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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 308, pp. 1--12.\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-2016/308\hfil Boundary behavior of solutions]
{Boundary behavior of the unique solution of
a one-dimensional problem}

\author[L. Mi \hfil EJDE-2016/308\hfilneg]
{Ling Mi}

\address{Ling Mi \newline
School of  Science,
Linyi University,
Linyi, Shandong 276005, China}
 \email{mi-ling@163.com}


\thanks{Submitted June 30, 2015. Published November 30, 2016.}
\subjclass[2010]{35J25, 35J60, 35J65}
\keywords{One-dimensional problems; uniqueness of the solution;
\hfill\break\indent boundary behavior}

\begin{abstract}
 In this article, we analyze the blow-up rate of the unique
 solution  to the singular boundary value problem
 \begin{gather*}
 u''(t) =b(t)f(u(t)), \quad u(t)>0, \; t>0, \\
 u(0)=\infty, \quad u(\infty)=0,
 \end{gather*}
 where  $f(u)$  grows more slowly than $u^p$ ($p > 1$) at infinity,
 and  $b \in C^{1}(0, \infty)$  which is positive and non-decreasing 
 (it may vanish at zero).
\end{abstract}

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

\section{Introduction and statement of main results}

 In this article, we  consider the blow-up rate of the
 unique solution  at zero of the singular  boundary-value problem
\begin{equation}\label{1.1}
\begin{gathered}
u''(t) =b(t)f(u(t)), \quad u(t)>0,\;  t>0, \\
u(0)=\infty, \quad u(\infty)=0,
\end{gathered}
\end{equation}
under the following assumptions on the functions $b$ and
$f$:
\begin{itemize}
\item[(A1)]   $b \in C^{1}(0, \infty)$ is non-decreasing
  and  $b(t)>0$  for $t>0$,

\item[(A2)]  $f\in C^1[0,\infty)$, $f(0)=f'(0)=0$, $f'(u)>0$ for any $u>0$,

\item[(A3)] the Keller-Osserman  \cite{KE,OS} condition
$$
   \Theta (r):= \int_r^\infty
\frac{ds}{\sqrt{2F(s)}}<\infty,  \quad \forall r>0,\quad
 F(s)=\int_0^s f(\tau) d\tau.
$$
\end{itemize}
Boundary blow-up problems rise in many branches of mathematics and
have been studied by many authors and in several contexts for a long
time. Generally, solutions of boundary blow-up problems are said to
be explosive solutions or large solutions. The pioneering research
work on boundary blow-up problems goes back to Keller-Osserman
\cite{KE,OS}, who
proved that the problem
\begin{equation}\label{e1}
\triangle u =f(u), \quad u>0,  \quad x\in \Omega,\quad  u|_{\partial \Omega}=\infty.
\end{equation}
has  one  solution $u\in C^2(\Omega)$ if and only if (A3) holds.

Loewner and Nirenberg \cite{LO}  showed that if $f(u) = u^{p_0}$
with $p_0 = \frac{N+2}{N-2 }$, $N > 2$, then problem \eqref{e1} has
a unique solution $u$ satisfying
\begin{equation*}
\lim_{d(x)\to 0} u(x)(d(x))^{(N-2)/2}
 =\Big(\frac{N(N-2) }{4}\Big)^{(N-2)/4}.
\end{equation*}

A function $f$ is weakly superlinear when
\begin{equation}\label{1.4}
 f(s)=\beta_1s(\ln s)^{\alpha}+\gamma_1s(\ln s)^{\alpha-1}[1+o(1)]
\quad\text{as }s \to \infty,
\end{equation}
with $\beta_1>0$, $\alpha>2$ and $\gamma_1\in(-\infty, +\infty)$.
This function grows more slowly at infinity than those variational functions
with index $p > 1$ or rapid ones.
 When $f$ is weakly superlinear, C\^irstea and Du \cite{CD1} consider the
first  order expansion of the blow-up solution of
\begin{equation}\label{e1.1}
   \triangle u=b(x)f(u), \quad u>0, \quad x \in \Omega,\quad
 u|_{\partial \Omega}=\infty,
\end{equation}
where  $\Omega$ is a bounded domain with smooth boundary in
$\mathbb{R}^N(N\geq2)$.

We point out that C\^{i}rstea and R\v{a}dulescu \cite{CR1}-\cite {CR4}, and
C\^{i}rstea and Du \cite {CD1} introduced a new
unified approach via the Karamata regular variation theory,
 to study the boundary behavior and uniqueness of
 solutions for boundary blow-up elliptic problems.
For singular elliptic problems, we refer the reader to the papers
\cite{CRV,GR}, \cite{VR}-\cite{DR}, \cite{Z1}-\cite{ZH1}
and the references therein.

