\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 02, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/02\hfil
 Global interval bifurcation and convex solutions]
{Global interval bifurcation and convex solutions for the
 Monge-Amp\`ere equations}

\author[W. Shen \hfil EJDE-2018/02\hfilneg]
{Wenguo Shen}

\address{Wenguo Shen \newline
Department of Basic Courses,
Lanzhou Institute  of Technology,
Lanzhou 730050, China}
\email{shenwg369@163.com}

\thanks{Submitted June 14, 2017. Published January 2, 2018.}
\subjclass[2010]{34B15, 34C10, 34C23}
\keywords{Global  bifurcation; interval bifurcation; convex solutions;
\hfill\break\indent  Monge-Amp\`ere equations}

\begin{abstract}
 In this article, we establish the global bifurcation result from the
 trivial solutions axis or from infinity for the Monge-Amp\`ere equations
 with non-differentiable nonlinearity.  By applying the above result,
 we shall determine the interval of $\gamma$,  in which there exist
 radial solutions for the following Monge-Amp\`ere equation
 \begin{gather*}
 \det(D^2u)= \gamma a(x)F(-u),\quad \text{in } B,\\
 u(x)=0,\quad \text{on }\partial B,
 \end{gather*}
 where $D^2u=(\partial^2u/\partial x_{i}\partial x_{j})$ is the Hessian
 matrix of $u$, where $B$ is the unit open ball of $\mathbb{R}^{N}$,
 $\gamma$ is a positive parameter. $a\in C(\overline{B}, [0,+\infty))$
 is a radially symmetric weighted function and $a(r):= a(|x|)\not\equiv0$ on
 any subinterval of $[0, 1]$ and the nonlinear term
 $F\in C(\mathbb{R}^+)$ but is not necessarily differentiable
 at the origin and infinity.
 We use  global interval  bifurcation techniques to prove our main results.
\end{abstract}

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

\section{Introduction}

The Monge-Amp\`ere equations are a type of important fully nonlinear elliptic
equations \cite{g1,t1}.  Historically, the study of Monge-Amp\`ere
equations is motivated by Minkowski problem \cite{c2,p1} and Weyl problem
\cite{g3,n1}.
Existence and regularity results of the Monge-Amp\`ere equations can be found
in \cite{c3,c4,g2,g4,l2,p1} and the reference therein.

We first consider the  real Monge-Amp\`ere equation
\begin{equation} \label{e1.1}
\begin{gathered}
\det(D^2u)=\lambda a(x)(-u)^{N} +g(x,-u,\lambda),\quad\text{in } B, \\
u(x)=0, \quad\text{on } \partial B,
\end{gathered}
\end{equation}
 where $D^2u = (\partial^2u/\partial x_{i}\partial x_{j})$ is the Hessian
 matrix of $u$, $B$ is the unit ball
of $\mathbb{R}^{N}$, $a(x)$ is a  weighted function, $\lambda$ is a positive
parameter and $g\in C(\overline{B}\times(\mathbb{R}^+)^2)$.
In  recent years, the study of the  problem \eqref{e1.1}  have attracted the
attention of many specialists in differential equations because of their
interesting applications.
For example, Caffarelli et al.\ \cite{c1} and Gilbarg et al.\ \cite{g1} 
have investigated  problem \eqref{e1.1} in general domains of $\mathbb{R}^{N}$.
Kutev \cite{k1} investigated the existence of strictly convex radial
solutions of problem \eqref{e1.1} with $a\equiv1$ and $g=0$.
Delano \cite{d6} treated the existence of
convex radial solutions of problem \eqref{e1.1}.

In \cite{h1,w1}, the authors have showed that problem \eqref{e1.1} 
can be reduced to the
boundary value problem
\begin{equation} \label{e1.2}
\begin{gathered}
((u')^{N})'=\lambda a(r)(-u)^{N} +g(r,-u,\lambda),\quad r\in(0,1), \\
u'(0)=u(1)=0.
\end{gathered}
\end{equation}
By a solution of problem \eqref{e1.2} we understand that it is a function 
which belongs to $C^2[0, 1]$ and satisfies \eqref{e1.2}.
It has been known that any negative solution of
problem \eqref{e1.2} is strictly convex in (0,1). 
Hu \cite{h1} and Wang \cite{w1}  (for $a(-u)^{N}=f(-u), g=0$) also established 
several criteria
for the existence, multiplicity and nonexistence of strictly convex solutions for
problem \eqref{e1.2} by using fixed index theorem. 
Lions \cite{l1} have proved the existence of the first eigenvalue $\lambda_1$ of
problem \eqref{e1.1} with $\lambda a(x)=\lambda^{N},g=0$ via constructive proof.
However, there is no information on the bifurcation points and the optimal 
intervals for the parameter $\lambda$ so as to
ensure existence of single or multiple convex solutions.

Recently, Dai et al.\ \cite{d1,d3} established a global bifurcation result for 
the Monge-Amp\`ere equations \eqref{e1.1} with 
$\lambda a(x)(-u)^{N} +g(x,-u,\lambda)$ equal  
$\lambda^{N} a(x)((-u)^{N} +g(-u))$ and $\lambda^{N}((-u)^{N} +g(-u))$ respectively.
Furthermore,  the radial solutions of the above problem in \cite{d1,d3} 
of \eqref{e1.1} is equivalent to the solutions of the  corresponding  
problem  \eqref{e1.2}, respectively. Where
$g:[0,+\infty)\to [0,+\infty)$ satisfies
$\lim_{s\to 0^{+}}g(s)/s^{N}=0$ and
\begin{itemize}
\item[(H0)] $a(x)\in C(\overline{B})$ is radially symmetric and 
$a(r)\geq0$, $a(r)\not\equiv0$ on any subinterval of
$[0, 1]$, where $r=|x|$ with $x\in\overline{B}$.
\end{itemize}

However, among the above papers, the nonlinearities are differentiable at the origin.
 Berestycki \cite{b1} established an important global bifurcation theorem from 
intervals for  a class of second-order problems involving non-differentiable 
nonlinearity.
In \cite{s1}, the result in \cite{b1} has been improved partially by Schmitt and Smith.
Recently,  Ma and Dai \cite{m1} improved Berestycki's result in \cite{b1} to show
a unilateral global bifurcation result for a class of second-order problems 
involving non-differentiable nonlinearity.
Later, Dai \cite{d2} considered similar problems with \cite{m1}, and  
Dai and Ma \cite{d4,d5} 
considered interval bifurcation problem for a class of $p$-Laplacian problems 
involving non-differentiable nonlinearity.

Motivated by above papers,  we shall establish a global bifurcation result  
from interval for the following Monge-Amp\`ere equations with nondifferentiable 
nonlinearity
\begin{equation} \label{e1.3}
\begin{gathered}
\det(D^2u)=\lambda a(x)(-u)^{N} +F(x,-u,\lambda),\quad\text{in } B, \\
u(x)=0, \quad\text{on } \partial B,
\end{gathered}
\end{equation}
where $\lambda$ is a positive parameter, 
$B$ is the unit open ball of $\mathbb{R}^{N}$, and the nonlinear term 
$F$ has the form $F=f+g$, where $f, g\in C(\overline{B}\times (\mathbb{R}^{+})^2)$
are radially symmetric with respect to $x$, where $\mathbb{R}^{+}=[0,\infty)$.

It is clear that the radial solutions of \eqref{e1.3} is equivalent to the 
solutions of the problem
\begin{equation} \label{e1.4}
\begin{gathered}
((u')^{N})'=\lambda Nr^{N-1}a(r)(-u)^{N} +Nr^{N-1}F(r,-u,\lambda),\quad r\in(0,1), \\
u'(0)=u(1)=0,
\end{gathered}
\end{equation}
where $a$ satisfies (H0), and  $F=f+g$, where 
$f, g\in C([0, 1]\times (\mathbb{R}^{+})^2)$, satisfying the following
conditions:
\begin{itemize}
\item[(H1)] $|\frac{f(r,s,\lambda)}{s^{N}}|\leq M_1$, for any $r\in(0,1)$,
 $0<s\leq1$ and  $\lambda\in \mathbb{R}$,
where $M_1$ is a positive constant.

