\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 328, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/328\hfil Regularly varying solutions]
{Regularly varying solutions with intermediate growth for  cyclic
differential systems of second order}

\author[J. Jaro\v{s}, K. Taka\^si, T. Tanigawa \hfil EJDE-2016/328\hfilneg]
{Jaroslav Jaro\v{s}, Kusano Taka\^si, Tomoyuki Tanigawa}

\address{Jaroslav Jaro\v{s} \newline
Department of Mathematical Analysis and Numerical Mathematics,
Faculty of Mathematics, Physics and Informatics,
Comenius University, 842 48 Bratislava, Slovakia}
\email{jaros@fmph.uniba.sk}

\address{ Kusano Taka\^si \newline
Department of Mathematics,
Faculty of Science,
Hiroshima University,
Higashi Hiroshima 739-8526, Japan}
\email{kusanot@zj8.so-net.ne.jp}

\address{Tomoyuki Tanigawa \newline
Department of Mathematics,
Faculty of Education,
Kumamoto University,
Kumamoto 860-8555, Japan}
\email{tanigawa@educ.kumamoto-u.ac.jp}

\thanks{Submitted December 30, 2015. Published December 22, 2016.}
\subjclass[2010]{34C11, 26A12}
\keywords{Systems of differential equations;
positive solutions;
\hfill\break\indent  asymptotic behavior; regularly varying functions}

\begin{abstract}
 In this article, we study the existence and accurate asymptotic
 behavior as $t \to \infty$ of positive solutions with
 intermediate growth for a class of cyclic systems of nonlinear
 differential equations of the second order
 $$
 (p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' +
 q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1} = 0, \quad i = 1,\ldots,n, \;
 x_{n+1} = x_1,
 $$
 where $\alpha_i$ and $\beta_i$, $i = 1,\dots,n$, are positive
 constants such that $\alpha_1{\dots}\alpha_n >\beta_1{\dots}\beta_n$ and
 $p_i, q_i: [a,\infty) \to (0,\infty)$ are continuous regularly varying
 functions (in the sense of Karamata). It is shown that the situation
 in which the system possesses regularly varying intermediate solutions can be
 completely characterized, and moreover that the asymptotic
 behavior of such solutions is governed by the unique formula
 describing their order of growth (or decay) precisely.
 The main results are applied to some classes of partial differential
 equations with radial symmetry including metaharmonic equations and
 systems involving $p$-Laplace operators on exterior domains.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article we analyze the differential system
\begin{equation}
(p_i(t)|x_i'|^{\alpha_i-1}x_i')' +
q_i(t)|x_{i+1}|^{\beta_i-1}x_{i+1} = 0, \quad i = 1,\dots,n, \;
x_{n+1} = x_1, \label{eA}
\end{equation}
where
\begin{itemize}
\item[(a)] $\alpha_i$ and $\beta_i$, $i = 1,\dots,n$, are positive
constants such that $\alpha_1{\dots}\alpha_n >
\beta_1{\dots}\beta_n$;

\item[(b)] $p_i$ and $q_i$ are continuous positive functions on
$[a,\infty)$, $a> 0$;

\item[(c)] all $p_i$ simultaneously satisfy either
\begin{equation}
\int_a^{\infty}p_i(t)^{-1/\alpha_i}dt = \infty, \quad i =
1,\dots,n, \label{e1.1}
\end{equation}
or
\begin{equation}
\int_a^{\infty}p_i(t)^{-1/\alpha_i}dt < \infty, \quad i =
1,\dots,n. \label{e1.2}
\end{equation}
\end{itemize}
By a positive solution of \eqref{eA} we mean a vector function
$(x_1,\dots,x_n)$ consisting of components $x_i$, $i = 1,\dots,n$,
which are positive and continuously differentiable together with
$p_i|x_i'|^{\alpha-1}x_i'$ on some interval $[T,\infty)$ and
satisfy system \eqref{eA} there.


Systems of the form \eqref{eA} with $p_i(t) = t^{N-1}$ and $q_i(t) =
t^{N-1}f_i(t)$, $N \geq 2$, $i = 1, \dots, n$, arise in the study of
positive radial solutions in exterior domains in $\mathbb{R}^N$
for the system of $p_i$-Laplacian equations
\begin{equation}
\begin{gathered}
\Delta_{p_i} u_i \equiv {\rm div} \big(|\nabla u_i|^{p_i-2} \nabla
u_i\big) + f_i(|x|)|u_{i+1}|^{\gamma_i-1} u_{i+1} = 0, i = 1, \dots, n,\\
  u_{n+1} = u_1,
\end{gathered} \label{eB}
\end{equation}
where $p_i > 1$ and $\gamma_i > 0, i = 1, \dots, n,$ are constants,
$|x|$ denotes the Euclidean norm of $x \in \mathbb{R}^N$ and $f_i$,
$i = 1,\dots,n$, are positive continuous functions on $[a,\infty)$.


Quasilinear elliptic system \eqref{eB} with negative $f_i, i = 1, \dots,
n$, and the exponents satisfying the super-homogeneity condition
$\gamma_1 \gamma_2 \dots \gamma_n > (p_1-1)(p_2-1)\dots (p_n-1)$
was studied by Teramoto \cite{T1}, while the problem of the existence and
precise asymptotic behavior as $|x| \to \infty$ of positive
strongly decreasing (resp. strongly increasing) radial solutions
of \eqref{eB} in the case $p_1 = \dots = p_n = p > 1$ under the
sub-homogeneity assumption $\gamma_1 \gamma_2 \dots \gamma_n <
(p-1)^n$ was investigated in \cite{JK3} (resp. \cite{JK4}).
(For the special case $p_1 = \dots = p_n = 2$ see also \cite{T2}.)


In this article we are concerned with positive solutions
$(x_1,\dots,x_n)$ of \eqref{eA} all components of which have the
intermediate growth (or decay) in the sense that they are
increasing to infinity as $t \to \infty$ and satisfy
\begin{equation}
\lim_{t \to \infty}p_i(t)|x_i'(t)|^{\alpha_i-1}x_i'(t) = 0,\quad i
= 1,\dots,n, \quad \text{in case \eqref{e1.1} holds}, \label{e1.3}
\end{equation}
or decreasing to zero as $t \to \infty$ and satisfy
\begin{equation}
\lim_{t \to \infty}p_i(t)|x_i'(t)|^{\alpha_i-1}x_i'(t) =
-\infty,\quad i = 1,\dots,n, \quad
\text{in case \eqref{e1.2} holds}. \label{e1.4}
\end{equation}
Note that this is equivalent to
\begin{equation}
\lim_{t \to \infty} x_i(t) = \infty, \quad
\lim_{t \to \infty}\frac{x_i(t)}{P_i(t)} = 0, \quad i = 1, \dots, n, \label{e1.5}
\end{equation}
if \eqref{e1.1} holds, where $P_i(t) = \int_a^t p_i(s)^{-1/\alpha_i}ds$,
or to
\begin{equation}
\lim_{t \to \infty}x_i(t) = 0, \quad
\lim_{t \to \infty}\frac{x_i(t)}{\pi_i(t)} = \infty, \quad i = 1, \dots, n,
\label{e1.6}
\end{equation}
if \eqref{e1.2} holds, where $\pi_i(t) = \int_t^\infty p_i(s)^{-1/\alpha_i}ds$.

In the scalar case, i.e., if \eqref{eA} reduces to a single equation of
the form
\begin{equation}
(p(t)|x'|^{\alpha-1}x')' + q(t)|x|^{\beta-1}x = 0, \label{eA1}
\end{equation}
where $\alpha$ and $\beta$ are positive constants such that
$\alpha > \beta$ and $p$ and $q$ are positive continuous functions
on $[a,\infty)$, necessary and sufficient conditions for the
existence of intermediate solutions of \eqref{eA1} have been
established for the case \eqref{e1.1} by Naito \cite{N} and for the case
\eqref{e1.2} by Kamo and Usami \cite{KU}.

It is to be noticed that system \eqref{eA} may possess also positive
solutions which have an extreme growth (or decay) in the sense
that if \eqref{e1.1} holds, then each component $x_i$ satisfies either
\begin{equation}
\lim_{t \to \infty}\frac{x_i(t)}{P_i(t)} = {\rm const} > 0,
\label{e1.7}
\end{equation}
or
\begin{equation}
\lim_{t \to \infty} x_i(t) = {\rm const} > 0, \label{e1.8}
\end{equation}
and if \eqref{e1.2} holds, then each component $x_i$ satisfies either
\eqref{e1.8} or
\begin{equation}
\lim_{t \to \infty} \frac{x_i(t)}{\pi_i(t)} = {\rm const} > 0.
\label{e1.9}
\end{equation}
Positive solutions of these types are not considered here.


Once the existence of intermediate solutions of \eqref{eA1} (or \eqref{eA})
has been confirmed, a natural question arises as to the
possibility of determining their asymptotic behavior at infinity
accurately. Partial answers to this question in the scalar case
have recently been given in the papers \cite{JKM,KMM} which are concerned
exclusively with regularly varying intermediate solutions of
equation \eqref{eA1} with regularly varying coefficients $p(t)$ and
$q(t)$. Restricting our consideration within the framework of
regular variation allows us to utilize basic theory of regular
variation to acquire thorough and precise information about the
existence, the asymptotic behavior and the structure of regularly
varying intermediate solutions of equation \eqref{eA1}. For the
definition of regularly varying functions see Section 2.


A prototype of the results we are going to prove says that if $f$
and $g$ are regularly varying functions of indices $\lambda$ and
$\mu$, respectively, and $p > N$, then the necessary and
sufficient condition for the existence of positive intermediate
radial solutions components of which are regularly varying
functions with indices in the interval $(0,\frac{p-N}{p-1})$ of
the system of two equations
\begin{equation}
\Delta_pu + f(|x|)v^\alpha = 0, \quad
\Delta_p v + g(|x|)u^\beta = 0, \label{eB2}
\end{equation}
where $\alpha\beta < (p-1)^2$, is the satisfaction of the system
of inequalities
\begin{gather*}
0 < p + \lambda + \frac{\alpha}{p-1}(p+\mu) < (p-N)\Big( 1 -
\frac{\alpha\beta}{(p-1)^2}\Big), \\
0 < \frac{\beta}{p-1}(p+\lambda) + p + \mu < (p-N)\Big(1 -
\frac{\alpha\beta}{(p-1)^2}\Big),
\end{gather*}
and if $p < N$, then the above two-dimensional system has
intermediate RV solutions with indices in $(\frac{p-N}{p-1}, 0)$
if and only if
$$
(p-N)\Big( 1 - \frac{\alpha\beta}{(p-1)^2}\Big) < p + \lambda +
\frac{\alpha}{p-1}(p+\mu) < 0,
$$
and
$$
(p-N)\Big(1 - \frac{\alpha\beta}{(p-1)^2}\Big) <
\frac{\beta}{p-1}(p+\lambda) + p + \mu < 0 .
$$

In both cases the indices $\rho$ and $\sigma$ of regular variation
of the components $u$ and $v$, respectively, are given (uniquely)
by
$$
\rho = \frac{p-1}{(p-1)^2-\alpha\beta}\Big[ p+\lambda +
\frac{\alpha}{p-1}(p+\mu)\Big], \quad
\sigma = \frac{p-1}{(p-1)^2-\alpha\beta }\Big[\frac{\beta}{p-1}(p+\lambda)
+ p + \mu \Big],
$$
and any such intermediate solution $(u,v)$ as $|x| \to \infty$
satisfies the asymptotic relation
\begin{gather*}
u(|x|) \sim |x|^\rho \Big[
\frac{\varphi(|x|)^\frac{1}{p-1}}{D(\rho)}
\Big(\frac{\psi(|x|)^\frac{1}{p-1}}{D(\sigma)}\Big)^\frac{\alpha}{p-1}
\Big]^\frac{(p-1)^2}{(p-1)^2-\alpha\beta},
\\
v(|x|) \sim |x|^\sigma \Big[
\Big(\frac{\varphi(|x|)^\frac{1}{p-1}}{D(\rho)}\Big)^\frac{\beta}{p-1}
\frac{\psi(|x|)^\frac{1}{p-1}}{D(\sigma)}\Big]^\frac{(p-1)^2}{(p-1)^2-\alpha\beta},
\end{gather*}
where $\varphi$ and $\psi$ are the slowly varying parts of $f$ and
$g$, respectively, and \\$D(\tau) =
\big(p-N-(p-1)|\tau|\big)^\frac{1}{p-1}|\tau|$.

