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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 213, pp. 1--9.\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-2013/213\hfil Pr\"ufer substitutions]
{Pr\"ufer substitutions on a coupled system involving the $p$-Laplacian}

\author[W.-C. Wang\hfil EJDE-2013/213\hfilneg]
{Wei-Chuan Wang}  % in alphabetical order

\address{Wei-Chuan Wang \newline
Center for General Education, National Quemoy
University, Kinmen, 892, Taiwan}
\email{wangwc72@gmail.com}


\thanks{Submitted July 12, 2013. Published September 25, 2013.}
\subjclass[2000]{34A55, 34B24, 47A75}
\keywords{Coupled system; $p$-Laplacian;  Pr\"ufer substitution}

\begin{abstract}
 In this article, we employ a modified Pr\"ufer substitution acting
 on a coupled system involving one-dimensional $p$-Laplacian equations.
 The basic properties for  the initial valued problem and some estimates are
 obtained. We also derive an analogous Sturmian theory and give a
 reconstruction formula for the potential function.
\end{abstract}

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

\section{Introduction}

 There has been recently  a lot of interest in the study of the 
$p$-Laplacian eigenvalue problem
\begin{gather*}
-\Delta_py+q|y|^{p-2}y=\lambda |y|^{p-2}y,\\
y|_{\partial \Omega}=0,
\end{gather*}
where $p>1$ and $q\in C(\Omega)$, $\Omega \subseteq \mathbb{R}^n$. 
This is a quasilinear partial differential equation when $p\neq 2$. 
The most cited application is the highly viscid fluid flow 
(cf. Ladyzhenskaya \cite{la} and Lions \cite{li}). 
When $p=2$, $q$ and $\lambda$ both vanish, it becomes  the Laplacian equation.
The $p$-Laplacian operator has the originally physical meaning, and can also
 be treated as a generalization of the Laplacian operator. 
For the one-dimensional case,
the $p$-Laplacian eigenvalue problem becomes, after scaling,
\begin{gather}
\label{eq1.4}
-(y'^{(p-1)})'=(p-1)(\lambda -q(x))y^{(p-1)}, \\
\label{eq1.5}
y(0)=y(1)=0,
\end{gather}
where $p>1$, $f^{(p-1)}\equiv |f|^{p-1}\operatorname{sgn} f=|f|^{p-2}f$, and $q$
is a continuous function defined on $[0,1]$. The following
Sturm-Liouville property for the one-dimensional $p$-Laplacian
operator is well-known now (cf. Binding \& Drabek \cite{bd}, 
Reichel \& Walter \cite{rw99}, Walter \cite{W98}, etc.).

\begin{theorem} \label{thm1.1}
For \eqref{eq1.4}-\eqref{eq1.5}, there exists a sequence 
of eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ such that
$$
-\infty<\lambda_1<\lambda_2<\lambda_3<\dots< \lambda_n< \dots\ \to\infty\,,
$$
and the eigenfunction corresponding to $\lambda_{n}$ has exactly $n-1$ zeros in
$(0,1)$.
\end{theorem}

In this article, we consider the coupled one-dimensional
 $p$-Laplacian problem 
\begin{equation}\label{eq1.1}
\begin{gathered}
(u'(x)^{(p-1)})'+(p-1)\lambda u(x)^{(p-1)}-(p-1)q(x)v(x)^{(p-1)}=0,\\
(v'(x)^{(p-1)})'+(p-1)\lambda v(x)^{(p-1)}+(p-1)q(x)u(x)^{(p-1)}=0,
\end{gathered}
\end{equation}
with the initial conditions
\begin{equation}
\label{eq1.2}
u(0)=v(0)=0,~u'(0)=v'(0)=\lambda^{1/p},
\end{equation}
where $\lambda$ is some positive parameter, $p>1$, and $q$ is a
continuous function defined in $\mathbb{R}$. When $p=2$,
\eqref{eq1.1} reduces to
\begin{equation}\label{eq1.3}
\begin{gathered}
u''(x)+\lambda u(x)-q(x)v(x)=0,\\
v''(x)+\lambda v(x)+q(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 $H(u,v)=\frac{\lambda}{2}u^2-\frac{\lambda}{2}v^2-q(x)uv$.
Then
$$
\frac{\partial H}{\partial u}=\lambda u-q(x)v,\quad 
-\frac{\partial H}{\partial v}=\lambda v+q(x)u.
$$
Equation \eqref{eq1.3} can be viewed as
a simplest model of diffusion systems with skew-gradient 
structure (cf.  \cite{Y02,Y021}).

