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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 61, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/61\hfil Sign-changing solutions]
{Sign-changing solutions of $p$-Laplacian equation with a
sub-linear nonlinearity at infinity}

\author[X. Xu, B. Xu  \hfil EJDE-2013/61\hfilneg]
{Xian Xu, Bin Xu} 

\address{Xian Xu  \newline
Department of Mathematics,
Jiangsu  Normal  University, Xuzhou,
Jiangsu, 221116, China}
\email{xuxian68@163.com}

\address{Bin Xu \newline
Department of Mathematics,
Jiangsu  Normal  University, Xuzhou,
Jiangsu, 221116, China}
\email{dream-010@163.com}

\thanks{Submitted October 24, 2012. Published February 27, 2013.}
\thanks{Supported by grant NSFC10971179}
\subjclass[2000]{34B15, 34B25}
\keywords{$p$-Laplacian equation; sign-changing solution;
 Leray-Schauder degree}

\begin{abstract}
 In this article we obtain some existence and multiplicity results
 for sign-changing solutions of a $p$-Laplacian equation.
 We use the method of lower and upper solutions and Leray-Schauder 
 degree theory. Moreover, the sign-changing  solutions are located 
 by using lower and upper solutions.

\end{abstract}

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

\section{Introduction}

In this article we present existence and multiplicity results
for sign-changing  solutions for the  problem
\begin{equation}
\begin{gathered}
 (\varphi_p(u'(t)))'+f(t,u(t), u'(t))=0\quad  \text{a.e. } t\in (0,1),\\
 u(0)=u(1)=0,
\end{gathered}\label{e1.1}
\end{equation}
where $\varphi_p(s)=|s|^{p-2}s$, $s\in \mathbb{R}^1$, $p> 1$,
$f:[0,1]\times \mathbb{R}^2\to \mathbb{R}^1$.

In recent years there have been many  studies on the existence of
non-zero solutions of  $p$-Laplacian differential boundary value
problems, especially the existence of positive solutions of the
$p$-Laplacian differential boundary value problems; see 
\cite{c1,d1,j1,k1,l1,l2,m1,m2,y1,z1,z2,z3}
and the references therein. Recently, there were some papers considered
the existence of sign-changing solutions of $p$-Laplacian
differential boundary value problems by using the Leray-Schauder
degree method, or the global bifurcation theorem or the variation
method. For instance, in \cite{l1} the authors studied the  $p$-Laplacian
differential boundary value problems of the form
\begin{equation}
\begin{gathered}
 (\varphi_p(u'(t))'+\lambda h(t)f(u(t))=0 \quad \text{a.e. } t\in (0,1),\\
u(0)=u(1)=0,
\end{gathered} \label{e1.2}
\end{equation}
 where $\lambda$ is a positive parameter, $h$ a nonnegative
 measurable function on $(0,1)$ and $f\in C(\mathbb{R}^1, \mathbb{R}^1)$.
By applying the global bifurcation
 theorem, the authors in paper \cite{l1} obtained existence results for positive
 solutions as well as sign-changing solutions of \eqref{e1.2}.

 Zhang  and Li  \cite{z1} studied the problem
\begin{equation}
\begin{gathered}
 -\Delta_p u=h(u)\quad \text{in } \Omega,\\
  u|_{\partial\Omega}=0,
\end{gathered} \label{e1.3}
\end{equation}
where $-\Delta_p u=-\operatorname{div}(|\nabla u|^{p-2})\nabla u$ is the
$p$-laplacian operator, $\Omega$  a smooth bounded domain in $\mathbb{R}^N$.
The authors of \cite{z1} assumed that the boundary value problems \eqref{e1.3}
are with jumping nonlinearities
at zero or infinity, then they get sign-changing solutions theorems
of the $p$-Laplacian boundary value problems \eqref{e1.3}.

 The main purpose of this paper is to obtain some existence and multiplicity
results for sign-changing solutions of \eqref{e1.1}.  We will employ the
lower and upper solutions method
 and the Leary-Schauder degree method to show the existence and multiplicity
results of sign-changing solutions of \eqref{e1.1}.
Some sub-linear conditions on the
 nonlinearity $f$ at infinity will be assumed.  To show the
 multiplicity results for sign-changing solutions a pair of well ordered
 strict lower and upper solutions also be assumed.
Then we will first construct another pair of well ordered (or non-well
  ordered) strict lower and upper solutions near the zero element
$\theta$ of the Banach space $C_0^1[0,1]$. Next by computing the
Leray-Schauder degree on different areas defined by the strict
lower and upper solutions, we obtain the existence and
multiplicity results for sign-changing solutions  as well as
positive and negative solutions of \eqref{e1.1}. The main feature of our
results is that we not only obtain multiplicity results for
sign-changing solutions of \eqref{e1.1}, but also give clear description
of the locations of the sign-changing solutions of \eqref{e1.1}  through the strict
lower and upper solutions.

  In recent years,  by using the method of invariant sets of the
descending flow corresponding to the functional of the nonlinear problems
some authors  studied the existence results of sign-changing solutions
of some partial differential boundary  value  problems, see
\cite{z1,z2,z3} and the references therein. To show their  results the 
authors always assumed the nonlinearities  satisfy some kinds of monotony 
properties and therefore they always assumed the nonlinearities
are without the gradient terms. For instance, Li  and Li  \cite{l2}
considered the elliptic equation with Neumann boundary condition
\begin{equation}
\begin{gathered}
 -\Delta u+a u=f(u),\quad x\in \Omega;\\
 \frac{\partial u}{\partial \nu}=0, \quad x\in \partial \Omega,
\end{gathered} \label{e1.4}
\end{equation}
where $\Omega$ is a bounded domain with smooth boundary. The authors
of \cite{l2} assumed that the nonlinearity $f$ satisfies some increasing
properties and obtained some multiplicity results for sign-changing
solutions of \eqref{e1.4} by using the method of invariant
sets of the descending  flow as well as the method of lower and upper solutions.
Since  we allow the nonlinearity   $f$ in \eqref{e1.1} are with $u'$,
generally speaking, any monotony type conditions can not be assumed
in \eqref{e1.1} and therefore our main results can not
be obtained by  the method in \cite{z1,z2,z3}.

  This paper is organized in the following way.
In the section 2, we   give general hypothesis and technical results about the
$p$-Laplacian differential boundary value problems. Then, we give
degree information in terms of the lower and upper solutions. In
the section 3, we will give existence and multiplicity results for
sign-changing solutions of \eqref{e1.1}.

\section{Some Lemmas}

 Let $N^+$ denote the set of natural numbers. Let
$C[0,1]$ and $C^1[0,1]$ be the usual Banach spaces with the norms
$\|\cdot\|_0$ and $\|\cdot\|$, respectively.
Let $C_0^1[0,1]=\{x\in C^1[0,1]|x(0)=x(1)=0\}$,
 $P_0=\{x\in C[0,1]|x(t)\geqslant 0$, $t\in [0,1]\}$ and
$P= P_0\cap C_0^1[0,1]$. Then $C_0^1[0,1]$ is also a
real Banach space with the norm $\|\cdot\|$, $P$ and $P_0$ are cones
of $C_0^1[0,1]$ and $C[0,1]$, respectively. Let $\leqslant$ denote
both the orderings induced by $P$ in $C_0^1[0,1]$ and $P_0$ in
$C[0,1]$. We write $x<y$ if $x\leqslant y$ and $x\neq  y$. Let
$e(t)=t(1-t)$ for all $t\in [0,1]$. For each $x, y\in C[0,1]$, we
denote by $x\prec y$ or $y\succ x$ if $y-x\geqslant \delta_0 e$ for some
$\delta_0>0$. For any $x_0\in C[0,1]$, let
$\Omega_1=\{x\in C_0^1[0,1]|x\succ x_0\}$ and
$\Omega_2=\{x\in C_0^1[0,1]|x\prec x_0\}$.
Then $\Omega_1$ and $\Omega_2$ are open subsets of $C_0^1[0,1]$.

Now we define the concepts of strict lower and upper solutions of
\eqref{e1.1} in a manner as that of \cite{d2}; see
\cite[Definition 5.4.47 and 5.4.48]{d2}.

\begin{definition} \label{def2.1}\rm
A function $u_0\in C^1[0,1]$ with $\varphi_p(u_0'(t))$ absolutely continuous
is called a lower solution of \eqref{e1.1} if
$$
u_0(0)\leqslant 0, \quad u_0(1)\leqslant 0
$$
and
$$
-(\varphi_p(u_0'(t))'\leqslant f(t, u_0(t), u_0'(t))\quad \text{for a. e. }
t\in (0,1).
$$
In an analogous way we define an upper solution of \eqref{e1.1}.
\end{definition}

 \begin{definition} \label{def2.2}\rm
 A lower solution $u_0$ is said
to be  strict if every possible solution $x$ of \eqref{e1.1} such that
$u_0\leqslant x$ satisfies $u_0\prec x$.
In an analogous way we define a strict upper solution of \eqref{e1.1}.
\end{definition}

\begin{remark} \label{rmk2.1}\rm
 Obviously, if $f$ has the form of $f(t, x)$ and
satisfies
$$
f(t, x_2)-f(t, x_1)\geqslant -M(x_2-x_1), \quad \forall x_2\geqslant
x_1
$$
for some $M>0$, $u_0\in C^2[0,1]$ satisfies
$u_0(0)\leqslant 0$, $u_0(1)\leqslant 0$ and
$$
u_0''+f(t, u_0(t))>0, \quad t\in (0,1),
$$
then $u_0$ will be a strict lower solution in the Definition \ref{def2.2} for
$p=2$ by the Maximum Principle.
\end{remark}

\begin{definition} \label{def2.3}\rm
Let $u_0$ and $v_0$ be strict lower and upper solutions of \eqref{e1.1},
respectively. Then $u_0$ and $v_0$ are called a pair of well-ordered
strict lower and upper solutions of \eqref{e1.1} if $u_0\prec v_0$.
\end{definition}

\begin{definition} \label{def2.4}\rm
A function $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}^1$ is
said to be a Carath\'{e}odory function, if $f(t, \cdot, \cdot)$ is
continuous on $\mathbb{R}^2$ for almost all $t\in [0,1]$; $f(\cdot, x,y)$ is
a measurable function on $[0,1]$ for all $(x,y)\in \mathbb{R}^2$; for every
$R>0$ there exists a real-valued function
$\Psi\equiv \Psi_R\in L^1(0,1)$ such that
$$
|f(t,x,y)|\leqslant \Psi(t)
$$
for a.e. $t\in [0,1]$ and for every $(x,y)\in \mathbb{R}^2$ with
$|x|+|y|\leqslant R$.
\end{definition}

Let $\alpha\in C^1[0,1]$. The function
$p:[0,1]\times \mathbb{R}^1\to \mathbb{R}^1$ be
defined by
$$
p(t, x)=\max\{ \alpha(t), x\}, \, \forall (t, x)\in [0,1]\times \mathbb{R}^1.
$$


The first result is Lemma \ref{lem2.1}, for which we omit the proof.
A similar result and its proof can be found in \cite{w1}.

\begin{lemma} \label{lem2.1}
 For each $u\in C^1[0,1]$, the next two properties hold:
\begin{itemize}
\item[(i)]  $\frac{d}{dt}p(t, u(t))$ exists for  a.e. $t\in I$.

