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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 87, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/876\hfil $p$-Laplacian systems]
{Nodal properties for $p$-Laplacian systems}

\author[Y.-H. Cheng, W.-C. Wang \hfil EJDE-2017/87\hfilneg]
{Yan-Hsiou Cheng, Wei-Chuan Wang}

\address{Yan-Hsiou Cheng \newline
 Department of Mathematics and Information Education,
 National Taipei University of Education, Taipei 106, Taiwan}
\email{yhcheng@tea.ntue.edu.tw}

\address{Wei-Chuan Wang \newline
Department of Civil Engineering and Engineering Management,
Center for General Education,
National Quemoy University, Kinmen 892, Taiwan}
\email{wangwc@nqu.edu.tw; wangwc72@gmail.com}

\dedicatory{Communicated by Pavel Drabek}

\thanks{Submitted January 18, 2017. Published March 28, 2017.}
\subjclass[2010]{34B15, 34A12}
\keywords{Quasilinear equation; $p$-Laplacian system; Pr\"{u}fer substitution}

\begin{abstract}
 We consider a system of differential equations involving the
 $p$-Laplacian. We prove the existence of oscillatory solutions
 with prescribed numbers of zeros, and show that the solutions satisfy
 the Dirichlet boundary conditions when the large parameters in the
 equations are suitable chosen. Our main tool in this work is a
 Pr\"{u}fer-type substitution.
\end{abstract}

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

\section{Introduction}

 The $p$-Laplacian operator
$ \Delta_pu=div (|\nabla u|^{p-2}\nabla u)$
attracts lots of attention  and arises in various fields, such as non-Newtonian
fluids and nonlinear diffusion problems.
The quantity $p$ is a characteristic of the medium.
 Media with $p>2$ are called dilatant fluids and those with $p<2$ are
called pseudoplastics. If $p=2$, they are Newtonian fluids.
For the above topics, the readers can refer to
\cite{D85,D01,la,li,PR74,SW91,Takeuchi12} and their bibliographies.

In this note we consider the one-dimensional $p$-Laplacian system
\begin{equation} \label{eq1.1}
\begin{gathered}
-(|u'(x)|^{p-2}u'(x))'=(\lambda-w(x))|u(x)|^{p-2}u(x)-|v(x)|^{p-2}v(x),\\
-(|v'(x)|^{p-2}v'(x))'=\lambda |v(x)|^{p-2}v(x)+|u(x)|^{p-2}u(x),
\end{gathered}
\end{equation}
with the initial conditions
\begin{equation} \label{eq1.2}
u(0)=v(0)=0,\quad u'(0)=\lambda^{1/p}, \quad v'(0)=\mu.
\end{equation}
Obviously, the first equation in \eqref{eq1.1} with $v\equiv 0$ can be regarded as
 a one-dimen\-sional $p$-Laplacian eigenvalue problem. Recently some results
related to $p$-Laplacian systems can be found, for example
\cite{HW07,L12,W13,Y11}. As a result of that, many authors have
studied the existence of positive solutions for $p$-Laplacian
boundary value problems, by using topological degree theory,
monotone iterative techniques, coincidence degree theory
\cite{GM77}, and the Leggett-Williams fixed point theorem
\cite{LW79} or its variants; see \cite{ACLO06,AN07,SL07,WZ06} and the references
therein. Note that as $p=2$,
\eqref{eq1.1} reduces to
\begin{equation} \label{eq1.3}
\begin{gathered}
u''(x)+(\lambda-w(x)) u(x)-v(x)=0,\\
v''(x)+\lambda v(x)+u(x)=0,
\end{gathered}
\end{equation}
which is a linear coupled system. One can treat \eqref{eq1.3} as a steady
state reaction diffusion model. Define
$H(u,v)=\frac{(\lambda-w(x))}{2}u^2-\frac{\lambda}{2}v^2-uv$.
Then
$$
\frac{\partial H}{\partial u}=(\lambda-w(x))u-v,
\quad -\frac{\partial H}{\partial v}=\lambda v+u.
$$
Equation \eqref{eq1.3} can be viewed as
a simple model of diffusion systems with skew-gradient structure
(cf. \cite{Y02,Y021}).