\item[(H2)] $g(r,s,\lambda)=o(s^{N})$ near
$s=0$ uniformly for $r\in(0,1)$ and $\lambda$ on bounded sets.

\item[(H3)] $|\frac{f(r,s,\lambda)}{s^{N}}|\leq M_2$ for any $r\in[0,1]$, $C<s$
 and $\lambda\in\mathbb{R}^{+}$,
where $M_2$ is a positive constant, $C$ is a positive constant which is
large enough.

\item[(H4)] $g(r,s,\lambda)=o(s^{N})$ near $s=+\infty$ uniformly for 
$r\in[0, 1]$ and on bounded $\lambda$ intervals.
\end{itemize}
Under the above assumptions, we shall establish the global bifurcation results
 for the problem \eqref{e1.4}, which bifurcates
from the trivial solutions axis or from infinity, respectively.

Following the above  theory (see Theorem \ref{thm3.1}, \ref{thm3.2}), we shall investigate the
existence of radial solutions for the problem
\begin{equation} \label{e1.5}
 \begin{gathered}
\det(D^2u)=\gamma a(x)F(-u),\quad\text{in } B, \\
u(x)=0,\quad \text{on } \partial B,
\end{gathered}
\end{equation}
where $\gamma$ is a positive parameter,
 the nonlinear term $F\in C(\mathbb{R}^+)$ but is not 
necessarily differentiable at the origin and infinity.

It is clear that the radial solutions of \eqref{e1.5} is equivalent to 
the solutions of the problem
\begin{equation} \label{e1.6}
 \begin{gathered}
((u')^{N})'= \gamma Nr^{N-1}a(r)F(-u),\quad r\in(0,1), \\
u'(0)=u(1)=0,
\end{gathered}
\end{equation}
where $a$ satisfying  condition (H0).
We assume that the nonlinear term $F$ has the form $F=f+g$, where $f$ 
and $g$ are continuous functions on $\mathbb{R}^{+}$, satisfying the following 
conditions:
\begin{itemize}
\item[(H5)] $|\frac{f(s)}{s^{N}}|\leq M_3$,
 $0<s\leq1$, where $M_3$ is a positive constant.

\item[(H6)] $|\frac{f(s)}{s^{N}}|\leq M_{4}$,
 $C<s$ for some positive constant $C$ large enough,
 where $M_{4}$ is a positive constant.

\item[(H7)] $g:[0,\infty)\to  [0,\infty)$ is a continuous
function and $g(s)>0$ for $s\in(0,\infty)$.

\item[(H8)] There exist $g_0,g_{\infty}\in (0,\infty)$ such that
$$
g_0= \lim_{s \to 0^{+}} \frac{g(s)}{s^{N}}, g_{\infty}
=\lim_{s\to +\infty} \frac{g(s)}{s^{N}}.
$$
\end{itemize}

 For the abstract global bifurcation theory,
we refer the reader to \cite{d1,d5,m1,r1,r2}
and the references therein.

Clearly, $F$ is
not necessarily differentiable at the origin because of the influence of the term
 $f$. So the bifurcation theory of \cite{d1,d3} can not be applied directly to 
obtain our results. Fortunately, using the global interval bifurcation 
(see Theorems \ref{thm3.1} and \ref{thm3.2}), we can obtain some results of 
the existence  of negative solutions which extend the corresponding results 
in \cite{d1,d3}.


The rest of this article is arranged as follows. 
In Section 2, we given some Preliminaries. In Section 3, we establish  
the global bifurcation results which bifurcates
from the trivial solutions axis or from infinity for  problem \eqref{e1.4}, 
respectively.
 In Section 4, on the basis of the  interval bifurcation result 
(see Theorems \ref{thm3.1}, \ref{thm3.2}), we give the intervals
for the parameter $\gamma$ which ensure existence of single or multiple 
strictly convex solutions for problem \eqref{e1.6} under the under the 
assumptions of (H5)--(H8).


\section{Preliminaries}

Following  \cite[Section 3-4]{d1}, we first consider the  problem
\begin{equation} \label{e2.1}
\begin{gathered}
((-v')^{N})'=h(r) ,\quad  r\in(0,1), \\
v'(0)=v(1)=0.
\end{gathered}
\end{equation}
Let us define the operator $ G_{N}(h): E\to  E$ by
\begin{equation} \label{e2.2}
 G_{N}(h)=\int_{t}^{1}\Big(\int_0^{s}(h(\tau))^{\frac{1}{N}}d\tau\Big)ds.
\end{equation}
For a given $h\in Y $,  $G_{N}(h) : Y\to  E$  is s completely continuous and 
 \eqref{e2.2} is equivalent to \eqref{e2.1}.

With a simple transformation $v=-u$, problem \eqref{e1.2} can be equivalently  
written as (see  \cite[Section 4-p.10]{d1}).
\begin{equation} \label{e2.3}
\begin{gathered}
((-v')^{N})'=\lambda Nr^{N-1}a(r)v^{N} +Nr^{N-1}g(r,v,\lambda),\quad r\in(0,1), \\
u'(0)=u(1)=0,
\end{gathered} 
\end{equation}
where $g\in C([0,1]\times (\mathbb{R}^{+})^2)$ satisfies
\begin{equation} \label{e2.4}
\lim_{s\to 0^{+}}\frac{g(r,s,\lambda)}{s^{N}}=0
\end{equation}
uniformly for  $r\in (0,1)$ and $\lambda$ on bounded sets.

Define the Nemitskii operator $H:\mathbb{R}\times E\to  Y$ by
$$
H(\mu, v)(r) :=\mu Nr^{N-1}a(r)v^{N}+Nr^{N-1}g(r,v,\mu).
$$
Then it is clear that $H$ is continuous (compact) operator and problem \eqref{e2.3} 
can be equivalently written as
$$
v=G_{N}\circ H(\mu,v):= F(\mu,v).
$$
Here $F$ is completely continuous in $\mathbb{R}\times E \to  E$ and 
$F(\mu,0)=0$ for all $\mu\in\mathbb{R}$.

Let $Y=C[0,1]$ with the norm $\|u\|_{\infty}=\max_{r\in[0,1]}|u(r)|$.
Let $E:=\{u(r) \in C^{1}(0, 1)|u'(0)=u(1)=0\}$ with the usual norm 
$\|u\|=\max\{\|u\|_{\infty},\|u'\|_{\infty}\}$.
Let $P^{+}=\{u\in E:u(r)>0, r\in(0, 1)\}$. Set $K^{+}=\mathbb{R}\times P^{+}$ 
under the product topology.

Now, we consider the eigenvalue problem
\begin{equation} \label{e2.5}
 \begin{gathered}
((-v')^{N})'=\lambda Nr^{N-1}a(r)v^{N} ,\quad r\in(0,1), \\
u'(0)=u(1)=0,
\end{gathered}
\end{equation}
By  \cite[(4.2) Section 4-p.11]{d1},
the same proof as in \cite[Theorem 1.1]{l1}, we can show that 
problem \eqref{e2.5} possesses the first
eigenvalue $\lambda_1$ which is positive, simple , the unique and
the corresponding eigenfunctions are positive in $(0,1)$ and concave on $[0, 1]$.