Now, let us return to problem \eqref{1.1}.
Cano-Casanova and L\'{o}pez-G\'{o}mez  \cite{SJ} studied the existence,
uniqueness and the blow-up rate of large solutions of
\begin{equation}\label{1.3}
u''(t) =b(t)f(u(t)), \quad t>0, \quad u(0)=+\infty, \quad u(+\infty)=0,
\end{equation}
where $f$ satisfies (A2), (A3) and $b$  satisfies
\begin{itemize}
\item[(A1')]   $b \in C[0, \infty)$   is non-decreasing
  and satisfies  $b(t)>0$  for  $t>0$,
\end{itemize}
Under the conditions (A2) and (A1'),   problem
\eqref{1.3} possesses a unique positive solution $\psi(t)$. Further,
assuming that the following conditions are satisfied
\begin{itemize}
\item[(i)]    $f^{*}(u)=f(u)/u$
is non-decreasing on $(0, \infty)$ and, for some $\sigma > 1$,
$c_0:=\lim_{u\to \infty} f(u)/u^{\sigma} \in (0, \infty)$;

\item[(ii)] the limit
$$
a_0:=\lim_{t\to 0^{+}}\frac{G(t)G''(t)}{[G'(t)]^{2}}
\in (0, \infty)
$$
is well defined for some $R > 0$, where $G(t)$
stands for the function
$$
G (t) = \int^{R}_{t} \frac{ds}{A(s)}, \quad
A(t) = \Big( \int^{t}_0(b(\tau))^{1/(\sigma+1)} d\tau \Big)^{(\sigma+1)/(\sigma-1)},
 \quad t \in (0, R],
$$
\end{itemize}
the unique large solution $\psi(t)$ of \eqref{1.3} satisfies
$$
\lim_{t\to 0^{+}} \frac{\psi(t)}{G (t)}
=a_0^{-\sigma/(\sigma-1)}\Big(\frac{\sigma+1}{\sigma-1}\Big)^{(\sigma+1)/(\sigma-1)}
c_0^{-1/(\sigma-1)}.
$$

Later, using the Karamata regular variation theory,  Zhang  et al.\
\cite{Z1} obtained the exact blow-up rate of the unique solution
$\psi(t)$ of  \eqref{1.3} for a more general nonlinear term $f$. Let
$b$ satisfy  (A1) and $\sqrt{b}\in \Lambda$ (see the definition
of $\Lambda$ below),  $f$ satisfy (A2) and
\begin{itemize}
\item[ (iii)]   $\int_0^{1}\frac{d\nu}{f(\nu)}=\infty$;

\item[(iv)]  $\lim_{s\to\infty} s f' (s)/f(s)  = \sigma > 1$.
\end{itemize}
Then,   the unique solution $\psi(t)$ of \eqref{1.3} satisfies
\begin{equation*}
\lim_{t \to 0^+} \frac{\psi(t)}{\varphi(K(t))}
=\Big(\frac{2(C_{k}(\sigma-1)+2)}{\sigma-1}\Big)^{\sigma-1},
\end{equation*}
where $K(t)=\int_0^{t}\sqrt{b(s)}ds$  and $\varphi$ is uniquely
determined by the problem
$$
\int_{\varphi(t)}^{\infty}\frac{d\nu}{f(\nu)}=t, \quad t>0.
$$

However,  there are  fewer results for  the exact blow-up rate of the
unique solution to \eqref{1.1} at zero  when $f (u)$ grows more slowly
than $u^{p}$ $(p >  1)$ at infinity. This case  is more difficult to
handle than those foregoing cases, since the blow-up behavior of the
solution depends more subtly on the behavior of $b(t)$ and $f (u)$.


Next we explain our assumption on $b(x)$.
Let $\Lambda$ denote the set of  positive non-decreasing functions in
$C^1(0,\delta_0)$ which satisfy
$$
\lim_{t \to 0^+} \frac
{d}{dt}\Big(\frac{K(t)}{k(t)}\Big): = C_{k}\in [0, \infty),\quad
K(t)=\int_0^t k(s)ds.
$$
We see that for each $k\in \Lambda$,
 $$
\lim_{t\to 0^+}\frac {K(t)}{k(t)}=0, \quad  C_{k}\in [0, 1]
$$
and
\begin{equation}\label{1.5}
\lim_{t \to 0^+} \frac{K(t)k'(t)}{k^2(t)}
=1-\lim_{t\to 0^+} \frac{d}{dt}\Big(\frac{K(t)}{k(t)}\Big)=1-C_{k}.
\end{equation}
The set $\Lambda$ was first introduced  by C\^{i}rstea and
R\v{a}dulescu \cite {CR1} for   studying  the boundary behavior and
uniqueness of solutions  of  problem \eqref{e1.1}.

Inspired by the above ideas, the main purpose of this article is to
establish blow-up rate of the unique solution  $l(t)$  at zero to
 \eqref{1.1} under appropriate conditions on the weight function
$b$ and the nonlinear term $f$. In this article,  we assume that $f$
growths more slowly than any $u^p$ ($p > 1$) at infinity. In
particular, we consider functions $f$ which satisfy
(A2) and (A3) and the following conditions hold:
\begin{itemize}
\item[(A4)] there exist  two  functions
$f_1\in C^1[S_0, \infty)$ for some large $S_0>0$ and $f_2$  such
that
$$
f(s): =f_1(s)+f_2(s),\quad s\geq S_0;
$$

\item[(A5)]
 \begin{equation}\label{e1.9}
\frac {f_1'(s)s}{f_1(s)}:=1+g(s), \quad s\geq S_0,
\end{equation}
 with   $g\in C^1[S_0, \infty)$ satisfying
 \begin{gather}\label{e1.10}
g(s)>0,\quad s\geq S_0,\quad \lim_{s\to \infty}g(s)=0,\\
\label{e1.11}
\lim_{s\to \infty}\frac {sg'(s)}{g(s)}=0, \quad
\lim_{s\to \infty}\frac {sg'(s)}{g^2(s)}=C_g\in \mathbb{R}, \quad
\lim_{s\to \infty} \frac {\sqrt{s/f_1(s)}}{g(s)}=0;
 \end{gather}

