{\text{'}}=-\frac{\mathrm{1}}{r^{(n-k)/k}}{\left[\overline{c}+{\int }_{\mathrm{0}}^r{c}_{n,k}^{-\mathrm{1}}{s}^{n-\mathrm{1}}K(s){\phi }^q(s)\mathrm{d}s\right]}^{\mathrm{1}/k}:=-\frac{\mathrm{1}}{r^{(n-k)/k}}{\overline{K}}^{\mathrm{1}/k}(r)$(5)

Not only do we need to use Taylor expansion for $K(r)$, but we also need to use Taylor expansion for $\overline{K}(r)$. The repeated application of Taylor expansion makes the calculation more complex. Inspired by Ref. [16], we replace weight term $K(r)=r^{\sigma }$ with $K(r)=r^{\lambda -\mathrm{2}}/(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}$. Since ${\mathrm{\Delta }}_p\phi =r^{\mathrm{1}-n}(r^{n-\mathrm{1}}|{\phi }^{\text{'}}{|}^{p-\mathrm{2}}{\phi }^{\text{'}}{)}^{\text{'}}$, we have the exponent of ${\phi }^{\text{'}}$ is $p-\mathrm{1}$, and then derive

${\phi }^{\text{'}}=-\frac{\mathrm{1}}{r^{(n-\mathrm{1})/(p-\mathrm{1})}}{\left[\tilde{c}+{\int }_{\mathrm{0}}^r{s}^{n-\mathrm{1}}K(s){\phi }^q(s)\mathrm{d}s\right]}^{\mathrm{1}/(p-\mathrm{1})}:=-\frac{\mathrm{1}}{r^{(n-\mathrm{1})/(p-\mathrm{1})}}{\overline{\overline{K}}}^{\mathrm{1}/(p-\mathrm{1})}(r)$(6)

In a similar manner with (5), we also need to use Taylor expansion for $\overline{\overline{K}}$ corresponding to $\overline{K}$ in (5), however, in this paper we require to use Taylor expansion for $K(r)$ once more. Second, we shall state that the precise ranges for $(n-\mathrm{2}k)/k$ and $\mathrm{\omega }$ are necessary in Theorem 1. Wang and Zhang^[16] obtained that $\phi$ is in the form of

$\phi =\frac{c}{r^{(n-p)/(p-\mathrm{1})}}\left[\mathrm{1}+{\displaystyle \sum _{i=\mathrm{1}}^{n_{\mathrm{0}}+\mathrm{1}}}{\widehat{a}}_i{r}^{i\mu }+o(r^{(n_{\mathrm{0}}+\mathrm{1})\mu })\right]:=\frac{c}{r^{(n-p)/(p-\mathrm{1})}}\left[\mathrm{1}+D_{\mathrm{1}}+o(r^{(n_{\mathrm{0}}+\mathrm{1})\mu })\right],$

where $\mu =N+\sigma -[(N-p)q]/(p-\mathrm{1})$, $\sigma >p$, and $n_{\mathrm{0}}$ is a positive integer.

When $n_{\mathrm{0}}\mu <(n-p)/(p-\mathrm{1})<(n_{\mathrm{0}}+\mathrm{1})\mu$, they split the term $r^{-(n-p)/(p-\mathrm{1})}{D}_{\mathrm{1}}$ into singular term (i.e., ${\displaystyle \sum _{i=\mathrm{1}}^{n_{\mathrm{0}}}}{\widehat{a}}_i{r}^{i\mu -(n-p)/(p-\mathrm{1})}$) and regular term (i.e., ${\widehat{a}}_{n{}_{\mathrm{0}}{}^{}+\mathrm{1}}{r}^{(n{}_{\mathrm{0}}{}^{}+\mathrm{1})\mu -(n-p)/(p-\mathrm{1})}$). In this paper, we obtain for $(n-\mathrm{2}k)/k<(n_{\mathrm{0}}+\mathrm{1})\omega$,

