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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 61, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/61\hfil 
Bifurcation curves for the perturbed Gelfand equation]
{Classification and evolution of bifurcation curves for the one-dimensional
perturbed Gelfand equation with mixed boundary conditions II}

\author[Y.-H. Liang, S.-H. Wang \hfil EJDE-2017/61\hfilneg]
{Yu-Hao Liang, Shin-Hwa Wang}

\address{Yu-Hao Liang (corresponding author) \newline
 Department of Applied Mathematics,
 National Chiao Tung University,
 Hsinchu 300, Taiwan}
\email{yhliang@nctu.edu.tw}

\address{Shin-Hwa Wang \newline
 Department of Mathematics,
 National Tsing Hua University,
 Hsinchu 300, Taiwan}
\email{shwang@math.nthu.edu.tw}


\dedicatory{Communicated by Paul H. Rabinowitz}

\thanks{Submitted November 30, 2016. Published February 28, 2017.}
\subjclass[2010]{34B18, 74G35}
\keywords{Multiplicity; positive solution; perturbed Gelfand equation;
\hfill\break\indent  $S$-shaped bifurcation curve; $\subset$-shaped bifurcation curve;
 time map}

\begin{abstract}
 In this article, we study the classification and evolution of bifurcation curves
 of positive solutions for the one-dimensional perturbed Gelfand equation with
 mixed boundary conditions,
 \begin{gather*}
 u''(x)+\lambda\exp\big( \frac{au}{a+u}\big) =0,\quad 0<x<1,\\
 u(0)=0,\quad u'(1)=-c<0,
 \end{gather*}
 where $4\leq a<a_1\approx4.107$. We prove that, for
 $4\leq a<a_1$, there exist two nonnegative $c_0=c_0(a)<c_1=c_1(a)$
 satisfying $c_0>0$ for $4\leq a<a^{\ast}\approx4.069$,
 and $c_0=0$ for $a^{\ast}\leq a<a_1$, such that, on the
 $(\lambda,\|u\|_{\infty})$-plane, (i) when $0<c<c_0$, the bifurcation curve is
 strictly increasing; (ii) when $c=c_0$, the bifurcation curve is monotone
 increasing; (iii) when $c_0<c<c_1$, the bifurcation curve is $S$-shaped;
 (iv) when $c\geq c_1$, the bifurcation curve is $\subset$-shaped. This work
 is a continuation of the work by Liang and Wang \cite{Liang-Wang} where
 authors studied this problem for $a\geq a_1$, and our results partially
 prove a conjecture on this problem for $4\leq a<a_1$ in \cite{Liang-Wang}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article, we study the classification and evolution of bifurcation curves
of positive solutions for the one-dimensional perturbed Gelfand equation with
mixed (or more precisely, Dirichlet-Neumann) boundary conditions given by
\begin{equation}
\begin{gathered}
u''(x)+\lambda\exp\big( \frac{au}{a+u}\big) =0,\quad 0<x<1,\\
u(0)=0,\quad u'(1)=-c<0,
\end{gathered} \label{eq:Model}
\end{equation}
where $\lambda>0$ is treated as a bifurcation parameter, $c>0$ is treated as
an evolution parameter, and constant $a$ satisfies $4\leq a<a_1\approx4.107$ 
where constant $a_1$ is defined in \cite[(3.23)]{Hung-Wang1}. 
The bifurcation curve of positive solutions of \eqref{eq:Model}
is defined by
\[
\tilde{S}_c=\{ (\lambda,\| u_{\lambda}\|_{\infty}
):\lambda>0\text{ and }u_{\lambda}\text{ is a positive solution of
\eqref{eq:Model}}\} .
\]


This work is a continuation of our previous work in \cite{Liang-Wang} where we
studied \eqref{eq:Model} for $a\geq a_1$. It is worthwhile noting that the
classification and evolution of bifurcation curves $\tilde{S}_c$ of
\eqref{eq:Model} is closely related to the one resulting from the same
differential equation in \eqref{eq:Model} with zero Dirichlet boundary
conditions \cite{Goddard-Shivaji-Lee,Hung-Wang-Yu,Liang-Wang}, that is,
\begin{equation}
\begin{gathered}
u''(x)+\lambda\exp\big( \frac{au}{a+u}\big) =0,\quad 0<x<1,\\
u(0)=0,\quad u(1)=0.
\end{gathered} \label{eq:DirichletModel}
\end{equation}
The bifurcation curve of positive solutions of \eqref{eq:DirichletModel} is
defined by
\[
S=\{ (\lambda,\| u_{\lambda}\|_{\infty}):\lambda>0\text{ and
}u_{\lambda}\text{ is a positive solution of \eqref{eq:DirichletModel}}\} .
\]


Before going into further discussions on problems \eqref{eq:Model} and
\eqref{eq:DirichletModel}, we first give some terminologies in this paper for
the shapes of bifurcation curves $\tilde{S}_c$ on the
$(\lambda,\|u\|_{\infty})$-plane (Following terminology also hold for $S$ if
$\tilde{S}_c$ is replaced by $S$.)

\begin{figure}[htb]
\begin{center}
\includegraphics[height=1.74in,width=4.4676in]{fig1}
\end{center}
\caption{Three different types of exactly $S$-shaped bifurcation curves
$\tilde{S}_c$ with $\lambda_0>0$ and $\| u_{\lambda_0}\|_{\infty
}>0$. (i) Type 1. (ii) Type 2. (iii) Type 3.}
\label{fig:1}
\end{figure}


\begin{description}
\item[$S$-shaped] The bifurcation curve $\tilde{S}_c$ on the
$(\lambda,\| u\|_{\infty})$-plane is said to be \emph{$S$-shaped} 
if $\tilde{S}_c$ has \emph{at least two} turning points, say
$(\lambda^{\ast},\|u_{\lambda^{\ast}}\|_{\infty})$ and 
$(\lambda_{\ast},\| u_{\lambda_{\ast}}\|_{\infty})$, satisfying 
$\lambda_{\ast}<\lambda^{\ast}$ and
$\| u_{\lambda^{\ast}}\|_{\infty}<\| u_{\lambda_{\ast}}\|_{\infty}$, and
\begin{enumerate}
\item[(i)] $\tilde{S}_c$ starts at some point
$(\lambda_0,\|u_{\lambda_0}\|_{\infty})$ and initially continues to the \emph{right},

\item[(ii)] at $(\lambda^{\ast},\| u_{\lambda^{\ast}}\|_{\infty})$,
$\tilde{S}_c$ turns to the \emph{left},

\item[(iii)] at $(\lambda_{\ast},\| u_{\lambda_{\ast}}\|_{\infty})$,
$\tilde{S}_c$ turns to the \emph{right},

\item[(iv)] $\tilde{S}_c$ tends to infinity as $\lambda\to\infty$.
That is, $\lim_{\lambda\to\infty}\| u_{\lambda}\|_{\infty
}=\infty$.
\end{enumerate}

\item[Exactly $S$-shaped] The bifurcation curve $\tilde{S}_c$ on the
$(\lambda,\| u\|_{\infty})$-plane is said to be \emph{exactly
$S$-shaped} if $\tilde{S}_c$ is $S$-shaped and it has \emph{exactly two}
turning points; see Figure \ref{fig:1}.