\item[(A6)] either there exists a constant $E_1\neq 0$   such that
 \begin{equation}\label{e1.12}
\lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=E_1
\end{equation}
or
\begin{equation}\label{e1.13}
\lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=0
\end{equation}
and there exists a constant   $\mu\leq 1$ such that
\begin{equation}\label{e1.14}
\lim_{s\to \infty}\frac{f_2(\xi s)}{f_2(s)}=\xi^\mu,\quad \forall \xi>0.
\end{equation}
\end{itemize}
Our main results are summarized as  follows.

 \begin{theorem} \label{thm1.1}
  Assume {\rm (A1)--(A6)} are satisfied.  If  $b(t)$ also satisfies
\begin{itemize}
\item[(A7)] there exist $k\in \Lambda$ and a positive
  constant $b_0$ such that
$$
\lim_{t \to 0^+} \frac{b(t)}{k^{2}(t)}=b_0^{2},
$$
\end{itemize}
   then the unique  solution $l(t)$ of \eqref{1.1} satisfies
\begin{equation}\label{1.6}
l(t)\sim \exp (\xi_0) \phi(b_0K(t)),
\end{equation}
where
    \begin{equation}\label{e1.17}
\begin{gathered}
\xi_0=\frac {1}{2}-E_2-(1-C_k)\big(\frac {1}{2}+C_g\big),\\
 E_2= \begin{cases}
E_1 &\text{if \eqref{e1.12} holds};\\
0, &\text{if \eqref{e1.13} and \eqref{e1.14} hold},
\end{cases}
\end{gathered}
\end{equation}
and $\phi$ is  the unique solution of the problem
\begin{equation}\label{e1.18}
 \int_{\phi(t)}^\infty \frac {ds} {\sqrt{sf_1(s)}}=t,  \quad \forall t>0.
\end{equation}
\end{theorem}

By $f_1(t) \sim  f_1(t)$  as $t \to t_0 \in
\bar{\mathbb{R}}$ we mean  $\lim_{t \to t_0}\frac{f_1(t)}{f_{2}(t)} = c$,
 where $c$ is a constant.

\section{Preliminaries}

Our approach relies on Karamata regular variation theory established
by Karamata in 1930  which is a basic tool in stochastic process
(see   Bingham,  Goldie and Teugels \cite{BGT},  Haan \cite {LD},
 Geluk and  Haan \cite {JL},   Maric \cite {MA}, Resnick \cite {RE},
Seneta \cite {SE} and the references therein.).
In this section, we present some bases of Karamata
regular variation theory which come from the Introductions and the Appendix
in Maric \cite {MA}, and Preliminaries in Resnick \cite {RE}, Seneta \cite {SE}.

% page 4

 \begin{definition}\label{def2.1}
A positive measurable function $f$ defined on $[a,\infty)$, for some $a>0$,
is called \emph{regularly varying at infinity} with index $\rho$, written
$f \in RV_\rho$, if for each $\xi>0$ and some $\rho \in \mathbb{R}$,
\begin{equation}\label{e2.1}
\lim_{t \to \infty} \frac{f(\xi t)}{f(t)}= \xi^\rho.
\end{equation}
 In particular, when $\rho=0$, $f$ is called
   \emph{slowly varying at infinity}.
   \end{definition}

Clearly, if $f\in RV_\rho$, then $L(t): =f(t)/{t^\rho}$ is slowly
varying at infinity.
Some basic  examples of slowly varying functions at infinity are
\begin{itemize}
\item[(i)]  every measurable function on $[a, \infty)$ which has a
positive limit at infinity;

\item[(ii)]  $(\ln t)^q$ and   $\big(\ln (\ln t)\big)^q$, $q\in \mathbb{R}$;

\item[(iii)]  $e^{(\ln t)^q}$,  $0<q<1$.
\end{itemize}
  We also say that a positive measurable
function $h$ defined on $(0,a)$  for some $a>0$,  is
\emph{regularly varying  at zero} with index $\rho$ (written $h \in RVZ_\rho$) if
$t\to h(1/t)$ belongs to $RV_{-\rho}$.

\begin{proposition}[Uniform convergence] \label{prop2.1}
If $f\in RV_\rho$, then  \eqref{e2.1}  holds uniformly
for $\xi \in [c_1, c_2]$ with $0<c_1<c_2$. Moreover, if $\rho<0$,
then uniform convergence holds on intervals of the form $(a_1,\infty)$ with
$a_1>0$; if $\rho>0$, then uniform convergence holds
on intervals $(0, a_1]$ provided $f$ is bounded on $(0, a_1]$ for
all $a_1>0$.
\end{proposition}

\begin{proposition}[Representation theorem] \label{prop2.2}
A function $L$ is slowly varying at infinity if and only if it can
be written in the form
\begin{equation}\label{e2.2}
L(t)=\varphi(t) \exp \Big( \int_{a_1}^t \frac {y(\tau)}{\tau} d\tau\Big), \quad
 t \geq a_1,
\end{equation}
for some $a_1\geq a$, where the functions $\varphi$ and $y$ are
measurable and as $t \to \infty$, $y(t)\to 0$ and
$\varphi(t)\to c_0$, with $c_0>0$.
\end{proposition}

We call
\begin{equation}\label{e2.3}
 \hat{L}(t)=c_0 \exp \Big( \int_{a_1}^t
\frac {y(\tau)}{\tau} d\tau \Big), \quad  t \geq a_1,
 \end{equation}
its \emph{normalized} slowly varying
 at infinity and
  \begin{equation}\label{e2.4}
  f(t)=t^\rho\hat{L}(t), \quad  t \geq a_1,
 \end{equation}
  its \emph{normalized} regularly varying at infinity with
 index $\rho$  (and write $f\in NRV_\rho$).