$\phi =\frac{c}{r^{(n-\mathrm{2}k)/k}}\left[\mathrm{1}+{\displaystyle \sum _{i=\mathrm{1}}^{n_{\mathrm{0}}+\mathrm{1}}}{\widehat{a}}_{i\mathrm{0}}{r}^{i\omega }+{\displaystyle \sum _{i=\mathrm{1}}^{n_{\mathrm{0}}}}{\displaystyle \sum _{j=\mathrm{1}}^{{\eta }_i}}{\widehat{a}}_{ij}{r}^{i\omega +\mathrm{2}j}+o(r^{(n_{\mathrm{0}}+\mathrm{1})\omega })\right]:=\frac{c}{r^{(n-\mathrm{2}k)/k}}\left[\mathrm{1}+D_{\mathrm{2}}+D_{\mathrm{3}}+o(r^{(n_{\mathrm{0}}+\mathrm{1})\omega })\right],$

where ${\eta }_i=(n_{\mathrm{0}}-i+\mathrm{1})k_{\mathrm{0}}+n_{\mathrm{0}}-i$ and ${\widehat{a}}_{ij}$ are some constants depending upon $c,\lambda ,q,k,k_{\mathrm{0}},n,$ and $n_{\mathrm{0}}$. The presence of the term $D_{\mathrm{3}}$ is due to the repeated use of Taylor expansion. In a similar manner with $r^{-(n-p)/(p-\mathrm{1})}{D}_{\mathrm{1}}$, we shall split the term $r^{-(n-\mathrm{2}k)/k}{D}_{\mathrm{2}}$ into singular and regular terms when $n_{\mathrm{0}}\omega <(n-\mathrm{2}k)/k<(n_{\mathrm{0}}+\mathrm{1})\omega$. But this range is no longer sufficient to divide the term $r^{-(n-\mathrm{2}k)/k}{D}_{\mathrm{3}}$ into singular and regular terms. To solve this problem, we require the following precise range of $(n-\mathrm{2}k)/k$:

$\mathrm{2}(n_{\mathrm{0}}+\mathrm{1})k_{\mathrm{0}}+\mathrm{2}{n}_{\mathrm{0}}<(n-\mathrm{2}k)/k<(n_{\mathrm{0}}+\mathrm{1})\omega .$

On the other hand, when $(n-\mathrm{2}k)/k=(n_{\mathrm{0}}+\mathrm{1})\omega$, since the fact that the size of the exponents $(n-\mathrm{2}k)/k$ and $i\omega +\mathrm{2}j$ could not be determined, we introduce a precise range on $\omega$, i.e., $\mathrm{2}{k}_{\mathrm{0}}+\mathrm{2}{n}_{\mathrm{0}}/(n_{\mathrm{0}}+\mathrm{1})<\omega \le \mathrm{2}(k_{\mathrm{0}}+\mathrm{1})$.

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 $k$-Hessian operator and maximal solution.

Definition 1   Let $\vartheta \in C^{\mathrm{2}}(\mathrm{\Omega }),\mathrm{1}\le k\le n,k\in \mathbb{N}$ and $\Lambda =({\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}},\cdots ,{\lambda }_n)$ be the eigenvalues of the Hessian matrix $(D^{\mathrm{2}}\vartheta )$. Then the $k$-Hessian operator is given by the formula $S_k(D^{\mathrm{2}}\vartheta )=P_k(\Lambda )=\sum _{\mathrm{1}\le i_{\mathrm{1}}<\cdots <i_k\le n}{\lambda }_{i_{\mathrm{1}}}\cdots {\lambda }_{i_k},$ where $P_k(\Lambda )$ is the $k$-th elementary symmetric polynomial in the eigenvalues $\Lambda$, see Ref. [33].