Here we intend to study the existence of sign-changing solutions
(or nodal solutions) of \eqref{eq1.1}-\eqref{eq1.2} and try to
derive an analog of Theorem \ref{thm1.1}. Employing the
information of solutions, a reconstruction formula for $q(x)$ is given. 
Such a procedure is called an inverse nodal problem.
An inverse problem of this type was designated by  McLaughlin \cite{M88} in 1988.
When one studies the inverse nodal problem of
\eqref{eq1.1}-\eqref{eq1.2}, an interesting observation arises. 
The asymptotic formula given in Theorem \ref{thm1.3} (see the following)
coincides with the one of the classical Sturm-Liouville eigenvalue problem
\begin{gather*}
-y''+w_0(x) y=\mu y,\\
y(0)=y(1)=0
\end{gather*}
(cf. \cite{M88,S88,LSY}). Besides, the Pr\"ufer substitution is
an efficient method in showing the oscillation property for
solutions (cf. \cite{BB}). In this article we utilize a modified
Pr\"ufer substitution to treat this problem. Fortunately we can
tackle the effect of the two coupled functions in
\eqref{eq1.1}-\eqref{eq1.2}, and obtain the detailed estimates of 
parameters $\lambda_m$ and nodal points. The following are our main results.

\begin{theorem} \label{thm1.2}
There exists a sequence of real parameters
$\{\lambda_k\}_{k=m}^{\infty}$ of the one-dimensional coupled
system \eqref{eq1.1}-\eqref{eq1.2}, where $m\in \mathbb{N}$ such that
$$
0<\lambda_m<\lambda_{m+1}<\lambda_{m+2}<\lambda_{m+3}<\dots\to \infty,
$$ 
and the corresponding solution $u(x,\lambda_k)$ has
exactly $k-1$ zeros in $(0,1)$ for $k\geq m$. 
In particular,
the solution pair $\{u(x,\lambda_k),v(x,\lambda_k)\}$ satisfies
the following boundary condition
$$u(0,\lambda_k)=v(0,\lambda_k)=0,~~u(1,\lambda_k)=0$$ for every
$k\geq m$.
\end{theorem}

Define the zero set (or nodal set) $\{x_i^{(k)}\}_{i=1}^{k-1}$ of
the solution $u(x,\lambda_k)$ to \eqref{eq1.1}-\eqref{eq1.2} and
the index $i_k(x)=\max \{i:x_i^{(k)}\leq x\}$. Let
$\ell_i^{(k)}=x_{i+1}^{(k)}-x_i^{(k)}$ for $0\leq i\leq k-1$,
where $x_0^{(k)}=0$ and $x_{k}^{(k)}=1$. We obtain an asymptotic
formula for the function $q(x)$.

\begin{theorem}\label{thm1.3}
Suppose that the above assumptions hold. Then an asymptotic
formula for $q(x)$ in \eqref{eq1.1} is
\begin{equation}
\label{eq1.6}
q(x)-\int_0^1q(t)dt=\lim_{m\to \infty}[p(m\pi_p)^p(m\ell_{i_m(x)}^{(m)}-1)],
\end{equation}
for all $x\in [0,1]$.
\end{theorem}

We remark that in Theorem \ref{thm1.2}, the right endpoint conditions
$v(1,\lambda_k)$ also vanish when $\lambda_k$ tends to the
infinity. Simultaneously, one can show an analogous Sturmian
theory for $v(x,\lambda)$. Then the data coming from
$v(x,\lambda_k)$ also make the asymptotic formula
\eqref{eq1.6} valid.


This article is organized as follows. After the introduction, we
employ a modified Pr\"ufer substitution to show the local solution 
of the initial value problem \eqref{eq1.1}-\eqref{eq1.2} is
unique and can be extended to the whole interval $[0,1]$. 
In section 3, we derive several lemmas to complete the proof 
of Theorem \ref{thm1.2}.
In section 4, some detailed estimates and the proof of 
Theorem \ref{thm1.3} are given.