Similarly, $h$ is called   \emph{normalized} regularly varying at
zero with index $\rho$, written $h \in NRVZ_\rho$ if $t\to
h(1/t)$ belongs to $NRV_{-\rho}$.

 A function $f\in RV_\rho$ belongs to $NRV_\rho$ if and only  if
 \begin{equation}\label{e2.5}
 f\in C^1[a_1, \infty),\quad\text{for some $a_1>0$ and }
 \lim_{t \to \infty}  \frac{tf'(t)}{f(t)}=\rho.
  \end{equation}

\begin{proposition}\label{prop2.3}
 If  functions $L, L_1$ are slowly varying at infinity, then
\begin{itemize}
 \item[(i)]   $L^\rho$ (for every $\rho\in \mathbb{R}$),
  $c_1 L+c_2L_1$  ($c_1\geq 0, c_2\geq0$ with $c_1+c_2>0$),
$L\circ L_1$  (if $L_1(t)\to +\infty$ as $t\to \infty$),
are also slowly varying at infinity.

 \item[(ii)]   For every $\rho >0$ and $t\to \infty$,
$$
t^{\rho} L(t)\to +\infty, \quad t^{-\rho} L(t)\to 0.
$$

 \item[(iii)]  For $\rho\in\mathbb{R}$ and $t\to \infty$,
$\ln (L(t))/{\ln t}\to 0$ and $\ln (t^\rho L(t))/{\ln t}\to \rho$.
\end{itemize}
\end{proposition}

\begin{proposition}\label{prop2.4}
If $f_1\in RV_{\rho_1}$, $f_2\in  {R}V_{\rho_2} $ with
$\lim_{t\to\infty} f_2 (t)=+\infty$, then $f_1\circ f_2\in {R}V_{\rho_1 \rho_2}$.
\end{proposition}

 \begin{proposition}[Asymptotic behavior] \label{prop2.5}
If a  function $L$ is slowly varying at infinity, then for $a\geq 0$
and $t\to \infty$,
\begin{itemize}
 \item[(i)] $\int_a^t s^{\rho}L(s)ds\cong (\rho+1)^{-1}t^{1+\rho}L(t)$,   for
$\rho>-1$'

 \item[(ii)]  $\int_t^\infty s^{\rho}L(s)ds\cong
(-\beta-1)^{-1}t^{1+\rho}L(t)$,    for   $\rho< -1$.
\end{itemize}
\end{proposition}

\section{Auxiliary results}

In this section, we  give some results to be used in the proof of Theorem \ref{thm1.1}.

\begin{lemma}[{\cite[Lemma 2.1]{ZH1}}]\label{lem3.1}
Let $k\in \Lambda$.
\begin{itemize}
\item[(i)]  When $C_k\in (0, 1)$,
$k$ is \emph{normalized} regularly varying at zero with  index
$(1-C_k)/{C_k}$;

\item[(ii)] when $C_k=1$, $k$  is  normalized slowly varying  at  zero;

\item[(iii)]  when $C_k=0$, $k$   grows faster  than any $t^p$ ($p > 1$)
 near  zero.
\end{itemize}
\end{lemma}

Denote
\begin{equation}\label{e2.6}
\Theta(r)=\int_r^\infty\frac {ds}{\sqrt{2F(s)}},\quad
\Theta_1(r)=\int_r^\infty\frac {ds}{\sqrt{s f_1(s)}}, \quad r>0.
\end{equation}
Then
\begin{equation}\label{e2.7}
 \Theta'(r)=-\frac {1}{\sqrt{2F(r)}},\quad
  \ \Theta_1'(r)=-\frac {1}{\sqrt{r f_1(r)}},\quad r>0.
\end{equation}

\begin{lemma}\label{lem3.2}
 Under the hypotheses of Theorem \ref{thm1.1}:
\begin{itemize}
\item[(i)]
$$
\int_{a}^{\infty}\frac{ds}{\sqrt{s f_1(s)}}<\infty,\quad \forall a >0;
$$
\item[(ii)]
$$
\lim_{r\to\infty} \frac {\Theta (\lambda r)}{\Theta ( r)} =
\lim_{r\to\infty} \frac {\Theta_1 (\lambda r)}{\Theta_1 (r)}=1,\quad \forall
 \lambda\in  (0, 1);
$$

\item[(iii)]
$$
\lim_{r\to \infty}  \frac {(r/ f_1(r))^{1/2} }
 {\Theta_1(r)g(r)}=\frac {1}{2}+C_g;
$$

\item[(iv)]
$$
\lim_{r\to \infty} \frac {\frac {f_1(\xi r)} {\xi f_1(r)}-1}{g(r)}= \ln \xi
$$
uniformly for $\xi \in [c_1, c_2]$ with $0<c_1<c_2$;