The main results of this paper will be presented in Section 4.
The existence of intermediate regularly varying solutions of \eqref{eA}
is proved by solving the system of integral equations
\begin{gather}
x_i(t) = c_{i} + \int_T^t \Bigl(\frac{1}{p_i(s)}\int_s^{\infty}
q_i(r)x_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds, \quad
i = 1,\dots,n, \quad \text{if \eqref{e1.1} holds,} \label{e1.10}
\\
x_i(t) =\int_t^{\infty}\frac{1}{p_i(s)}\Bigl(\int_T^s
q_i(r)x_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds, \quad
i = 1,\dots,n, \quad \text{if \eqref{e1.2} holds,} \label{e1.11}
\end{gather}
for some constants $T \geq a$ and $c_i > 0$ with the help of fixed
point techniques combined with basic theory of regularly varying
functions. Furthermore it is shown that the asymptotic behavior of
the obtained solutions is governed by the unique explicit law
describing their order of growth (in case \eqref{e1.1} holds) or decay
(in case of \eqref{e1.2} holds) accurately. To this end extensive use is
made of the knowledge derived through the analysis of the
following systems of asymptotic integral relations associated with
\eqref{e1.10} and \eqref{e1.11} by means of regular variation:
\begin{gather}
x_i(t) \sim \int_T^t \Bigl(\frac{1}{p_i(s)}\int_s^{\infty}
q_i(r)x_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds, \quad
t \to \infty, \quad i = 1,\dots,n, \label{e1.12} \\
x_i(t) \sim \int_t^{\infty}
 \Bigl(\frac{1}{p_i(s)}\int_T^s q_i(r)x_{i+1}(r)^{\beta_i}dr
\Bigr)^{1/\alpha_i}ds, \quad
t \to \infty, \quad i = 1,\dots,n.\label{e1.13}
\end{gather}
Here and hereafter the notation $f(t) \sim g(t)$ as $t \to \infty$
is used to mean
$$
\lim_{t \to \infty}\frac{g(t)}{f(t)} = 1.
$$
The details of the analysis of systems \eqref{e1.12} and \eqref{e1.13} in the
framework of regular variation is presented in Section 3, which is
preceded by Section 2 where the definition and some basic
properties of regularly varying functions are summarized for the
reader's convenience. The final Section 5 is designed to explain
the effective applicability of our results for \eqref{eA} to some classes
of partial differential equations with radial symmetry including
metaharmonic equations and systems involving $p$-Laplace operators
on exterior domains in $\mathbb{R}^N$.

The systematic study of differential equations in the framework of
regular variation was initiated by Mari\'c and Tomi\'c \cite{MT1, MT2, MT3}.
Since the publication of the monograph of Mari\'c \cite{Ma} in the year
2000 there has been an increasing interest in the study of
asymptotic properties of positive solutions of differential
equations by means of regularly varying functions, and it has
turned out that theory of regular variation combined with fixed
point techniques is so powerful as to cover a wide class of
ordinary differential equations including generalized Emden-Fowler
and Thomas-Fermi equations, and systems of such equations; see,
for example, \cite{CR,EV,JK1,JK2,JKM,JKT1,KM1,KM3,KMM,KMT}.


\section{Regularly varying functions}

For the reader's benefit we recall here the definition and basic
properties of regularly varying functions which will be used in
this paper.

\begin{definition} \label{def2.1} \rm
 A measurable function $f:[0,\infty) \to (0,\infty)$ is said to be
{\it regularly varying of index}
$\rho \in \mathbb{R}$ if
$$
\lim_{t \to \infty}\frac{f(\lambda t)}{f(t)}
= \lambda^{\rho} \quad \text{for all } \lambda > 0.
$$
The set of all regularly varying functions of index $\rho$ is
denoted by $\operatorname{RV}(\rho)$.
\end{definition}

Typical example of a function from
$\operatorname{RV}(\rho)$ is the power function $t^\rho$ or, more generally, any
function of the form $ct^\rho (1+ \varepsilon (t))$ where $c$ is a
positive constant and $\varepsilon (t)$ a measurable function on
$(0,\infty)$ such that $\varepsilon (t) \to 0$ as $t \to \infty$.
All these are examples of the so-called \emph{trivial}  regularly varying
functions of index $\rho$, i.e., functions $f \in$ $\operatorname{RV}(\rho)$
satisfying $\lim_{t \to \infty} f(t)/t^\rho = {\rm const} > 0$.
An example of a \emph{nontrivial}  regularly varying function of index
$\rho$ is $t^\rho \log (1+t)$.


We often use the symbol SV for $\operatorname{RV}(0)$ and call members of SV
\emph{slowly varying functions}. It is easy to check that any
$f \in$
$\operatorname{RV}(\rho)$ can always be represented in the form
$f(t) = t^\rho g(t)$, where $g \in {\rm SV}$. Thus, the properties of regularly varying
functions are easily deduced from the corresponding properties of
slowly varying functions.

One of the most important properties of regularly varying
functions is the following \emph{representation theorem}.


\begin{proposition} \label{prop2.1}
$f \in$ $\operatorname{RV}(\rho)$  if and only if $f(t)$ is
represented in the form
\begin{equation}
f(t) = c(t)\exp\Bigl\{\int_{t_0}^t \frac{\delta(s)}{s}ds\Bigr\},
\quad t \geq t_0, \label{e2.1}
\end{equation}
 for some $t_0 > 0$ and for some measurable functions $c(t)$ and
$\delta(t)$ such that
$$
\lim_{t \to \infty}c(t) = c_0 \in (0,\infty) \quad \text{and} \quad
\lim_{t \to \infty}\delta(t) = \rho.
$$
If in particular $c(t) \equiv c_0$ in \eqref{e2.1}, then $f(t)$
is referred to as a \emph{normalized} regularly varying function of
index $\rho$.
\end{proposition}

Examples of slowly varying functions include all functions tending to some
positive constants as $t \to \infty$, the logarithmic function,
its powers $\log^\gamma t, \gamma \in R$, iterated logarithms.
A more sophisticated example of a member from SV is the function
$\exp\{c\log ^\gamma t \cos^\delta(\log t)\}$, where $c > 0$ and
$\gamma, \delta \geq 0$ are such that $\gamma + \delta < 1$.
The following result illustrates operations which preserve slow variation.


\begin{proposition} \label{prop2.2}
 Let $L(t)$, $L_1(t)$, $L_2(t)$ be slowly varying. Then, $L(t)^{\alpha}$ for any
$\alpha \in \mathbb{R}$, $L_1(t)+L_2(t)$, $L_1(t)L_2(t)$ and
$L_1(L_2(t))$ (if $L_2(t) \to \infty$)  are slowly
varying.
\end{proposition}

A slowly varying function may grow to infinity or decay to $0$ as
$t \to \infty$. But its order of growth or decay is severely
limited as is shown in the following


\begin{proposition} \label{prop2.3}
 Let $f \in {\rm SV}$.  Then, for any $\varepsilon > 0$,
$$
\lim_{t \to \infty}t^{\varepsilon}f(t) = \infty, \quad
 \lim_{t \to \infty}t^{-\varepsilon}f(t) = 0.
$$
\end{proposition}

A simple criterion for determining the regularity of differentiable positive
functions follows.


\begin{proposition} \label{prop2.4}
 A differentiable positive function $f(t)$ is a normalized regularly varying
function of index $\rho$ if and only if
$$
\lim_{t \to \infty} t\frac{f'(t)}{f(t)} = \rho.
$$
\end{proposition}

The following proposition known as Karamata's integration theorem
will play an important role in this paper.


\begin{proposition} \label{prop2.5}
 Let $L(t)$ be a slowly varying function. Then:
\begin{itemize}
\item[(i)]  if $\alpha > -1$,
$$
\int_a^t s^{\alpha}L(s)ds \sim \frac{1}{\alpha+1}t^{\alpha+1}L(t), \quad
t \to \infty;
$$

\item[(ii)]  if $\alpha < -1$,
$$
\int_t^{\infty} s^{\alpha}L(s)ds \sim - \frac{1}{\alpha+1}t^{\alpha+1}L(t),
\quad t \to \infty;
$$

\item[(iii)] if $\alpha = -1$,
$$
l(t) = \int_a^t \frac{L(s)}{s}ds \in {\rm SV} ,
$$
 and if, in addition, $\int_a^\infty s^{-1}L(s) ds < \infty$,
then
$$
m(t) = \int_t^\infty \frac{L(s)}{s} ds \in {\rm SV}.
$$
\end{itemize}
\end{proposition}

\begin{definition} \label{def2.2} \rm
A vector function $(x_1(t),\dots,x_n(t))$ is said to be regularly varying of
index $(\rho_1,\dots,\rho_n)$ if $x_i \in$ $\operatorname{RV}(\rho_i$) for $i = 1,\dots,n$.
If all $\rho_i$ are positive (or negative), then $(x_1(t),\dots,x_n(t))$ is called
regularly varying of positive (or negative) index $(\rho_1,\dots,\rho_n)$.
The set of all regularly varying vector functions of
index $(\rho_1,\dots,\rho_n)$ is denoted by $\operatorname{RV}(\rho_1,\dots,\rho_n$).
\end{definition}

For a complete exposition of theory of regular variation
and its applications the reader is referred to the treatise of
Bingham, Goldie and Teugels \cite{BGT}. See also Seneta \cite{S}. A
comprehensive survey of results up to the year 2000 on the
asymptotic analysis of second order ordinary differential
equations by means of regular variation can be found in the
monograph of Mari\'c \cite{Ma}.


\section{Systems of asymptotic relations associated with \eqref{eA}}

We assume that $p_i \in\operatorname{RV}(\lambda_i)$ and
$q_i \in\operatorname{RV}(\mu_i$) and that they are represented as
\begin{equation}
p_i(t) = t^{\lambda_i}l_i(t), \quad
q_i(t) = t^{\mu_i}m_i(t),\quad
l_i,\; m_i \in {\rm SV} , \quad i = 1,\dots,n. \label{e3.1}
\end{equation}
In addition we require that $p_i(t)$ satisfy either \eqref{e1.1} or
\eqref{e1.2}. It is easy to see that \eqref{e1.1} (resp. \eqref{e1.2}) holds if and
only if
$$
\lambda_i < \alpha_i, \quad \text{or}\quad
\lambda_i = \alpha_i\quad \text{and}\quad
\int_a^{\infty}t^{-1}l_i(t)^{-1/\alpha_i}dt = \infty,
$$
resp.
$$
\lambda_i > \alpha_i, \quad \text{or}\quad \lambda_i = \alpha_i\quad
\text{and}\quad \int_a^{\infty}t^{-1}l_i(t)^{-1/\alpha_i}dt < \infty.
$$
Therefore, in case \eqref{e1.1} is satisfied, the functions
$P_i(t) =\int_a^t p_i(s)^{-1/\alpha_i}ds$, $i = 1,\dots,n$, are given
by
\begin{gather}
P_i(t) = \int_a^t s^{-1}l_i(s)^{-1/\alpha_i}ds \quad
\text{if } \lambda_i = \alpha_i, \label{e3.2} \\
P_i(t) \sim \frac{\alpha_i-\lambda_i}{\alpha_i}
 t^{\frac{\alpha_i-\lambda_i}{\alpha_i}}l_i(t)^{-1/\alpha_i},\quad
t \to \infty, \quad \text{if}\quad \lambda_i < \alpha_i,
\label{e3.3}
\end{gather}
and in case \eqref{e1.2} holds, the functions
$\pi_i(t) = \int_t^{\infty}
p_i(s)^{-1/\alpha_i}ds$, $i = 1,\dots,n$, are given by
\begin{gather}
\pi_i(t) = \int_t^{\infty} s^{-1}l_i(s)^{-1/\alpha_i}ds
\quad \text{if }  \lambda_i = \alpha_i, \label{e3.4}\\
\pi_i(t) \sim \frac{\lambda_i-\alpha_i}{\alpha_i}
 t^{\frac{\alpha_i-\lambda_i}{\alpha_i}}l_i(t)^{-1/\alpha_i},\quad
t \to \infty, \quad \text{if } \lambda_i > \alpha_i.
\label{e3.5}
\end{gather}
Our task in this section is to solve the following two problems.