By Rabinowitz \cite{r1}, using the same method to prove 
\cite[Theorems 4.1 and 4.2]{d1} with obvious changes, we may get the following  
global bifurcation result.

\begin{lemma}[{\cite[Theorem 4.2]{d1}}] \label{lem2.2})
 Assume that \eqref{e2.4} and {\rm (H0)} hold. Then $(\lambda_1, 0)$
is the unique bifurcation point of problem \eqref{e2.3} and there exists 
an unbounded continuum $C\subseteq(K^{+}\cup\{(\lambda_1, 0)\})$ of solutions
to  problem \eqref{e2.3} emanating from  $(\lambda_1, 0)$.
\end{lemma}

By \cite{d1}, to prove our main results, we need the following Sturm type 
comparison result.

\begin{lemma}[{\cite[Lemma 4.6]{d1}}]  \label{lem2.3} 
Let $b_{i}(r)\in C(0, 1)$, $i = 1, 2$ such that $b_2(r)\geq b_1(r)$
for $r\in(0, 1)$ and the inequality
 is strict on some subset of positive measure in $(0, 1)$.
 Also let $v_1, v_2$ be solutions of the  differential equations
\begin{equation} \label{e2.6}
\begin{gathered}
((-v')^{N})'=b_{i}(r)v^{N}, \quad r\in(0, 1),\; i = 1, 2, \\
v'(0)=v(1)=0,
\end{gathered} 
\end{equation}
respectively. If $v_1\neq0$ in $(0, 1)$, then $v_2$ has at least one
zero in $(0, 1)$.
\end{lemma}

Next, we give an important lemma which will be used later.

\begin{lemma} \label{lem2.4}
Let $I$ be an interval and if $y$ and $z$ are functions such that
$y$, $z$, $\varphi_{N}(y')$ and $\varphi_{N}(z')$ are differentiable on 
$I$ and $y(t)>0,z(t)>0,y'(t)<0,z'(t)<0$ for $t\in I$. 
Then we have the identity
\begin{equation} \label{e2.6b}
\begin{split}
&\frac{d}{dt}\Big\{\frac{y}{\varphi_{N}(z)}[\varphi_{N}(y)\varphi_{N}(-z')
 -\varphi_{N}(z)\varphi_{N}(-y')]\Big\}\\
&=\frac{y}{\varphi_{N}(z)}[\varphi_{N}(y)L_{N}[z]-\varphi_{N}(z)L_{N}[y]]\\
&\quad +\Big[(-y')^{N+1}+N\big(\frac{-yz'}{z}\big)^{N+1}+(N+1)y^{N}y'
\big(\frac{-z'}{z}\big)^{N}\Big],
\end{split}
\end{equation}
where
$\varphi_{N}(s)=s^{N}$, $L_{N}[y]=(\varphi_{N}(-y'))'$.
\end{lemma}

\begin{proof} The left-hand side of \eqref{e2.6b} equals
\begin{align*}
&\frac{d}{dt}\Big\{\frac{y^{N+1}(-z')^{N}}{z^{N}}-y(-y')^{N}\Big\}\\
&=\frac{[(N+1)y^{N}y'(-z')^{N}+y^{N+1}((-z')^{N})']z^{N}-y^{N+1}
 (-z')^{N}Nz^{N-1}z'}{z^{2N}} \\
&\quad -y'(-y')^{N}-y((-y')^{N})'\\
&=\frac{y}{\varphi_{N}(z)}\left[\varphi_{N}(y)L_{N}[z]-\varphi_{N}(z)L_{N}[y]\right]
\\
&\quad +\Big[(-y')^{N+1}+N\big(\frac{-yz'}{z}\big)^{N+1}+(N+1)y^{N}y'
\big(\frac{-z'}{z}\big)^{N}\Big].
\end{align*}
\end{proof}

\begin{remark} \label{rmk2.1} \rm
 In \eqref{e2.6b},  by Young's inequality, we obtain
\begin{equation} \label{e2.7}
\Big[(-y')^{N+1}+N\big(\frac{-yz'}{z}\big)^{N+1}+(N+1)y^{N}
y'\big(\frac{-z'}{z}\big)^{N}\Big]\geq0
\end{equation}
and the equality holds if and only if  $ \operatorname{sgn}y= \operatorname{sgn}z$ 
and  $|\frac{y'}{y}|^{N+1}=|\frac{z'}{z}|^{N+1}$.

We use Young's inequality
\begin{equation} \label{e2.8}
AB\leq\frac{A^{\alpha}}{\alpha}+\frac{B^{\beta}}{\beta},
\end{equation}
where $A,B\in \mathbb{R}^{+},\alpha,\beta>1,\frac{1}{\alpha}+\frac{1}{\beta}=1$.
Let $\alpha=N+1$, $\beta=\frac{N+1}{N}$, $A=-(N+1)^{\frac{1}{(N+1)}}y'$,
$B=(N+1)^{\frac{N}{(N+1)}}y^{N}\big(\frac{-z'}{z}\big)^{N}$ in \eqref{e2.8}.
We obtain that inequality \eqref{e2.7} holds.
\end{remark}

By Lemma \ref{lem2.4} and Remark \ref{rmk2.1}, we have the following result.

\begin{lemma} \label{lem2.5}
 In \eqref{e2.6b}, we have
$$
\int_0^{1}\frac{y}{\varphi_{N}(z)}\left(\varphi_{N}(y)L[z]
-\varphi_{N}(z)L[y]\right)\,dr\leq0.
$$
\end{lemma}

 \begin{proof} By Lemma \ref{lem2.4}, it follows that
\begin{equation} \label{e2.9}
\begin{split}
&\int_0^{1}\Big\{\frac{y}{\varphi_{N}(z)}\left[\varphi_{N}(y)\varphi_{N}(-z')
 -\varphi_{N}(z)\varphi_{N}(-y')\right]\Big\}'\,dr\\
&=\int_0^{1}\left[(-y')^{N+1}+N\big(\frac{-yz'}{z}\big)^{N+1}+(N+1)y^{N}y'
\big(\frac{-z'}{z}\big)^{N}\right]\,dr\\
&\quad +\int_0^{1}\frac{y}{\varphi_{N}(z)}\left[\varphi_{N}(y)L_{N}[z]
 -\varphi_{N}(z)L_{N}[y]\right]\,dr.
\end{split}
\end{equation}
As in the proof of  \cite[Lemma 4.5]{d1}, we can show that
the left-hand side of \eqref{e2.9} equals $0$. By  Remark \ref{rmk2.1}, We have the result.
\end{proof}

\section{Global bifurcation from an interval}

With a simple transformation $v=-u$, problem \eqref{e1.4} can be  equivalently 
 written as
\begin{equation} \label{e3.1}
 \begin{gathered}
((-v')^{N})'=\lambda Nr^{N-1}a(r)v^{N} +Nr^{N-1}F(r,v,\lambda),\quad 
r\in(0,1), \\
v'(0)=v(1)=0.
\end{gathered} 
\end{equation}
Let $\mathscr{S}$ denote the closure in $\mathbb{R}\times E$ of the set of 
nontrivial  solutions $(\lambda,v)$ of \eqref{e3.1} with $v\in P^{+}$.
By an argument similar to that of  \cite[Lemma 4.1]{d1} with obvious changes, 
we can show that the following existence and uniqueness theorem is valid 
for problem \eqref{e3.1}.

\begin{lemma}[{\cite[Lemma 4.1]{d1}}] \label{lem3.1}
If $(\lambda,v)$ is a solution of $\eqref{e3.1}$  under the assumptions of  
{\rm (H0)--(H2)} and $v$ has a double zero, then $u\equiv0$.
\end{lemma}

Our first main result for \eqref{e3.1} is the following theorem.

\begin{theorem} \label{thm3.1} 
Let {\rm (H0)--(H2)} hold. Let $d_1=M_1/a_0$, where $a_0=\min_{r\in[0,1]}a(r)$,
and let $I_1^{0}=[\lambda_1-d_1,\lambda_1+d_1]$.
 The component $\mathscr{C}$ of $\mathscr{S}\cup(I_1^{0}\times\{0\})$, 
containing $I_1^{0}\times \{0\}$ is unbounded and lies in
 $ K^{+}\cup(I_1^{0}\times\{0\})$.
\end{theorem}