\item[(v)]
 $$
\lim_{r\to \infty}\frac {f_2(\xi r)}{\xi g(r)f_1(r)}=E_2
$$
uniformly for $\xi \in [c_1, c_2]$ with $0<c_1<c_2$.
\end{itemize}
\end{lemma}

\begin{proof}
By \eqref{e1.9}, \eqref{e1.10} and \eqref{e2.5}, we see that
$f_1 \in NRV_1, $  hence,
$$
\sqrt{s f_1(s)}\in NRV_1
$$
Then, there exist $a_1 > 0$ and a function $\hat{L}$ which is
normalized slowly varying at infinity such that
\begin{equation}\label{e3.3}
\sqrt{s f_1(s)}= c_0s\hat{L}(s),\quad s\geq a_1.
 \end{equation}
(i) For arbitrary $\rho \in (1, \infty)$, it follows by
Proposition \ref{prop2.3} (ii) that
$$
\lim_{s\to \infty}\frac{\sqrt{sf_1(s)}}{s^{\rho}} =c_0\lim_{s\to \infty}
s^{1-\rho}\hat{L}(s)=\infty.
$$
Thus there exists $S_0> 0$ such that
$$
\sqrt{s f_1(s)}>s^{\rho}, \quad s \geq S_0,
$$
i.e.
$$
\frac{1}{\sqrt{s f_1(s)}} < \frac{1}{s^{\rho}}, \quad
s \geq S_0,
$$
and the results follow.
 The proof of (ii)--(v) can be found in \cite[Lemma 2.6]{ZH1}, we
omit here.
\end{proof}

\begin{lemma}[{\cite[Lemma 2.7]{ZH1}}] \label{lem3.3}  
Assume hypotheses of Theorem \ref{thm1.1}, and let  $\phi$ be  the solution 
to the problem
$$
\int_{\phi(t)}^\infty \frac {ds}{\sqrt{s f_1(s)}}=t,\quad \forall t>0.
$$
Then
\begin{itemize}
\item[(i)] $-\phi'(t)=\sqrt{\phi(t)f_1(\phi(t))}$,
 $\phi(t)>0$, $t>0$, $\phi(0):=\lim_{t\to 0^+}\phi(t)=\infty$,
$\phi''(t)=\frac {1}{2}\big(f_1(\phi(t))+\phi(t)f_1'(\phi(t))\big)$,
$t>0$;

\item[(ii)]
$$
\lim _{t\to 0}\big(g(\phi(t))\big)^{-1}
\Big(\frac {1}{2}\big( 1+\frac
{\phi(t)f_1'(\phi(t))}{f_1(\phi(t))}\big)-\frac {f_1(\xi
\phi(t))}{\xi f_1(\phi(t))}\Big)=\frac {1}{2}-\ln \xi;
$$

\item[(iii)]
$$
\lim _{t\to 0} \frac  {\sqrt{\phi(t)f_1(\phi(t))}}
 {t g(\phi(t))f_1(\phi(t))}=\frac {1}{2}+C_g;
$$

\item[(iv)]
$$
\lim _{t\to 0} \frac {f_2(\xi\phi(t))}{\xi
g(\phi(t))f_1(\phi(t))}=E_2
$$
 uniformly for $\xi \in [c_1, c_2]$ with $0<c_1<c_2$.
\end{itemize}
\end{lemma}

\section{Proof of Theorem \ref{thm1.1}}

Since the nonlinear term $f$ satisfies (A2) and (A3),
by \cite[Theorem 2.1]{SJ}, we obtain under the assumptions on
Theorem \ref{thm1.1},  that problem \eqref{1.1} has a unique positive
solution.

\begin{lemma}\label{lem4.1}
Under the assumptions on Theorem \ref{thm1.1}, there are
$\delta\in(0,\delta_0) $
 and $0 < \varsigma_0 < \lambda_0$ such that for every
$\varsigma \in (0, \varsigma_0]$ and
$\lambda\in [\lambda_0, \infty)$, $\bar{u} (t) = \lambda\exp(\xi_0) \phi(b_0K(t))$
and $\underline{u}(t) = \varsigma \exp(\xi_0)\phi(b_0K(t)) $  are a
supersolution and a subsolution, respectively, of the problem
\begin{equation}\label{4.1}
u''(t) =b(t)f(u(t)), \quad u(t)>0, \quad t>0, \quad u(0)=\infty, \quad
u(\delta)=l(\delta),
\end{equation}
where $l(t)$ denotes the unique solution of \eqref{1.1}.
\end{lemma}

\begin{proof}
 Let
\begin{align*}
\Upsilon_0(t)
&=\big(g(\phi(b_0K(t)))\big)^{-1}
\Big(\frac {1}{2}\big(1+\frac
{\phi(b_0K(t))f_1'(\phi(b_0K(t)))}{f_1(\phi(b_0K(t)))}\big) \\
&\quad -\frac{b(t)}{ b_0^{2}k^{2}(t)}\frac
{f_1(\omega\phi(b_0K(t)))}{\omega f_1(\phi(b_0K(t)))}\Big)
-\frac  {\sqrt{\phi(b_0K(t))f_1(\phi(b_0K(t)))}}
 {b_0K(t)g(\phi(b_0K(t)))f_1(\phi(b_0K(t)))}\frac
 {K(t)k'(t)}{k^2(t)},
\end{align*}
for $t\in (0,\delta_0)$, $\omega>0$;
and
 $$
\Upsilon_1(t) =\frac{b(t)}{b_0^{2}k^{2}(t)}\frac
{f_2(\omega\phi(b_0K(t)))} {\omega
g(\phi(b_0K(t)))f_1(\phi(b_0K(t)))} , \quad t\in (0,\delta_0),\; \omega>0.
$$
By \eqref{1.5}, Lemma \ref{lem3.3} and Proposition \ref{prop2.1},  we see  that
$$
\lim_{t\to 0^+}\Upsilon_0(t)=\theta_0:=
\frac {1}{2}-\ln \omega-(\frac {1}{2}+C_g)(1-C_{k}),
$$
and
$$
\lim_{t\to 0^+}\Upsilon_1(t)=E_2,
$$
which has uniform convergence on intervals
$(0,a_1]$ for all $a_1 > 0$ and $\omega\in(0, a_1]$.