\item[(ii)] If $u, u_m\in C^1[0,1]$ and $u_m\to u$ in $C^1[0,1]$, then
$$
\frac{d}{dt}p(t, u_m(t))\to \frac{d}{dt}p(t, u(t))\quad
\text{for a.e. } t\in [0,1].
$$
\end{itemize}
\end{lemma}

\begin{lemma} \label{lem2.2}
 Let $\alpha_1,\ \alpha_2\in C^1[0,1]$ and
$\bar\alpha(t)=\max\{\alpha_1(t), \alpha_2(t)\}$ for all $t\in [0,1]$. Then
the following conclusions hold.
\begin{itemize}
\item[(1)] $\bar\alpha'(t)=\alpha'_1(t)$ when $\alpha_1(t)>\alpha_2(t)$;

\item[(2)] $\bar\alpha'(t)=\alpha'_2(t)$ when $\alpha_2(t)>\alpha_1(t)$;

\item[(3)] $\bar\alpha'(t)=\alpha'_1(t)=\alpha'_2(t)$ when $\alpha_1(t)=\alpha_2(t)$ and
$\alpha'_1(t)=\alpha'_2(t)$;

\item[(4)] $\bar\alpha'_-(t)=\min\{\alpha'_1(t),\alpha'_2(t)\}$ and
$\bar\alpha'_+(t)=\max\{\alpha'_1(t),\alpha'_2(t)\}$ when $\alpha_1(t)=\alpha_2(t)$
and $\alpha'_1(t)\neq \alpha'_2(t)$;

\item[(5)] $\lim_{\tau\to t^-}\bar\alpha'(\tau)=\bar\alpha'_-(t)$ and
$\lim_{\tau\to t^+}\bar\alpha'(\tau)=\bar\alpha'_+(t)$ when
$\alpha_1(t)=\alpha_2(t)$ and $\alpha'_1(t)\neq \alpha'_2(t)$;


\item[(6)]
\begin{equation}
|\bar\alpha'(t)|\leqslant \max\{\|\alpha_1'\|_0, \|\alpha'_2\|_0\}\
\text{a.e. } t\in [0,1].\label{e2.1}
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
Let $I_1=\{t\in [0,1]|\alpha_1(t)>\alpha_2(t)\}$,
$I_2=\{t\in [0,1]|\alpha_2(t)>\alpha_1(t)\}$ and $I_3=[0,1]\backslash(I_1\cup I_2)$.
Assume without loss of generality that $I_i\neq \emptyset$ for $i=1,2,3$.
Obviously, we have $\bar\alpha'(t)=\alpha_1'(t)$ for each $t\in I_1$, and
$\bar\alpha'(t)=\alpha_2'(t)$ for each $t\in I_2$.
Let $I_{3,1}=\{t\in I|\alpha_1(t)=\alpha_2(t), \alpha'_1(t)=\alpha_2'(t)\}$ and
$I_{3,2}=\{t\in I|\alpha_1(t)=\alpha_2(t), \alpha'_1(t)\neq \alpha_2'(t)\}$.
 Then we have $I_3=I_{3,1}\cup I_{3,2}$.
Obviously, the conclusions (1) and (2) hold.
 Let $t_0\in I_{3}$.
Now for each $t>t_0$,  by the Mean-Value Theorem, there exists
$\xi_t$ and $\eta_t$ with $t_0<\xi_t<t$ and $t_0<\eta_t<t$ such that
\begin{gather*}
\alpha_1(t)=\alpha_1(t_0)+\alpha_1'(\xi_t)(t-t_0),\\
\alpha_2(t)=\alpha_2(t_0)+\alpha_2'(\eta_t)(t-t_0).
\end{gather*}
Then, we have for each $t>t_0$,
\begin{align*}
\bar \alpha(t)
&=\bar\alpha(t_0)+[\alpha_1'(\xi_t)\vee\alpha_2'(\eta_t)](t-t_0)\\
&=\bar\alpha(t_0)+\frac{\alpha_1'(\xi_t)+\alpha_2'(\eta_t)+|\alpha_1'(\xi_t)
-\alpha_2'(\eta_t)|}{2}(t-t_0).
\end{align*}
Consequently,
\begin{align*}
\bar\alpha'_+(t_0)
&=\lim_{t\to t_0^+}\frac{\bar\alpha(t)-\bar\alpha(t_0)}{t-t_0}\\
&=\lim_{t\to t_0^+}\frac{\alpha_1'(\xi_t)+\alpha_2'(\eta_t)
 +|\alpha_1'(\xi_t)-\alpha_2'(\eta_t)|}{2}\\
&=\frac{\alpha_1'(t_0)+\alpha_2'(t_0)+|\alpha_1'(t_0)-\alpha_2'(t_0)|}{2}\\
&=\begin{cases}\alpha'_1(t_0)=\alpha_2'(t_0),&\text{when } t_0\in I_{3,1};\\
\max\{\alpha'_1(t_0), \alpha'_2(t_0)\},&\text{when } t_0\in I_{3,2}.
\end{cases}
 \end{align*}
Similarly, we have
$$
\bar\alpha'_-(t_0)
=\lim_{t\to t_0^-}\frac{\bar\alpha(t)-\bar\alpha(t_0)}{t-t_0}
=\begin{cases}
 \alpha'_1(t_0)=\alpha_2'(t_0),&\text{when } t_0\in I_{3,1};\\
\min\{\alpha'_1(t_0), \alpha'_2(t_0)\},&\text{when } t_0\in I_{3,2}.
\end{cases}
$$
Therefore, $\bar\alpha$ is differentiable at $t_0\in I_{3,1}$
and $\bar\alpha'(t_0)=\alpha'_1(t_0)=\alpha_2'(t_0)$ for each
$t_0\in I_{3,1}$. Thus, the conclusion (3) and (4) hold.

Let $t_0\in I_{3,2}$. Assume without loss of generality that
$t_0\in (0,1)$ and $\alpha'_1(t_0)<\alpha'_2(t_0)$. Then there exists $\delta_0>0$
small enough such that $\alpha_1(t)>\alpha_2(t)$ for all
$t\in (t_0-\delta_0, t_0)$. Thus, we have $\bar\alpha(t)=\alpha_1(t)$
for all $t\in (t_0-\delta_0, t_0]$, and thus $\bar\alpha'(t)=\alpha_1'(t)$
for $t\in (t_0-\delta_0, t_0]$.
Therefore, $\lim_{t\to t_0^-}\bar\alpha'(t)=\alpha_1'(t_0)=\bar\alpha'_-(t_0)$.
Similarly, we have
$\lim_{t\to t_0^+}\bar\alpha'(t)=\alpha_2'(t_0)=\bar\alpha'_+(t_0)$.
This means that the conclusion (5) holds. The conclusion (6) follows
from (1)-(4). The proof is complete.
\end{proof}

 Let $h\in L^1(0,1)$. Consider the boundary-value problem
\begin{equation}
\begin{gathered}
(\varphi_p(u'(t)))'=h\quad \text{a.e. } t\in (0,1),\\
u(0)=u(1)=0.
\end{gathered} \label{e2.2}
\end{equation}
A function $u\in C_0^1[0,1]$
is called a solution of \eqref{e2.2}, if $\varphi_p (u'(t))$ is absolutely
continuous and satisfies \eqref{e2.2}. It is easy to see that \eqref{e2.2} is
equivalent to
\begin{equation}
u(t)=G_p(h)(t):=\int^t_0 \varphi_p^{-1}\Big(a(h)+\int^s_0
h(\tau)d\tau\Big)ds,\label{e2.3}
\end{equation}
where $a:L^1(0,1)\to \mathbb{R}^1$ is a continuous functional satisfying
$$
\int^1_0\varphi_p^{-1}\Big(a(h)+\int^s_0h(\tau)d\tau\Big)ds=0.
$$
 From \cite{p1}, we see that $G_p:L^1(0,1)\to C_0^1[0,1]$ is
 continuous and maps equi-integrable sets of $L^1(0,1)$ into
 relatively compact sets of $C_0^1[0,1]$. One may refer to
 Man\'{a}sevich and Mawhin \cite{m2} for more details.

 Next we consider the eigenvalues problem of the form \eqref{Elap}
\begin{equation}
 \begin{gathered}
(\varphi_p(u'(t)))'+\lambda \varphi_p(u(t))=0 \quad\text{ a.e. } t\in (0,1),\\
 u(0)=u(1)=0.
 \end{gathered} \label{Elap}
\end{equation}
 Define the operator $T_\lambda^p:C_0^1[0,1]\to C_0^1[0,1]$ by
 $$
(T_\lambda^p u)(t)=G_p(-\lambda  \varphi_p(u))(t)=\int^t_0\varphi_p^{-1}\Big(a(-\lambda
 \varphi_p(u))-\int^s_0 \lambda \varphi_p(u(\tau))d\tau\Big)ds.
$$
 Then $T^p_\lambda$ is completely continuous and problem \eqref{Elap} is
 equivalent to equation $u=T^p_\lambda u$.


 From \cite[Proposition 2.6, Lemmas 2.7 and 2.8]{l1},
we have the following Lemmas

\begin{lemma} \label{lem2.3}
  The following conditions hold:
\begin{itemize}
\item[(i)] the set of all eigenvalues of \eqref{Elap} is a countable set
$\{\mu_k(p)|k\in N^+\}$ satisfying
$$
0<\mu_1(p)<\mu_2(p)<\dots<\mu_k(p)<\dots\to \infty;
$$

\item[(ii)] for each $k$, $\ker(I-T_{\mu_k(p)}^p)$ is a subspace of
$C^1[0,1]$ and its dimension is 1;

\item[(iii)] let $\phi_k$ be a corresponding eigenfunction to $\mu_k(p)$,
then the number of interior zeros of $\phi_k$ is $k-1$.
\end{itemize}
\end{lemma}

\begin{lemma} \label{lem2.4}
 For each $k\in N^+$, $\mu_k(p)$  as a function of
$p\in (1,\infty)$ is continuous.
\end{lemma}

By Lemma \ref{lem2.4} and the method of homotopy along $p$ which developed in
\cite{p1}, we have the following Lemma.

\begin{lemma} \label{lem2.5}
  For fixed $p>1$ and all $r>0$, we have
$$
\deg (I- T_\lambda^p, B(\theta, r),\theta)
=\begin{cases} 1,&\text{when } \lambda<\mu_1(p);\\
(-1)^k,&\text{when } \lambda\in(\mu_k(p),\mu_{k+1}(p)).
\end{cases}
$$
\end{lemma}

 Now let us define the operator $F:C^1_0[0,1]\to
L^1[0,1]$ and $T_p:C^1_0[0,1]\to C_0^1[0,1]$ by
$$
(Fx)(t)=f(t, x(t), x'(t)),t\in [0,1]
$$
and $(T_px)(t)=(G_pFx)(t)$ for all $t\in [0,1]$. Then, $T_p$ is
completely continuous.