For the proof we introduce the  auxiliary approximate problem
\begin{equation} \label{e3.2}
\begin{gathered}
((-v')^{N})=\lambda Nr^{N-1}a(r)v^{N}+Nr^{N-1}f(r,v|v|^{\epsilon},\lambda) 
 +Nr^{N-1}g(r,v,\lambda),\\ r\in(0,1), \\
v'(0)=v(1)=0.
\end{gathered}
\end{equation}

The next lemma will play a key role in the proof of Theorem \ref{thm3.1}.

\begin{lemma} \label{lem3.2}
Let $\epsilon_n$, $0<\epsilon_n<1$, be a sequence converging to $0$.
If there exists a sequence $(\lambda_n,v_n)\in K^{+}$ such that
$(\lambda_n,v_n)$ is a nontrivial solution of problem \eqref{e3.2}
corresponding to $\epsilon=\epsilon_n$, and $(\lambda_n,v_n)$ converges
to $(\lambda,0)$ in $\mathbb{R}\times E$, then $\lambda\in I_1^{0}$.
\end{lemma}

\begin{proof}
Let $w_n=v_n/\|v_n\|$,
then $w_n$ satisfies
\begin{equation} \label{e3.3}
\begin{gathered}
\begin{aligned}
  ((-w_n')^{N})'
&=\lambda_n Nr^{N-1}a(r)w_n^{N}+\frac{Nr^{N-1}f(r,u_n|u_n|^{\epsilon},
 \lambda_n)}{\|u_n\|^{N}} \\
&\quad +\frac{Nr^{N-1}g(r,u_n,\lambda_n)}{\|u_n\|^{N}},\quad r\in(0,1),
\end{aligned} \\
w_n'(0)=w_n(1)=0,
\end{gathered}
\end{equation}
Let
\[
\overline{g}(r,v,\lambda)=\max_{0\leq|s|\leq v}
\big|g(r,s,\lambda)\big| \quad \text{ for all $r\in (0,1)$ and $\lambda$ on 
 bounded sets},
\]
then $\overline{g}$ is nondecreasing with respect to $v$ and
\begin{equation} \label{e3.4}
\lim_{v\to 0^{+}}\frac{\overline{g}(r,v,\lambda)}{v^{N}}=0
\end{equation}
uniformly for $r\in (0,1)$ and $\lambda$ on bounded sets. 
Further it follows from \eqref{e3.4} that
\begin{equation} \label{e3.5}
\frac{|g(r,v,\lambda)|}{\|v\|^{N}}
\leq\frac{\overline{g}(r,|v|,\lambda)}{\|v\|^{N}}
\leq\frac{\overline{g}(r,\|v\|_{\infty},\lambda)}{\|v\|^{N}}
\leq\frac{\overline{g}(r,\|v\|,\lambda)}{\|v\|^{N}}\to 0
\end{equation}
as $\|v\|\to 0$,
uniformly for $r\in(0,1)$ and $\lambda$ on bounded sets.

Clearly, (H1) implies
\begin{equation} \label{e3.6}
\begin{split}
\frac{|f(r,v_n|v_n|^{\epsilon_n},\lambda_n)|}{\|v_n\|^{N}}
&=\frac{|f(t,v_n|v_n|^{\epsilon_n},\lambda_n)|}{v_n^{N}|v_n|^{N\epsilon_n}}
\cdot\frac{v_n^{N}|v_n|^{N\epsilon_n}}{\|v_n\|^{N}}\\
&\leq M_1\cdot |v_n|^{N\epsilon_n}\to  M_1
\end{split}
\end{equation}
for all $r\in (0,1)$.

Note that $\|w_n\|=1$ implies $\|w_n\|_{\infty}\leq1$. Using this fact with 
\eqref{e3.5} and \eqref{e3.6}, we have 
$\lambda_nN r^{N-1}a(r)w_n^{N}
+Nr^{N-1}f(r,v_n|v_n|^{\epsilon_n},\lambda_n)/\|v_n\|^{N}
+Nr^{N-1}g(r,v_n,\lambda_n)/\|v_n\|^{N}$ 
is bounded in $E$ for $n$ large enough. The
compactness of $G_{N}$ implies that  $w_n$ is  convergence in $E$. 
Without loss of generality,  we may assume that $w_n\to  w$ in $E$ with 
$\|w\|=1$. Clearly, we have $w\in \overline{ P^{+}}$.

We claim that $w\in P^{+}$. On the contrary, suppose that $w\in \partial P^{+}$, 
by Lemma \ref{lem3.1}, then $w\equiv0$, which is a contradiction with $\|w\|=1$.


Now, we deduce the boundedness of $\lambda$. Let $\psi\in  P^{+}$
 be an eigenfunction of problem
\eqref{e2.5} corresponding to $\lambda_1$.
We know that $w_n$ satisfies
\begin{align*}
((-w_n')^{N})'
&=\lambda_n Nr^{N-1}a(r)w_n^{N}+Nr^{N-1}f(r,v_n|v_n|^{\epsilon_n},
\lambda_n)/\|v_n\|^{N} \\
&\quad +Nr^{N-1}g(r,v_n,\lambda_n)/
\|v_n\|^{N},
\end{align*}
 $r\in(0,1)$,
$w_n'(0)=w_n(1)=0$
and $\psi$ satisfies 
$((-\psi')^{N})'=\lambda_1 Nr^{N-1}a(r)\psi^{N}, r\in(0,1),\psi'(0)=\psi(1)=0$.

By Lemma \ref{lem2.5},
it follows that
\begin{equation} \label{e3.7}
\begin{split}
&\int_0^{1}\frac{w_n}{\varphi_{N}(\psi)}\left(\varphi_{N}(w_n)L[\psi]
 -\varphi_{N}(\psi)L[w_n]\right)\,dr\\
&=\int_0^{1}\Big[(\lambda_1-\lambda_n)a(r)
 -\frac{f(r,v_n|v_n|^{\epsilon_n},\lambda_n)}{\|v_n\|^{N}w_n^{N}}
 -\frac{g(r,v_n,\lambda_n)}{\|v_n\|^{N}w_n^{N}}\Big]r^{N-1}Nw_n^{N+1}dr\leq0.
\end{split}
\end{equation}
Similarly, we can also show that
\begin{equation} \label{e3.8}
\begin{aligned}
&\int_0^{1}\Big[(\lambda_n-\lambda_1)a(r) +\frac{f(r,v_n|v_n|^{\epsilon_n},
\lambda_n)}{\|v_n\|^{N}w_n^{N}} \\
&+\frac{g(r,v_n,\lambda_n)}{\|v_n\|^{N}w_n^{N}}\Big]r^{N-1}N\psi^{N+1}\,dr\leq0.
\end{aligned}
\end{equation}
If $\lambda\leq\lambda_1$, considering \eqref{e3.7}, (H1) and (H2), we have
\begin{align*}
&\int_0^{1}(\lambda_1-\lambda)a(r)N r^{N-1}w^{N+1} \,dr\\
& \leq\lim_{n\to \infty}\int_0^{1}\frac{f(r,v_n|v_n|^{\epsilon_n},
 \lambda_n)}{\|v_n\|^{N}w_n^{N}}Nr^{N-1}w_n^{N+1}\,dr\\
&\leq\lim_{n\to \infty}\int_0^{1}\frac{f(r,v_n|v_n|^{\epsilon_n},
 \lambda_n)}{v_n^{N}|v_n|^{N\epsilon_n}}|v_n|^{N\epsilon_n}Nr^{N-1}w_n^{N+1}\,dr
\\
&\leq\int_0^{1}M_1Nr^{N-1}w^{N+1}\,dr.
\end{align*}
Hence, we obtain
\[
\int_0^{1}(\lambda_1-\lambda)a_0N r^{N-1}w^{N+1}\,dr
 \leq\int_0^{1}M_1Nr^{N-1}w^{N+1}\,dr,
\]
which implies $\lambda\geq\lambda_1-d_1$.