Note that the $k$-Hessian operators are fully nonlinear for $k\ne \mathrm{1}$. Furthermore, they are not elliptic in general, unless they are restricted to the class

${\mathrm{\Phi }}_{\vartheta }^k(\mathrm{\Omega })=\left\{\vartheta \in C^{\mathrm{2}}(\mathrm{\Omega })\bigcap C^{\mathrm{1}}(\overline{\mathrm{\Omega }}):S_i(D^{\mathrm{2}}\vartheta )\ge \mathrm{0}\mathrm{i}\mathrm{n}\mathrm{\Omega },i=\mathrm{1},\cdots ,k\right\}$(7)

Observe that ${\mathrm{\Phi }}_{\vartheta }^k(\mathrm{\Omega })$ belongs to the class of subharmonic functions. Moreover, it follows from the maximum principle^[33] that the functions in ${\mathrm{\Phi }}_{\vartheta }^k(\mathrm{\Omega })$ are negative in $\mathrm{\Omega }$.

To investigate positive solutions of (1), under the change of variable $\phi =-\vartheta$, it is not hard to obtain $S_k(D^{\mathrm{2}}\vartheta )={(-\mathrm{1})}^k{S}_k(D^{\mathrm{2}}\phi )$ by the $k$-homogeneity of the $k$-Hessian operator ^[23,34].

Definition 2   A function $\phi =-\vartheta \in {\mathrm{\Phi }}_{\vartheta }^k(\mathrm{\Omega })$ is called a supersolution (resp. subsolution) of (1) if

$S_k(D^{\mathrm{2}}(-\phi ))\ge (\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.\le )\frac{|\mathit{x}{|}^{\lambda -\mathrm{2}}}{(\mathrm{1}+|\mathit{x}{|}^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}}{\phi }^q\mathrm{i}\mathrm{n}\mathrm{\Omega }$

Observe that the trivial function $\phi \equiv \mathrm{0}$ is always a subsolution.

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

Remark 1   Introduction of functional space ${\mathrm{\Phi }}_{\vartheta }^k(\mathrm{\Omega })$ is to ensure that the $k$-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 $K$ be a positive function in $C^{\mathrm{1}}({\mathbb{R}}^{+})$ such that $r^{(n-k)/k}K(r)$ is bounded as $r\to +\mathrm{\infty }$. Suppose that $\phi :(R_{-},R)\to (\mathrm{0},+\mathrm{\infty })$ is a maximal solution of the problem

$c_{n,k}{r}^{\mathrm{1}-n}(r^{n-k}(-{\phi }^{\text{'}}{)}^k{)}^{\text{'}}=K(r){\phi }^q,q>k>\mathrm{1},n>\mathrm{2}k$(8)

where $\mathrm{0}\le R_{-}<R<+\mathrm{\infty }$. Now we introduce the space of functions ${\mathrm{\Phi }}^k$ defined on $\phi =-\vartheta$ and $\mathrm{\Omega }=(R_{-},R)$ as in (7), for problem (8):

${\mathrm{\Phi }}^k=\left\{\phi \in C^{\mathrm{2}}((R_{-},R))\bigcap C^{\mathrm{1}}([R_{-},R]):(r^{n-i}(-{\phi }^{\text{'}}{)}^i{)}^{\text{'}}\ge \mathrm{0}\mathrm{i}\mathrm{n}(R_{-},R),i=\mathrm{1},\cdots ,k\right\}$

Note that the functions in ${\mathrm{\Phi }}^k$ are non-negative on $[R_{-},R]$. If $(r^{n-i}(-{\phi }^{\text{'}}{)}^i{)}^{\text{'}}>\mathrm{0}$$$for every $i=\mathrm{1},\cdots ,k$, then any function in ${\mathrm{\Phi }}^k$ is positive and strictly decreasing on $[R_{-},R]$. Let $r_{\mathrm{0}}\in (R_{-},R)$ and

$G(r):=-r_{\mathrm{0}}^{n-k}(-{\phi }^{\text{'}}(r_{\mathrm{0}}{))}^k-{\int }_{r_{\mathrm{0}}}^r{c}_{n,k}^{-\mathrm{1}}{s}^{n-\mathrm{1}}K(s){\phi }^q(s)\mathrm{d}s\mathrm{i}\mathrm{n}(R_{-},R)$(9)