For convenience, we make the following assumptions.
\begin{itemize}
\item[(H1)] $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}^1$ is  a Carath\'{e}odory
 function such that $xf(t, x, y)>0$ for all
$(t, y)\in [0,1]\times \mathbb{R}^1$ and $x\neq 0$, and  there exists
$\beta_\infty\geqslant 0$
 with
 $(2^p\beta_\infty)^{\frac{1}{p-1}}<1$ such that
 $$
\lim_{|x|+|y|\to \infty}\frac{|f(t, x, y)|}{\varphi_p(|x|+|y|)}
=\beta_\infty\quad \text{uniformly for } t\in  [0,1].
$$

\item[(H2)] There exists $R_*>0$ and $\beta_0>0$ such
 that
 $$\lim_{x\to 0}\frac{f(t,x,y)}{\varphi_p(x)}=\beta_0\quad \text{uniformly for }\
 t\in [0,1] \text{ and } y\in [-R_*,R_*].
$$

\item[(H3)] There exist sign-changing functions $u_1$, $v_1$ such that
$u_1$ and $v_1$ are a pair of strict lower and upper solutions of
 \eqref{e1.1}.
\end{itemize}
Let $f$ and $g$  be defined by
\begin{gather*}
f(t,x,y)=\begin{cases}
\beta_0\varphi_p(x)+[\varphi_p(x)]^2y^2, &x^2+y^2\leqslant 1;\\
10,&x^2+y^2\geqslant 2;\\
10(\sqrt{x^2+y^2}-1)+(2-\sqrt{x^2+y^2})g(x,y),&1<x^2+y^2<2,
\end{cases}
\\
g(x,y)=\beta_0\varphi_p\Big(\frac{x}{\sqrt{x^2+y^2}}\Big)
+\Big[\varphi_p\Big(\frac{x}{\sqrt{x^2+y^2}}\Big)\Big]^2
\Big(\frac{y}{\sqrt{x^2+y^2}}\Big)^2,\ 1<x^2+y^2<2.
\end{gather*}
Obviously, $f$ satisfies the conditions (H1) and (H2).


 \begin{lemma} \label{lem2.6}
 Suppose that {\rm (H1)} holds,  $\alpha_1, \alpha_2$ are
 strict lower solutions of \eqref{e1.1} such that $\alpha_1(t)\equiv \alpha_2(t)$
or the set $\{t\in [0,1]|\alpha_1(t)=\alpha_2(t), \alpha'_1(t)\neq \alpha'_2(t)\}$
contains at most finite elements. Then
there exists $R_0>0$ such that for
 each $R_1\geqslant R_0$, $T_p(\bar B(\theta, R_1))\subset B(\theta,
 R_1)$, $\alpha_1,\alpha_2\in B(\theta, R_1)$ and
 $$
\deg (I-T_p, \Omega, \theta)=1,
$$
 where $\Omega=\{x\in  B(\theta, R_1)|x\succ\alpha_1, x\succ  \alpha_2\}$ and
$B(\theta, R_1)=\{x\in C_0^1[0,1]:\|x\|<R_1\}$.
\end{lemma}

\begin{proof}
 We  consider only the case of $\alpha_1(t)\not\equiv  \alpha_2(t)$.
 Let $\bar\alpha(t)=\max\{\alpha_1(t), \alpha_2(t)\}$ for each $t\in [0,1]$.
Let $\beta'_\infty$ be such that $\beta'_\infty>\beta_\infty$
and $(2^p\beta'_\infty)^{\frac{1}{p-1}}<1$.
From (H1), there exists $R'>0$ such that
  $$
|f(t, x,y)|\leqslant \beta'_\infty\varphi_p(|x|+|y|),\forall t\in [0,1],
  |x|+|y|\geqslant R'.
$$
Since $f:[0,1]\times \mathbb{R}^2\to \mathbb{R}^1$ is a Carath\'{e}odory
function, then there exists $\Psi_{R'}\in L^1(0,1)$ such that
$$
|f(t, x, y)|\leqslant \Psi_{R'}(t)\quad \text{a.e. }
t\in [0,1], |x|+|y|\leqslant R'.
$$
Consequently, we have
  $$
|f(t,x, y)|\leqslant \beta'_\infty\varphi_p (|x|+|y|)+\Psi_{R'}(t), \quad
\text{a.e. } t\in   [0,1], (x,y)\in \mathbb{R}^2.
$$
Let
  $$
R_0>\max\Big\{\|\alpha_1\|, \|\alpha_2\|,\frac{\varphi_p^{-1}(2^p
  M_{R'})+2(\|\alpha_1\|+\|\alpha_2\|)}{1-(2^p\beta'_\infty)^{\frac{1}{p-1}}}\Big\}.
$$
  and $R_1\geqslant R_0$, where $M_{R'}=\|\Psi_{R'}\|_{L^1(0,1)}$.
For any $x\in \bar B(\theta, R_1)$, by the Rolle's Theorem there exists
  $t_x\in (0,1)$ such that $(T_px)'(t_x)=0$ and for all $t\in [0,1]$
  $$
|(T_px)'(t)|=\Big|\varphi_p^{-1}\Big(\int^{t_x}_t f(\tau, x(\tau),
  x'(\tau))d\tau\Big)\Big|
$$
  and
$$
|(T_px)(t)|=\Big|\int^t_0\varphi_p^{-1}\Big(\int^{t_x}_s f(\tau,
  x(\tau),x'(\tau))d\tau\Big)ds\Big|.
$$
  Then we have for all $t\in [0,1]$
\begin{align*}
  |(T_px)'(t)|
&\leqslant \varphi_p^{-1}\Big(\int^1_0|f(\tau, x(\tau),
  x'(\tau))|d\tau\Big)\\
&\leqslant \varphi_p^{-1}\Big(\int^1_0[\beta'_\infty \varphi_p
  (|x(\tau)|+|x'(\tau)|)+\Psi_{R'}(\tau)]d\tau\Big)\\
&\leqslant \varphi_p^{-1}(\beta'_\infty \varphi_p(\|x\|)+M_{R'})\\
&=\varphi_p^{-1}(\varphi_p((\beta'_\infty)^{\frac{1}{p-1}}\|x\|)+\varphi_p(\varphi_p^{-1}(M_{R'})))\\
&\leqslant
  \varphi_p^{-1}(2\varphi_p((\beta'_\infty)^{\frac{1}{p-1}}\|x\|+\varphi_p^{-1}(M_{R'})))\\
&=(2\beta'_\infty)^{\frac{1}{p-1}}\|x\|+\varphi_p^{-1}(2M_{R'})
\end{align*}
  Similarly, we have that for all $t\in [0,1]$,
\begin{align*}
  |(T_px)(t)|&\leqslant \int^t_0\varphi_p^{-1}\Big(\int^1_0|f(\tau,x(\tau),
  x'(\tau))|d\tau\Big)ds\\
  &\leqslant \varphi_p^{-1}\Big(\int^1_0|f(\tau, x(\tau),x'(\tau))| d\tau\Big)\\
  &\leqslant (2\beta'_\infty)^{\frac{1}{p-1}}\|x\|+\varphi_p^{-1}(2 M_{R'}).
\end{align*}
  Thus we have
\begin{align*}
  \|T_px\|&=\|(T_px)'\|_0+\|T_px\|_0\\
  &\leqslant   2[(2\beta'_\infty)^{\frac{1}{p-1}}\|x\|+\varphi_p^{-1}(2 M_{R'})]\\
  &=(2^p\beta'_\infty)^{\frac{1}{p-1}}\|x\|+\varphi_p^{-1}(2^pM_{R'}))\\
  &\leqslant (2^p\beta'_\infty)^{\frac{1}{p-1}} R_1+\varphi_p^{-1}(2^pM_{R'})
  <R_1.
  \end{align*}
  This implies that $T_p(\bar B(\theta, R_1))\subset B(\theta, R_1)$.

   Let the function $g:[0,1]\times \mathbb{R}^1\to \mathbb{R}^1$ be defined by
  $$
g(t,x)=\max\{\bar\alpha(t),x\}, \forall (t,x)\in [0,1]\times \mathbb{R}^1.
$$
  We denote by $\widetilde T_p:C_0^1[0,1]\to C_0^1[0,1]$ the solution
  operator of
\begin{equation}
\begin{gathered}
  (\varphi_p(y'(t)))'+f(t, g(t,x(t)),\frac{d}{dt}g(t, x(t))=0\quad \text{a.e. }
  t\in (0,1);\\
  y(0)=y(1)=0;
\end{gathered} \label{e2.4}
\end{equation}
  that is, for $x,y\in C_0^1[0,1]$,
  $$
y=\widetilde T_p x
$$
  if and only if  \eqref{e2.4} holds.
For  any $x\in  C_0^1[0,1]$ it follows by integration of \eqref{e2.4}
and the injectivity  of $\varphi(s)=|s|^{p-2} s$ that the operator $\widetilde T$
is well  defined. In fact, $\widetilde T_p=G_p\widetilde F$, where $\widetilde
  F:C^1_0[0,1]\to L^1[0,1]$ is defined by
$$
(\widetilde Fx)(t)=f(t, g(t,x(t)),\frac{d}{dt}g(t,  x(t))
$$
for a.e. $t\in [0,1]$. It follows from Lemma \ref{lem2.1} and \ref{lem2.2}
that $\widetilde F:C^1_0[0,1]\to L^1[0,1]$ is bounded and continuous,
and so $\widetilde T_p:C_0^1[0,1]\to C_0^1[0,1]$ is completely
continuous.