In \cite{W13}, we first considered a simple case related to a
coupled $p$-Laplacian equations and developed the existence of
oscillatory solutions with prescribed numbers of zeros. Motivated
by \cite{W13}, we study \eqref{eq1.1} and intend to extend the previous result.
We will show that the solutions with prescribed numbers of zeros solve
\eqref{eq1.1} and satisfy the Dirichlet boundary
conditions when the large parameters $\lambda$ are suitable
chosen. That is, the Dirichlet boundary value problem is solvable.
Essentially, the main method using in this work is a Pr\"{u}fer-type
substitution. Now for a solution $\{u,v\}$ we require $u,u',v$, and $v'$
are absolutely continuous. Throughout the paper we assume the following
conditions hold
\begin{itemize}
\item[(A1)] $p>1$ and $\lambda,\mu>0$;
\item[(A2)] $w\in C(\mathbb{R})$.
\end{itemize}
Now, we have the following result which is concerned with some oscillation
properties of solutions to \eqref{eq1.1}-\eqref{eq1.2}.


\begin{theorem}\label{thm1.1}
Assume the conditions {\rm (A1), (A2)} hold. Then there exists a sequence
of positive parameters $\{\lambda_k\}_{k=m}^{\infty}$ for the one-dimensional
coupled system \eqref{eq1.1}-\eqref{eq1.2}, where $m$ is some positive integer,
such that the corresponding solution $u(x;\lambda_k)$ has
exactly $k-1$ zeros in $(0,1)$ for $k\geq m$. For sufficiently large $\lambda_k$,
$u(x;\lambda_k)$ and $v(x;\lambda_k)$ have the same number of zeros in $(0,1)$ and
satisfy the right-endpoint conditions $u(1;\lambda_k)=v(1;\lambda_k)=0$ under
a suitable choice of the initial parameter $\mu$. That is,
$\{u(x,\lambda_k),v(x,\lambda_k)\}$ solves \eqref{eq1.1} and satisfies
the Dirichlet boundary conditions as $\mu=\lambda_k^{1/p}+o(1)$.
\end{theorem}

\section{Preliminaries and Proofs}

  Before to give the proof of Theorem \ref{thm1.1}, we first represent some
elementary results for the solutions  of the initial value problem
 \eqref{eq1.1}-\eqref{eq1.2}. Here we need the following lemma to discuss
the uniqueness of the local solution.

\begin{lemma}[{\cite[p.180]{W98}}] \label{lem2.1}
Let $W\in C^1(I)$, $x_0\in I$ and $W(x_0)=0$, where $I$ is a
compact interval containing $x_0$. Denote by $\|W\|_x$ the maximum
of $W$ in the interval from $x_0$ to $x$. Then
$|W'(x)|\leq K\| W\|_x$ in $I$  implies
\begin{equation} \label{eq2.1}
W=0 \quad \text{for }|x-x_0|\leq \frac{1}{K},\; x\in I.
\end{equation}
\end{lemma}

In \cite[Proposition 2.2]{W13}, the proof of the uniqueness gave some
inconsistencies. Here we refine the proof.

\begin{proposition} \label{prop2.2}
For any fixed $\lambda, \mu\in \mathbb{R}^+$, problem
\eqref{eq1.1}-\eqref{eq1.2} has a unique pair of solutions which
exist on an open interval $I$ containing zero.
\end{proposition}

\begin{proof}
System \eqref{eq1.1} can be written as
\begin{equation}\begin{gathered} \label{eq2.2}
u'=|U|^{p^*-2}U,\\
U'=|v|^{p-2}v-(\lambda-w(x))|u|^{p-2}u,\\
v'=|V|^{p^*-2}V,\\
V'=-|u|^{p-2}u-\lambda |v|^{p-2}v,
\end{gathered}
\end{equation}
with $u(0)=v(0)=0$, $U(0)=\lambda^{1/p^*}$ and $V(0)=\mu^{1/p^*}$, where
$p^*=\frac{p}{p-1}$ is the conjugate exponent of $p$. Then the
local existence of a solution is valid by the Cauchy-Peano
theorem. For the uniqueness, we define $M\equiv \max\{\lambda^{1/p},\mu\}$.
By \eqref{eq1.2}, we may find an interval I containing zero such that
\begin{equation} \label{eq2.3}
\frac{M}{2}|x-0|<|u(x)|, \quad |v(x)|<2M|x-0| \quad\text{for } x\in I.
\end{equation}

Suppose that $\{u_1(x),v_1(x)\}$ and $\{u_2(x),v_2(x)\}$ are two
distinct local solutions of \eqref{eq1.1}-\eqref{eq1.2}. For
$x\in I$ we can assume that
\begin{equation} \label{eq2.4}
\|v_1-v_2\|_x\leq c_1\|u_1-u_2\|_x
\end{equation}
for some constant $c_1$,
where the notation $\|\cdot\|_x$ is defined in the statement of
Lemma \ref{lem2.1} with $x_0=0$.