\item[Type 1/2/3 $S$-shaped] Assume that the bifurcation curve $\tilde{S}
_c$ is $S$-shaped on the $(\lambda,\| u\|_{\infty})$-plane. Let
$(\lambda_0,\| u_{\lambda_0}\|_{\infty})$ be the starting point of
$\tilde{S}_c$, and
\[
\bar{\lambda}_{\min}\equiv\min\{\lambda:(\lambda,\| u_{\lambda}
\|_{\infty})\text{ is a turning point of }\tilde{S}_c\}.
\]
Then $\tilde{S}_c$ is said to be type 1 (resp., type 2 and type 3)
$S$-shaped if $\lambda_0<\bar{\lambda}_{\min}$ (resp., $\lambda_0
=\bar{\lambda}_{\min}$ and $\lambda_0>\bar{\lambda}_{\min}$ ); see
Figure \ref{fig:1}(i) (resp., Figure \ref{fig:1}(ii) and \ref{fig:1}(iii)).

\item[$\subset$-shaped] The bifurcation curve $\tilde{S}_c$ on the
$(\lambda,\| u\|_{\infty})$-plane is said to be \emph{$\subset$-shaped}
if $\tilde{S}_c$ has \emph{at least one} turning point $(\lambda_{\ast
},\| u_{\lambda_{\ast}}\|_{\infty})$, and

\begin{enumerate}
\item[(i)] $\tilde{S}_c$ starts at some point $(\lambda_0,\|
u_{\lambda_0}\|_{\infty})$ and initially continues to the \emph{left},

\item[(ii)] at $(\lambda_{\ast},\| u_{\lambda_{\ast}}\|_{\infty}) $,
$\tilde{S}_c$ turns to the \emph{right},

\item[(iii)] $\lambda_{\ast}<\lambda_0$ and $\| u_{\lambda_0}
\|_{\infty}<\| u_{\lambda_{\ast}}\|_{\infty}$,

\item[(iv)] $\tilde{S}_c$ tends to infinity as $\lambda\to\infty$.
That is, $\lim_{\lambda\to\infty}\| u_{\lambda}\|_{\infty
}=\infty$.
\end{enumerate}

\item[Exactly $\subset$-shaped] The bifurcation curve $\tilde{S}_c$ on the
$(\lambda,\| u\|_{\infty})$-plane is said to be \emph{exactly $\subset
$-shaped} if $\tilde{S}_c$ is $\subset$-shaped and it has \emph{exactly one}
turning point; see Figure \ref{fig:2}.

\begin{figure}[htb]
\begin{center}
\includegraphics[height=1.4771in,width=1.3396in]{fig2}
\end{center}
\caption{Exactly $\subset$-shaped bifurcation curve $\tilde{S}_c$ with
$\lambda_0>0$ and $\left\| u_{\lambda_0}\right\| _{\infty}>0$.}
\label{fig:2}
\end{figure}


\item[Strictly/Monotone increasing] The bifurcation curve $\tilde{S}_c$ on
the $(\lambda,\| u\|_{\infty})$-plane is said to be \emph{strictly
(resp., monotone)} \emph{increasing} if $\lambda_1<\lambda_{2}$ (resp.,
$\lambda_1\leq\lambda_{2}$) for any two points $(\lambda_{i},\|
u_{\lambda_{i}}\|_{\infty})$, $i=1,2$, lying in $\tilde{S}_c$ with $\|
u_{\lambda_1}\|_{\infty}<\| u_{\lambda_{2}}\|_{\infty}$.
\end{description}

For \eqref{eq:DirichletModel}, it has been a long-standing conjecture
\cite{Brown-Ibrahim-Shivaji,Korman-Li,Shi} that there exists a positive
critical bifurcation value $a^{\ast}\approx4.07>4$ such that, on the
$(\lambda,\| u\|_{\infty})$-plane, the bifurcation curve $S$ is
strictly increasing for $0<a\leq a^{\ast}$ and is exactly type 1 $S$-shaped
for $a>a^{\ast}$. Very recently, Huang and Wang \cite{Huang-Wang1} gave a
rigorous proof of this conjecture for \eqref{eq:DirichletModel}. Their main
result is stated in the next theorem.

\begin{theorem}
[{\cite[Theorem 4 and Fig. 1]{Huang-Wang1}}]\label{thm:1_M}Consider
\eqref{eq:DirichletModel} with varying $a>0$.\ Then, on the $(\lambda,\|
u\|_{\infty})$-plane, the bifurcation curve $S$ of \eqref{eq:DirichletModel}
 is a continuous curve which starts at the origin and it tends to infinity as
$\lambda\to\infty$. Moreover, there exists a critical bifurcation
value $a^{\ast}\approx4.069$ satisfying $4<a^{\ast}<a_1\approx4.107$
such that the following assertions (i)--(iii) hold:

\begin{itemize}
\item[(i)] For $a>a^{\ast}$, the bifurcation curve $S$\ is exactly type 1
S-shaped on the $(\lambda,\left\| u\right\| _{\infty})$-plane. Moreover,
all positive solutions $u_{\lambda}$ are nondegenerate except that
$u_{\lambda_{\ast}}$ and $u_{\lambda^{\ast}}$ are degenerate for some positive
$\lambda_{\ast}<\lambda^{\ast}$.

\item[(ii)] For $a=a^{\ast}$, the bifurcation curve $S$ is strictly increasing on
the $(\lambda,\left\| u\right\| _{\infty})$-plane. Moreover, all
positive solutions $u_{\lambda}$ are nondegenerate except that $u_{\lambda
_0}$ is degenerate for some positive $\lambda_0$.