It follows that $G(r)=-r^{n-k}(-{\phi }^{\text{'}}{)}^k,\mathrm{i}.\mathrm{e}.,(-{\phi }^{\text{'}}{)}^k=-\frac{G(r)}{r^{n-k}}$ and $G^{\text{'}}(r)=-c_{n,k}^{-\mathrm{1}}{r}^{n-\mathrm{1}}K(r){\phi }^q<\mathrm{0}\mathrm{i}\mathrm{n}(R_{-},R)$. Hence the limit $G_{\mathrm{0}}:=\underset{r\to R_{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}G(r)\in (-\mathrm{\infty },+\mathrm{\infty }]$ exists.

We claim that $G_{\mathrm{0}}\le \mathrm{0}$. If not, there exists some $r^{\mathrm{*}}\in (R_{-},r_{\mathrm{0}})$ such that $G(r)>G(r^{\mathrm{*}})>\mathrm{0}$ in $(R_{-},r^{\mathrm{*}})$. Hence, $(-{\phi }^{\text{'}}{)}^k=-\frac{G(r)}{r^{n-k}}<\mathrm{0}\mathrm{i}\mathrm{n}(R_{-},r^{\mathrm{*}})$, which is impossible.

For $G_{\mathrm{0}}\le \mathrm{0}$, we claim that $R_{-}=\mathrm{0}$. We argue by contradiction. Suppose that $R_{-}>\mathrm{0}$. Then there exists $r_{\mathrm{*}}=(R_{-},r^{\mathrm{*}})$ such that $G(r_{\mathrm{*}})<\mathrm{0},{\phi }^{\text{'}}(r_{\mathrm{*}})<\mathrm{0},G(r)$ and ${\phi }^{\text{'}}(r)$ are bounded in $(R_{-},r_{\mathrm{*}})$. Therefore, $\phi (r)$ could be extended beyond $R_{-}$, which is a contradiction. Thus $R_{-}=\mathrm{0}$ and ${\phi }^{\text{'}}(r)<\mathrm{0}$ in $(\mathrm{0},R)$. Therefore, the limit $\underset{r\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\phi (r)\in (\mathrm{0},+\mathrm{\infty }]$ exists. In this case, we define $R_{\mathrm{0}}:=\mathrm{0}$ and have $R_{\mathrm{0}}=\mathrm{i}\mathrm{n}\mathrm{f}\left\{r\in (\mathrm{0},R)|{\phi }^{\text{'}}(r)<\mathrm{0}\right\}$.

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

(i) if $\underset{r\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\phi (r)=+\mathrm{\infty }$, then we call $\phi$ an M-solution;

(ii) if $\underset{r\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\phi (r)<+\mathrm{\infty }$, then we call $\phi$ an E-solution.

Remark 2   It is known Refs. [10,12] that when $G_{\mathrm{0}}>\mathrm{0}\Rightarrow R{}_{\mathrm{0}}{}^{}>R_{-}>\mathrm{0}$, there exists an F-solution. Because the functions are restricted to ${\mathrm{\Phi }}^k$, they are positive and strictly decreasing on $[R_{-},R]$. It follows that $G_{\mathrm{0}}\le \mathrm{0}$. Hence, the $k$-Hessian equation (8) has no F-solution.

In this paper, we set $K(r)=r^{\lambda -\mathrm{2}}/(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}$ with $\lambda >\mathrm{0}$, the equation (8) reduces to the radial form of (1) , i.e.,

$c_{n,k}{r}^{\mathrm{1}-n}(r^{n-k}(-{\phi }^{\text{'}}{)}^k{)}^{\text{'}}=\frac{r^{\lambda -\mathrm{2}}}{(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}}{\phi }^q,c_{n,k}=\left(\begin{array}{c}n\\ k\end{array}\right)/n$(10)

where $q>k>\mathrm{1},n>\mathrm{2}k$.

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 $p>[(n-\mathrm{2}k)q]/k$. The proof can be established by similar argument as in Ref. [4].