   For any $x\in \bar B(\theta, R_1)$ there exists
$\widetilde t_x\in (0,1)$ such that $(\widetilde T_px)'(\widetilde t_x)=0$ and  for all
$t\in [0,1]$,
\begin{equation}
|(\widetilde T_px)'(t)|=\Big|\varphi_p^{-1}\Big(\int^{\widetilde t_x}_t f(\tau, x(\tau),
  x'(\tau))d\tau\Big)\Big|\label{e2.5}
\end{equation}
and
\begin{equation}
|(\widetilde T_px)(t)|=\Big|\int^t_0\varphi_p^{-1}\Big(\int^{\widetilde t_x}_s f(\tau,
  x(\tau),x'(\tau))d\tau\Big)ds\Big|.\label{e2.6}
\end{equation}
It follows from Lemma \ref{lem2.2} that for each $x\in C_0^1[0,1]$,
 \begin{equation}
\big|\frac{d}{dt} g(t,x(t))\big|\leqslant \max\{\|\alpha_1'\|_0, \|\alpha'_2\|_0,
  \|x'\|_0\}\quad \text{a.e.}\ t\in (0,1)\label{e2.7}
\end{equation}
From \eqref{e2.5}-\eqref{e2.7} we have that for $t\in [0,1]$,
\begin{align*}
|(\widetilde T_p x)'(t)|&\leqslant \varphi_p^{-1}\Big(\int^1_0 |f(\tau,
g(\tau, x(\tau)),\frac{d}{d\tau}g(\tau,x(\tau)))|d\tau\Big)\\
&\leqslant \varphi_p^{-1}\Big(\int^1_0\big[\beta_\infty' \varphi_p(|g(\tau,
x(\tau))|+|\frac{d}{d\tau}g(\tau,x(\tau))|)+\Psi_{R'}(\tau)\big]d\tau\Big)\\
&\leqslant \varphi_p ^{-1}\Big(\int^1_0\big(\beta'_\infty\varphi_p
(\max\{\|x\|_0,
\|\alpha_1\|_0, \|\alpha_2\|_0\}\\
&\quad +\max\{\|x'\|_0, \|\alpha_1'\|_0, \|\alpha_2'\|_0\})
 +\Psi_{R'}(\tau)\big)d\tau\Big)\\
&\leqslant \varphi_p^{-1}(\beta'_\infty\varphi_p(\|x\|+\|\alpha_1\|+\|\alpha_2\|)+M_{R'})\\
&\leqslant
(2\beta'_\infty)^{\frac{1}{p-1}}(\|x\|+\|\alpha_1\|+\|\alpha_2\|)+\varphi_p^{-1}(2
M_{R'})\\
&\leqslant
(2\beta'_\infty)^{\frac{1}{p-1}}\|x\|+\|\alpha_1\|+\|\alpha_2\|+\varphi_p^{-1}(2
M_{R'}).
\end{align*}
Similarly, we have that for $t\in [0,1]$,
$$
|(\widetilde T_px)(t)|\leqslant
(2\beta'_\infty)^{\frac{1}{p-1}}\|x\|+\|\alpha_1\|+\|\alpha_2\|+\varphi_p^{-1}(2M_{R'}).
$$
Thus we have
\begin{align*}
\|\widetilde T_p x\|&\leqslant (2^p\beta'_\infty)^{\frac{1}{p-1}}\|x\|
 +2(\|\alpha_1\|+\|\alpha_2\|)+\varphi_p^{-1}(2^pM_{R'})\\
&\leqslant (2^p\beta_\infty')^{\frac{1}{p-1}}R_1
 +2(\|\alpha_1\|+\|\alpha_2\|)+\varphi_p^{-1}(2^p M_{R'})<R_1.
\end{align*}
This implies that $\widetilde T_p(\bar B(\theta,R_1))\subset B(\theta, R_1)$.
Consequently,
\begin{equation}
\deg (I-\widetilde T_p, B(\theta, R_1), \theta)=1.\label{e2.8}
\end{equation}
Then $\widetilde T_p$ has fixed points in $B(\theta, R_1)$. Now we show that
$x_0\in \Omega$ whenever $x_0\in \bar B(\theta, R_1)$ with $\widetilde T_p
x_0=x_0$. We need only to show that $x_0\succ  \alpha_1$ and
$x_0\succ \alpha_2$. Note that $\alpha_1$ and $\alpha_2$ are strict lower solutions of
\eqref{e1.1}, then we need to show
\begin{equation}
x_0(t)\geqslant \bar \alpha(t), \quad t\in [0,1].\label{e2.9}
\end{equation}
Assume on the contrary that \eqref{e2.9} does not hold. Then there exists
$t_0\in [0,1]$ such that
$$
\bar \alpha(t_0)-x_0(t_0)=\max_{t\in [0,1]}(\bar \alpha(t)-x_0(t))>0.
$$
Since $x_0$ is a fixed point of $\widetilde T_p$ and $\alpha_1, \alpha_2$
are strict lower solutions of \eqref{e1.1}, we
easily see that $t_0\in (0,1)$. Thus, there exists an interval
$I_+\subset (0,1)$ such that $\bar\alpha (t)>x_0(t)$ for all
$t\in I_+$ and $\bar\alpha(t)=x_0(t)$ $(\forall t\in \partial I_+)$. Let
$$
(\bar\alpha(t)-x_0(t))^*=\begin{cases}
\bar\alpha(t)-x_0(t),&\forall t\in I_+,\\
0,&\forall t\in [0,1]\backslash I_+. \end{cases}
$$
Then we have
\begin{equation}
\begin{aligned}
&\int_{[0,1]}\varphi_p(x_0'(t))\frac{d}{dt}(\bar\alpha(t)-x_0(t))^*dt\\
&=\int_{[0,1]}f(t, g(t,x_0(t)), \frac{d}{dt} g(t,
x_0(t))(\bar\alpha(t)-x_0(t))^*dt\\
&=\int_{[0,1]}f(t, \bar \alpha(t), \bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*dt.
\end{aligned}\label{e2.10}
\end{equation}
Since $\{t\in [0,1]|\alpha_1(t)=\alpha_2(t), \alpha'_1(t)\neq \alpha'_2(t)\}$ is
a subset of $[0,1]$ which contains at most finite elements,  for
simplicity we assume that $\{t\in [0,1]|\alpha_1(t)=\alpha_2(t),
\alpha'_1(t)\neq \alpha'_2(t)\}=\{t_1\}$, $t_1\in (0,1)$ and
$\alpha'_1(t_1)<\alpha_2'(t_1)$. Then we have $\bar\alpha(t)=\alpha_1(t)$ for
all $t\in [0,t_1]$, and $\bar\alpha(t)=\alpha_2(t)$ for all $t\in
[t_1,1]$.  From  Lemma \ref{lem2.2} we see that $\varphi_p(\bar\alpha'(t))$ is
absolutely continuous on $[0,t_1]$ and $[t_1,1]$, respectively.
Using the formula of Integrating by part, we have
\begin{equation}
\begin{aligned}
&\int_{[0,1]}\varphi_p(\bar\alpha'(t))\frac{d}{dt}(\bar\alpha(t)-x_0(t))^*dt\\
&=\varphi_p(\bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*\Big|_0^{t_1}
 +\varphi_p(\bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*\Big|_{t_1}^1\\
&\quad -\Big(\int_{[0,t_1)}+\int_{[t_1,1]}\Big)\frac{d}{dt}(\varphi_p
(\bar\alpha'(t))(\bar \alpha (t)-x_0(t))^*dt\\
&=(\bar\alpha(t_1)-x_0(t_1))^*[\varphi_p(\alpha'_1(t_1))-\varphi_p(\alpha'_2(t_1))]\\
&\quad -\Big(\int_{[0,t_1)}+\int_{[t_1,1]}\Big)\frac{d}{dt}(\varphi_p
(\bar\alpha'(t))(\bar \alpha (t)-x_0(t))^*dt\\
&\leqslant -\Big(\int_{[0,t_1)}+\int_{[t_1,1]}\Big)\frac{d}{dt}(\varphi_p
(\bar\alpha'(t))(\bar \alpha (t)-x_0(t))^*dt.
\end{aligned}\label{e2.11}
\end{equation}
Now since $\alpha_1$ is a strict lower solution of \eqref{e1.1},  we have
\begin{equation} \begin{aligned}
\int_{[0,t_1)}-\frac{d}{dt}\varphi_p(\bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*dt&=-\int_{[0,t_1)}\frac{d}{dt}\varphi_p(\alpha_1'(t))(\alpha_1(t)-x_0(t))^*dt\\
&\leqslant \int_{[0,t_1)}f(t, \alpha_1(t),
\alpha_1'(t))(\alpha_1(t)-x_0(t))^*dt\\
&=\int_{[0,t_1)}f(t,\bar\alpha(t),
\bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*dt.
\end{aligned}\label{e2.12}
\end{equation}
In the same way, we have
\begin{equation}
\int_{[t_1, 1]}-\frac{d}{dt}\varphi_p(\bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*dt\leqslant
\int_{[t_1,1]}f(t, \bar\alpha(t),
\bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*dt.\label{e2.13}
\end{equation}
From \eqref{e2.11}-\eqref{e2.13} it follows that
\begin{equation}
\int_{[0,1]}\varphi_p(\bar\alpha'(t))\frac{d}{dt}(\bar\alpha(t)-x_0(t))^*dt
\leqslant\int_{[0,1]}f(t,\bar\alpha(t),\bar\alpha'(t))(\bar\alpha(t)-x_0(t))^*dt.
\label{e2.14}
\end{equation}
By \eqref{e2.10} and \eqref{e2.14} we have
$$
\int_{[0,1]}[\varphi_p(x_0'(t))-\varphi_p(\bar\alpha'(t))](\bar\alpha'(t)-x_0'(t))dt
\geqslant 0.
$$
This is a contradiction to that $s\mapsto \varphi_p(s)$ is strictly
increasing, which proves that $x_0\succ \alpha_1$ and $x_0\succ \alpha_2$.
Now by the properties of the Leray-Schauder degree and \eqref{e2.8} we have
\begin{equation}
\deg (I-\widetilde T_p, \Omega, \theta)=1.\label{e2.15}
\end{equation}
The assertion now follows from the fact that $T_p$ and $\widetilde T_p$
coincides in $\bar\Omega$. The proof is complete. \vskip 0.1in
\end{proof}

As in the proof of Lemma \ref{lem2.6} we have the following result.

\begin{lemma} \label{lem2.7}
 Suppose that {\rm (H1)} holds,  $\beta_1, \beta_2$ are
 strict upper solutions of \eqref{e1.1} such that
$\beta_1(t)\equiv \beta_2(t)$ or the set
$\{t\in [0,1]|\beta_1(t)=\beta_2(t),
\beta'_1(t)\neq \beta'_2(t)\}$ contains at most finite elements.
Then there exists $R_0>0$ such that for
 each $R_1\geqslant R_0$, $T_p(\bar B(\theta, R_1))\subset B(\theta,
 R_1)$, $\beta_1,\beta_2\in B(\theta, R_1)$ and
 $$
\deg (I-T_p, \Omega, \theta)=1,
$$
 where $\Omega=\{x\in \bar B(\theta, R_1)|x\prec  \beta_1, x\prec  \beta_2\}$.
\end{lemma}

 \begin{lemma} \label{lem2.8}
 Suppose that {\rm (H1)} and {\rm (H2)} hold. Let $R_1>0$,
\[
S_{R_1}=\{x\in C_0^1[0,1]| x \text{ is a solution of \eqref{e1.1}
 and $\|x\|\leqslant R_1$}\},
\]
$S^+_{R_1}=\{x\in S_{R_1}|x>\theta\}$
 and $S_{R_1}^-=\{x\in S_{R_1}|x<\theta\}$. Then there exists
 $\zeta_{R_1}>0$ such that
 $$
S^+_{R_1}\geqslant \zeta_{R_1} e,\quad
S^-_{R_1}\leqslant-\zeta_{R_1} e.
$$
\end{lemma}

\begin{proof}
 Let $x_0\in S^+_{R_1}$ be fixed at present. Take
$t_0\in (0,1)$ such that $x_0(t_0)=\|x_0\|_0$. Then we have
\begin{equation}
\begin{gathered}
-(\varphi_p(x_0'(t)))'=f(t,x_0(t), x_0'(t)) \quad \text{a.e. } t\in (0,1),\\
x_0(1)=0,\quad x_0(t_0)=\|x_0\|_0.
\end{gathered} \label{e2.16}
\end{equation}
Assume that $v\in C^1[0,1]$ satisfying
\begin{equation}
\begin{gathered}
-(\varphi_p(v'(t)))'=0 \quad \text{a.e. } t\in (0,1),\\
v(1)=0,\quad  v(t_0)=\|x_0\|_0.
\end{gathered}\label{e2.17}
\end{equation}
Now we show that
\begin{equation}
 x_0(t)\geqslant v(t), \quad \forall t\in (t_0, 1).\label{e2.18}
\end{equation}
Assume \eqref{e2.18} is not true. Let $\omega(t)=x_0(t)- v(t)$ for all
$t\in [t_0,1]$. Then there exists  $t^*\in (t_0,1)$ such that
$\omega(t^*)=\min_{t\in [t_0, 1]}\omega(t)<0$. Take $[t_1,
t_2]\subset [t_0,1]$ such that $t^*\in (t_1, t_2)$,
$\omega(t_1)=\omega(t_2)=0$, and
\begin{equation}
\omega(t)<0,\quad \forall t\in (t_1, t_2).\label{e2.19}
\end{equation}
By \eqref{e2.16} and \eqref{e2.17} we have
\begin{equation}
(\varphi_p(x_0'(t)))'-(\varphi_p(v'(t)))'=-f(t,x_0(t), x_0'(t))\leqslant 0, \quad
\text{a.e. }\ t\in (t_1,t_2).\label{e2.20}
\end{equation}
By \eqref{e2.19} and \eqref{e2.20}, we have
\begin{equation}
\int^{t_2}_{t_1}[(\varphi_p(x_0'(t)))'-(\varphi_p(v'(t)))']\omega(t)dt>0.\label{e2.21}
\end{equation}
On the other hand, by \eqref{e2.19} and the inequality
\begin{equation}
(\varphi_p(b)-\varphi_p(a))(b-a)\geqslant 0, \forall b, a\in \mathbb{R}^1, \label{e2.22}
\end{equation}
we have
$$
\int^{t_2}_{t_1}[(\varphi_p(x_0'(t)))'-(\varphi_p(v'(t)))']\omega(t)dt
=-\int^{t_2}_{t_1}[(\varphi_p(x_0'(t)))-(\varphi_p(v'(t)))]\omega'(t)dt\leqslant 0,
$$
which contradicts to \eqref{e2.21}. This implies that \eqref{e2.18} holds.
Obviously, we have
$$
v(t)=\frac{\|x_0\|_0}{1-t_0} (1-t),\quad  t\in [t_0,1].
$$
Thus, we have
\begin{equation}
x_0(t)\geqslant \frac{\|x_0\|_0}{1-t_0} (1-t)\geqslant \|x_0\|_0t(1-t), \quad
 t\in [ t_0, 1].\label{e2.23}
\end{equation}
Similarly, we can show that
\begin{equation}
x_0(t)\geqslant \|x_0\|_0 e(t), \quad \forall t\in [0, t_0].\label{e2.24}
\end{equation}
By \eqref{e2.23} and \eqref{e2.24} we have $x_0\geqslant \|x_0\|_0 e$.
Thus,  we have
$x\geqslant \frac{\|x_0\|_0}{2}e$ for any $x\in B(x_0,
\frac{\|x_0\|}{4})$. Obviously, $\Big\{B(x,
\frac{\|x\|}{4})|x\in S_{R_1}^+\Big\}$ is an open cover of the
set $S_{R_1}^+$. Since $T_p(S_{R_1}^+)=S_{R_1}^+$ and
$T_p:C_0^1[0,1]\to C_0^1[0,1]$ is completely continuous, then
$S_{R_1}^+$ is a compact set. Therefore, there exist finite subsets
of $\big\{B(x, \frac{\|x\|}{4}):x\in S_{R_1}^+\big\}$, assume
without loss of generality that
$$
B\big(x_1, \frac{\|x_1\|}{4}\big), B\big(x_2, \frac{\|x_2\|}{4}\big),
\dots, B\big(x_n, \frac{\|x_n\|}{4}\big)
$$
such that
$$
\cup_{i=1}^n B\big(x_i, \frac{\|x_i\|}{4}\big)\supset S_{R_1}^+.
$$
Let
$$\varepsilon^+=\min\big\{\frac{\|x_1\|_0}{2}, \frac{\|x_2\|_0}{2}, \dots,
\frac{\|x_n\|_0}{2}\big\}>0.
$$
Then we have $S_{R_1}^+\geqslant \varepsilon^+ e$. Similarly, we can prove that
there exists $\varepsilon^->0$ such that
$S_{R_1}^-\leqslant-\varepsilon^-e$. Let $\zeta_{R_1}=\min\{\varepsilon^+, \varepsilon^-\}$. Then
the conclusion holds. The proof is complete.
\end{proof}