\section{Preliminaries - A modified Pr\"ufer substitution}

To discuss  the  existence and uniqueness of the
 local solution of \eqref{eq1.1}-\eqref{eq1.2}. We need the following lemma.

\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},\quad x\in I.
\end{equation}
\end{lemma}


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

\begin{proof}
System \eqref{eq1.1} can be written as
\begin{equation}\label{eq2.2}
\begin{gathered}
u'=U^{(p^*-1)},\\
U'=(p-1)[qv^{(p-1)}-\lambda u^{(p-1)}],\\
v'=V^{(p^*-1)},\\
V'=-(p-1)[qu^{(p-1)}-\lambda v^{(p-1)}],
\end{gathered}
\end{equation}
with $u(0)=v(0)=0$ and $U(0)=V(0)=\lambda^{1/p^*}$, where
$p^*=p/(p-1)$ is the conjugate exponent of $p$. Then the
local existence of a solution is valid by the Cauchy-Peano
theorem. Now it suffices to prove the uniqueness. By
\eqref{eq1.2}, we may assume that
\begin{equation} \label{eq2.3}
\frac{\lambda^{1/p}}{2}|x-0|<|u(x)|,~|v(x)|<2\lambda^{1/p}|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}. Without
loss of generality, we assume that $u_1(x)\geq u_2(x)$ and
$v_1(x)\geq v_2(x)$ in some small interval $I$ which contains zero. By
\eqref{eq2.2}, for $x\in I$ we have
\begin{align*}
&u_1'(x)^{(p-1)}-u_2'(x)^{(p-1)}\\
&=(p-1)\Big\{\int_0^xq(t)[v_1(t)^{(p-1)}-v_2(t)^{(p-1)}]dt
-\lambda\int_0^x[u_1(t)^{(p-1)}-u_2(t)^{(p-1)}]dt\Big\},
\end{align*}
\begin{align*}
&v_1'(x)^{(p-1)}-v_2'(x)^{(p-1)}\\
&=(1-p)\Big\{\int_0^xq(t)[u_1(t)^{(p-1)}-u_2(t)^{(p-1)}]dt
-\lambda\int_0^x[v_1(t)^{(p-1)}-v_2(t)^{(p-1)}]dt\Big\};
\end{align*}
i.e.,
\begin{equation}
\begin{aligned}
&|u_1'(x)^{(p-1)}-u_2'(x)^{(p-1)}+v_1'(x)^{(p-1)}-v_2'(x)^{(p-1)}| \\
&= (p-1)|\int_0^x(q(t)+\lambda)[v_1(t)^{(p-1)}-v_2(t)^{(p-1)}
 -u_1(t)^{(p-1)}+u_2(t)^{(p-1)}]dt|.
\end{aligned}\label{eq2.4}
\end{equation}
It follows from the mean value theorem, that for $a_1$ and $a_2$ of the
same sign,
\begin{equation} \label{eq2.5}
 a_1^{(p-1)}-a_2^{(p-1)}=(p-1)(a_1-a_2)|\bar{a}|^{p-2}\,,
\end{equation}
 where $\bar{a}$ lies between $a_1$, $a_2$. Note that there exists
 some $c_1$ such that the left hand side of \eqref{eq2.4}
is greater than or equal to
$c_1|u_1'(x)+v_1'(x)-u_2'(x)-v_2'(x)|$. On the other hand, by
\eqref{eq2.3} the right hand side of \eqref{eq2.4} is less than or
equal to
$(p+1)(\|q\|_x+\lambda)\int_0^x|u_1(t)+v_1(t)-u_2(t)-v_2(t)|\cdot
2\lambda^{1/p}t^{p-2}dt$, where the notation $\|\cdot\|_x$ is
defined as in  Lemma \ref{lem2.1}. Now set
$W(x)=u_1(x)+v_1(x)-u_2(x)-v_2(x)$. By Lemma \ref{lem2.1}, we can
obtain that $W(x)=0$ in $I$. This proves the uniqueness of the
local solution.
\end{proof}