\item[(iii)] For $0<a<a^{\ast}$, the bifurcation curve $S$ is strictly increasing
on the $(\lambda,\left\| u\right\| _{\infty})$-plane. Moreover, all
positive solutions $u_{\lambda}$ are nondegenerate.
\end{itemize}
\end{theorem}

For \eqref{eq:Model}, Liang and Wang \cite{Liang-Wang} proved the next theorem
with any fixed $a>a_1\approx4.107$.

\begin{theorem}
[{\cite[Theorem 2.4]{Liang-Wang} and see e.g., Figure \ref{fig:3} with $a=5$}
]\label{thm:2_M}Consider \eqref{eq:Model} with any fixed 
$a>a_1\approx4.107$. Then, on the $(\lambda,\| u\|_{\infty})$-plane, the
bifurcation curve $\tilde{S}_c$ of \eqref{eq:Model} is a continuous curve
which starts at some point $(\lambda_0,\| u_{\lambda_0}\|_{\infty
})$ with $\lambda_0>0$ and $\| u_{\lambda_0}\|_{\infty}>0$ and it
tends to infinity as $\lambda\to\infty$. Moreover, there exists
$c_1=c_1(a)>1.057$ such that the following two assertions (i) and (ii) hold:

\begin{enumerate}

\item[(i)] For $0<c<c_1$, the bifurcation curve $\tilde{S}_c$ is $S$-shaped on
the $(\lambda,\| u\|_{\infty})$-plane. More precisely, there exist three
positive $c_{1,1}\leq c_{1,2}\leq c_{1,3}$ on $\left( 0,c_1\right) $, all
depending on $a$, such that the $S$-shaped bifurcation curve $\tilde{S}_c$
belongs to type 1, type 2 and type 3 when $0<c<c_{1,1}$, $c=c_{1,2}$ and
$c_{1,3}<c<c_1$, respectively.

\item[(ii)] For $c\geq c_1$, the bifurcation curve $\tilde{S}_c$ is $\subset
$-shaped on the $(\lambda,\| u\|_{\infty})$-plane.
\end{enumerate}
\end{theorem}


\begin{figure}[ptbh]
\begin{center}
\includegraphics[height=2.8772in,width=3.6927in]{fig3}
\end{center}
\caption{Numerical simulations of bifurcation curves $S$ and $\tilde{S}_c$
for $a=5$ and varying $c>0$ on the $(\lambda,\| u\|_{\infty})$-plane of
the bi-logarithm coordinates. Here $c_{1,2}^{-}<c_{1,2}\approx0.488
<c_{1,2}^{+}<c_1\approx1.365<c_1^{+}<c_{2}\approx 7.718<c_{2}^{+}
<c_{3}\approx47.711<c_{3}^{+}$ (adopted from \cite[Fig. 4]{Liang-Wang}).}
\label{fig:3}
\end{figure}

This article is organized as follows: 
Section \ref{sec:Main_Results} contains
statements of the main result. Section \ref{sec:Proof} contains the proof of
the main result.

\section{Main result} \label{sec:Main_Results}

In this section, we give our main result (Theorem \ref{thm:3_M}) for problem
\eqref{eq:Model} with $4\leq a<a_1\approx4.107$, where classification
and evolution of bifurcation curves $\tilde{S}_c$ for \eqref{eq:Model} with
varying $c>0$ are studied. Theorem \ref{thm:3_M} with $4\leq a<a_1$ extends
Theorem \ref{thm:2_M} with $a\geq a_1$, and we obtain a more complicated
evolution of bifurcation curves $\tilde{S}_c$ with varying $c>0$. Note that
some basic properties and ordering properties of bifurcation curves $\tilde
{S}_c$ for positive $a$ and $c$, on the $(\lambda,\| u\|_{\infty})$-plane
have been discussed in \cite[Theorems 2.1 and 2.2]{Liang-Wang}.

\begin{theorem}[See Figure \ref{fig:4}]\label{thm:3_M}
 Consider \eqref{eq:Model} for any fixed
$a$ satisfying $4\leq a<a_1\approx4.107$. Then there exist two
nonnegative $c_0=c_0(a)<c_1=c_1(a)$ satisfying $c_0>0$ for
$4\leq a<a^{\ast}approx4.069$, $c_0=0$ for $a^{\ast}\leq a<a_1$, and
$c_1>1.057$ for $4\leq a<a_1$, such that the following assertions
(I)--(IV) hold:

\begin{enumerate}
\item[(i)] For $0<c<c_0$, the bifurcation curve $\tilde{S}_c$ is strictly
increasing on the $(\lambda,\| u_{\lambda}\|_{\infty})$-plane.
Moreover, there exists a positive $\lambda_0$ such that \eqref{eq:Model} has
no positive solution for $0<\lambda<\lambda_0$, and exactly one positive
solution for $\lambda\geq\lambda_0$.

\item[(ii)] For $c=c_0$, the bifurcation curve $\tilde{S}_c$ is monotone
increasing on the $(\lambda,\| u_{\lambda}\|_{\infty})$-plane.
Moreover, there exists a positive $\lambda_0$ such that \eqref{eq:Model} has
no positive solution for $0<\lambda<\lambda_0$, and at least one positive
solution for $\lambda\geq\lambda_0$.

\item[(iii)] (See Figure \ref{fig:1}.) For $c_0<c<c_1$, the bifurcation curve
$\tilde{S}_c$ is $S$-shaped on the $(\lambda,\| u_{\lambda}
\|_{\infty})$-plane. More precisely, there exist three positive
$c_{1,1}\leq c_{1,2}\leq c_{1,3}$ on $(c_0,c_1)$, all depending on $a$,
such that the following three assertions hold:

\begin{enumerate}

\item (See Figure \ref{fig:1}(i)) If $c_0<c<c_{1,1}$, then the bifurcation
curve $\tilde{S}_c$ is \emph{type 1} $S$-shaped on the $(\lambda,\|
u\|_{\infty})$-plane. Moreover, there exist three positive $\lambda
_0<\lambda_{\ast}<\lambda^{\ast}$ which are all strictly increasing
functions of $c$ on $(c_0,c_{1,1})$ such that \eqref{eq:Model} has no
positive solution for $0<\lambda<\lambda_0$, at least one positive solution
for $\lambda_0\leq\lambda<\lambda_{\ast}$ and $\lambda>\lambda^{\ast}$, at
least two positive solutions for $\lambda=\lambda_{\ast}$ and $\lambda
=\lambda^{\ast}$, and at least three positive solutions for $\lambda_{\ast
}<\lambda<\lambda^{\ast}$.

\item (See Figure \ref{fig:1}(ii)) If $c=c_{1,2}$, then the bifurcation curve
$\tilde{S}_c$ is \emph{type 2} $S$-shaped on the $(\lambda,\|
u\|_{\infty})$-plane. Moreover, there exist three positive $\lambda
_0=\lambda_{\ast}<\lambda^{\ast}$ such that \eqref{eq:Model} has no positive
solution for $0<\lambda<\lambda_0$, at least one positive solution for
$\lambda>\lambda^{\ast}$, at least two positive solutions for $\lambda
=\lambda_{\ast}$ and $\lambda=\lambda^{\ast}$, and at least three positive
solutions for $\lambda_{\ast}<\lambda<\lambda^{\ast}$.