If $\lambda\geq\lambda_1$, considering \eqref{e3.8}, (H1) and (H2), we have
\begin{align*}
&\int_0^{1}(\lambda-\lambda_1)a(r)N r^{N-1}\psi^{N+1}\,dr \\
&\leq\lim_{n \to \infty}\int_0^{1}
 \frac{-f(r,v_n|v_n|^{\epsilon_n},\lambda_n)}{\|v_n\|^{N}w_n^{N}}Nr^{N-1}
 \psi^{N+1}\,dr\\
&\leq\lim_{n\to \infty}\int_0^{1}\frac{-f(r,v_n|v_n|^{\epsilon_n},
 \lambda_n)}{v_n^{N}|v_n|^{N\epsilon_n}}|v_n|^{N\epsilon_n}Nr^{N-1}\psi^{N+1}\,dr\\
&\leq\int_0^{1}M_1Nr^{N-1}\psi^{N+1}\,dr.
\end{align*}
Hence, we obtain
\[
\int_0^{1}(\lambda-\lambda_1)a_0N r^{N-1}\psi^{N+1}
\leq\int_0^{1}M_1Nr^{N-1}\psi^{N+1}\,dr,
\]
which implies $\lambda\leq\lambda_1-d_1$.
Therefore, we have that $\lambda\in I_1^{0}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
 We divide the rest of proofs into two steps.
\smallskip

\noindent\textbf{Step 1.}
 We show that $\mathscr{C}\subset(K^{+}\cup(I_1^{0}\times\{0\}))$.
For any $(\lambda, v)\in \mathscr{C}$, there are two possibilities: 
(i) $v\in P^{+}$, or 
(ii) $v\in \partial P^{+}$. It is obvious that $(\lambda, v)\in  K^{+}$
 in the case of (i).
While, the case (ii) implies that $v$ has at least one double zero in $[0,1]$.
From Lemma \ref{lem3.1} it follows that $v\equiv0$.
Hence, there exists a sequence $(\lambda_n, v_n)\in  K^{+}$ such that 
$(\lambda_n, v_n)$ is a solution of problem
\eqref{e3.2} corresponding to $\epsilon= 0$,
and $(\lambda_n, v_n)$ converges to $(\lambda, 0)$ in $\mathbb{R}\times E$. 
By Lemma \ref{lem3.2}, we have $\lambda\in I_1^{0}$, i.e., 
$(\lambda, v)\in (I_1^{0}\times\{0\})$ in the case of 
(ii).
Hence, $\mathscr{C}\subset(K^{+}\cup(I_1^{0}\times\{0\}))$.
\smallskip

\noindent\textbf{Step 2.} We prove that $\mathscr{C}$ is unbounded.
Suppose on the contrary that $\mathscr{C}$ is bounded. Using the similar 
method to prove  \cite[Theorem 1]{b1} with obvious changes,
we can find a neighborhood $\mathcal{O}$ of $\mathscr{C}$ such that 
$\partial\mathcal{O}\cap\mathscr{S}=\emptyset$.

In order to complete the proof of this theorem, we consider  problem $\eqref{e3.2}$.
 For $\epsilon>0$, it is easy to show that nonlinear term 
$f(r,v|v|^{\epsilon},\lambda)+g(r,v,\lambda)$ satisfies the condition (H2). Let
$$
\mathscr{S}_{\epsilon}
=\overline{\{(\lambda,v):(\lambda,v)\text{ satisfies  \eqref{e3.2} and }
v \not\equiv  0\}}^{\mathbb{R}\times E}.
$$

By Lemma \ref{lem2.2}, there exists an unbounded continuum $\mathscr{C}_{\epsilon}$
of $\mathscr{S}_{\epsilon}$ bifurcating from $(\lambda_1, 0)$ such that
$$
\mathscr{C}_{\epsilon}\subset(K^{+}\cup\{(\lambda_1,0)\}).
$$
So there exists $(\lambda_{\epsilon},v_{\epsilon})
\in \mathscr{C}_{\epsilon}\cap\partial\mathcal{O}$ for all
 $\epsilon>0$. Since $\mathcal{O}$ is bounded in $K^{+}$, Equation \eqref{e3.2} 
shows that $(\lambda_{\epsilon},v_{\epsilon})$ is bounded in 
$\mathbb{R}\times C^2$ independently of $\epsilon$. By the compactness
of $G_{N}$, one can find a sequence $\epsilon_n\to 0$ such that 
$(\lambda_{\epsilon_n},v_{\epsilon_n})$ converges to a solution $(\lambda,v)$ 
of \eqref{e3.2}. So $v\in \overline{P^{+}}$. If $v\in \partial P^{+}$, 
then from Lemma \ref{lem3.1} follows that $v\equiv0$. 
By Lemma \ref{lem3.2}, $\lambda\in I_1^{0}$, which contradicts the definition
of $\mathcal{O}$. On the other hand, if $v\in P^{+}$, then 
$(\lambda,v)\in \mathscr{S}\cap\partial\mathcal{O}$ which contradicts 
$\mathscr{S}\cap\partial\mathcal{O}=\emptyset$.
\end{proof}

From Theorem \ref{thm3.1} and its proof, we can easily get  a corollary.

\begin{corollary} \label{coro3.1} 
There exists a unbounded sub-continua $\mathscr{D}$  of solutions of
 \eqref{e3.1} in $\mathbb{R}\times E$, bifurcating from $I_1^{0}\times\{0\}$, and
$\mathscr{D}\subset(K^{+}\cup(I_1^{0}\times\{0\}))$.
\end{corollary}

We add the points $\{(\lambda,\infty)|\lambda\in\mathbb{R}\}$ to space 
$\mathbb{R}\times E$. Let $\mathscr{T}$ denote the
closure in $\mathbb{R}\times E$ of the set of nontrivial solutions 
$(\lambda, v)$ of \eqref{e3.1} under conditions (H3) and (H4) with $v\in P^{+}$.
Let $S_{N}$ denote the
spectral set of problem \eqref{e2.5}. 
Let $\overline{I}_{\infty}=[\overline{\lambda}-d_2,\overline{\lambda}+d_2]$,
 where $\overline{\lambda}\in S_{N}\setminus \{\lambda_1\}$ and $d_2$ 
be given in Theorem \ref{thm3.2}.

By Rabinowitz \cite{r2}, our second main result for \eqref{e3.1} is the 
following theorem.

\begin{theorem} \label{thm3.2} 
Let {\rm (H0), (H3), (H4)} hold. Also let $d_2=M_2/a_0$, where
$a_0=\min_{t\in[0,1]}a(t)$, and let $I_1^{\infty}=[\lambda_1-d_2,\lambda_1+d_2]$.
There exists a connected component $\mathscr{D}$ of 
$\mathscr{T}\cup(I_1^{\infty}\times\{\infty\})$,  
containing $I_1^{\infty}\times\{\infty\}$. Moreover, if $\Lambda\subset\mathbb{R}$ 
is an interval such that 
$\Lambda\cap(\cup_{\overline{\lambda}\in S_{N}\setminus \{\lambda_1\}}
(\overline{I}_{\infty}\cup I_1^{\infty}))=I_1^{\infty}$ and 
$\mathscr{M}$ is a neighborhood of $I_1^{\infty}\times\{\infty\}$ 
whose projection on $\mathbb{R}$ lies in $\Lambda$ and whose projection on 
$E$ is bounded away from $0$, then either
\begin{itemize}
\item[(1)]  $\mathscr{D}-\mathscr{M}$ is bounded in $\mathbb{R}\times E$ 
in which case $\mathscr{D}-\mathscr{M}$ meets
 $\mathscr{R}=\{(\lambda,0)|\lambda\in\mathbb{R}\}$ or