\section{Main Results}

\begin{theorem} \label{thm3.1}
 Suppose that {\rm (H1)} and {\rm (H2)} hold,
$\beta_0\in (\mu_{2k_0}(p), \mu_{2k_0+1}(p))$ for some positive
integer $k_0$. Then \eqref{e1.1} has at least one sign-changing  solution.
Moreover, \eqref{e1.1} has at least one positive solution and one negative
solution.
\end{theorem}

\begin{proof}
 By (H1), Lemma \ref{lem2.6} and Lemma \ref{lem2.7}, there exists
$R_1>0$ such that $T_p(\bar B(\theta, R_1))\subset B(\theta, R_1)$, and so
\begin{equation}
\deg (I-T_p, B(\theta, R_1), \theta)=1.\label{e3.1}
\end{equation}
Let $\{\mu_{k}(p)|k\in N^+\}$ be the sequence of eigenvalues of the
problem \eqref{Elap} and $\phi_k$  the eigenfunction of \eqref{Elap}
corresponding to the eigenvalue $\mu_{k}(p)$.
From  (iii) of Lemma \ref{lem2.3}, $\phi_1$ is a non-negative function on $[0,1]$.
Take $\varepsilon_0>0$ small enough such that $\beta_0-\varepsilon_0>\mu_1(p)$.
By (H2), there exists $\delta_1>0$ such that
\begin{gather}
f(t, x, y)\geqslant (\beta_0-\varepsilon_0)\varphi_p (x), \quad t\in [0,1],\;
  |x|\leqslant \delta_1,\;  |y|\leqslant R_*,\;  x\geqslant 0,\label{e3.2}
\\
f(t, x, y)\leqslant (\beta_0-\varepsilon_0)\varphi_p (x), \quad
t\in [0,1],\; |x|\leqslant \delta_1,\; |y|\leqslant R_*,\; x\leqslant 0.
\label{e3.3}
\end{gather}
Assume that $R_*<R_1$, where $R_*$ as in (H2). Take $\delta_2>0$ small
enough such that $\|\delta\phi_1\|<\min\{\delta_1, R_*, R_1\}$ for each
$\delta\in (0, \delta_2]$. Then from \eqref{e3.2} we have for any
 $\delta\in (0,\delta_2)$,
\begin{equation}
\begin{aligned}
(\varphi_p(\delta\phi_1'))'+f(t, \delta\phi_1, \delta\phi_1')
&\geqslant (\varphi_p(\delta\phi_1'))'+(\beta_0-\varepsilon_0)\varphi_p(\delta\phi_1)\\
&=\delta^{p-1}[(\varphi_p(\phi_1'))'+(\beta_0-\varepsilon_0)\varphi_p(\phi_1)]\\
&=\delta^{p-1}(\beta_0-\varepsilon_0-\mu_1(p))\varphi_p(\phi_1)>0, \ \text{a.e.}\
t\in (0,1).
\end{aligned}\label{e3.4}
\end{equation}
and
\begin{equation}
\delta \phi_1(0)=\delta\phi_1(1)=0.\label{e3.5}
\end{equation}
 From \eqref{e3.4} and \eqref{e3.5}, we see that $\delta\phi_1$ is a lower
solution of \eqref{e1.1}. Similarly, by \eqref{e3.3} we can easily
see that $-\delta\phi_1$ is an upper solution of \eqref{e1.1} for each
$\delta\in (0, \delta_2)$. Let
$$
S_{R_1}=\{x\in C_0^1[0,1]|x \text{ is a solution of \eqref{e1.1} and }
\|x\|<R_1\},
$$
$S_{R_1}^+=\{x\in S_{R_1}|x>\theta\}$ and
$S_{R_1}^-=\{x\in S_{R_1}|x<\theta\}$. By Lemma \ref{lem2.8}, there exists
$\zeta_{R_1}>0$ such that
\begin{equation}
S_{R_1}^+\geqslant \zeta_{R_1}e,\quad
S_{R_1}^-\leqslant -\zeta_{R_1} e.\label{e3.6}
\end{equation}
Since $\phi_1\in C_0^1[0,1]$ satisfies
\begin{equation}
\begin{gathered}
(\varphi_p(\phi_1'))'+\mu_1(p)\varphi_p(\phi_1)=0 \quad \text{a.e. } t\in (0,1),\\
\phi_1(0)=\phi_1(1)=0,
\end{gathered}\label{e3.7}
\end{equation}
 by Rolle's Theorem, there exists $t^*\in (0,1)$ such that
$\phi_1'(t^*)=0$ and
\begin{equation}
\begin{aligned}
\phi_1(t)
&=\int^1_t\varphi_p^{-1}\Big(\int^s_{t^*}\mu_1(p)\varphi_p(\phi_1(\tau))d\tau\Big)ds\\
&\leqslant (1-t)\varphi_p^{-1}\Big(\mu_1(p)\int^1_0\varphi_p(\phi_1(\tau))d\tau\Big)\\
&\leqslant\frac{1}{t^*}e(t)\varphi_p^{-1}
 \Big(\mu_1(p)\int^1_0\varphi_p(\phi_1(\tau))d\tau\Big),
\forall t\in (t^*,1).
\end{aligned}\label{e3.8}
\end{equation}
Similarly, we can show that
\begin{equation}
\phi_1(t)\leqslant\frac{1}{1-t^*}e(t)\varphi_p^{-1}\Big(\mu_1(p)
\int^1_0\varphi_p(\phi_1(\tau))d\tau\Big),
\quad \forall t\in (0, t^*).\label{e3.9}
\end{equation}
By \eqref{e3.8} and \eqref{e3.9} we have
\begin{equation}
\phi_1(t)\leqslant \frac{1}{t^*(1-t^*)}e(t)\varphi_p^{-1}
\Big(\mu_1(p)\int^1_0\varphi_p(\phi_1(\tau))d\tau\Big),
\quad \forall t\in [0, 1].\label{e3.10}
\end{equation}
Take
$$
0<\delta_3<\min\Big\{\delta_2, t^*(1-t^*)\Big[\varphi_p^{-1}\big(\mu_1(p)
\int^1_0\varphi_p(\phi_1(\tau))d\tau\big)\Big]^{-1}\zeta_{R_1}\Big\}.
$$
Let $u_0=\delta_3\phi_1$ and $v_0=-\delta_3\phi_1$. Then by \eqref{e3.6} and
\eqref{e3.10}, we see that $u_0, v_0\in \bar B(\theta, R_1)$, $u_0$ and
$v_0$ are strict lower and upper solutions of \eqref{e1.1} in
 $\bar B(\theta, R_1)$, respectively.
Moreover, we have $S_{R_1}^+\succ u_0$ and $S_{R_1}^-\prec v_0$. Let
$\Omega_1=\{x\in \bar B(\theta, R_1)|x\succ u_0\}$ and
$\Omega_2=\{x\in \bar B(\theta, R_1)|x\prec v_0\}$.
By Lemmas \ref{lem2.6} and \ref{lem2.7} we have
\begin{gather}
\deg (I-T_p, \Omega_1, \theta)=1,\label{e3.11} \\
\deg (I-T_p,\Omega_2,\theta)=1.\label{e3.12}
\end{gather}
Let $h(t,x,y)=f(t,x,y)-\beta_0\varphi_p(x)$ for all
$(t,x,y)\in [0,1]\times \mathbb{R}^2$. By (H2) we have
\begin{equation}
\lim_{x\to 0}\frac{h(t, x, y)}{\varphi_p(x)}=0\quad
\text{uniformly for $t\in [0,1]$ and $ y\in[-R_*,R_*]$}.\label{e3.13}
\end{equation}
For each $\tau\in [0,1]$, denote by
 $H(\tau, \cdot):C_0^1[0,1]\to C_0^1[0,1]$ the solution operator of
\begin{equation}
 \begin{gathered}
 -(\varphi_p(y'(t)))'=\tau\beta_0\varphi_p(x(t))+(1-\tau)f(t, x(t), x'(t)) \quad
\text{a.e. } t\in (0,1)\\
 y(0)=y(1)=0;
 \end{gathered} \label{e3.14}
\end{equation}
 that is,  for $x, y\in C_0^1[0,1]$,
 $$
y=H(\tau, x)
$$
if and only if the equality in \eqref{e3.14} holds.
Then $H(\cdot,\cdot):C_0^1[0,1]\to C_0^1[0,1]$ is completely continuous.
Now we will show that  there exists $0<r_0<\min\{\|u_0\|_0, \|v_0\|_0\}$
such that
\begin{equation}
H(s, x)\neq  x,\quad  s\in [0,1], \; x\in \partial B(\theta,r_0).\label{e3.15}
\end{equation}
Assume that \eqref{e3.15} does not holds, then there exists
$\{\tau_n\}\subset [0,1]$, $\{x_n\}\subset C_0^1[0,1]$ with
$\|x_n\|>0$ for each $n=1,2,\dots$ and $\|x_n\|\to 0$ as
$n\to  \infty$ such that $H(\tau_n, x_n)=x_n$. Obviously,
$\|x_n\|_0>0$ for each $n=1,2,\dots$.  Assume without loss of
 generality that $\tau_n\to \tau_0$ as $n\to \infty$. Then we have
 for each $n=1,2,\dots$
\begin{gather}
\begin{aligned}
-(\varphi_p(x_n'(t)))'&=\tau_n\beta_0\varphi_p(x_n(t))+(1-\tau_n)f(t, x_n(t),
x_n'(t))\\
&=\beta_0 \varphi_p(x_n(t))+(1-\tau_n)h(t, x_n(t), x_n'(t)) \quad
\text{a.e. }t\in (0,1)
\end{aligned}\label{e3.16}
\\
x_n(0)=x_n(1)=0. \label{e3.17}
\end{gather}
Let $v_n(t)=\frac{x_n(t)}{\varphi_p(\|x_n\|_0)}$. Then by \eqref{e3.16} and
\eqref{e3.17} we have
\begin{equation}
 \begin{gathered}
-(\varphi_p(v'_n(t)))'=\beta_0(\varphi_p(v_n(t)))+(1-\tau_n)\frac{h(t,
x_n(t), x'_n(t))}{\varphi_p(\|x_n\|_0)}\quad \text{a.e.}\ t\in (0,1),\\
v_n(0)=v_n(1)=0.
 \end{gathered} \label{e3.18}
\end{equation}
 Let
$$
u_n(t)=\beta_0\varphi_p(v_n(t))+(1-\tau_n)\frac{h(t, x_n(t),
x_n'(t))}{\varphi_p(\|x_n\|_0)},\quad  t\in [0,1].
$$
By  \eqref{e3.13} and (H2)
we see that $\{u_n|n=1,2,\dots\}\subset L^1[0,1]$. By  \eqref{e3.18} and
 Rolle's Theorem, there exists $t_n\in (0,1)$ such that
$v_n'(t_n)=0$ for each $n=1,2,\dots$. Then we have by \eqref{e3.18}
$$
|v_n'(t)|=\Big|\varphi_p^{-1}(\int^{t_n}_t u_n(s)ds)\Big|\leqslant
\varphi_p^{-1}(\int^1_0|u_n(s)|ds),\quad  t\in [0,1].
$$
Thus,
$\{v_n'(t)|n=1,2,\dots\}$ is a bounded set. Consequently,
$\{v_n:n=1,2,\dots\}$ is a relatively compact set of $C[0,1]$.
Assume without loss of generality that
$v_n \to \bar v_0$ in $C[0,1]$  as $n\to \infty$.
From \eqref{e3.18} we have
\begin{equation}
v_n(t)=\int^t_0\varphi_p^{-1}\Big(\alpha(u_n)+\int^1_s u_n(\tau)d\tau\Big)ds,
\quad  t\in [0,1].\label{e3.19}
\end{equation}
where the continuous functional $\alpha(u_n)\in (0,1)$ satisfies
$$
\int^1_0\varphi_p^{-1}\Big(\alpha(u_n)+\int^1_s u_n(\tau)d\tau\Big)ds=0, \quad
n=1,2,\dots.
$$
Assume without loss of generality that $\alpha(u_n)\to a_0$ as $n\to \infty$.
Letting $n\to \infty$ in \eqref{e3.19}, by Lebesgue
dominated convergence theorem  we have
$$
\bar v_0(t)=\int^t_0\varphi_p^{-1}\Big(a_0+\int^1_s\beta_0\varphi_p(\bar
v_0(\tau))d\tau\Big)ds, \quad t\in [0,1].
$$
Consequently, $\bar v_0\in C^1[0,1]$. By direct computation we have
\begin{equation}
-(\varphi_p(\bar v_0'(t)))'=\beta_0\varphi_p (\bar v_0(t))\quad \text{a.e. }
 t\in (0,1).\label{e3.20}
\end{equation}
Obviously,
\begin{equation}
\bar v_0(0)=\bar v_0 (1)=0.\label{e3.21}
\end{equation}
By \eqref{e3.20} and \eqref{e3.21} we see that $ \beta_0$ is an eigenvalue of
\eqref{Elap} and $\bar v_0$ is the corresponding eigenfunction, which
is a contradiction. Therefore, there exists $r_0>0$ small enough
such that \eqref{e3.15} holds. Assume without loss of generality that $u_0,
v_0\not\in \bar B(\theta, r_0)$. By the properties of the
Leray-Schauder degree and Lemma \ref{lem2.5} we have
\begin{equation}
\begin{aligned}
\deg  (I-T_p, B(\theta, r_0), \theta)
&=\deg (I-H(0,\cdot), B(\theta, r_0), \theta)\\
&=\deg (I-H(1,\cdot), B(\theta, r_0), \theta)\\
&=\deg (I-T_{\beta_0}^p, B(\theta, r_0), \theta)\\
&=(-1)^{2k_0}=1.
\end{aligned}\label{e3.22}
\end{equation}
By \eqref{e3.1}, \eqref{e3.11}, \eqref{e3.12} and \eqref{e3.22}, we have
\begin{equation}
\deg  (T_p, \bar B(\theta, R_1)\backslash(\bar B(\theta, r_0)\cup
Cl_{\bar B(\theta, R_1)} \Omega_1\cup Cl_{\bar B(\theta, R_1)} \Omega_2),
\theta)=-1.\label{e3.23}
\end{equation}
It follows from \eqref{e3.11}, \eqref{e3.12} and \eqref{e3.23} that
$T_p$ has at least three  fixed points $x_1\in \Omega_1$,
$x_2\in \Omega_2$ and
$x_3\in \bar B(\theta, R_1)\backslash(\bar B(\theta, r_0)\cup
Cl_{\bar B(\theta, R_1)} \Omega_1\cup Cl_{\bar B(\theta, R_1)} \Omega_2)$.
Obviously $x_1$ is a positive solution of \eqref{e1.1}, $x_2$ is a negative
solution of \eqref{e1.1}. Since $S^+_{R_1}\succ u_0$ and
$S_{R_1}^-\prec v_0$, then $S^+_{R_1}\subset \Omega_1$
and $S_{R_1}^-\subset \Omega_2$.
Therefore, $x_3$ is a sign-changing solution of \eqref{e1.1}. The proof is
complete.
\end{proof}