\item (See Figure \ref{fig:1}(iii)) If $c_{1,3}<c<c_1$, then the bifurcation
curve $\tilde{S}_c$ is \emph{type 3} $S$-shaped on the $(\lambda,\|
u\|_{\infty})$-plane. Moreover, there exist three positive $\lambda_{\ast
}<\lambda_0<\lambda^{\ast}$ which are all strictly increasing functions of
$c$ on $(c_{1,3},c_1)$ such that \eqref{eq:Model} has no positive solution
for $0<\lambda<\lambda_{\ast}$, at least one positive solution for
$\lambda=\lambda_{\ast}$ and $\lambda>\lambda^{\ast}$, at least two positive
solutions for $\lambda^{\ast}<\lambda<\lambda_0$ and $\lambda=\lambda^{\ast
}$, and at least three positive solutions for $\lambda_0\leq\lambda
<\lambda^{\ast}$.
\end{enumerate}

\item[(iv)] (See Figure \ref{fig:2}) For $c\geq c_1$, the bifurcation curve
$\tilde{S}_c$ is $\subset$-shaped on the $(\lambda,\| u\|_{\infty}
)$-plane. Moreover, there exist two positive $\lambda_{\ast}<\lambda_0$ such
that \eqref{eq:Model} has no positive solution for $0<\lambda<\lambda_{\ast}$,
at least one positive solution for $\lambda=\lambda_{\ast}$ and $\lambda
>\lambda_0$, and at least two positive solutions for $\lambda_{\ast}
<\lambda\leq\lambda_0$.
\end{enumerate}
\end{theorem}

\begin{figure}[htb]
\begin{center}
\includegraphics[height=2.8478in,width=3.6633in]{fig4}
\end{center}
\caption{Numerical simulations of bifurcation curves $S$ and $\tilde{S}_c$
for $a=4$ and varying $c>0$ on the $(\lambda,\| u\|_{\infty})$-plane of
the bi-logarithm coordinates. Here $0<c_0^{-}<c_0\approx0.10<c_{1,2}
^{-}<c_{1,2}\approx0.85<c_{1,2}^{+}<c_1\approx1.39<c_{1_1}
^{+}<c_{1_{2}}^{+}$ (adopted from \cite[Fig. 7]{Liang-Wang}).}
\label{fig:4}
\end{figure}



\begin{remark} \label{rmk1}\rm 
By Theorem \ref{thm:3_M}, we conclude that, on the 
$(\lambda,\|u_{\lambda}\|_{\infty})$-plane, 
(i) For $4.069\approx a^{\ast}\leq a<a_1\approx4.107$,
 since $c_0=c_0(a)=0$, the bifurcation curve
$\tilde{S}_c$ evolves from an $S$-shaped curve to a $\subset$-shaped curve
as the evolution parameter varies from $0^{+}$ to $\infty$, which shows the
same evolution for $a\geq a_1$, as claimed in Theorem \ref{thm:2_M}. It then
implies, by Theorem \ref{thm:1_M}, that such evolution is persistent whenever
the bifurcation curve $S$ of \eqref{eq:DirichletModel} is exactly type 1
$S$-shaped on the $(\lambda,\| u_{\lambda}\|_{\infty})$-plane; (ii)
For $4\leq a<a^{\ast}$, since $c_0>0$, the bifurcation curve
$\tilde{S}_c$ evolves from a strictly increasing curve to a monotone
increasing curve, then to an $S$-shaped curve, and finally to a $\subset
$-shaped curve when $c$ varying from $0^{+}$ to $\infty$. It partially
verifies a conjecture on problem \eqref{eq:Model} for $4\leq a<a^{\ast}$
proposed in \cite[Theorem 2.3]{Liang-Wang} and shows the emergence of more
complicated evolution of bifurcation curves $\tilde{S}_c$ with varying $c>0$.
\end{remark}

\section{Proof of the main result\label{sec:Proof}}

To prove our main result (Theorem \ref{thm:3_M}) on problem \eqref{eq:Model},
we modify time-map technique (the quadrature method) used in
\cite{Goddard-Shivaji-Lee, Liang-Wang}. We shall recall some well-developed
results in \cite{Liang-Wang}. First, for fixed $a,c>0$, we define
\begin{equation}
\tilde{H}_c(\rho,q)=2\int_0^{\rho}\frac{ds}{\sqrt{F(\rho)-F(s)}}
-\int_0^{q}\frac{ds}{\sqrt{F(\rho)-F(s)}}-\frac{c}{\sqrt{F(\rho)-F(q)}} 
\label{eq:Htilde_Defi}
\end{equation}
for $0\leq q<\rho$, 
where $f(s)=\exp\big( \frac{as}{a+s}\big) $ and 
$F(s)=\int_0^{s}f(t)dt$;  see \cite[(3.6)]{Liang-Wang}. For fixed $a,c>0$, let 
$\rho_0=\rho_0(c)$ be the unique positive number such that 
$\tilde{H}_c(\rho_0,0)=0$, where the existence and uniqueness of $\rho_0$
are proved in \cite[Lemma 3.2(ii)]{Liang-Wang}. Then it can be proved that, for
fixed \thinspace$a,c>0$ and $\rho\geq\rho_0$, $\tilde{H}_c(\rho,q)$ has a
unique zero $q(\rho,c)$ on $[0,\rho)$; see \cite[Lemma 3.2(iv)]{Liang-Wang}. 
Moreover, the time map formula for mixed boundary value
problem \eqref{eq:Model} is defined as
\begin{equation}
H_c(\rho,q(\rho,c))\equiv\frac{c^{2}}{2\left[ F(\rho)-F(q(\rho,c))\right]
}\quad \text{for }\rho\geq\rho_0(c), \label{eq:HTimemap_Defi}
\end{equation}
see \cite[(3.26)]{Liang-Wang}. Then it can be easily derived, by similar
arguments as given in \cite[Theorem 3.3]{Goddard-Shivaji-Lee} or 
\cite[(3.26) and (3.27)]{Liang-Wang}, that positive solutions $u$
 of \eqref{eq:Model} correspond to
\begin{equation}
\| u\|_{\infty}=\rho\quad \text{and}\quad  H_c(\rho,q(\rho,c))=\lambda.
\label{eq:HTimemap_Defi_Notation}
\end{equation}
Thus studying the shape of the bifurcation curve $\tilde{S}_c$ of 
\eqref{eq:Model} for $a,c>0$
is equivalent to studying the shape of the time map $H_c(\rho,q(\rho,c))$
for $\rho\geq\rho_0$.

To prove Theorem \ref{thm:3_M}, we need the following Lemmas 
\ref{lemma:1_M}--\ref{lemma:4_M}. First, in Lemma \ref{lemma:1_M}, 
we record some results on
the time map formula $H_c(\rho,q(\rho,c))$ in \cite{Liang-Wang}.

\begin{lemma}\label{lemma:1_M} 
Fix $a\geq4$ and consider $H_c(\rho,q(\rho,c))$ for $c>0$
and $\rho\geq\rho_0$. Then the following assertions (i)--(ix) hold:
\begin{enumerate}
\item[(i)] \cite[Lemma 3.2(iv)]{Liang-Wang} For $c>0$, if
$0<\rho<\rho_0(c)$, then $\tilde{H}_c(\rho,q)$ has no zero $q$ on
$[0,\rho)$, while if $\rho\geq\rho_0(c)$, then $\tilde{H}_c(\rho,q)$ has a
unique zero $q(\rho,c)$ on $[0,\rho)$, that is,
\begin{equation}
\tilde{H}_c(\rho,q(\rho,c))=0. \label{eq:q_Defi}
\end{equation}
Moreover, $q(\rho,c)=0$ if and only if $\rho=\rho_0(c)$.