Problem (i): Under the condition \eqref{e1.1} characterize the situation
in which the system of asymptotic relations
\begin{equation}
x_i(t) \sim \int_T^t \Bigl(\frac{1}{p_i(s)}\int_s^{\infty}
q_i(r)x_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds, \quad
t \to \infty, \quad i = 1,\dots,n, \label{e3.6}
\end{equation}
possesses regularly varying solutions $(x_1,\dots,x_n)$ of positive
index $(\rho_1,\dots,\rho_n)$ satisfying
\begin{equation}
\lim_{t \to \infty}x_i(t) = \infty, \quad
\lim_{t \to \infty}\frac{x_i(t)}{P_i(t)} = 0, \quad i = 1,\dots,n. \label{e3.7}
\end{equation}

Problem (ii): Under the condition \eqref{e1.2} characterize the
situation in which the system of asymptotic relations
\begin{equation}
x_i(t) \sim \int_t^{\infty} \Bigl(\frac{1}{p_i(s)}\int_T^s
q_i(r)x_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds, \quad
t \to \infty, \quad i = 1,\dots,n, \label{e3.8}
\end{equation}
possesses regularly varying solutions $(x_1,\dots,x_n)$ of negative
index $(\rho_1,\dots,\rho_n)$ satisfying
\begin{equation}
\lim_{t \to \infty}x_i(t) = 0, \quad
\lim_{t \to \infty}\frac{x_i(t)}{\pi_i(t)} = \infty, \quad i = 1,\dots,n.
\label{e3.9}
\end{equation}

The positivity or negativity requirement for the regularity indices of
solutions excludes the possibility that $\lambda_i = \alpha_i$
for some or all $i$ which necessarily reduces the corresponding components
$x_i(t)$ to slowly varying functions ($\rho_i=0$)
(cf. \eqref{e3.2} and \eqref{e3.4}). The presence of slowly varying components
in the solutions seems to cause computational difficulty.

We begin with Problem (i). We assume that $\lambda_i < \alpha_i$,
$i = 1,\dots,n$, and seek solutions $(x_1,\dots,x_n)$ of \eqref{e3.6}
belonging to $\operatorname{RV}(\rho_1,\dots,\rho_n$) with all $\rho_i$ positive.
In view of \eqref{e3.3} each $\rho_i$ must satisfy $\rho_i >
\frac{\alpha_i-\lambda_i}{\alpha_i}$. Let $(x_1,\dots,x_n)$ be one
such solution on $[T,\infty)$. Suppose that $x_i$ are expressed in
the form
\begin{equation}
x_i(t) = t^{\rho_i}\xi_i(t), \quad \xi_i \in {\rm SV} , \quad i
= 1,\dots,n. \label{e3.10}
\end{equation}
Using \eqref{e3.1} and \eqref{e3.10}, we have
\begin{equation}
\int_t^{\infty}q_i(s)x_{i+1}(s)^{\beta_i}ds
=\int_t^{\infty}s^{\mu_i+\beta_i\rho_{i+1}}m_i(s)\xi_{i+1}(s)^{\beta_i}ds,
 \label{e3.11}
\end{equation}
for $t \geq T$ and $i = 1, \dots, n$.
The convergence of \eqref{e3.11} as $t \to \infty$ implies that
$\mu_i+\beta_i\rho_{i+1} \leq -1$, $i = 1,\dots,n$, but the equality should
be ruled out. In fact, if the equality holds for some $i$, then since
$$
\Bigl(\frac{1}{p_i(t)}\int_t^{\infty}q_i(s)x_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}
= t^{-\frac{\lambda_i}{\alpha_i}}l_i(t)^{-1/\alpha_i}
\Bigl(\int_t^{\infty}s^{-1}m_i(s)\xi_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i},
$$
from \eqref{e3.6} and Karamata's integration theorem we find that
$$
x_i(t) \sim \frac{\alpha_i}{\alpha_i-\lambda_i}t^{\frac{\alpha_i-\lambda_i}{\alpha_i}}
l_i(t)^{-1/\alpha_i}
\Bigl(\int_t^{\infty}s^{-1}m_i(s)\xi_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i} \in
\operatorname{RV}\Bigl(\frac{\alpha_i-\lambda_i}{\alpha_i}\Bigr)
$$
as $t \to \infty$. This implies that $\rho_i = \frac{\alpha_i-\lambda_i}{\alpha_i}$,
a contradiction. It follows that
$\mu_i+\beta_i\rho_{i+1} < -1$ for $i = 1,\dots,n$, and application of Karamata's
integration theorem to \eqref{e3.11} gives
\begin{equation}
\begin{aligned}
&\Bigl(\frac{1}{p_i(t)}\int_t^{\infty}q_i(s)x_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}\\
&\sim
\frac{t^{\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}}
l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}\xi_{i+1}(t)^{\beta_i/\alpha_i}}
{[-(\mu_i+\beta_i\rho_{i+1}+1)]^{1/\alpha_i}},
\end{aligned}\label{e3.12}
\end{equation}
as $t \to \infty$, $i = 1,\dots,n$.

 Because \eqref{e3.12} is not integrable on
$[T,\infty)$ we see that $(-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1)/\alpha_i \geq -1$,
$i = 1,\dots,n$. We claim that the equality can hold in none of these
inequalities. If the equality holds for some $i$, then integrating \eqref{e3.12}
from $T$ to $t$ yields
$$
x_i(t) \sim (\alpha_i-\lambda_i)^{-1/\alpha_i} \int_T^t
s^{-1}l_i(s)^{-1/\alpha_i}m_i(s)^{-1/\alpha_i}\xi_{i+1}(s)^{\beta_i/\alpha_i}ds
\in {\rm SV }, \quad t \to \infty,
$$
an impossibility. It holds therefore that
$(-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1)/\alpha_i > -1$ for all $i$, and hence
via application of Karamata's integration theorem to the integral of
\eqref{e3.12} on $[T,t]$ we conclude that
\begin{equation}
x_i(t) \sim
\frac{t^{\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}+1}
l_i(t)^{-1/\alpha_i}m_i(t)^{-1/\alpha_i}\xi_{i+1}(t)^{\beta_i/\alpha_i}}
{[-(\mu_i+\beta_i\rho_{i+1}+1)]^{1/\alpha_i}
\bigl(\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}+1\bigr)},
 \label{e3.13}
\end{equation}
as $t \to \infty$, $i = 1,\dots,n$.
This implies
$$
\rho_i = \frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}+1, \quad
i = 1,\dots,n, \quad  \rho_{n+1} = \rho_1
$$
or equivalently
\begin{equation}
\rho_i - \frac{\beta_i}{\alpha_i}\rho_{i+1} =
\frac{\alpha_i-\lambda_i+\mu_i+1}{\alpha_i}, \quad i = 1,\dots,n,
\quad \rho_{n+1} = \rho_1 \,.\label{e3.14}
\end{equation}
The coefficient matrix
\begin{equation}
 A = A\Bigl(\frac{\beta_1}{\alpha_1},\dots,\frac{\beta_n}{\alpha_n}\Bigr)
=   \begin{pmatrix}
     1           & -\frac{\beta_1}{\alpha_1} & 0           & \ldots & 0      & 0             \\
     0           & 1           & -\frac{\beta_2}{\alpha_2} & \ldots & 0      & 0             \\
     \vdots      & \vdots      & \ddots      & {}     & \vdots & \vdots        \\
     \vdots      & \vdots      & {}          & \ddots & \vdots & \vdots        \\
     0           & 0           & 0      & \ldots & 1      & -\frac{\beta_{n-1}}{\alpha_{n-1}} \\
     -\frac{\beta_n}{\alpha_n} & 0           & 0           & \ldots & 0      & 1
    \end{pmatrix}  \label{e3.15}
\end{equation}
of the algebraic linear system \eqref{e3.14} is nonsingular because
\begin{equation}
\det(A) = 1 - \frac{\beta_1\beta_2 \dots
\beta_n}{\alpha_1\alpha_2 \dots \alpha_n} > 0 \label{e3.16}
\end{equation}
because of condition (a). Thus, $A$ is invertible and the explicit
calculation gives
\begin{equation}
A^{-1} = \frac{A_n}{A_n-B_n}
         \begin{pmatrix}
         1 & \frac{\beta_1}{\alpha_1}   & \frac{\beta_1 \beta_2}{\alpha_1 \alpha_2} & \ldots    & \ldots & \frac{\beta_1 \beta_2{\dots}\beta_{n-1}}{\alpha_1 \alpha_2{\dots}\alpha_{n-1}} \\
        {} & 1          & \frac{\beta_2}{\alpha_2}           & \frac{\beta_2\beta_3}{\alpha_2\alpha_3}    & \ldots & \frac{\beta_2 \beta_3{\dots}\beta_{n-1}}{\alpha_2 \alpha_3{\dots}\alpha_{n-1}} \\

        {} & {}         & 1                    & \frac{\beta_3}{\alpha_3}& \ldots & \frac{\beta_3{\dots}\beta_{n-1}}{\alpha_3{\dots}\alpha_{n-1}} \\