\item[(2)]  $\mathscr{D}-\mathscr{M}$ is unbounded.
\end{itemize}
If (2) occurs and $\mathscr{D}-\mathscr{M}$ has a bounded projection on 
$\mathbb{R}$, then $\mathscr{D}-\mathscr{M}$ meets $\overline{I}_{\infty}$.
Moreover, there exists a neighborhood $\mathscr{N}\subset \mathscr{M}$ of 
$I_1^{\infty}\times\{\infty\}$ such that 
$(\mathscr{D}\cap \mathscr{N})\subset (K^{+}\cup(I_1^{\infty}\times\{\infty\}))$.
\end{theorem}

\begin{proof}
 The idea is similar to the one in the proof of \cite[Theorem 1.6]{r2}, 
but we give a rough sketch of the proof for readers convenience.
If $(\lambda,v)\in \mathscr{T}$ with $\|v\|\neq0$, dividing \eqref{e3.1} 
by $\|v\|^2$ and setting $w=v/\|v\|^2$ yield
\begin{equation} \label{e3.9}
\begin{gathered}
((-w')^{N})'=\lambda Nr^{N-1}a(r)w^{N}+\frac{Nr^{N-1}F(r,v,\lambda)}{\|v\|^{2N}},
\quad  r\in(0,1), \\
w'(0)=w(1)=0,
\end{gathered}
\end{equation}
Define
\begin{gather*}
\widetilde{f}(r,w,\lambda)
=\begin{cases}
\|w\|^{2N}f(r,\frac{w}{\|w\|^2},\lambda), &\text{if }  w\neq0, \\
0,&\text{if } w=0; 
\end{cases} \\
\widetilde{g}(r,w,\lambda)
= \begin{cases}
\|w\|^{2N}g(r,\frac{w}{\|w\|^2},\lambda),&\text{if } w\neq0, \\
0,&\text{if } w=0. 
\end{cases} 
\end{gather*}
Clearly, \eqref{e3.9} is equivalent to
\begin{equation} \label{e3.10}
\begin{gathered}
\begin{aligned}
((-w')^{N})'
&=\lambda Nr^{N-1}a(r)w^{N}+Nr^{N-1}\widetilde{f}(r,w,\lambda) \\
\quad +Nr^{N-1}\widetilde{g}(r,w,\lambda),\quad r\in(0,1),
\end{aligned} \\
w'(0)=w(1)=0.
\end{gathered}
\end{equation}

It is obvious that $(\lambda, 0)$ is always the solution of \eqref{e3.10}. 
By simple computation, we can show that assumptions (H3) and (H4) imply
\begin{itemize}
\item[(H9)] $|\frac{\widetilde{f}(r,w,\lambda)}{w^{N}}|\leq M_2$ for all 
$r\in[0,1]$, $0<w\leq1$ and $\lambda\in\mathbb{R}^+$,
 where $M_2$ is a positive constant.

\item[(H10)] $\widetilde{g}(r,w,\lambda)=o(w^{N})$ near $w=0$, uniformly 
for all $r\in(0, 1)$ and on bounded $\lambda$ intervals.
\end{itemize}

Now applying Theorem \ref{thm3.1} to  \eqref{e3.10}, we have a connected 
component $\mathscr{C}$ of $\mathscr{S}\cup(I_1^{0}\times\{0\})$. 
Under the inversion $w \to  \frac{w}{\|w\|^2}=v$, $\mathscr{C}\to  \mathscr{D}$
satisfying  problem \eqref{e3.1}.
Clearly, $\mathscr{D}$ satisfy the conclusions of this theorem.

Finally, We  show that there exists a neighborhood $\mathscr{N}\subset \mathscr{M}$ 
of $I_1^{\infty}\times\{\infty\}$ such that 
$(\mathscr{D}\cap \mathscr{N})\subset (K^{+}\cup(I_1^{\infty}\times\{\infty\}))$. 
Clearly, the inversion $w\to  w/\|w\|^2=v$ turns $I_1^{0}\times\{0\}$ into
$I_1^{\infty}\times\{\infty\}$. Let $\mathscr{O}$ be a bounded neighborhood of 
$I_1^{0}\times\{0\}$.
 Then $(\mathscr{C}\cap(\mathscr{O}\setminus(I_1^{0}\times\{0\})))\subset K^{+}$, 
containing $I_1^{0}\times\{0\}$ is unbounded and lies in 
$K^{+}\cup(I_1^{\infty}\times\{\infty\})$. While, by the inversion 
$w\to  w/\|w\|^2=v,\mathscr{C}\cap(\mathscr{O}\setminus(I_1^{0}\times\{0\}))$
 is translated to a deleted neighborhood $\mathscr{N}^{0}$ of 
$I_1^{\infty}\times\{\infty\}$. It is obvious that 
$(\lambda,w)\in \mathscr{C}\cap(\mathscr{O}\setminus(I_1^{0}\times\{0\}))$  
implies that there exists a constant $c_0$ such that $0<\|w\|\leq c_0$.
It follows that $(\lambda,v)\in \mathscr{N}^{0}$ implies that 
$1/c_0\leq\|v\|<+\infty$.
It follows that
$(\mathscr{D}\cap \mathscr{N})\subset(K^{+}\cup(I_1^{\infty}\times\{\infty\}))$
 by taking $\mathscr{N}:=\mathscr{N}^{0}\cup (I_1^{\infty}\times\{\infty\})$.
\end{proof}

\section{Applications}

In this section, we shall investigate the existence and multiplicity of 
convex solutions of problem \eqref{e1.6}. With a simple transformation $v=-u$, 
problem \eqref{e1.6} can be written as
\begin{equation} \label{e4.1}
 \begin{gathered}
((-v')^{N})'=\gamma Nr^{N-1}a(r)F(v),r\in(0,1), \\
v'(0)=v(1)=0,
\end{gathered}
\end{equation}
The main results of this section are the following theorems.

\begin{theorem} \label{thm4.1}
 Let $a_0=\min_{r\in[0,1]}a(r)$, $a^{0}=\max_{r\in[0,1]}a(r)$.
 Let {\rm (H0), (H5)--(H8)} hold. If $g_0a_0>M_3a^{0}$ and
$g_{\infty}a_0>M_{4}a^{0}$, either
\begin{equation} \label{e4.2}
\frac{\lambda_1}{g_0-M_3a^{0}/a_0}<\gamma
<\frac{\lambda_1}{g_{\infty}+M_{4}a^{0}/a_0}
\end{equation}
or
\begin{equation} \label{e4.3}
\frac{\lambda_1}{g_{\infty}-M_{4}a^{0}/a_0}<\gamma
<\frac{\lambda_1}{g_0+M_3a^{0}/a_0},
\end{equation}
then  problem \eqref{e1.6} has at least one solution $u$ such that it is 
negative, and strictly convex in $(0, 1)$.
\end{theorem}

\begin{theorem} \label{thm4.2} 
Let {\rm (H0)--(A2), (H7), (H8)} hold.  
If  $g_0a_0>M_3a^{0}$ but $g_{\infty}a_0\leq M_{4}a^{0}$, for
\[
\frac{\lambda_1}{g_0-M_3a^{0}/a_0}<\gamma
<\frac{\lambda_1}{g_{\infty}+M_{4}a^{0}/a_0},
\]
then  problem \eqref{e1.6} has at least one solution $u$ such that it is negative, 
strictly convex in $(0, 1)$.
\end{theorem}

 \begin{theorem} \label{thm4.3} 
Let {\rm (H0), (H5)--(H8)} hold. If  $g_0a_0\leq M_3a^{0}$ but
$g_{\infty}a_0>M_{4}a^{0}$, for
\[
\frac{\lambda_1}{g_{\infty}-M_{4}a^{0}/a_0}<\gamma
<\frac{\lambda_1}{g_0+M_3a^{0}/a_0},
\]
then  problem $\eqref{e1.6}$ has at least one solution $u$ such that it is negative, 
strictly convex in $(0, 1)$.
\end{theorem}

\begin{remark} \label{rmk4.1} \rm
From (H8), we can see that there exists a positive constant $M_5$
such that $g(s)/s^{N}\geq M_{5}$ for all $s\neq0$.
\end{remark}

\begin{remark} \label{rmk4.2} \rm
 Note that if $M_{i}\equiv0$  $(i=3, 4)$, then the cases of 
Theorems \ref{thm4.2} and \ref{thm4.3} do not occur and
 Theorem \ref{thm4.1} is equivalent to  \cite[Theorem 4.1]{d3} or 
\cite[Theorem 5.1]{d1}.
In this sense, Theorem \ref{thm4.1} is also a generalization of 
\cite[Theorem 4.1]{d3} or \cite[Theorem 5.1]{d1}.
\end{remark}