\item[(ii)] \cite[Lemma 3.2(vi)]{Liang-Wang} For $c>0$ and
$\rho\geq\rho_0$,
\begin{equation}
0<\rho-q(\rho,c)\leq\frac{c^{2}e^{a}}{4\rho}. \label{eq:rho_q_differences}
\end{equation}


\item[(iii)]  \cite[Lemma 3.2(vii)]{Liang-Wang} $\rho_0(c)\in
C(0,\infty)$ is a strictly increasing function of $c$ on $(0,\infty)$.

\item[(iv)]  \cite[Lemma 3.2(viii)]{Liang-Wang} For $\rho>0$,
$q(\rho,c)\in C(0,\hat{c}]\cap C^{1}(0,\hat{c})$ is a strictly decreasing
function of $c$ on $(0,\hat{c}]$. Here $\hat{c}=\sqrt{2F(\rho)}\,G(\rho)$.

\item[(v)] \cite[Lemma 3.4(i)]{Liang-Wang} For any two
positive numbers $\tilde{c}_1<\tilde{c}_{2}$, $H_{\tilde{c}_1}(\rho
,q(\rho,\tilde{c}_1))<H_{\tilde{c}_{2}}(\rho,q(\rho,\tilde{c}_{2}))$ for
$\rho\geq\rho_0(\tilde{c}_{2})$.

\item[(vi)]  \cite[Lemma 3.5(i)]{Liang-Wang} There exists a
unique positive $c_1=c_1(a)$ such that
\begin{equation}
\lim_{\rho\to\rho_0(c)^{+}}\frac{d}{d\rho}H_c(\rho,q(\rho,c))
\begin{cases}
>0 & \text{when }c\in(0,c_1),\\
=0 & \text{when }c=c_1,\\
<0 & \text{when }c\in(c_1,\infty).
\end{cases} \label{eq:rho0_Defi}
\end{equation}


\item[(vii)] \cite[Lemma 3.5(ii)]{Liang-Wang} For $c\geq c_1$,
there exists $\bar{\rho}(c)>\rho_0(c)$ such that $\frac{d}{d\rho}H_c
(\rho,q(\rho,c))<0$ for $\rho_0(c)<\rho<\bar{\rho}(c)$.

\item[(viii)] \cite[Lemma 3.5(iii)]{Liang-Wang} For $0<c<c_1$
and $\rho_0(c)<\rho<\rho_0(c_1)$, $\frac{d}{d\rho}H_c(\rho
,q(\rho,c))>0$.
\end{enumerate}
\end{lemma}

On the other hand, for zero Dirichlet boundary value problem
\eqref{eq:DirichletModel}, its time map formula is defined as
\begin{equation}
G(\rho)\equiv\sqrt{2}\int_0^{\rho}\frac{ds}{\sqrt{F(\rho)-F(s)}
}\quad \text{for }\rho>0, \label{eq:GTimemap_Defi}
\end{equation}
see \cite{Brown-Ibrahim-Shivaji,Hung-Wang1,Laetsch}. Then positive solutions
$u$ of \eqref{eq:DirichletModel} correspond to
\begin{equation}
\| u\|_{\infty}=\rho\quad\text{and}\quad G(\rho)=\sqrt{\lambda}.
\label{eq:rho_defi}
\end{equation}
Thus studying the shape of the bifurcation curve of 
\eqref{eq:DirichletModel} for $a>0$\ is equivalent to studying the shape of
the time map $G(\rho)$ on $[0,\infty)$. It is worthwhile to point out that the
first term of $\tilde{H}_c(\rho,q)$ defined in the right hand side of
\eqref{eq:Htilde_Defi} is equal to $\sqrt{2}G(\rho)$, which implies that
$G(\rho)$ has an influence on $H_c(\rho,q(\rho,c))$ (or say that the shape
of the bifurcation curve $\tilde{S}_c$ of \eqref{eq:Model} is correlated
with the shape of the bifurcation curve $S$ of \eqref{eq:DirichletModel}.)

In the next Lemma \ref{lemma:2_M}, we record some results on the relationship
between $H_c(\rho,q(\rho,c))$ and $G(\rho)$ in \cite{Liang-Wang}.

\begin{lemma}\label{lemma:2_M}
 Fix $a>0$ and consider $G(\rho)$ for $\rho>0$ and
$H_c(\rho,q(\rho,c))$ for $\rho\geq\rho_0$ and $c>0$. Then the following two
assertions hold:
\begin{enumerate}
\item[(i)]  \cite[Lemma 3.3(i)]{Liang-Wang} For $c>0$ and
$\rho\geq\rho_0$, $H_c(\rho,q(\rho,c))\leq\left[ G(\rho)\right] ^{2}$,
and the equality holds if and only if $\rho=\rho_0$.

\item[(ii)] \cite[Lemma 3.6]{Liang-Wang} If $G'
(\rho)\leq0$ for some $\rho>0$, then $\frac{d}{d\rho}H_c(\rho,q(\rho,c))<0$
for $0<c<\hat{c}$.
\end{enumerate}
\end{lemma}

In the next lemma  we record the sign of derivatives of the
time map formula $G(\rho)$ for $\rho>0$ in \cite{Huang-Wang1}.

\begin{lemma}[{\cite[Theorem 4]{Huang-Wang1}}]\label{lemma:3_M}
Consider \eqref{eq:DirichletModel} with varying $a>0$.
There exists a critical bifurcation value $a^{\ast}\approx4.069$ 
satisfying $4<a^{\ast}<a_1\approx4.107$ such that the following three
assertions hold:
\begin{enumerate}

\item[(i)]  For $0<a<a^{\ast}$, $G'(\rho)>0$ for all
$\rho>0$.

\item[(ii)]  For $a=a^{\ast}$, there exist a unique positive
$\rho^{\ast}$ such that $G'(\rho^{\ast})=0$ and $G'(\rho)>0$
for all $\rho>0$ and $\rho\neq\rho^{\ast}$.