Now we will give some multiplicity results for sign-changing
solutions of \eqref{e1.1}.

\begin{theorem} \label{thm3.2}
 Suppose that {\rm (H1)--(H3)} hold,
$\beta_0>\mu_1(p)$, $\beta_0\neq \mu_k(p)$ for each $k=1,2,\dots$.
Moreover, there exists $\bar\delta_0>0$ such that both
$\{t\in [0,1]|\delta\phi_1(t)=u_1(t)\}$ and
$\{t\in [0,1]|-\delta\phi_1(t)=v_1(t)\}$ contain at most finite elements for
each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least four
sign-changing solutions. Moreover, \eqref{e1.1} has at least one positive
solution and one negative solution.
\end{theorem}

\begin{proof}
From (H1), there exists $R_1>0$ such that
 $T_p(\bar B(\theta, R_1))\subset B(\theta, R_1)$ and so \eqref{e3.1} holds. Since
$\beta_0>\mu_1(p)$ and $\beta_0\neq \mu_k(p)$ for each
$k=1,2,\dots$, in the same way as the proof of Theorem \ref{thm3.1}, we see
that there exists $0<\delta_2<\bar\delta_0$ such that for any
$\delta\in (0,\delta_2)$, $\delta\phi_1$ is a lower solution of \eqref{e1.1} and
$-\delta\phi_1$ is an upper solution of \eqref{e1.1}. Let $S^+_{R_1}$ and
$S_{R_1}^-$ be defined as Theorem \ref{thm3.1}. Then by Lemma \ref{lem2.8}, there
exists $\zeta_{R_1}>0$ such that \eqref{e3.6} holds. In the same way as the
proof of Theorem \ref{thm3.1}, we can take $\delta_3>0$ small enough such that
$u_0\in \bar B(\theta, R_1)$ and
$v_0\in \bar B(\theta, R_1)$, where $u_0:=\delta_3\phi_1$ and
$v_0:=-\delta_3\phi_1$.  Moreover,
$u_0$ and $v_0$ are strict lower and upper solutions of \eqref{e1.1},
respectively,  and $S_{R_1}^+\succ u_0$, $S_{R_1}^-\prec v_0$. Also,
assume $\delta_3>0$ small enough such that $u_0\not\geqslant u_1$ and
$v_0\not\leqslant v_1$. Define the subsets $\Omega_1, \Omega_2, \Omega_3$ and
$\Omega_4$ of $C_0^1[0,1]$ by
\begin{gather*}
\Omega_1=\{x\in  B(\theta, R_1):x\succ u_0\},\quad
\Omega_2=\{x\in  B(\theta, R_1):x\prec v_0\},\\
\Omega_3=\{x\in  B(\theta, R_1):x\prec v_1\},\quad
\Omega_4=\{x\in  B(\theta, R_1):x\succ u_1\}.
\end{gather*}
Then $\Omega_1, \Omega_2,\Omega_3,\Omega_4$ are four  closed convex subsets of
$C_0^1[0,1]$. Let
$$
O_{2,3}=\Omega_2\cap \Omega_3,\quad
\Omega_{3,4}=\Omega_3\cap\Omega_4,\quad
 O_{4,1}=\Omega_4\cap \Omega_1.
$$
By Lemmas \ref{lem2.6} and \ref{lem2.7} we have
\begin{gather}
\deg (I-T_p, \Omega_i, \theta)=1, \quad i=1,2,3,4,\label{e3.24}\\
\deg (I-T_p, O_{2,3}, \theta)=1,\label{e3.25} \\
\deg (I-T_p, O_{3,4}, \theta)=1,\label{e3.26} \\
\deg (I-T_p, O_{4,1}, \theta)=1.\label{e3.27}
\end{gather}
Since $\beta_0>\mu_1(p), \beta_0\neq \mu_k(p), k=1,2, \dots$, then
by a similar way as that of the proof of Theorem \ref{thm3.1} we see that,
there exists $r_0>0$ small enough such that
$B(\theta, r_0)\cap \Omega_i=\emptyset( i=1,2,3,4)$ and
\begin{equation}
\deg (I-T_p, B(\theta, r_0), \theta)=(-1)^{k_0}=\pm 1,\label{e3.28}
\end{equation}
where $k_0$ is the sum of all algebraic multiplicities
of all eigenvalues $\mu_k(p)$ of (E$_\lambda^p$) with
$\beta_0>\mu_k(p)$. Let
\begin{gather*}
O_1=\Omega_3\backslash(Cl_{\bar B(\theta, R_1)}O_{2,3}\cup Cl_{\bar B(\theta,
R_1)}O_{3,4}),\\
O_2=\Omega_4\backslash(Cl_{\bar B(\theta, R_1)}O_{3,4}\cup Cl_{\bar B(\theta,
R_1)}O_{4,1}).
\end{gather*}
Then, by \eqref{e3.24}-\eqref{e3.27} we have
\begin{gather}
\deg (I-T_p, O_1, \theta)=1-1-1=-1,\label{e3.29} \\
\deg (I-T_p, O_2, \theta)=1-1-1=-1.\label{e3.30}
\end{gather}
It follows from \eqref{e3.1}, \eqref{e3.24}, \eqref{e3.28}-\eqref{e3.30} that
\begin{equation}
\begin{aligned}
\deg \Big(&I-T_p, \bar B(\theta, R_1)\backslash
(Cl_{\bar B(\theta, R_1)}\Omega_1\cup Cl_{\bar B(\theta, R_1)}O_1\cup Cl_{\bar B(\theta,
R_1)}\Omega_{3,4}\cup Cl_{\bar B(\theta, R_1)}O_2\\
&\cup Cl_{\bar B(\theta, R_1)}\Omega_2\cup \bar B(\theta, r_0)),
\theta\Big)=1-1-(-1)-1-(-1)-1-(\pm 1)=\mp 1.
\end{aligned}\label{e3.31}
\end{equation}
It follows from \eqref{e3.24}, \eqref{e3.26}, \eqref{e3.29}, \eqref{e3.30}
and \eqref{e3.31} that $T_p$ has fixed points $x_1\in \Omega_1$,
$x_2\in \Omega_2$, $x_3\in O_1$
$x_4\in O_2$, $x_5\in \Omega_{3,4}$ and $ x_6\in \bar B(\theta, R_1)\backslash
(Cl_{\bar B(\theta, R_1)}\Omega_1\cup Cl_{\bar B(\theta, R_1)}O_1\cup
Cl_{\bar B(\theta, R_1)}\Omega_{3,4}\cup Cl_{\bar B(\theta, R_1)}O_2 \cup
Cl_{\bar B(\theta, R_1)}\Omega_2\cup \bar B(\theta, r_0))$. It is easy to see
that $x_1$ is a positive solution of \eqref{e1.1}, $x_2$ is a negative
solution of \eqref{e1.1}, $x_3, x_4, x_5, x_6$ are four sign-changing
solutions of \eqref{e1.1}. The proof is complete.
\end{proof}

\begin{remark} \label{rmk3.1}\rm
To show multiplicity results for sign-changing solutions of  \eqref{e1.1}
in Theorem \ref{thm3.2}  we constructed a pair of lower and upper solutions
$u_0$ and $v_0$ which satisfy $u_0\not\leqslant v_0$.
We call this pair of lower and upper solutions is non-well
ordered. For other discussions concerning the non-well ordered upper
and lower solutions, the reader is refereed to \cite[5.4B]{d2}.
\end{remark}