Then, for $x\in I$ one has
\begin{align*}
&|u_1'(x)|^{p-2}u_1'(x)-|u_2'(x)|^{p-2}u_2'(x) \\
&=\int_0^x[|v_1(t)|^{p-2}v_1(t)-|v_2(t)|^{p-2}v_2(t)]dt \\
&-\int_0^x(\lambda-w(t))[|u_1(t)|^{p-2}u_1(t)-|u_2(t)|^{p-2}u_2(t)]dt.
\end{align*}
The following is a version of the mean value theorem, that for $a_1$ and
$a_2$ of the same sign
\begin{equation*}
 |a_1|^{p-2}a_1-|a_2|^{p-2}a_2=(p-1)(a_1-a_2)|\bar{a}|^{p-2},
\end{equation*}
 where $\bar{a}$ lies between $a_1$, $a_2$. So, for $x\in I$ one can obtain
\begin{align*}
&[u_1'(x)-u_2'(x)]|\bar{u'}|^{p-2}\\
&=\int_0^x[v_1(t)-v_2(t)]|\bar{v}|^{p-2}dt
 -\int_0^x(\lambda-w(t))[u_1(t)-u_2(t)]|\bar{u}|^{p-2}dt,
\end{align*}
for some $\bar{u'}$, $\bar{v}$ and $\bar{u}$.
By \eqref{eq2.3}-\eqref{eq2.4}, for $x\in I$ one can get
\begin{equation*}
c_2|u_1'(x)-u_2'(x)|
\leq \Big((c_1+\lambda+\|w\|_x)\|u_1-u_2\|_x\int_0^x(2M)^{p-2}t^{p-2}dt\Big),
\end{equation*}
where $c_2$ is some constant. Now, set $W(x)=u_1(x)-u_2(x)$.
By Lemma \ref{lem2.1}, one can obtain that $W(x)=0$ near $x=0$.
Applying the similar arguments on $v_1'-v_2'$, one can prove the uniqueness
for $v(x)$.
\end{proof}