Now we introduce a modified Pr\"ufer substitution for the
local solution $\{u(x),v(x)\}$ using the generalized sine 
function $S_p(x)$. The $S_p(x)$ function is well known 
now (cf. \cite{bd,e79,rw99}),
and satisfies
 \begin{equation}
 \label{eq2.6}
 |S_p(x)|^{p}+|S'_p(x)|^{p}=1,
 \end{equation}
 and
 \begin{equation}
 \label{eq2.7}
 (S_p)''= \frac{-S_p^{(p-1)} S'_p}{(S'_p)^{(p-1)}}
=\frac{-S_p^{(p-1)}}{|S'_p|^{p-2}}.
 \end{equation}
Thus one has $S_p(\pi_p/2)=1$, and by \eqref{eq2.6}, $S'_p(0)=1$,
 $S'_p(\pi_p/2)=0$.
Define
\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}
Then, we obtain 
\begin{equation}
\label{eq2.10}
\lambda
R(x,\lambda)^p=\lambda|u(x,\lambda)|^p+|u'(x,\lambda)|^p,\quad
\lambda r(x,\lambda)^p=\lambda|v(x,\lambda)|^p+|v'(x,\lambda)|^p,
\end{equation}
where $R(x,\lambda)$ and $r(x,\lambda)$ are the Pr\"ufer
amplitude functions; and $\theta(x,\lambda)$ and $\phi(x,\lambda)$
are the Pr\"ufer phase angles of $\{u(x),v(x)\}$, respectively.
By a direct computation, we have the following lemma.

\begin{lemma}\label{lem2.3}
For the modified Pr\"ufer substitution \eqref{eq2.8}-\eqref{eq2.9}, one has
\begin{gather}
\theta'(x,\lambda)=1-\frac{q(x)}{\lambda}(\frac{r(x,\lambda)}{R(x,\lambda)}
)^{p-1}S_p (\lambda^{1/p}\theta(x,\lambda))
 S_p(\lambda^{1/p}\phi(x,\lambda))^{(p-1)},\label{eq2.11}
\\
R'(x,\lambda)=\frac{q(x)}{\lambda^{\frac{p-1}{p}}}\frac{r(x,\lambda)^{p-1}}
{R(x,\lambda)^{p-2}}
S_p(\lambda^{1/p}\phi(x,\lambda))^{(p-1)}S_p'
(\lambda^{1/p}\theta(x,\lambda)),\label{eq2.12}
\\
\phi' (x,\lambda)=1+\frac{q(x)}{\lambda}(\frac{R(x,\lambda)}
{r(x,\lambda)})^{p-1}S_p
(\lambda^{1/p}\phi(x,\lambda))S_p
(\lambda^{1/p}\theta(x,\lambda))^{(p-1)},\label{eq2.13}
\\
r'(x,\lambda)=\frac{-q(x)}{\lambda^{\frac{p-1}{p}}}\frac{R(x,\lambda)^{p-1}}{r(x,\lambda)^{p-2}}S_p
(\lambda^{1/p}
\theta(x,\lambda))^{(p-1)}S_p'
(\lambda^{1/p}\phi(x,\lambda)),\label{eq2.14}
\end{gather}
where $'=\frac{d}{dx}$.
\end{lemma}

\begin{proof}
Here we prove the first two equations, and the rest is similar.
For the sake of simplicity, we drop the function variable
$\lambda$ in the proof. By \eqref{eq2.8},
$$
\frac{u'(x)^{(p-1)}}{u(x)^{(p-1)}}
=\frac{\lambda^{\frac{p-1}{p}}S_p'(\lambda^{1/p}\theta(x))^{(p-1)}}
{S_p(\lambda^{1/p}\theta(x))^{(p-1)}}.
$$
Differentiating the above equation on both sides and applying
\eqref{eq1.1} and \eqref{eq2.7}, we obtain
$$
[\lambda+|\frac{u'(x)}{u(x)}|^p-q(x)(\frac{v(x)}{u(x)})^{(p-1)}]
= \lambda \theta'(x)[1+|\frac{S_p'(\lambda^{1/p}\theta(x))}
{S_p(\lambda^{1/p}\theta(x))}|^p].
$$ 
Multiplying by
$|S_p(\lambda^{1/p}\theta(x))|^p$,  from \eqref{eq2.6}, it follows that
 \eqref{eq2.11} holds.