Lemma 1   Assume that $\beta \in \mathbb{R}$ and $f(r,\mathrm{\Theta }):(\mathrm{0},\mathrm{\infty })\times \mathbb{R}\to \mathbb{R}$ satisfies the following conditions:

(i) $f(r,\mathrm{\Theta })\in C\left((\mathrm{0},\mathrm{\infty })\times \mathbb{R}\right)$;

(ii) $f(r,\beta )\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{1}}[\mathrm{0},\mathrm{\infty })$;

(iii) there exist a number $\delta >\mathrm{0}$ and a function

$L_{\beta }:(\mathrm{0},\delta )\to [\mathrm{0},\mathrm{\infty }]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{L}_{\beta }(r)\in L^{\mathrm{1}}[\mathrm{0},\delta ]$

such that for every $r\in (\mathrm{0},\delta )$ and ${\mathrm{\Theta }}_{\mathrm{1}},{\mathrm{\Theta }}_{\mathrm{2}}\in [\beta -\delta ,\beta +\delta ]$,

$|f(r,{\mathrm{\Theta }}_{\mathrm{1}})-f(r,{\mathrm{\Theta }}_{\mathrm{2}})|\le L_{\beta }(r)|{\mathrm{\Theta }}_{\mathrm{1}}-{\mathrm{\Theta }}_{\mathrm{2}}|$

Then the initial value problem:

${\mathrm{\Theta }}^{\text{'}}=f(r,\mathrm{\Theta }),\mathrm{\Theta }(\mathrm{0})=\beta$

admits a unique solution $\mathrm{\Theta }$ on $(\mathrm{0},R)$ with $\mathrm{\Theta }(\mathrm{0}):=\underset{r\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{\Theta }=\beta$.

1.2 Transformation to Lotka-Volterra System

In this subsection we discuss the solutions $\phi$ of (8) when $r\in (\mathrm{0},R)$. Inspired by Refs. [23,24], we adopt a more general transformation

$u(t):=r^k\frac{K(r){\phi }^q}{c_{n,k}(-{\phi }^{\text{'}}{)}^k},v(t):=r\frac{-{\phi }^{\text{'}}}{\phi },r:={\mathrm{e}}^t$(11)

where ${\phi }^{\text{'}}=\mathrm{d}\phi /\mathrm{d}r$. Set $J_{\phi }:=(\mathrm{0},R)$ and $I_{\varphi }:=\mathrm{l}\mathrm{n}{J}_{\phi }$. We find that $\varphi :=(u,v):I_{\varphi }\to {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$ is a maximal solution of a non-autonomous Lotka-Volterra system

$\{\begin{array}{l}\dot{u}=u\left[p(t)-u-qv\right],\\ \dot{v}=v\left[-\frac{n-\mathrm{2}k}{k}+\frac{u}{k}+v\right],\end{array}$(12)

where $"\cdot "$ denotes differentiation with respect to $t$, ${\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$ is an invariant set of system (12), i.e., the positive $u$- and $v$- axes are invariant, and $p(t)=n+r\frac{K^{\text{'}}(r)}{K(r)}$.

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

$\phi ={\left[\frac{u(\mathrm{l}\mathrm{n}r)v{(\mathrm{l}\mathrm{n}r)}^k}{c_{n,k}{r}^{\mathrm{2}k}K(r)}\right]}^{\mathrm{1}/(q-k)}$(13)

In particular, when $K(r)=r^{\lambda -\mathrm{2}}/(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}$, we have $p(t)=n-\mathrm{2}+\lambda -\frac{\lambda {\mathrm{e}}^{\mathrm{2}t}}{\mathrm{1}+{\mathrm{e}}^{\mathrm{2}t}}$.