Now we introduce a Pr\"{u}fer-type substitution for the solution of
\eqref{eq1.1}-\eqref{eq1.2} by using the generalized sine function $S_p(x)$.
The generalized sine function $S_p$ has been well studied in the literature (see
Lindqvist \cite{lind95} or \cite{BD03,e79,RW99} with a minor
difference in setting). Here we outline some properties for the
readers' convenience. The function $S_p$ satisfies
\begin{gather}  \label{eq4.1}
 |S_p'(x)|^{p}+\frac{|S_p(x)|^{p}}{p-1}=1, \\
 \label{eq4.2}
 (|S_p'|^{p-2}S_p')'+|S_p|^{p-2}S_p=0.
 \end{gather}
Moreover,
$$
\pi_p\equiv 2\int_{0}^{(p-1)^{1/p}}\frac{dt}{(1-\frac{t^p}{p-1})^{1/p}}
=\frac{2(p-1)^{1/p}\pi}{p\sin(\pi/p)}
$$
is the first zero of $S_p$ in the positive real axis. Similarly,
one has
$$
S_p(\frac{\pi_p}{2})=\sqrt[p]{p-1},~~S'_p(0)=1\quad \text{and}\quad
S'_p(\frac{\pi_p}{2})=0.
$$
With the help of the generalized sine function and denoting $'=\frac{d}{dx}$,
we introduce phase-plane coordinates $R,~r>0$ and $\theta,~\phi$ for the solution
$\{u(x;\lambda),v(x;\lambda)\}$ as follows:
\begin{gather}
u(x;\lambda)=R(x;\lambda)S_p(\lambda^{1/p}\theta(x;\lambda)), \quad
u'(x;\lambda)=\lambda^{1/p}R(x;\lambda)S_p'
(\lambda^{1/p}\theta(x;\lambda)),\label{eq2.8}\\
v(x;\lambda)=r(x;\lambda)S_p(\lambda^{1/p}\phi(x;\lambda)), \quad
v'(x;\lambda)=\lambda^{1/p}r(x;\lambda)S_p'
(\lambda^{1/p}\phi(x;\lambda)).\label{eq2.9}
\end{gather}
with $\theta(0;\lambda)=\phi(0;\lambda)=0$. Then
\begin{equation}
\label{eq2.10}
\lambda R(x;\lambda)^p=\frac{\lambda|u(x;\lambda)|^p}{p-1}+|u'(x;\lambda)|^p,\quad
\lambda r(x;\lambda)^p=\frac{\lambda|v(x;\lambda)|^p}{p-1}+|v'(x;\lambda)|^p
\end{equation}
with $R(0;\lambda)=1$ and $r(0;\lambda)=\frac{\mu}{\lambda^{1/p}}$. Moreover,
\begin{gather*}
\frac{|u'(x;\lambda)|^{p-2}u'(x;\lambda)}{|u(x;\lambda)|^{p-2}u(x;\lambda)}
= \frac{\lambda^{\frac{p-1}{p}}|S_p'(\lambda^{1/p}\theta(x;\lambda))|^{p-2}
 S_p'(\lambda^{1/p}\theta(x;\lambda))}
{|S_p(\lambda^{1/p}\theta(x;\lambda))|^{p-2}S_p(\lambda^{1/p}\theta(x;\lambda))}, \\
\frac{|v'(x;\lambda)|^{p-2}v'(x;\lambda)}{|v(x;\lambda)|^{p-2}v(x;\lambda)}=
\frac{\lambda^{\frac{p-1}{p}}|S_p'(\lambda^{1/p}\phi(x;\lambda))|^{p-2}S_p'
 (\lambda^{1/p}\phi(x;\lambda))}
{|S_p(\lambda^{1/p}\phi(x;\lambda))|^{p-2}S_p(\lambda^{1/p}\phi(x;\lambda))}.
\end{gather*}

Differentiating both sides of the above two identities with respect to $x$
and employing \eqref{eq1.1}, one can obtain the following result.

\begin{lemma} \label{lem2.3}
For the sake of simplicity, write $\theta(x)=\theta(x;\lambda)$,
$\phi(x)=\phi(x;\lambda)$, $R(x)=R(x;\lambda)$ and
$r(x)=r(x;\lambda)$. Then, for $x\in I$,
\begin{gather}
\begin{aligned}
\theta'(x)&= 1-\frac{w(x)}{(p-1)\lambda}|S_p(\lambda^{1/p}\theta(x))|^p \\
&\quad-\frac{1}{(p-1)\lambda}\big[\frac{r(x)}{R(x)}\big]^{p-1}
 |S_p(\lambda^{1/p}\phi(x))|^{p-2}S_p(\lambda^{1/p}\phi(x))
S_p(\lambda^{1/p}\theta(x)),
\end{aligned}\label{eq2.12}\\
\begin{aligned}
&\phi'(x) \\
&=1+\frac{1}{(p-1)\lambda}\big[\frac{R(x)}{r(x)}\big]^{p-1}|
S_p(\lambda^{1/p}\theta(x))|^{p-2}
S_p(\lambda^{1/p}\theta(x))S_p(\lambda^{1/p}\phi(x)),
\end{aligned}\label{eq2.13}\\
\begin{aligned}
&R'(x) \\
&=\frac{w(x)R(x)}{(p-1)\lambda^{1-1/p}}|S_p(\lambda^{1/p}\theta(x))|^{p-2}
S_p(\lambda^{1/p}\theta(x))S_p'(\lambda^{1/p}\theta(x)) \\
&\quad +\frac{1}{(p-1)\lambda^{1-1/p}}\big[\frac{r(x)^{p-1}}{R(x)^{p-2}}\big]
 |S_p(\lambda^{1/p}\phi(x))|^{p-2} S_p(\lambda^{1/p}\phi(x))
 S_p'(\lambda^{1/p}\theta(x)),
\end{aligned} \label{eq2.14}\\
\begin{aligned}
&r'(x) \\
&=-\frac{1}{(p-1)\lambda^{1-1/p}}\big[\frac{R(x)^{p-1}}{r(x)^{p-2}}\big]
|S_p(\lambda^{1/p}\theta(x))|^{p-2}
S_p(\lambda^{1/p}\theta(x))S_p'(\lambda^{1/p}\phi(x)).
\end{aligned}\label{eq2.15}
\end{gather}
\end{lemma}