Next, differentiate $u(x)=R(x)S_p(\lambda^{1/p}\theta(x))$
with respect to $x$ and employ \eqref{eq2.11}, to obtain 
\eqref{eq2.12}.
\end{proof}

Applying Lemma \ref{lem2.3}, we find that
$\{u(x),v(x); \lambda\}$ is  a solution of
\eqref{eq1.1}-\eqref{eq1.2} if and only if  $\{\theta(x), R(x),
\phi(x), r(x); \lambda\}$
 is  a solution of \eqref{eq2.11}-\eqref{eq2.14} coupled with the
 following conditions
\begin{equation}
\label{eq2.15}
\theta(0,\lambda)=\phi(0,\lambda)=0,~~and~~R(0,\lambda)=r(0,\lambda)=1.
\end{equation}

 Next we derive some properties for the phase and amplitude
functions.

\begin{lemma}\label{lem2.4}
(i) For $x>0$, the amplitude functions satisfy that
\begin{equation} \label{eq2.16}
2\exp[-c_1\lambda^{\frac{1-p}{p}}x]\leq R(x,\lambda)^{p-1}+r(x,\lambda)^{p-1}\leq
2\exp[c_2\lambda^{\frac{1-p}{p}}x],
\end{equation}
where $c_1,~c_2$ are some
positive constants.

(ii) For fixed $x>0$ and sufficiently large $\lambda$, we have
\begin{equation} \label{eq2.17}
\frac{r(x,\lambda)}{R(x,\lambda)}=1+o(1).
\end{equation}
Moreover, $\frac{R(x,\lambda)}{r(x,\lambda)}$ has the same 
asymptotic estimate as in \eqref{eq2.17}.
\end{lemma}

\begin{proof}
(i) By assumption and \eqref{eq2.12} and \eqref{eq2.14},
there exist some positive constants $c_1$ and $c_2$ such that
\begin{align*}
&-c_1\lambda^{\frac{1-p}{p}}[R(x)^{p-1}+r(x)^{p-1}]\\
&\leq R(x)^{p-2}R'(x)+r(x)^{p-2}r'(x)\leq
c_2\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{eq2.15}, we obtain the inequality \eqref{eq2.16}.

(ii) As in $(i)$, there exists some positive
constant $c_3$ such that 
$$
\frac{R(x)r'(x)-r(x)R'(x)}{R(x)^2}\leq
c_3\lambda^{\frac{1-p}{p}}[\frac{R(x)^{p-2}}{r(x)^{p-2}}+\frac{r(x)^p}{R(x)^p}].
$$
Letting $y(x)=\frac{r(x)}{R(x)}$, we have 
$$
y'(x)\leq c_3\lambda^{\frac{1-p}{p}}[y(x)^{2-p}+y(x)^p].
$$ 
Note that
$$
\frac{dy}{dx}\leq c_3\lambda^{\frac{1-p}{p}}(\frac{1+y^{2p-2}}{y^{p-2}});\quad
\text{i.e., }
\frac{y^{p-2}dy}{1+y^{2p-2}}\leq c_3\lambda^{\frac{1-p}{p}}dx.
$$ 
Let $z=y^{p-1}$ and integrate the
above inequality; we obtain 
$$
\tan^{-1}(y(x)^{p-1})-\tan^{-1}(y(0)^{p-1})\leq
(p-1)c_3\lambda^{\frac{1-p}{p}}x;
$$ 
i.e.,
$$
0< \tan^{-1}(y(x)^{p-1})\leq \tan^{-1}(1)+(p-1)c_3\lambda^{\frac{1-p}{p}}x.
$$ 
So
\begin{equation} \label{eq2.18}
y(x)^{p-1}\leq 1+o(1)
\end{equation}
as $\lambda$ is sufficiently large. This completes the proof.
\end{proof}

From Proposition \ref{prop2.2} and \eqref{eq2.16}, we have
the following property.