\item[(iii)]  For $a>a^{\ast}$, there exist two positive 
$\bar{\rho}_1<\bar{\rho}_{2}$ such that
\begin{equation}
G'(\rho)\begin{cases}
<0 & \text{when }\rho\in(\bar{\rho}_1,\bar{\rho}_{2}),\\
=0 & \text{when }\rho=\bar{\rho}_1\text{ or }\bar{\rho}_{2},\\
>0 & \text{when }\rho\in(0,\bar{\rho}_1)\cup(\bar{\rho}_{2},\infty).
\end{cases} \label{eq:diff_G_a_larger_a_star}
\end{equation}
\end{enumerate}
\end{lemma}

\begin{lemma} \label{lemma:4_M} 
Fix $a\geq4$ and consider $H_c(\rho,q(\rho,c))$ for
$\rho\geq\rho_0$ and $c>0$. Then the following three assertions  hold:
\begin{enumerate}

\item[(i)]  For any $c>0$, there exists a positive $\rho
_{M}=\rho_{M}(a,c)\geq\rho_0$ such that $\frac{d}{d\rho}H_c
(\rho,q(\rho,c))>0$ for $\rho\geq\rho_{M}$.

\item[(ii)]  For any two positive numbers $\tilde{c}_1<\tilde
{c}_{2}$ and $\rho\geq\rho_0(\tilde{c}_{2})$, if $\frac{d}{d\rho}
H_{\tilde{c}_{2}}(\rho,q(\rho,\tilde{c}_{2}))\geq0$, then $\frac{d}{d\rho
}H_{\tilde{c}_1}(\rho,q(\rho,\tilde{c}_1))>0$.

\item[(iii)]  If there exist two positive numbers $\tilde{\rho}
_1<\tilde{\rho}_{2}$ such that $G'(\rho)>0$ for $\tilde{\rho}
_1\leq\rho\leq\tilde{\rho}_{2}$, then there exists a positive $\tilde
{c}=\tilde{c}(a)$ such that $\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$ for
$\tilde{\rho}_1\leq\rho\leq\tilde{\rho}_{2}$ and $0<c<\tilde{c}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Note first that, as computed in \cite[(3.3), (3.30), (3.31)
and the last equation in the proof of Lemma 3.6]{Liang-Wang},
\begin{equation}
\begin{aligned}
&\frac{d}{d\rho}H_c(\rho,q(\rho,c)) \\
&=\frac{c^{2}f(q(\rho,c))}{2\left[
F(\rho)-F(q(\rho,c))\right] ^{1/2}\left\{ 2\left[ F(\rho)-F(q(\rho
,c))\right] +cf(q(\rho,c))\right\} }\Psi(\rho,q(\rho,c))
\end{aligned}\label{eq:diff_H_c_Psi}
\end{equation}
where
\begin{align*}
\Psi(\rho,q(\rho,c)) 
& =\sqrt{2}G'(\rho)-2\int_{q(\rho,c)}^{\rho
}\frac{f'(s)f(\rho)}{[f(s)]^{2}\sqrt{F(\rho)-F(s)}}ds\\
& =\int_0^{\rho}\frac{\theta(\rho)-\theta(s)}{\rho\left[ F(\rho
)-F(s)\right] ^{3/2}}ds-2\int_{q(\rho,c)}^{\rho}\frac{f'(s)f(\rho
)}{[f(s)]^{2}\sqrt{F(\rho)-F(s)}}ds
\end{align*}
and $\theta(\rho)=2F(\rho)-\rho f(\rho)$. Hence studying the sign of 
$\frac{d}{d\rho}H_c(\rho,q(\rho,c))$ is equivalent to studying that of 
$\Psi(\rho,q(\rho,c))$.

(I) We prove Lemma~\ref{lemma:4_M}(i). 
For fixed $c>0$, it can be verified easily that there exists a sufficiently 
large $\rho_{M}>c^{2} e^{a}$ such that, for $\rho>\rho_{M}$, the following 
three inequalities hold:
\begin{gather}
\theta(\rho)-\theta(s)>0 \quad\text{for } 0\leq s<\rho, \label{eq:condA}\\
\big[ \frac{3}{2}F(\rho)-\rho f(\rho)\big] 
-\big[ \frac{3}{2}F(s)-sf(s)\big] >0\quad\text{for } 0\leq s<\rho, 
 \label{eq:condB} \\
\rho f(\rho)\frac{f'(s)}{[f(s)]^{2}}<\frac{1}{4} \quad\text{for }
\rho-1<s<\rho. \label{eq:condC}
\end{gather}
The proofs of \eqref{eq:condA}--\eqref{eq:condC} are omitted since they are
trivial. Then, for $\rho>\rho_{M}$, we have that $\rho-q(\rho,c)<1$ by
\eqref{eq:rho_q_differences}, and
\begin{align*}
&\Psi(\rho,q(\rho,c)) \\
& =\int_0^{\rho}\frac{\theta(\rho)-\theta(s)}
{\rho\left[ F(\rho)-F(s)\right] ^{3/2}}ds-2\int_{q(\rho,c)}^{\rho}
\frac{f'(s)f(\rho)}{[f(s)]^{2}\sqrt{F(\rho)-F(s)}}\,ds\\
& >\int_{q(\rho,c)}^{\rho}\frac{2[ 1-\rho f(\rho)\frac{f'
(s)}{[f(s)]^{2}}] [F(\rho)-F(s)]-[\rho f(\rho)-sf(s)]}
{\rho[F(\rho)-F(s)] ^{3/2}}\,ds\quad \text{(by \eqref{eq:condA})}\\
& >\int_{q(\rho,c)}^{\rho}\frac{\frac{3}{2}[F(\rho)-F(s)]-[\rho
f(\rho)-sf(s)]}{\rho\left[ F(\rho)-F(s)\right] ^{3/2}}\,ds\;\;\text{(by
\eqref{eq:condC})}\\
& >0
\end{align*}
by \eqref{eq:condB}. So Lemma~\ref{lemma:4_M}(i)  holds.

(II) We prove Lemma~\ref{lemma:4_M}(ii). Let $\tilde{c}
_1<\tilde{c}_{2}$ be arbitrary two positive numbers and suppose that
$\frac{d}{d\rho}H_{\tilde{c}_{2}}(\rho,q(\rho,\tilde{c}_{2}))\geq0$ for some
$\rho\geq\rho_0(\tilde{c}_{2})$. Then, since
\[
\frac{\partial}{\partial q}\Psi(\rho,q)=2\frac{f'(q)f(\rho
)}{[f(q)]^{2}\sqrt{F(\rho)-F(q)}}>0
\]
and $q(\rho,\tilde{c}_1)>$ $q(\rho,\tilde{c}_{2})$ for all $\rho\geq\rho
_0(\tilde{c}_{2})$ by Lemma~\ref{lemma:1_M}(iv),  we have 
\[
\Psi(\rho,q(\rho,\tilde{c}_1))>\Psi(\rho,q(\rho,\tilde{c}_{2}))\geq0.
\]
Consequently, $\frac{d}{d\rho}H_{\tilde{c}_1}(\rho,q(\rho,\tilde{c}
_1))>\frac{d}{d\rho}H_{\tilde{c}_{2}}(\rho,q(\rho,\tilde{c}_{2}))$ by
\eqref{eq:diff_H_c_Psi}. So Lemma~\ref{lemma:4_M}(ii)  holds.