        {} & {}         & {}                   & \ddots    & \ddots & \vdots \\
        {} & {}         & {}                   & {}        & 1      & \frac{\beta_{n-1}}{\alpha_{n-1}} \\
         *& {}         & {}                   & {}        & {}     & 1 \\
        \end{pmatrix},  \label{e3.17}
\end{equation}
where $A_n = \alpha_1\alpha_2{\dots}\alpha_n$,
$B_n = \beta_1\beta_2{\dots}\beta_n$, and the lower triangular elements
are omitted for economy of notation. Let $(M_{ij})$ denote the
matrix on the right-hand side of \eqref{e3.17}. It is easy to see that
the $i$-th row of ($M_{ij}$) is obtained by shifting the vector
$$
\Bigl(1,\frac{\beta_i}{\alpha_i},\frac{\beta_{i}\beta_{i+1}}{\alpha_{i}\alpha_{i+1}},
\dots, \frac{\beta_{i}\beta_{i+1}{\dots}\beta_{i+(n-2)}}{\alpha_{i}\alpha_{i+1}
{\dots}\alpha_{i+(n-2)}}\Bigr)
\quad \alpha_{n+k} = \alpha_k,\; \beta_{n+k} = \beta_k \text{ for } k = 1, 2, \dots
$$
($i-1$)-times to the right cyclically, so that the lower
triangular elements $M_{ij}$ for $j < i$, satisfy the relations
$$
M_{ij}M_{ji} = \frac{\beta_1\beta_2{\dots}
\beta_n}{\alpha_1\alpha_2{\dots}\alpha_n},\quad i > j, \quad i = 1,2,\dots,n.
$$
Then the unique solution $\rho_i$, $i = 1,\dots,n$, of \eqref{e3.14}
is given explicitly by
\begin{equation}
\rho_i = \frac{A_n}{A_n-B_n}\sum_{j=1}^n
M_{ij}\frac{\alpha_j-\lambda_j+\mu_j+1}{\alpha_j}, \quad i =
1,\dots,n, \label{e3.18}
\end{equation}
from which it follows that all $\rho_i$ satisfy $0 < \rho_i <
\frac{\alpha_i-\lambda_i}{\alpha_i}$ if and only if
\begin{equation}
0 < \sum_{j=1}^n
M_{ij}\frac{\alpha_j-\lambda_j+\mu_j+1}{\alpha_j} <
\frac{\alpha_i-\lambda_i}{\alpha_i}\Big(1-\frac{B_n}{A_n}\Big),
\quad i = 1,\dots,n. \label{e3.19}
\end{equation}
We note that \eqref{e3.13} can be expressed in the form
\begin{equation}
x_i(t) \sim
\frac{t^{\frac{\alpha_i+1}{\alpha_i}}p_i(t)^{-1/\alpha_i}q_i(t)^{1/\alpha_i}
x_{i+1}(t)^{\beta_i/\alpha_i}}
{D_i}, \quad t \to \infty, \label{e3.20}
\end{equation}
where
\begin{equation}
D_i = (\alpha_i-\lambda_i -
\alpha_i\rho_i)^{1/\alpha_i}\rho_i, \label{e3.21}
\end{equation}
for $i = 1,\dots,n.$  This is a cyclic system of asymptotic
relations, from which one can derive without difficulty the
following independent explicit asymptotic formula for each $x_i$:
\begin{equation}
x_i(t) \sim \Bigl[\prod_{j=1}^n
\Bigl(\frac{t^{\frac{\alpha_j+1}{\alpha_j}}p_j(t)^{-1/\alpha_j}q_j(t)^{1/\alpha_j}}{D_j}\Bigr)^{M_{ij}}
\Bigr]^{\frac{A_n}{A_n-B_n}}, \quad t \to \infty, \quad i =
1,\dots,n. \label{e3.22}
\end{equation}
This represents the unique law describing precisely the growth
order of all possible regularly varying solutions of positive
indices of system \eqref{e3.6} satisfying \eqref{e3.7}.
 Note that \eqref{e3.22} is rewritten in the form
\begin{equation}
x_i(t) \sim
t^{\rho_i}\Bigl[\prod_{j=1}^n\Bigl(\frac{l_j(t)^{-1/\alpha_j}
m_j(t)^{1/\alpha_j}}{D_j}\Bigr)^{M_{ij}}
\Bigr]^{\frac{A_n}{A_n-B_n}}, \quad t \to \infty, \quad i =
1,\dots,n. \label{e3.23}
\end{equation}

Now we assume that \eqref{e3.19} is satisfied and define
$\rho_i \in (0,\frac{\alpha_i-\lambda_i}{\alpha_i})$ and $D_i$ by
\eqref{e3.18} and \eqref{e3.21}, respectively.  Let $X_i \in$ $\operatorname{RV}(\rho_i$)
denote the functions
\begin{equation}
X_i(t) = \Bigl[\prod_{j=1}^n
\Bigl(\frac{t^{\frac{\alpha_j+1}{\alpha_j}}p_j(t)^{-1/\alpha_j}q_j(t)^{1/\alpha_j}}{D_j}\Bigr)^{M_{ij}}
\Bigr]^{\frac{A_n}{A_n-B_n}}, \quad i = 1,\dots,n. \label{e3.24}
\end{equation}
Then the $X_i$'s satisfy the system of asymptotic relations \eqref{e3.6}, i.e.,
\begin{equation}
\int_b^t \Bigl(\frac{1}{p_i(s)}\int_s^{\infty}
q_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds \sim
X_i(t), \quad t \to \infty, \; i = 1,,,,n, \label{e3.25}
\end{equation}
for any $b \geq a$, where $X_{n+1}(t) = X_1(t)$. In fact, noting that $X_i(t)$
are expressed as
$$
X_i(t) = t^{\rho_i}\Xi_{i}(t), \quad \Xi_{i}(t)
= \Bigl[\prod_{j=1}^n\Bigl(\frac{l_j(t)^{-1/\alpha_j}
m_j(t)^{1/\alpha_j}}{D_j}\Bigr)^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
$$
and using Karamata's integration theorem, we obtain
$$
\Bigl(\frac{1}{p_i(t)}\int_b^t
q_i(s)X_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i} \sim
\frac{t^{\rho_i-1}l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}\Xi_{i+1}(t)^{\beta_i/\alpha_i}}
{(\alpha_i-\lambda_i-\alpha_i\rho_i)^{1/\alpha_i}},
$$
and
\begin{equation}
\begin{aligned}
&\int_b^t \Bigl(\frac{1}{p_i(s)}\int_b^t
q_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds\\
& \sim
\frac{t^{\rho_i}l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}
\Xi_{i+1}(t)^{\beta_i/\alpha_i}}
{D_i}, \quad \text{as }t \to \infty.
\end{aligned} \label{e3.26}
\end{equation}
 A simple calculation with the help of the relations
\begin{equation}
M_{i+1,i}\frac{\beta_i}{\alpha_i} = \frac{B_n}{A_n}, \quad
M_{i+1,j}\frac{\beta_i}{\alpha_i} = M_{ij}, \quad
\text{for } j \neq i, \label{e3.27}
\end{equation}
(with the convection that $M_{n+1,j} = M_{1,j}$) between the
$i$-th and the $(i+1)$-th rows of the matrix $A$ shows that
\begin{align*}
&\frac{l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}}{D_i}
 \Xi_{i+1}(t)^{\beta_i/\alpha_i} \\
&= \frac{l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}}{D_i}\Bigl[\prod_{j=1}^n
\Bigl(\frac{l_j(t)^{-1/\alpha_j}m_j(t)^{1/\alpha_j}}{D_j}\Bigr)^{M_{i+1,j}
\frac{\beta_i}{\alpha_i}}\Bigr]^{\frac{A_n}{A_n-B_n}} \\
&= \Bigl[\prod_{j=1}^n\Bigl(\frac{l_j(t)^{-1/\alpha_j}m_j(t)^{1/\alpha_j}}{D_j}\Bigr)^{M_{ij}}\Bigr]^
{\frac{A_n}{A_n-B_n}}
= \Xi_{i}(t).
\end{align*}
From \eqref{e3.26} we conclude that the $X_{i}$'s satisfy \eqref{e3.25} as desired.

Summarizing the above observations, we obtain the following result which provides
complete information about the existence
and asymptotic behavior of regularly varying solutions with positive indices
for system \eqref{e3.6}.

\begin{theorem} \label{thm3.1}
 Let $p_i \in \operatorname{RV}(\lambda_i)$  and
$q_i \in \operatorname{RV}(\mu_i)$, and suppose that
$\lambda_i <\alpha_i$, $i = 1,\dots,n$.  Then system of asymptotic relations
\eqref{e3.6}  has regularly varying solutions
$(x_1,\dots,x_n) \in \operatorname{RV}(\rho_1,\dots,\rho_n)$ with
$\rho_i \in (0,\frac{\alpha_i-\lambda_i}{\alpha_i})$, $i = 1,\dots,n$, if and only
if \eqref{e3.19} holds  in which case $\rho_i$ are uniquely
determined by \eqref{e3.18}  and the asymptotic behavior of any such
solution is governed by the unique formula \eqref{e3.22}.
\end{theorem}

Our next task is to study Problem (ii). We assume that
$\lambda_i> \alpha_i$, $i = 1,\dots,n$, and seek solutions
$(x_1,\dots,x_n) \in \operatorname{RV}(\rho_i)$ with all $\rho_i$ negative. In
view of \eqref{e3.5} each $\rho_i$ must satisfy
$\frac{\alpha_i-\lambda_i}{\alpha_i} < \rho_i < 0$. Our solution
to this problem is formulated below with the help of the matrix
\eqref{e3.15} and its inverse \eqref{e3.17}.

\begin{theorem} \label{thm3.2}
Let $p_i \in \operatorname{RV}(\lambda_i)$  and
$q_i \in \operatorname{RV}(\mu_i)$ and suppose that
$\lambda_i >\alpha_i$, $i = 1,\dots,n$. System of asymptotic relations
\eqref{e3.8}  has regularly varying solutions
$(x_1,\dots,x_n) \in \operatorname{RV}(\rho_1,\dots,\rho_n)$  with
$\rho_i \in(\frac{\alpha_i-\lambda_i}{\alpha_i},0)$, $i = 1,\dots,n$, if
and only if
\begin{equation}
\frac{\alpha_i-\lambda_i}{\alpha_i}\Big(1-\frac{B_n}{A_n}\Big) <
\sum_{j=1}^n M_{ij}\frac{\alpha_j-\lambda_j+\mu_j+1}{\alpha_j} <
0 \label{e3.28}
\end{equation}
in which case $\rho_i$ are given by \eqref{e3.18}  and the
asymptotic behavior of any such solution $(x_1,\dots,x_n)$ is
governed by the unique formula
\begin{equation}
x_i(t) \sim \Bigl[\prod_{j=1}^n
\Bigl(\frac{t^{\frac{\alpha_j+1}{\alpha_j}}p_j(t)^{-1/\alpha_j}q_j(t)
^{1/\alpha_j}}
{\Delta_j}\Bigr)^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}}, \quad
t \to \infty, \; i = 1,\dots,n. \label{e3.29}
\end{equation}
 where
\begin{equation}
\Delta_i = (\lambda_i - \alpha_i +
\alpha_i\rho_i)^{1/\alpha_i}(-\rho_i), \quad i = 1,\dots,n.
\label{e3.30}
\end{equation}
\end{theorem}

\begin{proof}
 Let $(x_1,\dots,x_n)$ be one such solution on
$[T,\infty)$. Using \eqref{e3.1} and \eqref{e3.10} we obtain
\begin{equation}
\int_T^t q_i(s)x_{i+1}(s)^{\beta_i}ds = \int_T^t
s^{\mu_i+\beta_i\rho_{i+1}}\mu_i(s)\xi_{i+1}(s)^{\beta_i}ds, \label{e3.31}
\end{equation}
for $t \geq T$ and $i = 1, \dots, n$,
all of which are required to diverge as $t \to \infty$.
Therefore $\mu_i+\beta_i\rho_{i+1} \geq -1$ for all $i$. If the equality
holds for some $i$, then noting that
$$
\Bigl(\frac{1}{p_i(t)}\int_T^t q_i(s)x_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}
= t^{-\frac{\lambda_i}{\alpha_i}}l_i(t)^{-1/\alpha_i}
\Bigl(\int_T^t s^{-1}m_i(s)\xi_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i},
$$
and integrating this from $t$ to $\infty$, we obtain via
Karamata's integration theorem,
$$
x_i(t) \sim \frac{\alpha_i}{\lambda_i-\alpha_i}
t^{\frac{\alpha_i-\lambda_i}{\alpha_i}}l_i(t)^{-1/\alpha_i}
\Bigl(\int_T^t s^{-1}m_i(s)\xi_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i} \in
\operatorname{RV}\Bigl(\frac{\alpha_i-\lambda_i}{\alpha_i}\Bigr),
$$
which is a contradiction. It follows that $\mu_i+\beta_i\rho_{i+1}> -1$
for all $i$. Applying Karamata's integration theorem to
\eqref{e3.31}, we have
\begin{equation}
\begin{aligned}
&\Bigl(\frac{1}{p_i(t)}\int_T^t
q_i(s)x_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i} \\
& \sim \frac{t^{\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}}
l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}\xi_{i+1}(t)^{\beta_i/\alpha_i}}
{(\mu_i+\beta_i\rho_{i+1}+1)^{1/\alpha_i}}, \quad t \to
\infty. \label{e3.32}
\end{aligned}
\end{equation}
Since \eqref{e3.32} is integrable over $[T,\infty)$, we see that
$\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i} \leq -1$
for all $i$. Note that all of these inequalities should be strict,
because if the equality holds for some $i$, then integrating on
$[t,\infty)$, we have
$$
x_i(t) \sim (\lambda_i-\alpha_i)^{-1/\alpha_i}
\int_t^{\infty}s^{-1}l_i(s)^{-1/\alpha_i}m_i(s)^{1/\alpha_i}\xi_{i+1}(s)^{\beta_i/\alpha_i}ds
\in {\rm SV},
$$
a contradiction. It follows that
$\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i} < -1$ for
all $i$, in which case integration of \eqref{e3.32} on $[t,\infty)$
yields
\begin{equation}
x_i(t) \sim \frac{t^{\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}+1}
l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}\xi_{i+1}(t)^{\beta_i/\alpha_i}}
{(\mu_i+\beta_i\rho_{i+1}+1)^{1/\alpha_i}
\Bigl[-\Bigl(\frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}+1\Bigr)\Bigr]},
\label{e3.33}
\end{equation}
as $t \to \infty$ and $i = 1,\dots,n$.
This implies
$$
\rho_i =  \frac{-\lambda_i+\mu_i+\beta_i\rho_{i+1}+1}{\alpha_i}+1, \quad
i = 1,\dots,n,
$$
which is equivalent to the linear algebraic system \eqref{e3.14} in
$\rho_i$. From this point on one can proceed exactly as in the
proof of the ``only if'' part of Theorem \ref{thm3.1}, asserting that system
\eqref{e3.8} may have regularly varying solutions of negative indices
$\rho_i \in (\frac{\alpha_i-\lambda_i}{\alpha_i},0)$ only if
\eqref{e3.28} is fulfilled.