Thus for each $m_0 \in (0, 1), M_0 \in  (1,\infty)$ and
$\omega>0$, there exists $\delta \in (0, \delta_0)$ such that
\begin{gather*}
m_0\theta_0<\Upsilon_0(t)< M_0\theta_0,\quad \forall t \in (0, \delta); \\
m_0E_2<\Upsilon_1(t)< M_0E_2,\quad \forall t \in (0, \delta).
\end{gather*}
Let $\lambda$ and $\varsigma$ be positive constants satisfying
\begin{gather*}
\lambda\geq\lambda_0:=\max\Big\{\frac{l(\delta)\exp(-\xi_0)}{\phi(b_0K(\delta))},
\exp\Big(E_2-\frac{m_0}{M_0}E_2\Big)\Big\}, \\
\varsigma\leq\varsigma_0:=\min\Big\{\frac{l(\delta)\exp(-\xi_0)}{\phi(b_0K(\delta))},
\exp\Big(E_2-\frac{M_0}{m_0}E_2\Big)\Big\}.
\end{gather*}
By a direct computation, we have
\begin{gather*}
\bar{u}''(t) \leq b(t)f(\bar{u}(t)), \quad  t\in (0, \delta), \quad
\bar{u}(0)=\infty, \quad \bar{u}(\delta)\geq l(\delta); \\
\underline{u}''(t) \geq b(t)f(\underline{u}(t)), \quad t\in (0, \delta),
\quad \underline{u}(0)=\infty, \quad  \underline{u}(\delta)\leq
l(\delta).
\end{gather*}
i.e., $\bar{u}$ is a supersolution and $\underline{u}$ is a
subsolution to  \eqref{4.1}.
\end{proof}

\begin{lemma}\label{lem4.2}
Let $\delta>0$, $\varsigma_0>0$  and $\lambda_0>0$
be the positive constants given by Lemma \ref{lem4.1}. Then,
for every $\varsigma \in (0, \varsigma_0]$ and $\lambda\in [\lambda_0,
\infty)$,
$$
\varsigma \exp(\xi_0)\phi(b_0K(t))\leq l(t)
\leq \lambda \exp(\xi_0)\phi(b_0K(t)), \quad t\in(0, \delta),
$$
where $l(t)$ denotes the unique solution of  \eqref{1.1} and $\phi$
is defined by \eqref{e1.18}.
\end{lemma}