Applying Lemma \ref{lem2.3}, we find that
$\{u(x;\lambda),v(x;\lambda)\}$ is  a solution of
\eqref{eq1.1}-\eqref{eq1.2} if and only if  $\{\theta(x;\lambda), R(x;\lambda),
\phi(x;\lambda), r(x;\lambda)\}$
 is  a solution of \eqref{eq2.12}-\eqref{eq2.15} coupled with the
 following conditions
\begin{equation}
\label{eq2.16}
\theta(0;\lambda)=\phi(0;\lambda)=0,\quad \text{and}\quad
R(0;\lambda)=1,\quad r(0;\lambda)=\frac{\mu}{\lambda^{1/p}}.
\end{equation}

 Next we derive some properties for the radial
functions $R(x;\lambda)$ and $r(x;\lambda)$.

\begin{lemma} \label{lem2.4}
Write $R(x)=R(x;\lambda)$ and $r(x)=r(x;\lambda)$.
\begin{itemize}
\item[(i)] For $x\in I$, the radial functions satisfy
\begin{equation} \label{eq2.17}
\begin{aligned}
\Big(1+\frac{\mu^{p-1}}{\lambda^{\frac{p-1}{p}}}\Big)
\exp[-c_1\lambda^{\frac{1-p}{p}}x]
&\leq R(x)^{p-1}+r(x)^{p-1} \\
&\leq \Big(1+\frac{\mu^{p-1}}{\lambda^{\frac{p-1}{p}}}\Big)
 \exp[c_1\lambda^{\frac{1-p}{p}}x],
\end{aligned}
\end{equation}
where $c_1$ is a positive constant.

\item[(ii)] For fixed $x\in I$ and sufficiently large $\lambda$, we can choose
$\mu=\lambda^{1/p}$ and obtain that
\begin{equation} \label{eq2.19}
\frac{r(x)}{R(x)}=1+o(1).
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
(i) By \eqref{eq2.14}-\eqref{eq2.15},
there exists some positive constant $c_1$ such that
\begin{align*}
-c_1\lambda^{\frac{1-p}{p}}[R(x)^{p-1}+r(x)^{p-1}]
&\leq (p-1)\left[R(x)^{p-2}R'(x)+r(x)^{p-2}r'(x)\right] \\
&\leq c_1\lambda^{\frac{1-p}{p}}[R(x)^{p-1}+r(x)^{p-1}].
\end{align*}
Solving the above differential inequality and applying the initial condition
\eqref{eq2.16}, we obtain the inequality \eqref{eq2.17}.

(ii) As in (i), there exists a  positive constant $c_2$ such that
\begin{equation*}
 \frac{R(x)r'(x)-r(x)R'(x)}{R(x)^2}
\leq c_2\lambda^{\frac{1-p}{p}}\big[\frac{R(x)^{p-2}}{r(x)^{p-2}}+\frac{r(x)}{R(x)}
+\frac{r(x)^p}{R(x)^p}\big].
\end{equation*}
Letting $y(x)=\frac{r(x)}{R(x)}$, we have
$$
y'(x)\leq c_2\lambda^{\frac{1-p}{p}}[y(x)^{2-p}+y(x)+y(x)^p].
$$
Note that
$$
\frac{dy}{dx}\leq c_2\lambda^{\frac{1-p}{p}}(\frac{1+y^{p-1}+y^{2p-2}}{y^{p-2}}),
\text{i.e.,}\quad
\frac{y^{p-2}dy}{1+y^{p-1}+y^{2p-2}}\
leq c_2\lambda^{\frac{1-p}{p}}dx.
$$
Letting $z=y^{p-1}$ and integrating the above inequality, we obtain
$$
\frac{2}{\sqrt{3}}\Big[\tan^{-1}\frac{2z+1}{\sqrt{3}}\Big]_{y(0)}^{y(x)^{p-1}}
\leq (p-1)c_2\lambda^{\frac{1-p}{p}}x.
$$
i.e.,
$$
\frac{2}{\sqrt{3}}\Big[\tan^{-1}\frac{2z+1}{\sqrt{3}}\Big]_{y(0)}^{y(x)^{p-1}}=o(1).
$$
 Then,
$$
0< \tan^{-1}\frac{2y(x)^{p-1}+1}{\sqrt{3}}=\frac{\pi}{3}+o(1).
$$
So $y(x)^{p-1}= 1+o(1)$  as $\lambda$ is sufficiently large.
This completes the proof.
\end{proof}