(III) We prove Lemma~\ref{lemma:4_M}(ii). Suppose there exist two
positive numbers $\tilde{\rho}_1<\tilde{\rho}_{2}$ such that 
$G'(\rho)>0$ for $\tilde{\rho}_1\leq\rho\leq\tilde{\rho}_{2}$. Then there
exists $\epsilon>0$ such that $G'(\rho)\geq\epsilon$ for 
$\tilde{\rho}_1\leq\rho\leq\tilde{\rho}_{2}$. By \eqref{eq:rho_q_differences}, 
there exists $\tilde{c}>0$ such that $\rho-q(\rho,c)<\frac{\epsilon^{2}}{16e^{4a}}$
for $\tilde{\rho}_1\leq\rho\leq\tilde{\rho}_{2}$ and $0<c\leq\tilde{c}$.
This implies that
\[
\Psi(\rho,q(\rho,c))\geq\sqrt{2}\epsilon-2\int_{q(\rho,c)}^{\rho}\frac{e^{2a}
}{\sqrt{\rho-s}}\,ds=\sqrt{2}\epsilon-4e^{2a}\sqrt{\rho-q(\rho,c)}>0
\]
for $\tilde{\rho}_1\leq\rho\leq\tilde{\rho}_{2}$ and $0<c\leq\tilde{c}$. So
Lemma~\ref{lemma:4_M}(iii) holds.
The proof is  complete.
\end{proof}

We are now in a position to prove Theorem \ref{thm:3_M}.

\begin{proof}[Proof of Theorem \ref{thm:3_M}]
\textbf{Case 1.} $4\leq a<a^{\ast }\approx4.069$. Define set
\begin{equation}
I=\{ c>0:\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0\text{ \ on }(\rho
_0(c),\infty)\} . \label{eq:S_defi}
\end{equation}
We first show that $I$ is nonempty. In fact, let $c_1$ be defined in
\eqref{eq:rho0_Defi} and $\tilde{\rho}_1=\rho_0(c_1)$. Then, by
Lemma~\ref{lemma:1_M}(viii), we have that, for $0<c<c_1$,
\begin{equation}
\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0\quad \text{on }(\rho_0(c),\tilde{\rho
}_1). \label{eq:diff_H_posti_rho_small}
\end{equation}
On the other hand, by Lemma~\ref{lemma:4_M}(i)--(ii)
and letting $\tilde{\rho}_{2}=\rho_{M}(a,c_1)$, we have that, for
$0<c<c_1,$
\begin{equation}
\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0\quad \text{on }\left[ \tilde{\rho}
_{2},\infty\right) . \label{eq:diff_H_posti_rho_large}
\end{equation}
Moreover, by Lemma~\ref{lemma:3_M}(i)  and 
Lemma~\ref{lemma:4_M}(iii),
there exists a positive $\tilde{c}_0<c_1$ such that, for
$0<c<\tilde{c}_0$, $\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$ on $\left[
\tilde{\rho}_1,\tilde{\rho}_{2}\right] $. Hence, for $0<c<\tilde{c}_0$,
$\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$ on $(\rho_0(c),\infty)$ and hence
$(0,\tilde{c}_0)\subset I$. So $I$ is nonempty.

Next, we show that $I$ is a finite connected interval. Note that, by Lemma
\ref{lemma:1_M}(vii), when $c\geq c_1$, $\frac{d}{d\rho}
H_c(\rho,q(\rho,c))<0$ for $\rho$ slightly larger than $\rho_0(c)$. Hence
$I\subset(0,c_1)$. Moreover, if there exist $\bar{c}\in(0,c_1)$ such that
$\bar{c}\not \in I$, then there exists $\bar{\rho}>\rho_0(\bar{c})$ such
that $\frac{d}{d\rho}H_{\bar{c}}(\bar{\rho},q(\bar{\rho},\bar{c}))\leq0$.
Then, by \eqref{eq:diff_H_posti_rho_small},
we have that
$\bar{\rho}>\tilde{\rho}_1$. It implies, by Lemma \ref{lemma:4_M}(ii),
 that, for $c\in(\bar{c},c_1)$, $\bar{\rho}\,(>\tilde{\rho
}_1 = \rho_0(c_1))>$ $\rho_0(c)$ and $\frac{d}{d\rho}H_c(\bar{\rho},q(\bar{\rho
},c))<0$. Consequently, $(\bar{c},c_1)\not \in I$ and hence $I$ is a finite
connected interval.

By the definition of $I$, above arguments and Lemma \ref{lemma:1_M}(vii),
 we obtain that there exists a positive $c_0<c_1$ such
that
\begin{equation}
I=(0,c_0). \label{eq:S_interval_c0}
\end{equation}
Moreover, when $c=c_0$,
\begin{equation}
\frac{d}{d\rho}H_{c_0}(\rho,q(\rho,c_0))\geq0\text{ \ on }(\rho_0
(c_0),\infty), \label{eq:H_diff_posti_c_c0}
\end{equation}
and there exists $\tilde{\rho}>\rho_0(c_0)$ such that 
$\frac{d}{d\rho }H_{c_0}(\tilde{\rho},q(\tilde{\rho},c_0))=0$. 
Indeed, such $\tilde{\rho}>\tilde{\rho}_1$ by \eqref{eq:diff_H_posti_rho_small}. 
It follows that, by Lemma \ref{lemma:4_M}(ii), for $c_0<c<c_1$, 
$\tilde{\rho }\,(>\tilde{\rho}_1)>\rho_0(c)$ and
\begin{equation}
\frac{d}{d\rho}H_c(\tilde{\rho},q(\tilde{\rho},c))<0.
\label{eq:H_diff_neg_some_rho}
\end{equation}


By the relationship between bifurcation curves $\tilde{S}_c$ and the time
map $H_c$ from \eqref{eq:HTimemap_Defi} and \eqref{eq:HTimemap_Defi_Notation},
 we have the following conclusions:

\textbf{Case (I).} For $0<c<c_0$, that is, $c\in I$, the bifurcation curve
$\tilde{S}_c$ is strictly increasing on the $(\lambda,\| u\|
_{\infty})$-plane since $\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$ on $(\rho
_0(c),\infty)$.

\textbf{Case (II).} For $c=c_0$, the bifurcation curve $\tilde{S}_c$ is
monotone increasing on the $(\lambda,\| u\|_{\infty})$-plane by
\eqref{eq:H_diff_posti_c_c0}.

\textbf{Case (III).} For $c_0<c<c_1$, the bifurcation curve $\tilde{S}
_c$ is $S$-shaped on the $(\lambda,\| u\|_{\infty})$-plane since
$\lim_{\rho\to\rho_0(c)^{+}}\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$
by \eqref{eq:rho0_Defi}, $\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$ on $[\tilde{\rho}_2, \infty)$
by \eqref{eq:diff_H_posti_rho_large}, and
$\frac{d}{d\rho}H_c(\tilde{\rho},q(\tilde{\rho},c))<0$ by
\eqref{eq:H_diff_neg_some_rho}.