\begin{proof}
 According to \cite[Remark 1]{SJ}, $l(t)$ provides us with the unique
positive solution of
\begin{equation}\label{4.2}
u''(t) = b(t)f(u(t)), \quad  t\in (0, \delta), \quad
 u(0)=\infty, \quad  u(\delta)= l(\delta).
\end{equation}
Subsequently, given $\varsigma \in (0, \varsigma_0]$ and
$\lambda\in [\lambda_0, \infty)$, for each natural number
$n > \delta^{-1}$  we consider the boundary value problem
\begin{equation} \label{4.3}
\begin{gathered}
u''(t) = b(t)f(u(t)), \quad t\in (n^{-1}, \delta), \\
u(n^{-1})=\frac{\varsigma+\lambda}{2}\exp(\xi_0)\phi(b_0K(n^{-1})),
\quad u(\delta)= l(\delta).
\end{gathered}
\end{equation}
Set $\underline{u}(t)=\varsigma \exp(\xi_0)\phi(b_0K(t))$ and
$\bar{u}(t)=\lambda\exp(\xi_0) \phi(b_0K(t))$. By Lemma \ref{lem4.1},
$(\underline{u},  \bar{u})$ provides us with an ordered
sub-supersolution pair of \eqref{4.3}. Thus, this problem possesses
a solution $u_{n}$ such that
$$
\underline{u}(t)\leq u_{n}(t) \leq \bar{u} (t), \quad
t\in[n^{-1}, \delta].
$$
By a standard compactness argument, we
can extract a subsequence of $u_{n}, $ say ${u_{n}}_{m}$,
$m \geq 1$, approximating to a solution of \eqref{4.2};
 necessarily $l$, by uniqueness. Therefore, passing to the
limit as $m\to \infty$ in the estimates
$$
\underline{u}(t)\leq {u_{n}}_{m}(t) \leq \bar{u} (t),
\quad t\in[n_{m}^{-1}, \delta],
$$
we can get the result easily.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 We consider the auxiliary function
$$
h(t)=\frac{l(t)}{\exp(\xi_0)\phi(b_0K(t))}, \quad t\in(0, \delta].
$$
By  Lemma \ref{lem4.2}, $h(t)$ satisfies the estimate
$$
\varsigma \leq h(t) \leq \lambda, \quad t\in(0, \delta],
$$
and, hence,
$$
0 < \varsigma \leq \underline{h} := \liminf_{t\to 0^+}
 h(t) \leq \bar{h} := \limsup_{t\to 0^+} h(t) \leq\lambda.
$$
To show the existence of $\lim_{t\to 0^+} h(t)$, we
argue by contradiction. Suppose $\underline{h}< \bar{h}$. Then,
there exist two sequences $t_{n}, s_{n}, n \geq 1$, such that
$$
\lim_{n\to\infty} t_{n} = \lim_{n\to\infty} s_{n} = 0,\quad
\lim_{n\to\infty}h(t_{n})=\bar{h},\quad
\lim_{n\to\infty}h(s_{n})=\underline{h},
$$
and, for each $n \geq 1$,
\begin{equation} \label{4.4}
h'(t_{n})=h'(s_{n})=0,\quad h''(t_{n})\leq0, \quad
 h''(s_{n})\geq0.
\end{equation}
 Clearly,
\[
l'(t)=\exp(\xi_0)\big(h'(t)\phi(b_0K(t))+b_0h(t)\phi'(b_0K(t))
k(t)\big),
\]
 and
\begin{align*}
 l''(t)
&= \exp(\xi_0)\big(h''(t)\phi(b_0K(t))+2b_0h'(t)\phi'(b_0K(t))k(t) \\
&\quad +b_0^{2}h(t) \phi''(b_0K(t))k^{2}(t)
 +b_0h(t)\phi'(b_0K(t))k'(t)\big).
\end{align*}
Since $l''(t)=b(t)f(l(t))$, we have
\begin{equation} \label{4.5}
\begin{aligned}
&\exp(\xi_0)\big(h''(t)\phi(b_0K(t))+ 2b_0h'(t)\phi'(b_0K(t))k(t) \\
&+b_0^{2}h(t)\phi''(b_0K(t))k^{2}(t)
+b_0h(t)\phi'(b_0K(t))k'(t)\big) \\
&=b(t)f(l(t)), \quad  t\in (0, \delta].
\end{aligned}
\end{equation}
By \eqref{1.5},  Proposition \ref{prop2.1},  Lemma \ref{lem3.3} and Lemma \ref{lem4.1}, 
we have
\begin{align*}
&\lim_{t\to 0}\big(g(\phi(b_0K(t)))\big)^{-1}
\Big(\frac{b_0^{2}\phi''(b_0K(t))}
{k^{2}(t)f_1(\phi(b_0K(t)))} +\frac{b_0\phi'(b_0K(t))k'(t) }
{k^{2}(t)f_1(\phi(b_0K(t)))} \\
&\quad -\frac{b(t)f_1(l(t))}
  {\exp(\xi_0)h(t)k^{2}(t)f_1(\phi(b_0K(t)))}\Big)>0.
\end{align*}
On the other hand, $k^{2}(t)>0$, $g(\phi(b_0K(t)))>0 $ and
$f_1(\phi(b_0K(t)))>0$ for all $t\in(0,\delta ]$.
Hence, there exists $\delta_1\in (0, \delta)$ such that
\begin{align*}
&b_0^{2} \phi''(b_0K(t))k^{2}(t) +b_0\phi'(b_0K(t))k'(t)
-\frac{b(t)f_1(l(t))}{\exp(\xi_0)h(t)}\\
&= k^{2}(t)f_1(\phi(b_0K(t)))
\Big(\frac{b_0^{2}\phi''(b_0K(t))}
{k^{2}(t)f_1(\phi(b_0K(t)))} +\frac{b_0\phi'(b_0K(t))k'(t) }
{k^{2}(t)f_1(\phi(b_0K(t)))} \\
&\quad -\frac{\exp(-\xi_0)b(t)f_1(l(t))}
  {h(t)k^{2}(t)f_1(\phi(b_0K(t)))}\Big)
 >0, \quad t\in(0, \delta_1].
\end{align*}
Thus, by \eqref{4.4} and \eqref{4.5}, we obtain that, for any
$n\geq 1$,
\begin{align*}
&h(t_{n}) \\
&\geq h(t_{n})+h''(t_{n})\frac{\phi(b_0K(t_{n}))}
{b_0^{2} \phi''(b_0K(t_{n}))k^{2}(t_{n})
+b_0\phi'(b_0K(t_{n}))k'(t_{n})
-\frac{b(t_{n})f_1(l(t_{n}))}{\exp(\xi_0)h(t_{n})}}\\
&=\frac{b(t_{n})f_{2}(l(t_{n}))}{b_0^{2}
\phi''(b_0K(t_{n}))k^{2}(t_{n})
+b_0\phi'(b_0K(t_{n}))k'(t_{n})
-\frac{b(t_{n})f_1(l(t_{n}))}{\exp(\xi_0)h(t_{n})}},
\end{align*}
and
\begin{align*}
&h(s_{n}) \\
&\geq h(s_{n})+h''(s_{n})\frac{\phi(b_0K(s_{n}))}
{b_0^{2} \phi''(b_0K(s_{n}))k^{2}(s_{n})
+b_0\phi'(b_0K(s_{n}))k'(s_{n})
-\frac{b(s_{n})f_1(l(s_{n}))}{\exp(\xi_0)h(s_{n})}}\\
&=\frac{b(s_{n})f_{2}(l(s_{n}))}{b_0^{2}
\phi''(b_0K(s_{n}))k^{2}(s_{n})
+b_0\phi'(b_0K(s_{n}))k'(s_{n})
-\frac{b(s_{n})f_1(l(s_{n}))}{\exp(\xi_0)h(s_{n})}}.
\end{align*}
Therefore, passing to the limit as
$n\to \infty$ in these inequalities, it follows from
(A7),  \eqref{1.5}, \eqref{e1.17}  and Lemma \ref{lem3.3} that
$$
\bar{h}\geq\frac{ E_{2}\bar{h}}{E_{2}-\ln \bar{h}}, \quad
\text{and}\quad 
\underline{h}\leq\frac{ E_{2}\underline{h}}{E_{2}-\ln \underline{h}}.
$$
Consequently,
$\bar{h}=\underline{h}=1$, which contradicts the assumption
$\underline{h}<\bar{h}$. Therefore, the following limit exists
$$
h_0:=\lim_{t\to 0^+}\frac{l(t)}{\exp(\xi_0)\phi(b_0K(t))}
\in [\varsigma, \lambda],
$$
i.e.
$l(t)\sim\exp(\xi_0)\phi(b_0K(t))$. 
The proof is complete.
\end{proof} 