\begin{remark} \rm
From the proof of Lemma \ref{lem2.4} (ii), one can obtain the boundedness
of $\frac{r(x)}{R(x)}$ and $\frac{R(x)}{r(x)}$. Then, the right-hand sides
of \eqref{eq2.12}-\eqref{eq2.15} satisfy the generalized Lipschitz continuous
in $\theta$, $\phi$, $R$ and $r$ respectively. Thus, the existence of the
unique absolutely continuous solutions is valid. The above can be referred
to \cite{W983} (pp.121-123).
\end{remark}

\begin{remark} \label{rmk2.6} \rm
For sufficiently large $\lambda$, one can choose $\mu$ closed to
$\lambda^{1/p}$; that is, $\mu=\lambda^{1/p}+o(1)$,
and the asymptotic estimate \eqref{eq2.19}   is still valid.
\end{remark}

From Proposition \ref{prop2.2} and Lemma \ref{lem2.4} (i), we have the following
result.

\begin{proposition} \label{prop2.7}
For fixed $\lambda,\mu>0$, the uniquely local solution
$\{u(x;\lambda),v(x;\lambda)\}$ can be extended to the whole real axis.
\end{proposition}

Now we derive some properties related to the phase functions $\theta(x;\lambda)$
and $\phi(x;\lambda)$.

\begin{lemma} \label{lem2.8}
For $\lambda>0$, the phase functions $\theta(x;\lambda)$ and $\phi(x;\lambda)$
satisfy the following properties.
\begin{itemize}
\item[(i)] $\theta(\cdot;\lambda)$ and $\phi(\cdot;\lambda)$ are continuous in
$\lambda$ and satisfy $\theta(0;\lambda)=\phi(0;\lambda)=0$.

\item[(ii)] If $\lambda^{1/p}\theta(x_n;\lambda)=n\pi_p$ for some $x_n\in (0,1)$,
then $\lambda^{1/p}\theta(x;\lambda)>n\pi_p$ for every $x>x_n$.

\item[(iii)] For sufficiently large $\lambda$,
\begin{equation} \label{eq3.2}
\lambda^{1/p}\theta(1;\lambda)=\lambda^{1/p}+O(\frac{1}{\lambda^{1-\frac{1}{p}}}).
\end{equation}
Moreover, $\lambda^{1/p}\phi(1;\lambda)$ has the same estimate as
\eqref{eq3.2}.

\item[(iv)] For sufficiently large $\lambda$, a suitable initial parameter
$\mu=\lambda^{1/p}+o(1)$ can be chosen such that
$\theta(1;\lambda)=\phi(1;\lambda)$.
\end{itemize}
\end{lemma}