To prove Theorem \ref{thm4.1}, we need the following results.

\begin{lemma} \label{lem4.1}
 Let {\rm (H0), (H5)--(H8)} hold. If $g_0a_0>M_3a^{0}$ and
$g_{\infty}a_0>M_{4}a^{0}$, either \eqref{e4.2} or \eqref{e4.3} hold, then
\begin{itemize}
\item[(i)] There is a distinct unbounded component
$\mathscr{D}_0$ of $\mathscr{S}\cup(I_1^{0}\times\{0\})$, containing
$I_1^{0}\times \{0\}$  and lying in $K^{+}\cup(I_1^{0}\times\{0\})$.

\item[(ii)] There is a distinct unbounded component
$\mathscr{D}_{\infty}$ of $\mathscr{T}\cup(I_1^{\infty}\times\{\infty\})$, 
which satisfy the alternates of Theorem \ref{thm3.2}. Moreover, there exists a neighborhood
$\mathscr{N}\subset \mathscr{M}$ of $I_1^{\infty}\times\{\infty\}$ such that 
$(\mathscr{D}_{\infty}\cap\mathscr{N})\subset(K^{+}
\cup(I_1^{\infty}\times\{\infty\}))$.
\end{itemize}
\end{lemma}

\begin{proof} 
Firstly, we study the bifurcation phenomena for the following eigenvalue problem
\begin{equation} \label{e4.4}
\begin{gathered}
((-v')^{N})'=\lambda \gamma Nr^{N-1}a(r)g(v)+\gamma  Nr^{N-1}a(r)f(v),\quad
 r\in(0,1), \\
v'(0)=v(1)=0,
\end{gathered} 
\end{equation}
where $\lambda>0$ is a parameter.

(i) Clearly, condition (H5) implies 
\begin{equation} \label{e4.5}
|\frac{a(r)f(s)}{s^{N}}|\leq M_3a^{0}, \quad 0<s\leq1.
\end{equation}
Let $\zeta\in C(\mathbb{R}\setminus\mathbb{R}^- ,\mathbb{R}\setminus\mathbb{R}^- )$ 
be such that
\begin{equation} \label{e4.6}
g(s)=g_0s^{N}+\zeta(s)
\end{equation}
with $\lim_{s \to  0^+}\zeta(s)/s^{N}=0$. Let
$\overline{\zeta}(v)=\max_{0\leq|s|\leq v}|\zeta(s)|$,
then  $\overline{\zeta}(v)$ is nondecreasing and
\begin{equation} \label{e4.7}
\lim_{s\to 0^{+}}\frac{\overline{\zeta}(s)}{s^{N}}=0.
\end{equation}
Further it follows from \eqref{e4.7} that
\begin{equation} \label{e4.8}
\frac{|\zeta(v(r))|}{\|v\|^{N}}
\leq\frac{\overline{\zeta}(|v(r)|)}{\|v\|^{N}}
\leq\frac{\overline{\zeta}(\|r\|_{\infty})}{\|v\|^{N}}
\leq\frac{\overline{\zeta}(\|r\|)}{\|v\|^{N}}\quad
\text{as } \|v\|\to 0.
\end{equation}
Hence, \eqref{e4.5}, \eqref{e4.6} and \eqref{e4.8} imply that conditions 
(H1) and (H2) hold. Moreover, let $d_3=M_3a^{0}/g_0a_0$ and
$I_1^{0}=[\frac{\lambda_1}{\gamma g_0}-d_3,\frac{\lambda_1}{\gamma g_0}+d_3]$.
By Theorem \ref{thm3.1}, the result follows.

(ii) Clearly, condition (H6) implies 
\begin{equation} \label{e4.9}
|\frac{a(t)f(s)}{s^{N}}|\leq M_{4}a^{0}, \quad C<s.
\end{equation}
Let $\xi\in C(\mathbb{R}\setminus\mathbb{R}^- ,\mathbb{R}\setminus\mathbb{R}^- )$ 
be such that
\begin{equation} \label{e4.10}
g(s)=g_{\infty}s^{N}+\xi(s)
\end{equation}
with $\lim_{s \to +\infty}\xi(s)/s^{N}=0$. 
Let $\overline{v}=\max_{0\leq|s|\leq v}|\xi(s)|$,
then  $\overline{\xi}$ is nondecreasing and
\begin{equation} \label{e4.11}
\lim_{s\to +\infty}\frac{\overline{\xi}(s)}{s^{N}}=0.
\end{equation}
Further it follows from \eqref{e4.11} that
\begin{equation} \label{e4.12}
\frac{|\xi(v(r))|}{\|v\|^{N}}
\leq\frac{\overline{\xi}(|v(r)|)}{\|v\|^{N}}
\leq\frac{\overline{\xi}(\|v\|_{\infty})}{\|v\|^{N}}
\leq\frac{\overline{\xi}(\|v\|)}{\|v\|^{N}}\quad \text{as } \|v\|\to +\infty.
\end{equation}
Hence, \eqref{e4.9}, \eqref{e4.10} and \eqref{e4.12} imply that conditions
(H3) and (H4) hold.
 Moreover, let $d_{4}=M_{4}a^{0}/g_{\infty}a_0$ and
$I_1^{\infty}=[\frac{\lambda_1}{\gamma g_{\infty}}-d_{4},
\frac{\lambda_1}{\gamma g_{\infty}}+d_{4}]$.
Using Theorem \ref{thm3.2}, the result follows.
\end{proof}

\begin{lemma} \label{lem4.2} 
If $\mathscr{D}_0$ and $\mathscr{D}_{\infty}$ are defined as in Lemma \ref{lem4.1},
then  $\mathscr{D}_0=\mathscr{D}_{\infty}$.
\end{lemma}

\begin{proof}
(i) We shall prove that (1) of Theorem \ref{thm3.2} occurs.
It suffices to show that $\mathscr{D}_{\infty}$ meets some point 
$(\lambda_{\ast},0)$ of $\mathscr{R}$. In fact, if this occurs, we can 
show that $\lambda_{\ast}\in I_1^{0}$.
Suppose on the contrary that $\lambda_{\ast}\not\in I_1^{0}$, hence 
$\lambda_{\ast}\in \overline{I}_0$.
So $(\mathscr{D}_{\infty}\cap\mathscr{N})\subset \mathscr{D}_{\infty}
\subset \overline{\mathscr{D}}_0
\subset((\mathbb{R}\times \overline{P})\cup(\overline{I}_0\times\{0\}))$,
noting
$(\mathscr{D}_{\infty}\cap\mathscr{N})\cap(\mathbb{R}\times\{0\})=\emptyset$, 
which contradicts $(\mathscr{D}_{\infty}\cap\mathscr{N})
\subset(K^{+}\cup(I_{\infty}\times\{\infty\}))$.
Where $\overline{\mathscr{T}}_0$ denote the
closure in $\mathbb{R}\times E$ of the set of nontrivial solutions 
$(\lambda, v)$ of \eqref{e3.1} under conditions (H5), (H7) and (H8) 
with $v\in \overline{P}$,
where $ \overline{P}=\{v|(\overline{\lambda},v)\in( S_{N}\setminus 
\{\lambda_1\})\times E\}$ and 
$\overline{I}_0=[\frac{\overline{\lambda}}{\gamma g_0}-d_3,
\frac{\overline{\lambda}}{\gamma g_0}+d_3]$.
 $\overline{\mathscr{D}}_0$ is a connected component of 
$\overline{\mathscr{T}}_0\cup(\overline{I}_0\times\{0\})$,
 containing $\overline{I}_0\times\{0\}$.
Hence $\lambda_{\ast}\in I_1^{0}$, it follows that 
$\mathscr{D}_0=\mathscr{D}_{\infty}$.