Now we assume that \eqref{e3.28} holds. Define $\rho_i \in
(0,\frac{\alpha_i-\lambda_i}{\alpha_i})$ by \eqref{e3.18} and let $X_i
\in$ $\operatorname{RV}(\rho_i$) denote the functions
\begin{equation}
X_i(t) = \Bigl[\prod_{j=1}^n
\Bigl(\frac{t^{\frac{\alpha_j+1}{\alpha_j}}p_j(t)^{-1/\alpha_j}q_j(t)^{1/\alpha_j}}
{\Delta_j}\Bigr)^{M_{ij}}
\Bigr]^{\frac{A_n}{A_n-B_n}}, \quad i = 1,\dots,n. \label{e3.34}
\end{equation}
Then the $X_i$'s satisfy the system of asymptotic relations \eqref{e3.8} (with
$T = b$), i.e.,
\begin{equation}
\int_t^{\infty} \Bigl(\frac{1}{p_i(s)}\int_b^s
q_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds \sim
X_i(t), \quad t \to \infty, \; i = 1,,,,n, \label{e3.35}
\end{equation}
for any $b \geq a$, where $X_{n+1}(t) = X_1(t)$. In fact, using the
following expression for $X_i(t)$,
$$
X_i(t) = t^{\rho_i}\Xi_{i}(t), \quad \Xi_{i}(t)
= \Bigl[\prod_{j=1}^n\Bigl(\frac{l_j(t)^{-1/\alpha_j}
m_j(t)^{1/\alpha_j}}{\Delta_j}\Bigr)^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
$$
we obtain
$$
\Bigl(\frac{1}{p_i(t)}\int_b^t q_i(s)X_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}
\sim \frac{t^{\rho_i-1}l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}\Xi_{i+1}
(t)^{\beta_i/\alpha_i}}
{(\alpha_i-\lambda_i+\alpha_i\rho_i)^{1/\alpha_i}},
$$
and
\begin{equation}
\int_t^{\infty} \Bigl(\frac{1}{p_i(s)}\int_b^s
q_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds
\sim \frac{t^{\rho_i}l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}
\Xi_{i+1}(t)^{\beta_i/\alpha_i}}{\Delta_i}, \label{e3.36}
\end{equation}
as $t \to \infty$. Since it can be verified with the help of
\eqref{e3.27} that
$$
\frac{l_i(t)^{-1/\alpha_i}m_i(t)^{1/\alpha_i}}{\Delta_i}\Xi_{i+1}
(t)^{\beta_i/\alpha_i} = \Xi_{i}(t),
$$
the desired relation \eqref{e3.35} immediately follows from \eqref{e3.36}. This
completes the proof.
\end{proof}

\section{Regularly varying intermediate solutions of \eqref{eA}}

We are now in a position to state and prove our main results on
the existence and the precise asymptotic behavior of regularly
varying intermediate solutions of system \eqref{eA} with regularly
varying coefficients $p_i$ and $q_i$. Use is made of the notation
and properties of the matrix \eqref{e3.15} and \eqref{e3.17}.

\begin{theorem} \label{thm4.1}
 Let $p_i \in\operatorname{RV}(\lambda_i)$  and
$q_i \in\operatorname{RV}(\mu_i)$, $i = 1,\dots,n$.
Suppose that $\lambda_i < \alpha_i$ for $i = 1,\dots,n$. System \eqref{eA}
possesses intermediate solutions in $\operatorname{RV}(\rho_1,\dots,\rho_n)$
with $\rho_i \in (0,\frac{\alpha_i-\lambda_i}{\alpha_i})$,
$i =1,\dots,n$,  if and only if \eqref{e3.19}  holds, in which case
$\rho_i$ are given by \eqref{e3.18}  and the asymptotic behavior of
any such solution $(x_1,\dots,x_n)$ is governed by the unique
formula \eqref{e3.22}.
\end{theorem}

\begin{theorem} \label{thm4.2}
 Let $p_i \in\operatorname{RV}(\lambda_i$)  and
$q_i \in\operatorname{RV}(\mu_i)$, $i = 1,\dots,n$. Suppose that
$\lambda_i > \alpha_i$ for $i = 1,\dots,n$. System \eqref{eA}
possesses intermediate solutions in
$\operatorname{RV}(\rho_1,\dots,\rho_n)$ with
$\rho_i \in (\frac{\alpha_i-\lambda_i}{\alpha_i},0)$, $i =1,\dots,n$,
if and only if \eqref{e3.28} holds, in which case
$\rho_i$ are given by \eqref{e3.18}  and the asymptotic behavior of
any such solution $(x_1,\dots,x_n)$ is governed by the unique
formula \eqref{e3.29}.
\end{theorem}

We remark that the ``only if'' parts of these theorems follow
immediately from the corresponding parts of Theorems \ref{thm3.1}
 and \ref{thm3.2}
because any solution $(x_1,\dots,x_n)$ of \eqref{eA} with the indicated
property satisfies the systems of asymptotic relations \eqref{e3.6} plus
\eqref{e3.7} or \eqref{e3.8} plus \eqref{e3.9}. The ``if''
 parts are proved by way of the
following existence theorems for intermediate solutions for system
\eqref{eA} with nearly regularly varying coefficients $p_i(t)$ and
$q_i(t)$ in the sense defined below.

\begin{definition} \label{def4.1} \rm
Let $f(t)$ be a regularly varying function
of index $\sigma$ and suppose that $g(t)$ satisfies $kf(t) \leq
g(t) \leq Kf(t)$ for some positive constants $k$ and $K$ and for
all large $t$. Then $g(t)$ is said to be a \emph{nearly regularly
varying function of index} $\sigma$. Such a relation between
$f(t)$ and $g(t)$ is denoted by $g(t) \asymp f(t)$ as $t \to
\infty$.
\end{definition}

\begin{theorem} \label{thm4.3}
 Let $p_i$ and $q_i$ be nearly regularly
varying of indices $\lambda_i$ and $\mu_i$, respectively; that is,
there exist $\tilde{p}_i \in \operatorname{RV}(\lambda_i)$  and
$\tilde{q}_i \in\operatorname{RV}(\mu_i)$  such that
\begin{equation}
p_i(t) \asymp \tilde{p}_i(t), \quad q_i(t) \asymp \tilde{q}_i(t),
\quad t \to \infty, \; i = 1,\dots,n. \label{e4.1}
\end{equation}
 Suppose in addition that $\lambda_i < \alpha_i$, $i =1,\dots,n$, and that
\eqref{e3.19}  holds. Then, system \eqref{eA} possesses intermediate solutions
$(x_1,\dots,x_n)$ which are nearly regularly varying of positive index
$(\rho_1,\dots,\rho_n)$ with $\rho_i \in (0,\frac{\alpha_i-\lambda_i}{\alpha_i})$
in the sense that
\begin{equation}
x_i(t) \asymp
\Bigl[\prod_{j=1}^n\Bigl(\frac{t^{\frac{\alpha_j+1}{\alpha_j}}\tilde{p}_j(t)^{-1/\alpha_j}
\tilde{q}_j(t)^{1/\alpha_j}}{D_j}\Bigr)^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
\label{e4.2}
\end{equation}
for $t \to \infty$ and $i = 1,\dots,n$,
where $\rho_i$ and $D_i$ are defined by \eqref{e3.18} and
\eqref{e3.21},  respectively.
\end{theorem}