\begin{remark} \label{rmk3.2}\rm
In Theorem \ref{thm3.2} we  obtained not only multiplicity
results for sign-changing solutions of \eqref{e1.1} but also the existence
results for positive solutions as well as negative solution of
\eqref{e1.1}.
\end{remark}

\begin{theorem} \label{thm3.3}
 Suppose that {\rm (H1)--(H3)} hold,
$\beta_0<\mu_1(p)$. Moreover, there exists $\bar\delta_0>0$ such that
both $\{t\in [0,1]|\delta\phi_1(t)=v_1(t)\}$ and
$\{t\in [0,1]|-\delta\phi_1(t)=u_1(t)\}$ contain at most finite elements for
each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least four
sign-changing solutions. Moreover, \eqref{e1.1} has at least two positive
solutions and two negative solutions.
\end{theorem}

\begin{proof}
By (H1), there exists $R_1>0$ such that
$T_p(\bar B(\theta, R_1))\subset B(\theta, R_1)$ and so \eqref{e3.1} holds.
Let $S_{R_1}^+$ and $S_{R_1}^-$ be defined as Theorem \ref{thm3.1}.
By Lemma \ref{lem2.8}, there exists $\zeta_{R_1}>0$ such that \eqref{e3.6} holds.
Since $\beta_0<\mu_1(p)$, in the same way as that of Theorem \ref{thm3.1}
 we can show that, there exists $\bar\delta_0>\delta_2>0$ such that for any
$\delta\in (0, \delta_2)$, $-\delta\phi_1$ is a lower solution of \eqref{e1.1} and
$\delta\phi_1$ is an upper solution of \eqref{e1.1}.  Also by a similar
argument as the proof of \eqref{e3.15}  we can show that, there exists
$r_0>0$ small enough such that $\theta$ is the unique fixed point of
$T_p$ in $\bar B(\theta, r_0)$, and for any $0<r\leqslant r_0$,
\begin{equation}
\deg (I-T_p, B(\theta, r), \theta)=1.\label{e3.32}
\end{equation}
Let
\begin{gather*}
S_i=\{ x\in C_0^1[0,1]:x(t) \text{ has exactly $i-1$ simple zeros
on }  (0,1)\},
\\
S_i^+=\{x\in S_i:\lim_{t\to 0^+}\text{sign} x(t)=1\}, \quad
S_i^-=S_i\backslash S_i^+,\quad  i=1,2,\dots.
\end{gather*}
Then  we have $S_{R_1}^+=S_1^+\cap \bar B(\theta, R_1)$,
$S_{R_1}^-=S_1^-\cap \bar B(\theta, R_1)$ and
$S_{R_1}\subset \big(\cup_{i=1}^\infty S_i\big)\cap \bar B(\theta, R_1)$.
 Moreover, for each
$i=1,2,\dots$, $S_i$ is an open subset of $C_0^1[0,1]$. We say that
there exists $\delta_3\in (0,\delta_2)$ small enough such that
\begin{gather}
\{x\in C_0^1[0,1]:-\delta_3\phi_1\leqslant x, \|x\|\leqslant R_1\}\cap
(S_{R_1}\backslash\{\theta\})\cap \Big((\cup_{i=2}^\infty
S_i)\cup S^-_{R_1}\Big)=\emptyset, \label{e3.33}
\\
\{x\in C_0^1[0,1]:x\leqslant\delta_3\phi_1, \|x\|\leqslant R_1\}\cap
(S_{R_1}\backslash\{\theta\})\cap \Big((\cup_{i=2}^\infty
S_i)\cup S^+_{R_1}\Big)=\emptyset. \label{e3.34}
\end{gather}
We  prove only \eqref{e3.33}. In a
similar way we can prove \eqref{e3.34}.
If \eqref{e3.33} does not hold, then there
exists a sequence of positive numbers $\{\bar\delta_n\}$ with
$\bar\delta_n\to 0$ as $n\to \infty$ such that for each $n=1,2,\dots$,
$$
\{x\in C_0^1[0,1]:-\delta_n\phi_1\leqslant x, \|x\|\leqslant R_1\}\cap
(S_{R_1}\backslash\{\theta\})\cap \Big((\cup_{i=2}^\infty
S_i)\cup S^-_{R_1}\Big)\neq \emptyset.
$$
For each $n=1,2,\dots$, take
$$
x_n\in \{x\in C_0^1[0,1]:-\delta_n\phi_1\leqslant x, \|x\|\leqslant R_1\}\cap
(S_{R_1}\backslash\{\theta\})\cap \Big((\cup_{i=2}^\infty
S_i)\cup S^-_{R_1}\Big).
$$
Obviously, $\|x_n\|\geqslant r_0$ for
each $ n=1,2,\dots$. Let $D=\{x_n|n=1,2,\dots\}$. Then we have
$D=T_p(D)$. Therefore, $D$ is a relatively compact subset of
$C_0^1[0,1]$. Assume without loss of generality that $x_n\to x_0$ as
$n\to \infty$ for some $x_0\in C_0^1[0,1]$. Obviously, $x_0$ is a
solution of \eqref{e1.1} and $\|x_0\|\geqslant r_0$,  and thus
$x_0\in \big(\cup_{i=1}^\infty S_i\big)\cap \bar B(\theta, R_1)$.
Note that $-\bar \delta_n\phi_1\leqslant x_n$, letting $n\to\infty$ then we
have $x_0\in S_1^+\cap \bar B(\theta, R_1)$. Since $S_1^+$ is an open
subset of $C_0^1[0,1]$, then there exists $r_1>0$ such that $B(x_0,
r_1)\subset S_1^+$. Now since $x_n\to x_0$ as $n\to +\infty$, then
we can take $n_0$ large enough such that
$x_{n_0}\in B(x_0, r_1)\subset S_1^+$,  which contradicts to
$$
x_{n}\in (\cup_{i=2}^\infty S_i)\cup S^-_{R_1}
$$
for each
$n=1,2,\dots$. Therefore, \eqref{e3.33} and \eqref{e3.34} hold. Take
$0<\delta_4<\delta_3$. Then $-\delta_4\phi_1$ is a strict lower solution of
\eqref{e1.1} and $\delta_4\phi_1$ is a strict upper solution of \eqref{e1.1}.  Also,
assume that $\delta_4>0$ small enough such that
$-\delta_4 \phi_1\not\leqslant v_1$, $\delta_4\phi_1\not\geqslant u_1$ and
$-\delta_4\phi_1, \delta_4\phi_1\in \bar B(\theta, R_1)$. Let
$u_0=-\delta_4\phi_1$ and $v_0=\delta_4\phi_1$. Let the subsets $\Omega_1,
\Omega_2, \Omega_3,\Omega_4$ of $C_0^1[0,1]$ be defined by
\begin{gather*}
\Omega_1=\{x\in \bar B(\theta, R_1):x\succ u_0\},\quad
\Omega_2=\{x\in \bar B(\theta, R_1):x\prec v_0\},\\
\Omega_3=\{x\in \bar B(\theta, R_1):x\prec v_1\},\quad
\Omega_4=\{x\in \bar B(\theta, R_1):x\succ u_1\}.
\end{gather*}
Let $O_{1,2}=\Omega_1\cap \Omega_2$, $O_{2,3}=\Omega_2\cap\Omega_3$,
$O_{3,4}=\Omega_3\cap\Omega_4$ and $O_{4,1}=\Omega_4\cap\Omega_1$. Then $\Omega_1,
\Omega_2, \Omega_3, \Omega_4$ and $O_{1,2}, O_{2,3}, O_{3,4}, O_{4,1}$ are
nonempty open subsets  of $\bar B(\theta, R_1)$.
It follows from Lemmas \ref{lem2.6} and \ref{lem2.7} that
\begin{gather}
\deg (I-T_p, \Omega_1, \theta)=1, \label{e3.35}\\
\deg (I-T_p, \Omega_2, \theta)=1, \label{e3.36}\\
\deg (I-T_p, \Omega_3, \theta)=1, \label{e3.37}\\
\deg (I-T_p, \Omega_4, \theta)=1, \label{e3.38}\\
\deg (I-T_p, O_{1,2}, \theta)=1, \label{e3.39}\\
\deg (I-T_p, O_{2,3}, \theta)=1, \label{e3.40}\\
\deg (I-T_p, O_{3,4}, \theta)=1, \label{e3.41}\\
\deg (I-T_p, O_{4,1}, \theta)=1. \label{e3.42}
\end{gather}
Let
\begin{gather*}
O_1=\Omega_1\backslash(Cl_{\bar B(\theta, R_1)}O_{1,2}\cup Cl_{\bar B(\theta,
 R_1)}O_{4,1}), \\
O_2=\Omega_2\backslash(Cl_{\bar B(\theta, R_1)}O_{1,2}\cup
Cl_{\bar B(\theta, R_1)}O_{2,3}),
\\
O_3=\Omega_3\backslash(Cl_{\bar B(\theta, R_1)}O_{2,3}\cup Cl_{\bar B(\theta,
R_1)}O_{3,4}), \\
O_4=\Omega_4\backslash(Cl_{\bar B(\theta, R_1)}O_{3,4}\cup
Cl_{\bar B(\theta, R_1)}O_{4,1}).
\end{gather*}
Then by \eqref{e3.35}-\eqref{e3.42} we have
\begin{gather}
\deg (I-T_p, O_1, \theta)=-1,\label{e3.43}\\
\deg (I-T_p, O_2, \theta)=-1,\label{e3.44}\\
\deg (I-T_p, O_3, \theta)=-1,\label{e3.45}\\
\deg (I-T_p, O_4, \theta)=-1.\label{e3.46}
\end{gather}
It follows from \eqref{e3.36}, \eqref{e3.38}, \eqref{e3.43},\eqref{e3.45} that
\begin{equation}
\begin{aligned}
&\deg \Big(I-T_p, \bar B(\theta, R_1)\backslash 
\Big(Cl_{\bar B(\theta, R_1)}O_1\cup Cl_{\bar B(\theta, R_1)}O_3\cup Cl_{\bar B(\theta,
R_1)}\Omega_2\\
& \cup Cl_{\bar B(\theta, R_1)}\Omega_4\Big),\theta\Big)\\
&=1-(-1)-(-1)-1-1=1.
\end{aligned}\label{e3.47}
\end{equation}
From \eqref{e3.35}-\eqref{e3.47}, $T_p$ has
fixed points $x_1\in O_{3,4}$, $x_2\in O_4$, $x_3\in O_3$,
$$
x_4\in \bar B(\theta, R_1)\backslash (Cl_{\bar B(\theta, R_1)}O_1\cup Cl_{\bar B(\theta,
R_1)}O_3\cup Cl_{\bar B(\theta, R_1)}\Omega_2\cup Cl_{\bar B(\theta, R_1)}\Omega_4).
$$
Then $x_1, \dots, x_4$ are four sign-changing
solutions of \eqref{e1.1}. From \eqref{e3.42} and \eqref{e3.43},
$T_p$ has fixed points
$x_5\in O_{4,1}$, $x_6\in O_1$. Obviously, $x_5\geqslant u_0$,
$x_6\geqslant u_0$ and $x_5\neq \theta$, $x_6\neq \theta$. Then we
see from \eqref{e3.33} that $x_5$ and $x_6$ are two positive solutions of
\eqref{e1.1}. Similarly we can show that there exist $x_7\in O_{3,4}$ and
$x_8\in O_2$, and $x_7$, $x_8$ are two negative solutions of \eqref{e1.1}.
The proof is complete.
\end{proof}

Now we study the existence and multiplicity of sign-changing
solutions of \eqref{e1.1}  when  $f$ has jumping nonlinearity at zero. Let
us first introduce the following conditions.
\begin{itemize}
\item[(H4)]  There exist $R_*, \beta_+>0$ such
 that
 $$
\lim_{x\to 0^+}\frac{f(t,x,y)}{\varphi_p(x)}=\beta_+\quad
 \text{uniformly for $t\in [0,1]$ and $y\in [-R_*, R_*]$}.
$$