(ii) We shall show that (2) of Theorem \ref{thm3.2} does not occur.
Suppose on the contrary that (2) of Theorem \ref{thm3.2} occurs, then we shall 
deduce a contradiction. We divide the proof into two steps.
\smallskip

\noindent\textbf{Step 1.} We show that $\mathscr{D}_{\infty}-\mathscr{M}$ 
has a bounded projection on $\mathbb{R}$.
Firstly, we show that $\mathscr{D}_{\infty}\subset K^{+}$. 
If $(\mathscr{D}_{\infty}-(\mathscr{D}_{\infty}\cap\mathscr{N}))\not\subset K^{+}$, 
then there exists 
$(\mu,v)\in(\mathscr{D}_{\infty}-(\mathscr{D}_{\infty}\cap\mathscr{N}))
\cap(\mathbb{R}\times\partial P^{+})$. 
Since $v\in\partial P^{+}$, by Lemma \ref{lem3.1}, $v\equiv0$, i.e.\ (1) of 
Theorem \ref{thm3.2} occurs, 
which is a contradiction.

On the contrary, we suppose that $(\mu_n, v_n)\in \mathscr{D}_{\infty}-\mathscr{M}$ 
such that
$$
\lim_{n\to \infty}\mu_n= +\infty.
$$
It follows that
\begin{equation} \label{e4.13}
\begin{gathered}
((-v_n')^{N})'=\mu_n \gamma  Nr^{N-1}a(r)g(v_n)
+ \gamma  Nr^{N-1}a(r)f(v_n),\quad r\in(0,1), \\
v_n'(0)=v_n(1)=0,
\end{gathered}
\end{equation}

In view of Remark \ref{rmk4.1}, (H5) and (H6),we have that 
\[
\lim_{n\to \infty}(\mu_n Nr^{N-1}a(r)\frac{g(v_n)}{v_n^{N}}
+ Nr^{N-1}a(r)\frac{f(v_n)}{v_n^{N}})=+\infty
\]
 for any $r\in(0,1)$. By the Sturm Comparison Lemma \ref{lem2.3},
 we get that $v_n$ has more change its sign in $(0,1)$ for $n$ large enough, 
and this contradicts the fact that  
$(\mu_n, v_n)\in \mathscr{D}_{\infty}^{+}-\mathscr{M}$.
\smallskip

\noindent\textbf{Step 2.} We show that the case of
 $\mathscr{D}_{\infty}-\mathscr{M}$ meeting $\overline{I}_{\infty}\times\{\infty\}$ 
is impossible.
Assume on the contrary that $\mathscr{D}_{\infty}-\mathscr{M}$ meets 
$\overline{I}_{\infty}\times\{\infty\}$. So there exists a neighborhood 
$\widetilde{\mathscr{N}}\subset \widetilde{\mathscr{M}}$ of 
$\overline{I}_{\infty}\times\{\infty\}$ such that 
$(\mathscr{D}_{\infty}-\mathscr{M})\cap(\widetilde{\mathscr{N}}
\setminus(\overline{I}_{\infty}\times\{\infty\}))
\subset (\mathbb{R}\times \overline{P})$,
where $\widetilde{\mathscr{M}}$ is a neighborhood of 
$\overline{I}_{\infty}\times\{\infty\}$ which satisfies the assumptions 
of Theorem \ref{thm3.2}, which contradicts $\mathscr{D}_{\infty}\subset \overline{P}$,
where $ \overline{P}=\{v|(\overline{\lambda},v)\in( S_{N}
\setminus \{\lambda_1\})\times E\}$ and
 $\overline{I}_{\infty}=[\frac{\overline{\lambda}}{\gamma g_{\infty}}-d_{4},
 \frac{\overline{\lambda}}{\gamma g_{\infty}}+d_{4}]$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.1}]
 It suffices to prove that problem \eqref{e4.1} has at least one solution $v$ 
such that it is positive, strictly concave in $(0, 1)$.

By Lemmas \ref{lem4.1} and \ref{lem4.2}, we write  $\mathscr{D}=\mathscr{D}_0=\mathscr{D}_{\infty}$ 
for simplicity. It is clear that any solution of \eqref{e4.4} of the form 
$(1,v)$ yields a solution $v$ of \eqref{e4.1}.
In this case, $d_3<1, d_{4}<1$. By \eqref{e4.2}, we obtain
\begin{equation} \label{e4.14}
\frac{\lambda_1}{\gamma g_0}+d_3<1,\quad
\frac{\lambda_1}{\gamma g_{\infty}}-d_{4}>1.
\end{equation}
By \eqref{e4.3}, we have
\begin{equation} \label{e4.15}
\frac{\lambda_1}{\gamma g_{\infty}}+d_{4}<1,\quad
\frac{\lambda_1}{\gamma g_0}-d_3>1
\end{equation}
From $I_1^{0}=[\frac{\lambda_1}{\gamma g_0}-d_3,\frac{\lambda_1}{\gamma g_0}+d_3]$
and 
$I_1^{\infty}=[\frac{\lambda_1}{\gamma g_{\infty}}-d_{4},
\frac{\lambda_1}{ \gamma g_{\infty}}+d_{4}]$, 
it follows that the subsets $I_1^{0}\times E$ and $I_1^{\infty}\times E$ of 
$\mathbb{R}\times E$ can  be separated by the hyperplane $\{1\}\times E$. 
Furthermore, we have  $\mathscr{D}$ cross the hyperplane 
$\{1\}\times E$ in $\mathbb{R}\times E$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.2}]
 The proof is similar to that of Theorem \ref{thm4.1}. In the case, 
$d_3<1, d_{4}\geq1$, which follows that \eqref{e4.14} hold.
By $d_{4}\geq1$, it follows that \eqref{e4.15}  is impossible.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.3}]
 The proof is similar to that of Theorem \ref{thm4.1}.
In the case, $d_3\geq1, d_{4}<1$, which follows that \eqref{e4.15} hold.
By $d_3\geq1$, it follows that $\eqref{e4.14}$  is impossible.
\end{proof}

\begin{remark} \label{rmk4.3} \rm
Note that if $d_3\geq1,d_{4}\geq1$, \eqref{e4.14} and \eqref{e4.15}
  are impossible, it follows that the subsets $I_1^{0}\times E$ and 
$I_1^{\infty}\times E$ of $\mathbb{R}\times E$ can not be separated 
by the hyperplane $\{1\}\times E$. In this case, we cannot give a suitable 
interval of $\gamma$ in which there exist positive solutions for \eqref{e4.1}. 
It would be interesting to have more information about this case.
\end{remark}

\subsection*{Acknowledgments}
The authors want to thank the editors and the reviewers for their
constructive remarks that led to the improvement of this article.
This research was supported by the NSFC (no. 11561038), and  
the Gansu Provincial National Science
Foundation of China (no. 145RJZA087).

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\end{document}