\begin{theorem} \label{thm4.4}
Let $p_i$ and $q_i$ be nearly regularly
varying of indices $\lambda_i$ and $\mu_i$, respectively; that is,
there exist $\tilde{p}_i \in \operatorname{RV}(\lambda_i)$  and
$\tilde{q}_i \in \operatorname{RV}(\mu_i)$  satisfying \eqref{e4.1}.
Suppose that $\lambda_i > \alpha_i$, $i = 1,\dots,n$, and that
\eqref{e3.28} holds. Then, system \eqref{eA} possesses intermediate
solutions solutions $(x_1,\dots,x_n)$ which are nearly regularly
varying of negative index $(\rho_1,\dots,\rho_n)$ with
$\rho_i \in(\frac{\alpha_i-\lambda_i}{\alpha_i},0)$, $i = 1,\dots,n$,  and
satisfy
\begin{equation}
x_i(t) \asymp
\Bigl[\prod_{j=1}^n\Bigl(\frac{t^{\frac{\alpha_j+1}{\alpha_j}}\tilde{p}_j(t)^{-1/\alpha_j}
\tilde{q}_j(t)^{1/\alpha_j}}{\Delta_j}\Bigr)^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
\quad t \to \infty, \; i = 1,\dots,n, \label{e4.3}
\end{equation}
where $\rho_i$ and $\Delta_i$ are defined by \eqref{e3.18}
and \eqref{e3.30}.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{thm4.3}]
We assume that the regularly varying functions $\tilde{p}_i(t)$ and
$\tilde{q}_i(t)$ are expressed in the form
\begin{equation}
\tilde{p}_i(t) = t^{\lambda_i}\tilde{l}_i(t), \quad
\tilde{q}_i(t) = t^{\mu_i}\tilde{m}_i(t), \quad
\tilde{l}_i, \tilde{m}_i \in {\rm SV} , \quad i = 1,\dots,n. \label{e4.4}
\end{equation}
By \eqref{e4.1} there exist positive constants $h_i$, $H_i$, $k_i$ and $K_i$
such that
\begin{equation}
h_i\tilde{p}_i(t) \leq p_i(t) \leq H_i\tilde{p}_i(t), \quad
k_i\tilde{q}_i(t) \leq q_i(t) \leq K_i\tilde{q}_i(t), \label{e4.5}
\end{equation}
for $t \geq a$ and $i = 1,\dots,n$.
Define the functions $X_i \in \operatorname{RV}(\rho_i)$ by
\begin{equation}
X_i(t) =
t^{\rho_i}\Bigl[\prod_{j=1}^n\Bigl(\frac{\tilde{l}_j(t)^{-1/\alpha_j}
\tilde{m}_j(t)^{1/\alpha_j}} {D_j}\Bigr)^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
\quad t \geq a, \; i = 1,\dots,n. \label{e4.6}
\end{equation}
It is known that
\begin{equation}
\int_b^t\Bigl(\frac{1}{\tilde{p}_i(s)}
\int_s^{\infty}\tilde{q}_i(r)X_{i+1}(s)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds
\sim X_i(t), \quad t \to \infty, \; i = 1,\dots,n, \label{e4.7}
\end{equation}
for any $b \geq a$, from which it follows that there exists $T >b$ such that
\begin{equation}
\int_T^t\Bigl(\frac{1}{\tilde{p}_i(s)}
\int_s^{\infty}\tilde{q}_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds
\leq  2X_i(t), \quad t \geq T, \; i = 1,\dots,n. \label{e4.8}
\end{equation}
Without loss of generality we may assume that each $X_i(t)$ is increasing on
$[T,\infty)$ because it is known that any regularly varying
function of positive index is asymptotically equivalent to an increasing
RV function of the same index (cf. \cite[Theorem 1.5.3]{BGT}).
Since \eqref{e4.7} holds for $b = T$ it is possible to choose
$T_1 > T$ so large that
\begin{equation}
\int_T^t\Bigl(\frac{1}{\tilde{p}_i(s)}\int_s^{\infty}\tilde{q}_i(r)
 X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds
\geq  \frac{1}{2}X_i(t) \label{e4.9}
\end{equation}
for $t \geq T_1$, $i = 1,\dots,n$.
We define the positive constants $l_i$ and $L_i$ ($l_i \leq L_i$) by
\begin{equation}
l_i =
\Bigl[\prod_{j=1}^n\Bigl\{\frac{1}{2}\Bigl(\frac{k_j}{H_j}
\Bigr)^{1/\alpha_j}\Bigr\}^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},\quad
L_i = \Bigl[\prod_{j=1}^n\Bigl\{4\Bigl(\frac{K_j}{h_j}\Bigr)^{1/\alpha_j}
\Bigr\}^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}}, \label{e4.10}
\end{equation}
$i = 1,\dots,n$.
As is easily verified, $l_i$ and $L_i$ satisfy the cyclic systems
of equalities
\begin{gather*}
l_i = \frac{1}{2}\Bigl(\frac{k_i}{H_i}\Bigr)^{1/\alpha_i}l_{i+1}^{\beta_i/\alpha_i},
\quad L_i = 4\Bigl(\frac{K_i}{h_i}\Bigr)^{1/\alpha_i}L_{i+1}^{\beta_i/\alpha_i},
\quad i = 1,\dots,n, \\
 L_{n+1} = L_1, \quad l_{n+1} = l_1.
\end{gather*}
Since
$$
\frac{L_i}{l_i} =
\Bigl[\prod_{j=1}^n\Bigl\{8\Bigl(\frac{H_jK_j}{h_jk_j}
\Bigr)^{1/\alpha_j}\Bigr\}^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
$$
the constants $h_i$, $H_i$, $k_i$ and $K_i$ can be chosen so that
$L_i/l_i \geq 2X_i(T_1)/X_i(T)$; that is,
\begin{equation}
2l_iX_i(T_1) \leq L_iX_i(T), \quad i = 1,\dots,n, \label{e4.11}
\end{equation}
because these constants are independent of $X_i(t)$ and the choice of
$T$ and $T_1$.

Let $\mathcal{X}$ denote the set consisting of continuous vector
functions $(x_1,\dots,x_n)$ on $[T,\infty)$ satisfying
\begin{equation}
l_iX_i(t) \leq x_i(t) \leq L_iX_i(t), \quad t \geq T, \; i =
1,\dots,n. \label{e4.12}
\end{equation}
It is clear that $\mathcal{X}$ is a closed convex subset of the locally convex
space $C[T,\infty)^n$. We consider the integral
operators $\mathcal{F}_i$ given by
\begin{equation}
\mathcal{F}_ix(t) = c_i + \int_T^t
\Bigl(\frac{1}{p_i(s)}\int_s^{\infty}q_i(r)x(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds,
\quad t \geq T, \; i = 1,\dots,n, \label{e4.13}
\end{equation}
where $c_i$ are positive constants such that
\begin{equation}
l_iX_i(T_1) \leq c_i \leq \frac{1}{2}L_iX_i(T), \quad i = 1,\dots,n,
\label{e4.14}
\end{equation}
and define the mapping $\Phi: \mathcal{X} \to C[T,\infty)^n$ by
\begin{equation}
\begin{gathered}
\Phi(x_1,x_2,\dots,x_n)(t) =
(\mathcal{F}_1x_2(t),\mathcal{F}_2x_3(t),\dots,\mathcal{F}_nx_{n+1}(t)),\\
t \geq T, \quad (x_{n+1}(t) = x_1(t)).
\end{gathered} \label{e4.15}
\end{equation}

We will show that the Schauder-Tychonoff fixed point theorem is applicable
to $\Phi$.

(i) $\Phi$ maps $\mathcal{X}$ into itself. Let $(x_1,\dots,x_n) \in \mathcal{X}$.
Then, using \eqref{e4.8}--\eqref{e4.15}, we see that
\begin{align*}
\mathcal{F}_i x_{i+1}(t)
&\leq \frac{1}{2}L_iX_i(T) +
\Bigl(\frac{K_iL_{i+1}^{\beta_i}}{h_i}\Bigr)^{1/\alpha_i}
\int_T^t \Bigl(\frac{1}{\tilde{p}_i(s)}
\int_s^{\infty}\tilde{q}_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds \\
& \leq \frac{1}{2}L_iX_i(T) +
2\Bigl(\frac{K_iL_{i+1}^{\beta_i}}{h_i}\Bigr)^{1/\alpha_i}X_i(t) \\
& \leq \frac{1}{2}L_iX_i(t) + \frac{1}{2}L_iX_i(t) = L_iX_i(t) \quad
\text{for } t \geq T,
\end{align*}
and
\begin{gather*}
\mathcal{F}_ix_{i+1}(t) \geq c_i \geq l_iX_i(T_1) \geq l_iX_i(t)
\quad \text{for } T \leq t \leq T_1, \\
\begin{aligned}
\mathcal{F}_ix_{i+1}(t)
&\geq \Bigl(\frac{k_il_{i+1}^{\beta_i}}{H_i}\Bigr)^{1/\alpha_i}
\int_T^t\Bigl(\frac{1}{\tilde{p}_i(s)}\int_s^{\infty}
\tilde{q}_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds \\
&\geq \frac{1}{2}\Bigl(\frac{k_il_{i+1}^{\beta_i}}{H_i}\Bigr)^{1/\alpha_i}X_i(t)
= l_iX_i(t)\quad t \geq T_1.
\end{aligned}
\end{gather*}
This shows that $\Phi(x_1,\dots,x_n) \in \mathcal{X}$; that is,
 $\Phi$ is a self-map on $\mathcal{X}$.

(ii) $\Phi(\mathcal{X})$ is relatively compact. From the inclusion
$\Phi(\mathcal{X}) \subset \mathcal{X}$ proven above
it follows that $\Phi(\mathcal{X})$ is locally uniformly bounded on
$[T,\infty)$. From the inequalities
$$
0 \leq (\mathcal{F}_ix_{i+1})'(t) \leq L_i^{\beta_i/\alpha_i}
\Bigl(\frac{1}{p_i(t)}\int_t^{\infty}q_i(s)X_{i+1}(s)^{\beta_i}ds
\Bigr)^{1/\alpha_i}, \quad t \geq T, \; i = 1,\dots,n,
$$
holding for all $(x_1,\dots,x_n) \in \mathcal{X}$ we see that
$\Phi(\mathcal{X})$ is locally equicontinuous on $[T,\infty)$. The
relative compactness of $\Phi(\mathcal{X})$ is an immediate
consequence of the Arzela-Ascoli lemma.

(iii) $\Phi$ is continuous. Let $\{(x_1^{\nu}(t),\dots,x_n^{\nu}(t))\}$
be a sequence in $\mathcal{X}$ converging as $\nu \to \infty$
to $(x_1(t),\dots,x_n(t)) \in \mathcal{X}$ uniformly on compact subintervals
of $[T,\infty)$. Using \eqref{e4.13} we obtain
\begin{equation}
|\mathcal{F}_{i}x_{i+1}^{\nu}(t)-\mathcal{F}_{i}x_{i+1}(t)| \leq
\int_T^t p_i(s)^{-1/\alpha_i}F_{i}^{\nu}(s)ds, \quad t
\geq T, \label{e4.16}
\end{equation}
where
$$
F_i^{\nu}(t)
= \Bigl|\Bigl(\int_t^{\infty} q_i(s)x_{i+1}^{\nu}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}
- \Bigl(\int_t^{\infty} q_i(s)x_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}\Bigr|.
$$
It is easy to see that
\begin{equation}
F_i^{\nu}(t) \leq \Bigl(\int_t^{\infty}
q_i(s)\Bigl|x_{i+1}^{\nu}(s)^{\beta_i}-x_{i+1}(s)^{\beta_i}
\Bigr|ds\Bigr)^{1/\alpha_i},
\label{e4.17}
\end{equation}
if $\alpha_i \geq 1$ and
\begin{equation}
\begin{aligned}
F_i^{\nu}(t)
& \leq \frac{1}{\alpha_i}\Bigl(L_{i+1}^{\beta_i}\int_t^{\infty}
q_i(s)X_{i+1}(s)^{\beta_i}ds\Bigr)^{\frac{1}{\alpha_i}-1} \\
&\quad\times \int_t^{\infty}
q_i(s)\Bigl|x_{i+1}^{\nu}(s)^{\beta_i}-x_{i+1}(s)^{\beta_i}\Bigr|ds,
\end{aligned} \label{e4.18}
\end{equation}
if $\alpha_i < 1$. Combine \eqref{e4.16} with \eqref{e4.17} or \eqref{e4.18}
and apply the Lebesgue dominated convergence theorem. Then we conclude that
$$
\lim_{\nu \to \infty}\mathcal{F}_{i}x_{i+1}^{\nu}(t)
= \mathcal{F}_{i}x_{i+1}(t)
$$
uniformly on any compact subset of
$[T,\infty)$, $i = 1,\dots,n$,
proving the continuity of $\Phi$.