\begin{proof}
Item (i) is valid by \eqref{eq2.12}-\eqref{eq2.13} and \eqref{eq2.16}.
For (ii), if
$\lambda^{1/p}\theta(x_n;\lambda)=n\pi_p$ for some $x_n\in (0,1)$, then by
\eqref{eq2.12} and Lemma \ref{lem2.4} (ii), we have
\begin{equation} \label{eq3.31}
\theta'(x_n;\lambda)=1>0.
\end{equation}
For (iii), integrating \eqref{eq2.12} and \eqref{eq2.13} over
$[0,1]$ and applying $(i)$ and Lemma \ref{lem2.4} (ii), one can obtain
the asymptotic estimates \eqref{eq3.2} as $\lambda$
is sufficiently large. This proves $(iii)$.
Besides, write $\theta'(x;\lambda)=F(x;\lambda;\theta;\phi)$ and
 $\phi'(x;\lambda)=H(x;\lambda;\theta;\phi)$.
Then, for $x\in [0,1]$,
\begin{align*}
&\theta(x;\lambda)-\phi(x;\lambda) \\
&=\int_0^x(F(t;\lambda;\theta;\phi)-H(t;\lambda;\theta;\phi))dt\\
&=\int_0^x\Big[F(t;\lambda;\theta;\phi)-F(t;\lambda;\theta;\theta)
 +F(t;\lambda;\theta;\theta)-H(t;\lambda;\theta;\theta) \\
&\quad +H(t;\lambda;\theta;\theta)-H(t;\lambda;\theta;\phi)\Big]dt\\
&=\int_0^x\frac{\partial}{\partial \phi}F(t;\lambda;\theta;\xi)
 [\phi(t;\lambda)-\theta(t;\lambda)]dt
 -\frac{1}{(p-1)\lambda} \int_0^xw(t)|S_p(\lambda^{1/p}\theta(t;\lambda))|^pdt\\
&\quad -\frac{1}{(p-1)\lambda}\int_0^x
 \Big(\big[\frac{r(x)}{R(x)}\big]^{p-1}+\big[\frac{R(x)}{r(x)}\big]^{p-1}\Big)
 |S_p(\lambda^{1/p}\theta(t;\lambda))|^pdt\\
&\quad +\int_0^x\frac{\partial}{\partial \phi}H(t;\lambda;\theta;\eta)
 [\phi(t;\lambda)-\theta(t;\lambda)]dt,
\end{align*}
where $\xi(t;\lambda)$ and $\eta(t;\lambda)$ are between $\phi(t;\lambda)$
and $\theta(t;\lambda)$.
Note that $|\frac{\partial}{\partial \phi}F(t;\lambda;\theta;\xi)|$ and
$|\frac{\partial}{\partial \phi}H(t;\lambda;\theta;\eta)|$
are uniformly bounded by some constant $K$ for all $t\in [0,1]$.
Then, by Lemma \ref{lem2.4} (ii)
for any $\delta>0$ there exists sufficiently large $\lambda$ such that
$$
|\theta(x;\lambda)-\phi(x;\lambda)|
\leq \delta+\int_0^x 2K|\phi(t;\lambda)-\theta(t;\lambda)|dt.
$$
By the Gronwall inequality, we obtain
\begin{equation} \label{eq2.30}
|\theta(x;\lambda)-\phi(x;\lambda)|\leq \delta e^{2K}.
\end{equation}
By \eqref{eq2.12}-\eqref{eq2.13}, Remark \ref{rmk2.6} and \eqref{eq2.30},
one can choose suitable $\mu$ satisfying
$\mu=\lambda^{1/p}+o(1)$ such that the two Pr\"{u}fer phases
are identical at the right end-point as $\lambda$ is sufficiently large.
That is, $\theta(1;\lambda)=\phi(1;\lambda)$.
Now the proof of $(iv)$ is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.1}]
By Lemma \ref{lem2.8} (i) and (iii), the modified phase
$\lambda^{1/p}\theta(1;\lambda)$ tends to infinity as $\lambda\rightarrow \infty$.
Hence, for every sufficiently large
$k\in\mathbb{N}$ there exists $\lambda_k >0$ satisfies
$\lambda_k^{1/p}\theta(1;\lambda_k)=k\pi_p$. This implies that
there exists $m\in\mathbb{N}$ such that
$\lambda_k^{1/p}\theta(1;\lambda_k)=k\pi_p$ for every $k\geq m$.
Furthermore, the remaining results are valid by Lemma \ref{lem2.8} (iii) and
(iv).
\end{proof}

\begin{thebibliography}{99}

\bibitem{ACLO06} R. P. Agarwal, D. Cao, H. Lu, D. O'Regan;
\emph{Existence and mulplicity of positive solutions for singular semipositone
 $p$-Laplacian equations}, Canad. J. Math.
\textbf{58} (2006), 449--475.

\bibitem{AN07} P. Amster, P. De Napoli;
\emph{Landesman-Lazer type conditions for a system of $p$-Laplacian like
operators}, J. Math. Anal. Appl., \textbf{326} (2007), 1236--1243.

\bibitem{BD03} P. Binding, P. Drabek;
\emph{Sturm--Liouville theory for the $p$-Laplacian}, Studia Scientiarum
Mathematicarum Hungarica, \textbf{40} (2003), 373-396.

\bibitem{D85} J. I. Diaz;
\emph{Nonlinear Partial Differential Equations and Free Boundaries},
vol. I. Elliptic Equations, Res. Notes Math., vol. 106, Pitman, Boston, MA, 1985.

\bibitem{D01} J. I. Diaz;
\emph{Qualitative study of nonlinear parabolic equations: an introduction},
Extracta Math., \textbf{16} (2001), 303-341.