We next show that the $S$-shaped bifurcation curve $\tilde{S}_c$ could be of
either type 1, type 2 or type 3 for some value $c$ on $( c_0,c_1) $.

\textbf{Case (III)(a).} The existence of type 1 $S$-shaped bifurcation curves
$\tilde{S}_c$. Since $\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$ on $\left[
\tilde{\rho}_{2},\infty\right) $ by \eqref{eq:diff_H_posti_rho_large}, we
have that, for $c_0<c<c_1$,
\begin{equation}
\begin{aligned}
\min_{\rho\geq\tilde{\rho}_1}H_c(\rho,q(\rho,c)) & =\min_{\tilde{\rho
}_1\leq\rho\leq\tilde{\rho}_{2}}H_c(\rho,q(\rho,c)) \\
& >\min_{\tilde{\rho}_1\leq\rho\leq\tilde{\rho}_{2}}H_{c_0}(\rho
,q(\rho,c_0))\;\;\text{(by Lemma~\ref{lemma:1_M}(v))} \\
& =H_{c_0}(\tilde{\rho}_1,q(\tilde{\rho}_1,c_0))
\end{aligned} \label{eq:H_c_11_east}
\end{equation}
by \eqref{eq:H_diff_posti_c_c0}. On the other hand, by
\eqref{eq:diff_H_posti_rho_small} and Lemma~\ref{lemma:1_M}(v),
we have that
\begin{align*}
H_{c_0}(\rho_0(c),q(\rho_0(c),c_0))
& <H_{c_0}(\tilde{\rho}_1,q(\tilde{\rho}_1,c_0))\\
& <H_{c_1}(\tilde{\rho}_1,q(\tilde{\rho}_1,c_1))=H_{c_1}(\rho
_0(c),q(\rho_0(c),c_1)).
\end{align*}
Consequently, by the intermediate value theorem, there exists $c_{1,1}
\in\left( c_0,c_1\right) $ such that
\begin{equation}
H_{c_{1,1}}(\rho_0(c_{1,1}),q(\rho_0(c_{1,1}),c_{1,1}))=H_{c_0}
(\tilde{\rho}_1,q(\tilde{\rho}_1,c_0)). \label{eq:difi_c_11}
\end{equation}
Hence, for $0<c<c_{1,1}$,
\begin{align*}
 H_c(\rho_0(c),q(\rho_0(c),c))
&= G(\rho_0(c)) \;\;\text{(by Lemma~\ref{lemma:2_M}(i))}\\
 &< G(\rho_0(c_{1,1})) \;\;\text{(by Lemma~\ref{lemma:3_M}(i) 
 and Lemma~\ref{lemma:1_M}(iii))}\\
 &= H_{c_{1,1}}(\rho_0(c_{1,1}),q(\rho_0(c_{1,1}),c_{1,1})) \quad 
\text{(by Lemma \ref{lemma:2_M}(i))}\\
 &= H_{c_0}(\tilde{\rho}_1,q(\tilde{\rho}_1,c_0)) \;\;\text{(by \eqref{eq:difi_c_11})}\\
 &< \min_{\rho\geq\tilde{\rho}_1}H_c(\rho,q(\rho,c))
\end{align*}
by \eqref{eq:H_c_11_east}. It then follows, by \eqref{eq:diff_H_posti_rho_small},
that
\[
H_c(\rho_0(c),q(\rho_0(c),c))<H_c(\rho,q(\rho,c))
\]
for $\rho>\rho_0(c)$. It implies that, for $0<c\leq c_{1,1}$, the $S$-shaped
bifurcation curve $\tilde{S}_c$ is of type 1 on the
$(\lambda,\|u\|_{\infty})$-plane.

\textbf{Case (III)(b).} The existence of type 3 $S$-shaped bifurcation curves
$\tilde{S}_c$. The proof of this part is the same as that given in
\cite[Proof of Theorem 2.4, Cases (i)(b)]{Liang-Wang} and hence the proof is omitted.

\textbf{Case (III)(c).} The existence of a type 2 $S$-shaped bifurcation curve
$\tilde{S}_c$. The proof of this part is the same as that given in
\cite[Proof of Theorem 2.4, Case (i)(c)]{Liang-Wang} and hence the proof is omitted.

\textbf{Case (IV).} For $c>c_1$, the bifurcation curve $\tilde{S}_c$ is
$\subset$-shaped on the $(\lambda,\| u\|_{\infty})$-plane since
$\lim_{\rho\to\rho_0(c)^{+}}\frac{d}{d\rho}H_c(\rho,q(\rho,c))<0$
by \eqref{eq:rho0_Defi} and since $\frac{d}{d\rho}H_c(\rho,q(\rho,c))>0$ for
$\rho\geq\rho_{M}(a,c)$ by Lemma~\ref{lemma:4_M}(i).
\smallskip

\textbf{Case 2.} $a=a^{\ast}\approx4.069$. Let $\rho^{\ast}$ be the unique
positive number such that $G'(\rho^{\ast})=0$ as defined in
Lemma~\ref{lemma:3_M}(ii). Then, for $c>0$, $\frac{d}{d\rho}
H_c(\rho^{\ast},q(\rho^{\ast},c))<0$ by Lemma~\ref{lemma:2_M}(ii).
Hence the bifurcation curve $\tilde{S}_c$ must
not be monotone increasing on the $(\lambda,\| u\|_{\infty})$-plane.
Or equivalently, $c_0=0$ if we similarly define $I=(0,c_0)$ as in
\eqref{eq:S_defi} and \eqref{eq:S_interval_c0} in Case 1. The remaining parts
of the proof in this case followed by similar arguments stated in above Case 1
and hence they are omitted here.
\smallskip

\textbf{Case 3.} $a^{\ast}<a<a_1$. Note that, by Lemma~\ref{lemma:3_M}(iii),
 Equation \eqref{eq:diff_G_a_larger_a_star} holds for all
$a>a^{\ast}$. Thus the proof of this part followed by same arguments given as
in \cite[Proof of Theorem 2.4]{Liang-Wang} and hence the proof is omitted here.

Finally, we remark that the proof of the estimation of $c_1>1.057$ for
$4\leq a<a_1$ is the same as the one computed in \cite[Proof of Theorem 2.4,
part (III)]{Liang-Wang} and the multiplicity result of positive solutions for
\eqref{eq:Model} in each case follows immediately from the definition of
shapes of bifurcations curves, see e.g., Figures \ref{fig:1} and \ref{fig:2}.
The proof is complete.
\end{proof}

\subsection*{Acknowledgements}
This work is partially supported by the Ministry of Science and Technology of
the Republic of China under grant No. MOST 103-2115-M-007-001-MY2.

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\end{document}