Therefore, all the hypotheses of the Schauder-Tychonoff fixed point theorem
are fulfilled and $\Phi$ has a fixed point
$(x_1,\dots,x_n) \in \mathcal{X}$, which satisfies
\begin{equation}
\begin{aligned}
x_i(t) &= \mathcal{F}_i x_{i+1}(t) \\
&= c_i + \int_T^t
\Bigl(\frac{1}{p_i(s)}\int_s^{\infty}
q_i(r)x_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds, \quad
t \geq T, \; i = 1,\dots,n.
\end{aligned} \label{e4.19}
\end{equation}
This shows that $(x_1,\dots,x_n)$ is a solution of system \eqref{eA} on
$[T,\infty)$. Since the obtained solution is a member of
$\mathcal{X}$, it is nearly regularly varying of positive index
$(\rho_1,\dots,\rho_n)$ and hence is an intermediate solution of
\eqref{eA}. This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.4}]
Assume that $\tilde{p}_i(t)$ and
$\tilde{q}_i(t)$ are expressed in the form \eqref{e4.4} and satisfy \eqref{e4.5}
for positive constants $h_i$, $H_i$, $k_i$ and $K_i$. Suppose that
\eqref{e3.28} holds. Define $\rho_i$ and $\Delta_i$ by \eqref{e3.18}
and \eqref{e3.30}, respectively, and consider the regularly varying functions of
indices $\rho_i$
\begin{equation}
Y_i(t) = t^{\rho_i}\Bigl[\prod_{j=1}^n
\Bigl(\frac{\tilde{l}_j(t)^{-1/\alpha_j}\tilde{m}_j(t)^{1/\alpha_j}}{\Delta_j}
\Bigr)^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
\quad i = 1,\dots,n. \label{e4.20}
\end{equation}
Since $Y_i(t)$ satisfy the asymptotic relations
\begin{equation}
\int_t^{\infty}\Bigl(\frac{1}{\tilde{p}_i(s)}\int_b^s
\tilde{q}_i(r)Y_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_j}ds
\sim Y_i(t), \quad t \to \infty, \; i = 1,\dots,n, \label{e4.21}
\end{equation}
one can choose $T > a$ so that
\begin{equation}
\frac{1}{2}Y_i(t) \leq
\int_t^{\infty}\Bigl(\frac{1}{\tilde{p}_i(s))}\int_b^s
\tilde{q}_i(r)Y_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_j}ds
\leq 2Y_i(t), \label{e4.22}
\end{equation}
for $t \geq T$ and $i = 1,\dots,n$.
All of $Y_i(t)$ may be assumed to be decreasing on $[T,\infty)$ because
it is known \cite[Theorem 1.5.3]{BGT} that a regularly varying function of
negative index is asymptotic to a decreasing RV function of the same index.
Denote by $\mathcal{Y}$ the set consisting of continuous vector
functions $(x_1,\dots,x_n)$ such that
\begin{equation}
l_iY_i(t) \leq x_i(t) \leq L_iY_i(t), \quad t \geq T, \;
 i = 1,\dots,n, \label{e4.23}
\end{equation}
where
\begin{equation}
l_i =\Bigl[\prod_{j=1}^n\Bigl\{\frac{1}{2}\Bigl(\frac{k_j}{H_j}\Bigr)^{1/\alpha_j}
\Bigr\}^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}}, \quad
L_i = \Bigl[\prod_{j=1}^n\Bigl\{2\Bigl(\frac{K_j}{h_j}\Bigr)^{1/\alpha_j}
\Bigr\}^{M_{ij}}\Bigr]^{\frac{A_n}{A_n-B_n}},
 \label{e4.24}
\end{equation}
for $i = 1,\dots,n$, which satisfy the cyclic systems of equalities
\begin{equation}
l_i =\frac{1}{2}\Bigl(\frac{k_i}{H_i}\Bigr)^{1/\alpha_i}l_{i+1}^{\beta_i/\alpha_i},
\quad
L_i = 2\Bigl(\frac{K_i}{h_i}\Bigr)^{1/\alpha_i}L_{i+1}^{\beta_i/\alpha_i},
\quad l_{n+1}=l_1,\; L_{n+1}=L_1. \label{e4.25}
\end{equation}

We now consider the mapping $\Psi: \mathcal{Y} \to C[T,\infty)^n$ defined by
\begin{equation}
\Psi(x_1,\dots,x_n)(t)
= (\mathcal{G}_1x_2(t),\mathcal{G}_2x_3(t),\dots,\mathcal{G}_nx_{n+1}(t)),
\label{e4.26}
\end{equation}
for $t \geq T$ and $x_{n+1}(t) = x_1(t)$, 
where $\mathcal{G}_i$ denotes the integral operator
\begin{equation}
\mathcal{G}_ix(t) =
\int_t^{\infty}\Bigl(\frac{1}{\tilde{p}_i(s)}\int_T^s
\tilde{q}_i(r)x(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds,
\quad t \geq T, \quad i = 1,\dots,n. \label{e4.27}
\end{equation}

It is a matter of straightforward calculation to verify that
$\Psi$ is a self-map on $\mathcal{Y}$ and sends $\mathcal{Y}$ into
a relatively compact subset of $C[T,\infty)^n$. The details may
be omitted. Therefore, the Schauder-Tychonoff fixed point theorem
ensures the existence of a fixed point $(x_1,\dots,x_n) \in
\mathcal{Y}$ of $\Psi$. This fixed point gives rise to an
intermediate solutions of \eqref{eA} which is nearly regularly varying of
negative index $(\rho_1,\dots,\rho_n)$. This completes the proof.
\end{proof}

To complete the proof of the ``if'' parts of Theorems \ref{thm4.1}
 and \ref{thm4.2} it
suffices to show that if $p_i(t)$ and $q_i(t)$ are assumed to be
regularly varying, then the nearly regularly varying solutions
obtained in Theorems \ref{thm4.3} and \ref{thm4.4} actually become regularly varying
of the same specified indices. For this purpose use is made of the
following generalized L'H$\hat{\rm o}$pital's rule. See, for
example, Haupt and Aumann \cite{HA}.

\begin{lemma} \label{lem4.1}
 Let $f(t), g(t) \in C^{1}[T,\infty)$ and suppose that
$$
\lim_{t \to \infty} f(t) = \lim_{t \to \infty} g(t) = \infty \quad
\text{and} \quad g'(t) > 0\quad\text{for all large } t,
$$
or
$$
\lim_{t \to \infty} f(t) = \lim_{t \to \infty} g(t) = 0 \quad
\text{and} \quad g'(t) < 0\quad\text{for all large } t.
$$
 Then
$$
\liminf_{t \to \infty}\frac{f'(t)}{g'(t)} \leq  \liminf_{t \to
\infty}\frac{f(t)}{g(t)}, \quad
\limsup_{t \to \infty}\frac{f(t)}{g(t)} \leq  \limsup_{t \to
\infty}\frac{f'(t)}{g'(t)}.
$$
\end{lemma}

\begin{proof}[Proof of the ``if'' parts of Theorem \ref{thm4.1}]
 Suppose that $p_i \in$ $\operatorname{RV}(\lambda_i$) and
$q_i \in$ $\operatorname{RV}(\mu_i$). Suppose in
addition that $\lambda_i < \alpha_i$.  Define the positive
constants $\rho_i$ and $D_i$ by \eqref{e3.18} and \eqref{e3.21}, respectively,
and let $X_i \in$ $\operatorname{RV}(\rho_i$) denote the functions on the
right-hand side of \eqref{e4.2} with $\tilde{p}_i(t)$ and
$\tilde{q}_i(t)$ replaced with $p_i(t)$ and $q_i(t)$,
respectively. Then, by Theorem \ref{thm4.3} system \eqref{eA} has a nearly
regularly varying solution $(x_1,\dots,x_n)$ such that
$x_i(t) \asymp X_i(t)$ as $t \to \infty$, $i = 1,\dots,n$. Notice that
$x_i(t)$ satisfy the system of integral equations \eqref{e4.19}.

It remains to verify that $x_i$ are regularly varying functions of
indices $\rho_i$, $i = 1,\dots,n,$ respectively. We define
\begin{equation}
u_i(t) = \int_T^t \Bigl(\frac{1}{p_i(s)}\int_s^{\infty}
q_i(r)X_{i+1}(r)^{\beta_i}dr\Bigr)^{1/\alpha_i}ds, \quad
i = 1,\dots,n, \label{e4.28}
\end{equation}
and put
$$
\omega_i = \liminf_{t \to \infty} \frac{x_i(t)}{u_i(t)}, \quad
\Omega_i = \limsup_{t \to \infty} \frac{x_i(t)}{u_i(t)}.
$$
Since $x_i(t) \asymp X_i(t)$ and
\begin{equation}
u_i(t) \sim X_i(t), \quad t \to \infty, \; i = 1,\dots,n,
\label{e4.29}
\end{equation}
it follows that $0 < \omega_i \leq \Omega_i < \infty$,
$i = 1,\dots,n$. Using Lemma \ref{lem4.1} we obtain
\begin{align*}
\omega_i &\geq \liminf_{t \to \infty}\frac{x_i'(t)}{u_i'(t)}
=\liminf_{t \to \infty}\frac{\Bigl(\frac{1}{p_i(t)}\int_t^{\infty}
q_i(s)x_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}}
{\Bigl(\frac{1}{p_i(t)}\int_t^{\infty}
q_i(s)X_{i+1}(s)^{\beta_i}ds\Bigr)^{1/\alpha_i}} \\
&= \liminf_{t \to \infty} \Bigl(\frac{\int_t^{\infty}
q_i(s)x_{i+1}(s)^{\beta_i}ds}{\int_t^\infty
q_i(s)X_{i+1}(s)^{\beta_i}ds}\Bigr)^{1/\alpha_i} \\
&= \Bigl(\liminf_{t \to \infty} \frac{\int_t^{\infty}
q_i(s)x_{i+1}(s)^{\beta_i}ds}{\int_t^\infty
q_i(s)X_{i+1}(s)^{\beta_i}ds}\Bigr)^{1/\alpha_i} \\
&\geq \Bigl(\liminf_{t \to
\infty}\frac{q_i(t)x_{i+1}(t)^{\beta_i}}{q_i(t)X_{i+1}
(t)^{\beta_i}}\Bigr)^{1/\alpha_i}
= \liminf_{t \to
\infty}\Bigl(\frac{x_{i+1}(t)}{X_{i+1}(t)}\Bigr)^{\beta_i/\alpha_i}
= \omega_{i+1}^{\beta_i/\alpha_i},
\end{align*}
where \eqref{e4.29} has been used in the last step. Thus, $\omega_i$
satisfy the cyclic system of inequalities
\begin{equation}
\omega_i \geq \omega_{i+1}^{\beta_i/\alpha_i}, \quad i =
1,\dots,n, \quad \omega_{n+1} = \omega_1. \label{e4.30}
\end{equation}
Likewise, by taking the upper limits instead of the lower limits we
are led to the cyclic inequalities
\begin{equation}
\Omega_i \leq \Omega_{i+1}^{\beta_i/\alpha_i}, \quad i =
1,\dots,n, \quad \Omega_{n+1} = \Omega_1. \label{e4.31}
\end{equation}
From \eqref{e4.30} and \eqref{e4.31} we easily see that
$$
\omega_i \geq
\omega_i^{\frac{\beta_1{\dots}\beta_n}{\alpha_1{\dots}\alpha_n}},
\quad \Omega_i \leq
\Omega_i^{\frac{\beta_1{\dots}\beta_n}{\alpha_1{\dots}\alpha_n}},
$$
whence, because of the hypothesis
$\beta_1{\dots}\beta_n/\alpha_1{\dots}\alpha_n < 1$, we find
that $\omega_i \geq 1$ and $\Omega_i \leq 1$. It follows therefore
that $\omega_i = \Omega_i = 1$ or $\lim_{t \to
\infty}x_i(t)/u_i(t) = 1$ for $i = 1,\dots,n.$  This combined with
\eqref{e4.29} implies that $x_i(t) \sim u_i(t) \sim X_i(t)$ as $t \to
\infty$, which shows that each $x_i(t)$ is a regularly varying
function of index $\rho_i$. Thus the proof of the ``if" part of
Theorem \ref{thm4.1} is complete.

In essentially the same way one can complete the proof of the
``if" part of Theorem \ref{thm4.2}.
\end{proof}


\section{Applications to partial differential equations}

The purpose of the final section is to demonstrate that our results on cyclic
systems of ordinary differential equations \eqref{eA}
can be applied to some classes of partial differential equations to provide
new information about the existence and asymptotic
behavior of their radial positive solutions.
Throughout this section $x = (x_1,\dots,x_N)$ represents the space variable in
$\mathbb{R}^N$, $N \geq 2$, and $|x|$ denotes the Euclidean length of $x$.
All partial differential equations will be considered in
an exterior domain $\Omega_R = \{x \in \mathbb{R}^N: |x| \geq R\}$, $R > 0$.