Define $p:=n-\mathrm{2}+\lambda$. Thus $\underset{t\to -\mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}p(t)=p$. Then the limiting system of system (12) as $t\to -\mathrm{\infty }$ could be written as

$\{\begin{array}{l}\dot{u}=u\left[p-u-qv\right],\\ \dot{v}=v\left[-\frac{n-\mathrm{2}k}{k}+\frac{u}{k}+v\right].\end{array}$(14)

2 Singular Solutions

In this section, we mainly study singular solutions $\phi$ in $(\mathrm{0},R)$ and their corresponding solutions $\varphi$ in $(-\mathrm{\infty },T)$, where $\mathrm{0}<R<+\mathrm{\infty }$ and $T=\mathrm{l}\mathrm{n}R$. To establish more precisely asymptotic expansions of singular solutions (i.e., the M-solutions) near the origin, we need to divide $p>[(n-\mathrm{2}k)q]/k$ into the following three subcases:

(a) $[(n-\mathrm{2}k)q]/k<p<[(n-\mathrm{2}k)(q+\mathrm{1})]/k$;

(b) $p=[(n-\mathrm{2}k)(q+\mathrm{1})]/k$;

(c) $p>[(n-\mathrm{2}k)(q+\mathrm{1})]/k$.

For the case $p\le [(n-\mathrm{2}k)q]/k$ 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 $p>[(n-\mathrm{2}k)q]/k$.

Lemma 2   Let $p>[(n-\mathrm{2}k)q]/k$. Then the following statements are equivalent:

(i) $\phi$ is an M-solution.

(ii) There exists a constant $c>\mathrm{0}$ such that $\phi =\frac{c}{r^{(n-\mathrm{2}k)/k}}[\mathrm{1}+o(\mathrm{1})],r\to \mathrm{0}$.

(iii) There exists a constant $c>\mathrm{0}$ such that $u(t)=\frac{k^k{c}^{q-k}}{{(n-\mathrm{2}k)}^k{c}_{n,k}}{\mathrm{e}}^{\left\{p-[(n-\mathrm{2}k)q]/k\right\}t}[\mathrm{1}+o(\mathrm{1})],v(t)=\frac{n-\mathrm{2}k}{k}[\mathrm{1}+o(\mathrm{1})],t\to -\mathrm{\infty }$.

Moreover, $\phi$ satisfies

$r^{n-k}(-{\phi }^{\text{'}}{)}^k=\overline{c}+{\int }_{\mathrm{0}}^r{c}_{n,k}^{-\mathrm{1}}{s}^{n-\mathrm{1}}K(s){\phi }^q\mathrm{d}s$(15)

where $\overline{c}={\left(\frac{(n-\mathrm{2}k)c}{k}\right)}^k$ and $c$ 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 $\phi$ has a splitting form $\phi =S+\mathrm{\Theta }$, where $S$ is the singular term in the form of $S=cP/r^{(n-\mathrm{2}k)/k}$ with $P=\mathrm{1}+o(\mathrm{1})$ as $r\to \mathrm{0}$, and $\mathrm{\Theta }$ is the regular term which satisfies

$\{\begin{array}{l}{\mathrm{\Theta }}^{\text{'}}=-\frac{\mathrm{1}}{r^{(n-k)/k}}{\left[\overline{c}+{\int }_{\mathrm{0}}^r{c}_{n,k}^{-\mathrm{1}}{s}^{n-\mathrm{1}}K(s){(\mathrm{\Theta }+S)}^q\mathrm{d}s\right]}^{\mathrm{1}/k}-S^{\text{'}},\mathrm{0}<r<R,\\ \mathrm{\Theta }(\mathrm{0}):=\underset{r\to {\mathrm{0}}^{+}}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{\Theta }=\beta \in \mathbb{R}.\end{array}$(16)