\section{Examples}

In this section,  we shw some basic cases of the nonlinear term
$f$, and apply our results to this examples. 


\begin{example} \label{examp6.1} \rm
$f(s)=C_1^2s(\ln s)^{2\alpha}+f_2(s)$, where $\alpha>1$, $s>S_0$,
\begin{gather*}
g(s)=2\alpha (\ln s)^{-1}; \quad 
\lim_{s\to \infty} \frac{\sqrt{s/f_1(s)}}{g(s)}
= \frac {1}{2\alpha C_1}\lim_{s\to \infty}(\ln s)^{-(\alpha-1)}=0;
\\
 \frac {sg'(s)} {g^2(s)}\equiv C_g=-\frac {1}{2\alpha};\quad
\lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=\frac {1}{2\alpha C_1^2}\lim_{s\to
\infty}\frac {f_2(s)}{s(\ln s)^{2\alpha-1}}=E_2;
\\
\phi(t)=\exp\big(C_1(\alpha-1) t\big)^{-1/(\alpha-1)}.
\end{gather*}
Then  
\[
l(t)\sim \exp\Big(\frac{1}{2}-E_2-\frac{(1-C_k)(\alpha-1)}{2\alpha}\Big)
\exp\big(C_1(\alpha-1) b_0K(t)\big)^{-1/(\alpha-1)}
\]
 as $t\to 0^{+}$.

In particular, when $f_2(s)=C_2 s^\mu (\ln s)^\beta$ with 
$\beta\leq 2\alpha-1$,  $E_1=0$ for $\mu<1$ or $\mu=1$ and 
$\beta< 2\alpha-1$, and $E_1=\frac {C_2}{2\alpha C_1^2}$ for $\mu=1$
 and $\beta= 2\alpha-1$.
\end{example} 

\begin{example} \label{examp6.2} \rm
$f(s)=C_1^2 s e^{(\ln s)^{q}}+f_{2}(s)$, where 
$q\in (0, 1)$, $s>S_0$,
 \begin{gather*}
g(s)=q(\ln s)^{-(1-q)};\quad 
\lim_{s\to \infty} \frac {\sqrt{s/f_1(s)}} {g(s)}
= \frac {1}{q C_1}\lim_{s\to \infty}
\frac { \exp (-\frac {1}{2}(\ln s)^q)}{(\ln s)^{-(1-q)}}=0;
\\
\lim_{s\to \infty}\frac {sg'(s)}{g^2(s)}
=-\frac {1-q}{q} \lim_{s\to\infty} (\ln s)^{-q}=C_g=0;
\\
\lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)}=\frac {1}{q
C_1^2}\lim_{s\to \infty}\frac {f_2(s)}{s(\ln s)^{-(1-q)}
\exp ((\ln s)^q)}=E_2;
\end{gather*}
Then
$$
l(t)\sim \exp\big(\frac{C_k}{2}-E_2\big) \phi( b_0K(t))\quad\text{as }
t\to 0^{+},
$$ 
where $\phi(t)$ is defined by
 $$
\int_{\ln (\phi(t))}^\infty\exp(-s^q/2)ds=C_1 t.
$$
\end{example} 

\begin{example} \label{examp6.3} \rm
$f(s)=C_1^2 s(\ln s)^{2}(\ln (\ln s))^{2\alpha}+f_2(s)$,
 where  $\alpha>1$, $s>S_0$,
\begin{gather*}
g(s)=2(\ln s)^{-1}\big(1+\alpha (\ln (\ln s))^{-1}\big);\\
\lim_{s\to \infty}  \frac {\sqrt{s/f_1(s)}} {g(s)}
= \frac {1}{2 C_1}\lim_{s\to \infty}\frac {(\ln (\ln s))^{-\alpha}}{1+\alpha(\ln
(\ln s))^{-1}}=0;
\\
 \lim_{s\to \infty} \frac {sg'(s)}{g^2(s)}
=-\lim_{s\to \infty} \frac {1+\alpha(\ln (\ln s))^{-1} +\alpha(\ln
(\ln s ))^{-2}} {2(1+\alpha(\ln (\ln s ))^{-1})^2}
=C_g =-\frac{1}{2};
\\
\lim_{s\to \infty}\frac {f_2(s)}{ g(s)f_1(s)} =\frac {1}{2
C_1^2}\lim_{s\to \infty}\frac {f_2(s)}{s \ln s (\ln (\ln
s))^{2\alpha}(1+\alpha(\ln (\ln s ))^{-1})}=E_2;
\\
\phi(t)=\exp\big(\exp\big(C_1(\alpha-1)t\big)^{-1/(\alpha-1)}\big).
\end{gather*}
Then
$$
l(t)\sim
\exp\big(\frac{1}{2}-E_2\big)
\exp\big(\exp\big(C_1(\alpha-1)b_0K(t)\big)^{-1/(\alpha-1)}\big).
$$
\end{example}


\subsection*{Acknowledgments}
This work was partially supported by the NSF of China (no.
11301250) and the  NSF of Shandong Province (no. ZR2013AQ004).


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\end{document}