\subsection{Systems of $p$-Laplacian equations}

We are concerned with the system of nonlinear $p$-Laplacian equations
\begin{equation}
\operatorname{div}\Bigl(|\nabla u_i|^{p-2}\nabla u_i\Bigr) +
f_i(|x|)|u_{i+1}|^{\gamma_i-1}u_{i+1} = 0, \quad i = 1,\dots,n,
\quad (u_{n+1} = u_1) \label{e5.1}
\end{equation}
where $p > 1$ and $\gamma_i > 0$ are constants, and $f_i(t)$ are positive
continuous functions on $[a,\infty)$ which are regularly
varying of indices $\nu_i$, $i = 1,\dots,n.$  Our attention will be focused on
radial solutions $(u_1(|x|),\dots,u_n(|x|))$ of \eqref{e5.1}
defined in $\Omega_R$, $R > a$. A radial vector function $(u_1(|x|),\dots,u_n(|x|))$
is a solution of \eqref{e5.1} in $\Omega_R$ if and only if
$(u_1(t),\dots,u_n(t))$ is a solution of the system of ordinary differential
equations
\begin{equation}
\begin{gathered}
(t^{N-1}|u_{i}'|^{p-2}u_{i}')' +
t^{N-1}f_i(t)|u_{i+1}|^{\gamma_i-1}u_{i+1} = 0, \quad t \geq a,
\quad i = 1,\dots,n, \\
 u_{n+1} = u_1
\end{gathered} \label{e5.2}
\end{equation}
which is a special case of system \eqref{eA} with
\begin{gather*}
\alpha_1 = \dots = \alpha_n = p-1, \quad \beta_i = \gamma_i, \quad i = 1,\dots,n;\\
\lambda_1 = {\dots} = \lambda_n = N-1, \quad
\mu_i = N-1+\nu_i, \quad  i = 1,\dots,n.
\end{gather*}
It is always assumed that
\begin{equation}
\gamma_1{\dots}\gamma_n < (p-1)^n. \label{e5.3}
\end{equation}
We need the matrix $A\Bigl(\frac{\gamma_1}{p-1},{\dots},\frac{\gamma_n}{p-1}\Bigr)$
associated with \eqref{e5.2} and its inverse
(cf. \eqref{e3.15} and \eqref{e3.17}). We define
\begin{equation}
(M_{ij}) =
\frac{(p-1)^n-\gamma_1{\dots}\gamma_n}{(p-1)^n}A
\Bigl(\frac{\gamma_1}{p-1},{\dots},\frac{\gamma_n}{p-1}\Bigr)^{-1}.
\label{e5.4}
\end{equation}

To analyze \eqref{e5.2} it is necessary to distinguish the two cases $p > N$
and $p < N$.

(i) Suppose that $p > N$. In this case applying Theorem \ref{thm4.1} to \eqref{e5.2},
 we conclude that system \eqref{e5.1} possesses increasing
radial solutions $(u_1(|x|),\dots,u_n(|x|))$ such that
$u_i \in\operatorname{RV}(\rho_i$), $0 < \rho_i < \frac{p-N}{p-1}$,
 $i = 1,\dots,n$,
if and only if
\begin{equation}
0 < \sum_{j=1}^n M_{ij}(p+\nu_j) < (p-N)\Big(1 -
\frac{\gamma_1\gamma_2 \dots \gamma_n}{(p-1)^n}\Big), \quad i =
1,\dots,n. \label{e5.5}
\end{equation}
In this case the $\rho_i$'s are uniquely determined by
\begin{equation}
\rho_i = \frac{(p-1)^{n-1}}{(p-1)^n -
\gamma_1{\dots}\gamma_n}\sum_{j=1}^n M_{ij}(p+\nu_j), \quad i =
1,\dots,n, \label{e5.6}
\end{equation}
and moreover the asymptotic behavior of any such solution as $|x| \to \infty$
is governed by the unique growth law
\begin{equation}
u_i(|x|) \sim |x|^{\rho_i}\Bigl[\prod_{j=1}^n
\Bigl(\frac{\varphi_j(|x|)}{(p-N-(p-1)\rho_j)\rho_j^{p-1}}\Bigr)^{M_{ij}}
\Bigr]^{\frac{(p-1)^{n-1}}
{(p-1)^n - \gamma_1{\dots}\gamma_n}}, \quad |x| \to \infty,
\label{e5.7}
\end{equation}
for $i = 1,\dots,n$, where $\varphi_i \in {\rm SV}$ is the slowly varying
part of $f_i$; that is, $f_i(t) = t^{\nu_i}\varphi_i(t)$.


(ii) Suppose that $p < N$. In this case from Theorem \ref{thm4.2} applied
to \eqref{e5.2} it follows that system \eqref{e5.1} possesses decreasing radial
solutions $(u_1(|x|),\dots,u_n(|x|))$ such that
$u_i \in \operatorname{RV}(\rho_i)$, $\frac{p-N}{p-1} < \rho_i < 0$,
$i = 1,\dots,n$, if and only if
\begin{equation}
(p-N)\Big(1 - \frac{\gamma_1\gamma_2 \dots
\gamma_n}{(p-1)^n}\Big) < \sum_{j=1}^nM_{ij}(p + \nu_j) < 0,
\quad i = 1,\dots,n. \label{e5.8}
\end{equation}
In this case $\rho_i$ are uniquely determined by \eqref{e5.6} and the
asymptotic behavior of any such solution as $|x| \to \infty$ is
governed by the unique decay law
\[
u_i(|x|) \sim |x|^{\rho_i}\Bigl[\prod_{j=1}^n
\Bigl(\frac{\varphi_j(|x|)}{(N-p+(p-1)\rho_j)(-\rho_j)^{p-1}}\Bigr)^{M_{ij}}\Bigr]^{\frac{(p-1)^{n-1}}
{(p-1)^n - \gamma_1{\dots}\gamma_n}},
\]
as $|x| \to \infty$,
for $i = 1,\dots,n$, where $\varphi_i$ is the slowly varying part
of $f_i$.

Consider the particular case of \eqref{e5.1} in which $f_i(t) \equiv c_i > 0$, i.e.,
\begin{equation}
\operatorname{div}\Bigl(|\nabla u_i|^{p-2}\nabla u_i\Bigr) + c_i
|u_{i+1}|^{\gamma_i-1}u_{i+1} = 0, \quad i = 1,\dots,n, \quad
u_{n+1} = u_1. \label{e5.9}
\end{equation}
In this case $\nu_i = 0$ for all $i$, and so \eqref{e5.5} and \eqref{e5.8}
are always violated. Therefore system \eqref{e5.9} cannot admit
intermediate radial solutions $(u_1(|x|),\dots,u_N(|x|))$ such that
$u_i \in$ $\operatorname{RV}(\rho_i$), where $\rho_i$ satisfy
\begin{gather*}
0 < \rho_i < \frac{p-N}{p-1}, \quad i = 1,\dots,n,\quad \text{if } p > N; \\
\frac{p-N}{p-1} < \rho_i < 0, \quad i = 1,\dots,n,\quad  \text{if } p < N.
\end{gather*}

\subsection{Nonlinear metaharmonic equations}

Now we consider the nonlinear metaharmonic equation
\begin{equation}
\Delta^{m}u = (-1)^m g(|x|)|u|^{\gamma-1}u, \quad x \in \Omega_R,
\label{e5.10}
\end{equation}
 where $N \geq 3$, $m \geq 2$ and $\gamma > 0$ are constants, and $g(t)$
is a positive continuous function
on $[a,\infty)$ which is regularly varying of index $\nu$.
We are interested in radial positive solutions $u$ of \eqref{e5.10} such that
$u$ and $(-1)^{i}\Delta^{i}u$, $i = 1,\dots,m-1$, are regularly varying of
negative indices. It is clear that seeking such solutions
of \eqref{e5.10} is equivalent to seeking radial regularly varying solutions
of negative indices of the system
\begin{equation}
\begin{gathered}
\Delta u_i + u_{i+1} = 0, \quad i = 1,\dots,m-1, \\
\Delta u_{m} + g(|x|)|u_{m+1}|^{\gamma-1}u_{m+1} = 0, \quad x \in \Omega_R,
\end{gathered} \label{e5.11}
\end{equation}
where $u_{m+1} = u_1$. This system is equivalent to the system of ordinary
 differential equations
\begin{equation}
\begin{gathered}
(t^{N-1}u_{i}')' + t^{N-1}u_{i+1} = 0, \quad i = 1,\dots,m-1, \\
(t^{N-1}u_m')' + t^{N-1}g(t)|u_{m+1}|^{\gamma-1}u_{m+1} = 0, \quad
t \geq R,
\end{gathered} \label{e5.12}
\end{equation}
which is a special case of \eqref{eA} with
\begin{gather*}
\alpha_1 ={\dots}=\alpha_m = 1, \quad \beta_i = {\dots} = \beta_{m-1} =1, \quad
 \beta_m = \gamma, \\
\lambda_1 ={\dots} = \lambda_m = N-1, \quad \mu_1 = {\dots} = \mu_{m-1}= N-1,
\quad \mu_{m} = N-1+\nu.
\end{gather*}

We assume that $\gamma < 1$. The $m{\times}m$-matrix \eqref{e3.15}
associated with \eqref{e5.12} reads $A(1,\dots,1,\gamma)$. Define the
matrix $(M_{ij})$ by
\begin{equation}
(M_{ij}) = (1-\gamma)A(1,\dots,1,\gamma)^{-1}. \label{e5.13}
\end{equation}
As is easily checked, $M_{ij} = 1$ for $1 \leq i \leq j \leq m$ and
 $M_{ij} = \gamma$ for $1 \leq j < i \leq m$.

Since $\lambda_i = N-1 > 1 = \alpha_i$ for all $i$, Theorem \ref{thm4.2}
can be utilized to determine the structure of decreasing regularly
varying solutions $(u_1,\dots,u_m) \in \operatorname{RV}(\rho_1,\dots,\rho_m)$,
$2-N < \rho_i < 0$, of the cyclic system \eqref{e5.12}. The regularity
indices $\rho_i$ should be given by \eqref{e3.18} which in the present
situation reduce to
\begin{equation}
\rho_i = \frac{2m+\nu}{1-\gamma} - 2(i-1), \quad i = 1,\dots,m,
\label{e5.14}
\end{equation}
from which we see that all $\rho_i$ are admissible if and only if
\begin{equation}
2m-N < \frac{2m+\nu}{1-\gamma} < 0 \; \Longleftrightarrow \;
-2m+(2m-N)(1-\gamma) < \nu < -2m. \label{e5.15}
\end{equation}
Clearly, \eqref{e5.15} makes sense only if $N > 2m$, in which case it is
concluded that equation \eqref{e5.10} possesses radial positive solutions
$u(|x|) \in\operatorname{RV}(\rho_1)$, where
\begin{equation}
\rho_1 = \frac{2m+\nu}{1-\gamma} \in (2-N, 0), \label{e5.16}
\end{equation}
such that $(-1)^{i}\Delta^{i}u(|x|) \in \operatorname{RV}(\rho_{i+1})$,
 $2-N < \rho_{i+1} < 0$, for $i = 1,\dots,m-1$. Furthermore, the asymptotic
behavior of any such solution $u(|x|)$ is governed by the formula
\begin{equation}
u(|x|) \sim
|x|^{\rho_1}\Bigl[\frac{\psi(|x|)}{[\prod_{j=1}^{i-1}(N-2+\rho_j)(-\rho_j)]^{\gamma}
\prod_{j=i}^{m}(N-2+\rho_j)(-\rho_j)}\Bigr]^{\frac{1}{1-\gamma}}, \label{e5.17}
\end{equation}
as $|x| \to \infty$,
where $\psi(t)$ denotes the slowly varying function such that
$g(t) = t^{\nu}\psi(t)$.
We remark that the particular case of \eqref{e5.10},
$$
\Delta^{m}u = (-1)^m c|u|^{\gamma-1}u, \quad x \in \Omega_R\,,
$$
where $c > 0$ is a constant, can by no means possess radial solutions
$u(|x|) \in \operatorname{RV}(\rho_1$) with $2-N < \rho_1 < 0$.


\subsection*{Acknowledgments}
The authors would like to express their
sincere thanks to the anonymous referees for their valuable comments and
suggestions.

The first author was supported by the grant No.1/0071/14 of the
Slovak Grant Agency VEGA.



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\end{document}