It follows that

$\begin{array}{l}{\mathrm{\Theta }}^{\text{'}}=-\frac{\mathrm{1}}{r^{(n-k)/k}}{\left[\overline{c}+{\int }_{\mathrm{0}}^r{c}_{n,k}^{-\mathrm{1}}{s}^{n-\mathrm{1}}K(s){\left(\mathrm{\Theta }+\frac{cP}{s^{(n-\mathrm{2}k)/k}}\right)}^q\mathrm{d}s\right]}^{\mathrm{1}/k}-{\left(\frac{cP}{r^{(n-\mathrm{2}k)/k}}\right)}^{\mathrm{\text{'}}}\\ =-\frac{(n-\mathrm{2}k)c}{k{r}^{(n-k)/k}}{\left[\mathrm{1}+\frac{c^q}{\overline{c}{c}_{n,k}}{\int }_{\mathrm{0}}^r{s}^{\omega -\lambda +\mathrm{1}}K(s)P^q{\left(\mathrm{1}+\frac{s^{(n-\mathrm{2}k)/k}}{cP}\mathrm{\Theta }\right)}^q\mathrm{d}s\right]}^{\mathrm{1}/k}+\frac{(n-\mathrm{2}k)c}{k{r}^{(n-k)/k}}P-\frac{c}{r^{(n-\mathrm{2}k)/k}}{P}^{\text{'}}\\ =-\frac{(n-\mathrm{2}k)c}{k{r}^{(n-k)/k}}{\left[\mathrm{1}+\frac{c^q}{\overline{c}{c}_{n,k}}{\int }_{\mathrm{0}}^r{s}^{\omega -\lambda +\mathrm{1}}K(s)P^q\left(\mathrm{1}+O\left(\frac{s^{(n-\mathrm{2}k)/k}}{P}\mathrm{\Theta }\right)\right)\mathrm{d}s\right]}^{\mathrm{1}/k}+\frac{(n-\mathrm{2}k)c}{k{r}^{(n-k)/k}}P-\frac{c}{r^{(n-\mathrm{2}k)/k}}{P}^{\text{'}}\\ =-\frac{(n-\mathrm{2}k)c}{k{r}^{(n-k)/k}}{\left[\mathrm{1}+\frac{c^q}{\overline{c}{c}_{n,k}}{\int }_{\mathrm{0}}^r{s}^{\omega -\lambda +\mathrm{1}}K(s)P^{q}ds+O(r^{\omega +(n-\mathrm{2}k)/k})\right]}^{\mathrm{1}/k}+\frac{(n-\mathrm{2}k)c}{k{r}^{(n-k)/k}}P-\frac{c}{r^{(n-\mathrm{2}k)/k}}{P}^{\text{'}}\\ =:f(r,\mathrm{\Theta })\end{array}$(17)

It is clear that if $f(r,\mathrm{\Theta })$ satisfies the assumptions in Lemma 1, then the problem (16) admits a unique solution. In the following, we will discuss three different subcases $p=[(n-\mathrm{2}k)(q+\mathrm{1})]/k$,$p>[(n-\mathrm{2}k)(q+\mathrm{1})]/k$ and $[(n-\mathrm{2}k)q]/k<p<[(n-\mathrm{2}k)(q+\mathrm{1})]/k$, and establish the expansions of $S$ and $\mathrm{\Theta }$ near the origin, respectively. In order to do so, we need the following Taylor expansions

$\begin{array}{l}{(\mathrm{1}+r)}^{\alpha }=\mathrm{1}+\alpha r+\frac{\alpha (\alpha -\mathrm{1})}{\mathrm{2}!}{r}^{\mathrm{2}}+\cdots +\frac{\alpha (\alpha -\mathrm{1})\cdots (\alpha -h+\mathrm{1})}{h!}{r}^h+o(r^h),\\ \frac{\mathrm{1}}{(\mathrm{1}+r^{\mathrm{2}}{)}^{\lambda /\mathrm{2}}}=\mathrm{1}-\frac{\lambda }{\mathrm{2}}{r}^{\mathrm{2}}+\cdots +{(-\mathrm{1})}^{k_{\mathrm{0}}}\frac{\lambda (\lambda +\mathrm{2})\cdots (\lambda +\mathrm{2}{k}_{\mathrm{0}}-\mathrm{2})}{(\mathrm{2}{k}_{\mathrm{0}})!!}{r}^{\mathrm{2}{k}_{\mathrm{0}}}+O(r^{\mathrm{2}{k}_{\mathrm{0}}+\mathrm{2}}),\end{array}$