\bibitem{e79} A. Elbert;
\emph{A half-linear second order differential equation},
Colloqia mathematica Societatis J\'{o}nos Bolyai, \textbf{30},
 Qualitiative Theory of Differential Equations, Szeged
(Hungary) (1979), 153-180.

\bibitem{GM77} R. E. Gaines, J. L. Mawhin;
\emph{Coincidence Degree and Nonlinear Differential Equations},
Springer-Verlag, Berlin, 1977.

\bibitem{HW07} D. D. Hai, H. Wang;
\emph{Nontrivial solutions for $p$-Laplacian systems}, J. Math. Anal. Appl.,
 \textbf{330} (2007), 186--194.

\bibitem{la} O. A. Ladyzhenskaya;
\emph{The Mathematical Theory of Viscous Incompressible Flow}, 2nd
ed., Gordon and Breach (New York), 1969.

\bibitem{LW79} R. W. Leggett, L. R. Williams;
\emph{Multiple positive fixed points of a nonlinear operators on ordered Banach
spaces},  Indiana Univ. Math. J., \textbf{28} (1979), 673--688.

\bibitem{lind95} P. Lindqvist;
\emph{Some remarkable sine and cosine functions}, Ricerche Mat.,
 \textbf{44} (1995), 269-290.

\bibitem{li}  J. L. Lions;
\emph{Quelques methodes de resolution des problemes aux limites non
lineaires}, Dunod (Paris) 1969.

\bibitem{L12} D. L\"{u};
\emph{Multiple solutions for $p$-Laplacian systems with critical homogeneous
nonlinearity}, Boundary Value Problems (2012), 2012:27.

\bibitem{PR74} M. C. Pelissier, M. L. Reynaud;
\emph{Etude d?n modele mathematique d?coulement de glacier},
C. R. Acad. Sci. Paris Ser. I Math., \textbf{279} (1974), 531-534.

\bibitem{RW99} W. Reichel,  W. Walter;
\emph{Sturm-Liouville type problems for the $p$-Laplacian under asymptotic
non-resonance conditions}, J. Differential Equations \textbf{156} (1999), 50-70.

\bibitem{SW91} R. E. Showalter, N. J. Walkington;
\emph{Diffusion of fluid in a fissured medium with microstructure}, SIAM J. Math. Anal.
 \textbf{22} (1991), 1702-1722.

\bibitem{SL07} H. Sun, W. Li;
\emph{Existence theory for positive solutions to one-dimensional $p$-Laplacian
boundary value problems}, J. Differential Equations, \textbf{240} (2007), 217--248.

\bibitem{Takeuchi12} S. Takeuchi;
\emph{Generalized Jacobian elliptic functions and their application to bifurcation
problems associated with $p$-Laplacian}, J. Math. Anal. Appl. \textbf{385} (2012),
 24-35.

\bibitem{W98} W. Walter;
\emph{Sturm-Liouville theory for the radial $\Delta_p$-operator},
Math. Z. \textbf{227} (1998), 175-185.

\bibitem{W983} W. Walter;
\emph{Ordinary Differential Equations}, Springer-Verlag, 1998.

\bibitem{W13} W. C. Wang;
\emph{Pr\"{u}fer substitutions on a coupled system involving $p$-Laplacian},
Electronic J. Differential Equations, \textbf{2013} (213) (2013), 1-9.

\bibitem{WZ06} Z. Wang, J. Zhang;
\emph{Positive solutions for one-dimensional $p$-Laplacian boundary value
problems with dependence on the first order derivative},
J. Math. Anal. Appl., \textbf{314} (2006), 618--630.

\bibitem{Y02} E. Yanagida;
\emph{Mini-maximizers for reaction-diffusion systems with skew-gradient structure},
J. Differential Equations \textbf{179} (2002), 311-335.

\bibitem{Y021} E. Yanagida;
\emph{Standing pulse solutions in reaction-diffusion systems with skew-gradient
structure}, J. Dynam. Differential Equations \textbf{14} (2002), 189-205.

\bibitem{Y11} Z. Yang;
\emph{Positive solutions for a system of $p$-Laplacian boundary value
problems}, Computers and Mathematics with Applications \textbf{62}
(2011), 4429-4438.

\end{thebibliography}

\end{document}