\item[(H5)] There exist $R_*, \beta_->0$ such
 that
 $$
\lim_{x\to 0^-, \ x<0}\frac{f(t,x,y)}{\varphi_p(x)}=\beta_-\quad
 \text{uniformly for $t\in [0,1]$ and $y\in [-R_*, R_*]$}.
$$
\end{itemize}
In the same way as the proof of Theorems \ref{thm3.1}, \ref{thm3.2}
and \ref{thm3.3}, we can prove the
following Theorems \ref{thm3.4}--\ref{thm3.10}.
 For brevity, we only give the sketch of  the proof of  
Theorem \ref{thm3.4}.

\begin{theorem} \label{thm3.4}
 Suppose that {\rm (H1),  (H3), (H4)} hold, and
$\beta_+>\mu_1(p)$. Moreover, there exists $\bar\delta_0>0$ such that
 $\{t\in [0,1]|\delta\phi_1(t)=u_1(t)\}$ contains at most finite
elements for each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least
two sign-changing solutions. Moreover, \eqref{e1.1} has at least one
positive solution.
\end{theorem}

\begin{theorem} \label{thm3.5}
 Suppose that {\rm (H1),  (H3), (H4)} hold,
$\beta_+<\mu_1(p)$. Moreover, there exists $\bar\delta_0>0$ such that
 $\{t\in [0,1]|\delta\phi_1(t)=v_1(t)\}$  contains at most finite
elements for each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least
two sign-changing solutions. Moreover, \eqref{e1.1} has at least one
negative solution.
\end{theorem}

\begin{theorem} \label{thm3.6}
 Suppose that {\rm (H1),  (H3), (H5)} hold,
$\beta_->\mu_1(p)$. Moreover, there exists $\bar\delta_0>0$ such that
 $\{t\in [0,1]|-\delta\phi_1(t)=v_1(t)\}$  contains at most finite
elements for each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least
two sign-changing solutions. Moreover, \eqref{e1.1} has at least one
negative solution.
\end{theorem}

\begin{theorem} \label{thm3.7}
 Suppose that {\rm (H1),  (H3), (H5)} hold,
$\beta_-<\mu_1(p)$. Moreover, there exists $\bar\delta_0>0$ such that
 $\{t\in [0,1]|-\delta\phi_1(t)=u_1(t)\}$ contains at most finite
elements for each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least
two sign-changing solutions. Moreover, \eqref{e1.1} has at least one
positive solution.
\end{theorem}

\begin{theorem} \label{thm3.8}
 Suppose that {\rm (H1),  (H3), (H4), (H5)} hold,
$\beta_->\mu_1(p)$, and $\beta_+>\mu_1(p)$.
 Moreover, there exists $\bar\delta_0>0$ such that both
 $\{t\in [0,1]:-\delta\phi_1(t)=v_1(t)\}$ and
$\{t\in [0,1]:\delta\phi_1(t)=u_1(t)\}$ contain at most finite elements for
each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least three
sign-changing solutions. Moreover, \eqref{e1.1} has at least one positive
solution and one negative solution.
\end{theorem}

\begin{theorem} \label{thm3.9}
Suppose that {\rm (H1),  (H3), (H4), (H5)} hold, $\beta_-<\mu_1(p)$,
$\beta_+<\mu_1(p)$. Moreover,
there exists $\bar\delta_0>0$ such that both
$\{t\in [0,1]|\delta\phi_1(t)=v_1(t)\}$ and
$\{t\in [0,1]|-\delta\phi_1(t)=u_1(t)\}$ contain at most finite elements for
each $\delta\in (0, \bar\delta_0)$. Then \eqref{e1.1} has at least four
sign-changing solutions. Moreover, \eqref{e1.1} has at least two positive
solutions and two negative solutions.
\end{theorem}

\begin{theorem} \label{thm3.10}
 Suppose that {\rm (H3)} holds, $f$ is a
Carath\'{e}odory function. Then \eqref{e1.1} has at least one sign-changing
solution.
\end{theorem}

\begin{proof}[Sketch of  the Proof of  Theorem \ref{thm3.4}]
 By assumption (H1), there exists $R_1>0$ such that
$T_p(\bar B(\theta, R_1))\subset B(\theta, R_1)$. Let $S_{R_1}^+$
 be defined as Theorem \ref{thm3.1}. By Lemma \ref{lem2.8}, there
exists $\zeta_{R_1}>0$ such that $S_{R_1}^+\geqslant \zeta_{R_1} e$. Since
$\beta_+>\mu_1(p)$, there exists $\bar\delta_0>\delta_2>0$ such that for any
$\delta\in (0, \delta_2)$, $\delta\phi_1$ is a lower solution of \eqref{e1.1}.
Take a $\delta_3\in(0,\delta_2)$ small enough such that
$u_0:=\delta_3 \phi_1$ is a strict lower solution of \eqref{e1.1} in
$\bar B(\theta, R_1)$, $S^+_{R_1}\geqslant u_0, u_1\not\leqslant u_0$.
 Let us define the sets $\Omega_1, \Omega_3,\Omega_4, O_{3,4}, O_{4,1}$ and
$O_4$ as in Theorem \ref{thm3.3}.
Then \eqref{e3.38}, \eqref{e3.41}, \eqref{e3.42} and \eqref{e3.46} hold.
Therefore, $T_p$ has fixed points $x_1\in O_{3,4}, x_2\in O_{4}$
and $x_3\in \Omega_1$. Obviously, $x_1$ and $x_2$ are two sign-changing
solutions of \eqref{e1.1}, and $x_3$ is a positive solution of \eqref{e1.1}.
The proof is complete.
\end{proof}

\begin{remark} \label{rmk3.3}\rm
 We should point out, the condition that $f$ is
sub-linear at infinity can be substituted by a pair of well ordered
lower and upper solutions $u_3$ and $v_3$ such that $u_1$ and $v_1$
belongs to the ordered interval $[u_3, v_3]$. However, in those
cases we need a condition of Nagumo type, see \cite{x1,x3}. Also, in
those case we can study the multiplicity of sign-changing solutions
when $f$ both has jumping nonlinearity at zero and infinity.
\end{remark}

\begin{remark} \label{rmk3.4}\rm
 In Theorem \ref{thm3.3} the two pairs of well ordered lower
and upper solutions $u_0$ and $v_0$, $u_1$ and $v_1$ satisfy
\begin{equation}
u_0\not\leqslant v_1, \quad \ u_1\not\leqslant v_0.\label{e3.48}
\end{equation}
We say two pairs of well ordered lower and upper solutions $u_0$ and
$v_0$, $u_1$ and $v_1$ are parallelled to each other when \eqref{e3.48}
holds. The concept of parallelled pairs of well ordered lower and
upper solutions is put forward by Sun Jingxian. For other
discussions concerning parallelled pairs of well ordered lower and
upper solutions, the reader is refereed to \cite{x2}.
\end{remark}

\begin{remark} \label{rmk3.5}\rm
 In Theorems \ref{thm3.2} and \ref{thm3.3}, we employed a pair of
sign-changing strict lower and upper solutions. Generally speaking,
it is difficult to construct a pair of sign-changing strict lower
and upper solutions. However, we can use the method of \cite{x3} to
give an example of this kind strict lower and upper solutions; see
\cite[Example 3.1]{x3}.
\end{remark}


\begin{thebibliography}{00}

\bibitem{c1} Alberto Cabada, Rordrigo L. Pouso;
\emph{Existence results for the problem $(\phi(u'))'=f(t,u,u')$
with nonlinear boundary conditions}, Nonlinear Analysis, 35(1999), 221-231.

\bibitem{d1} Pavel Dr\'{a}bek, Marta Garc\'{\i}a-Huidobro, Raul
Man\'{a}sevich;
\emph{Positive solutions for a class of equations with a
$p$-Laplace like operator and weights},  Nonlinear Analysis,
71(2009), 1281-1300.

\bibitem{d2} Pavel Dr\'{a}bek, Jaroslav Milota;
\emph{Methods of Nonlinear Analysis Applications to  Differential Equations}, 
Birkh\"auser, Basel - Boston - Berlin, 2007.

\bibitem{j1} Tadeusz Jankowski;
\emph{Positive solutions for three-point one-dimensional p-Laplacian
boundary value problems with advanced arguments}, Applied Mathematics
and Computation, 215(2009), no. 1, 125-131.

\bibitem{k1} Chan-Gyun Kim;
\emph{Existence of positive solutions for singular boundary value problems
involving the one-dimensional $p$-Laplacian},  Nonlinear Analysis,
70(2009),  4259-4267.

\bibitem{l1} Yong-Hoon Lee, Inbo Sim;
\emph{Global bifurcation phenomena for singular one-dimensional
p-Laplacian}, J. Differential Equations, 229 (2006) 229-256.

\bibitem{l2} Chong Li, Shujie Li;
\emph{Multiple solutions and sign-changing solutions
of a class of nonlinear elliptic equations with Neumann boundary
condition}, J. Math. Anal. Appl.,  298 (2004) 14-32.

\bibitem{m1} Ahmed Mohammed;
\emph{Positive solutions of the p-Laplace equation with singular
nonlinearity}, J. Math. Anal. Appl.,
 352(2009), 234-245.

\bibitem{m2} R. Man\'{a}sevich, J. Mawhin;
\emph{Boundary value problems for nonlinear perturbations of vector 
$p$-Laplacian-like operators}, J. Korean Math. Soc.,  37 (2000) 665-685.

\bibitem{p1} M. del Pino, M. Elgueta, R. Man\'{a}sevich;
\emph{A homotopic deformation along p of a Leray-Schauder degree result 
and existence for $(|u'|^{p-2}u')'+ f (t,u) = 0$, $u(0) = u(T ) = 0$, $p >1$}, 
J. Differential Equations, 80 (1989), 1-13.

\bibitem{w1} M. X. Wang, A. Cabada, J. J. Nieto;
\emph{Monotone method for nonlinear second order periodic boundary
value problems with Carath\'{e}odory functions},
Ann.Polon. Math. 58(1993),221-235.

\bibitem{x1} Xian Xu, Donal O'Regan, Zhang Ruifang;
\emph{Existence and location results for sign-changing solutions for
three-point boundary value problems using Leray-Schauder degree},
Monatsh Math., 158(2009), no.4, 413-439.

\bibitem{x2} Xian Xu, Jingxian Sun;
\emph{Solutions for an operator equation under the conditions of pairs of
paralleled lower and upper solutions}, Nonlinear Analysis, 69 (2008),
2251-2266.

\bibitem{x3} Xian Xu, Donal O'Regan;
\emph{Multiplicity of sign-changing solutions for some four-point boundary
value problem}, Nonlinear Analysis, 69 (2008), 434-447.

\bibitem{y1} Jingbao Yang, Zhongli Wei, Ke Liu;
\emph{Existence of symmetric positive solutions for a class of
Sturm-Liouville-like boundary value problems}, Applied Mathematics
and Computation, 214(2009), no. 2, 424-432.

\bibitem{z1} Zhitao Zhang, Shujie Li;
\emph{On sign-changing andmultiple solutions of the $p$-Laplacian},
Journal of Functional Analysis, 197 (2003) 447-468.

\bibitem{z2} Zhitao Zhang, Kanishka Perera;
\emph{Sign changing solutions of Kirchhoff
type problems via invariant sets of descent flow}, J. Math. Anal.
Appl., 317 (2006) 456-463.

\bibitem{z3} Zhitao Zhang, Jianqing Chen, Shujie Li;
\emph{Construction of pseudo-gradient vector field and sign-changing
multiple solutions involving $p$-Laplacian}, J. Differential Equations,
201 (2004) 287-303.

\end{thebibliography}

\end{document}