\begin{proposition} \label{prop2.5}
For any fixed $\lambda\in \mathbb{R}^+$,  problem \eqref{eq1.1}-\eqref{eq1.2}
 has a unique solution which exists over the whole interval $[0,1]$.
\end{proposition}


\section{The Sturmian property}

In this section, we first derive the following lemma for the proof of 
Theorem \ref{thm1.2}.

\begin{lemma} \label{lem3.1}
For $\lambda>0$, the phase angle function $\theta(x,\lambda)$
satisfies the following properties.
\begin{itemize}
\item[(i)] $\theta(\cdot,\lambda)$ is continuous in $\lambda$ 
and satisfies $\theta(0,\lambda)=0$.
\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)] \begin{equation}
\label{eq3.2}
\lim_{\lambda \to \infty}\lambda^{1/p}\theta(1,\lambda)=\infty.
\end{equation}
\end{itemize}
\end{lemma}

\begin{proof}
For (i), $\theta(\cdot,\lambda)$ is continuous in $\lambda$ followed by
the continuous dependence on parameters. 
And $\theta(0,\lambda)=0$ is valid by \eqref{eq2.15}.
Also if
$\lambda^{1/p}\theta(x_n,\lambda)=n\pi_p$ for some 
$x_n\in (0,1)$, then by \eqref{eq2.11} and Lemma \ref{lem2.4}, we have
\begin{equation} \label{eq3.1}
\theta'(x_n,\lambda)=1>0.
\end{equation}
This proves (ii). For (iii), integrating \eqref{eq2.11} over
$[0,1]$ and applying (i), one obtains
\begin{equation} \label{eq3.3}
\lambda^{1/p}\theta(1,\lambda)
=\lambda^{1/p}-\lambda^{\frac{1-p}{p}}\int_0^1q(t)
(\frac{r(t,\lambda)}{R(t,\lambda)})^{p-1}S_p
(\lambda^{1/p}\theta(t,\lambda))S_p(\lambda^{1/p}\phi(t,\lambda))^{(p-1)}dt.
\end{equation}
By \eqref{eq2.17}, one has
$$\lambda^{1/p}\theta(1,\lambda)=\lambda^{1/p}+O(\frac{1}{\lambda^{1-\frac{1}{p}}})$$
for sufficiently large $\lambda$. This completes the proof.
\end{proof}

We remark that using  \eqref{eq2.13} and Lemma \ref{lem2.4}, 
one can apply the similar arguments as in the above proof to obtain 
the  conclusions  in Lemma \ref{lem3.1} for 
the phase function $\phi(x,\lambda)$.


\begin{proof}[Proof of Theorem \ref{thm1.2}]
By Lemma \ref{lem3.1}, 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 $k\geq m$.
 In this case,  $\lambda_m <  \lambda_{m+1}<\dots< \lambda_{k+1}
<\dots\to\infty$, and
 $\{\theta(x,\lambda_{k}),\phi(x,\lambda_{k}) \}_{k\geq m}$ satisfy \eqref{eq2.11}-\eqref{eq2.15}. Hence, $\{u(x,\lambda_{k}),v(x,\lambda_{k}) \}_{k\geq m}$
are solutions of \eqref{eq1.1}-\eqref{eq1.2} and satisfy
$$u(0,\lambda_k)=v(0,\lambda_k)=0,~~u(1,\lambda_k)=0\quad
\mbox{ for every } k\geq m.$$ This completes the proof.
\end{proof}

\section{Some detailed estimates - Proof of Theorem \ref{thm1.3}}
 
\begin{theorem} \label{thm4.1}
The parameter $\lambda_m$ of \eqref{eq1.1}-\eqref{eq1.2} satisfies
\begin{equation}
\label{eq4.1}
\lambda_m^{1/p}=m\pi_p+\frac{1}{p(m\pi_p)^{p-1}}
\int_0^1q(t)dt+o(\frac{1}{m^{p-1}})
\end{equation}
as $m\to \infty$.
\end{theorem}

\begin{proof}
First, integrating \eqref{eq2.11} over $[0,x]$, with the associated 
$\lambda_m$,
\begin{equation}\label{eq4.2}
\begin{aligned}
&\theta(x,\lambda_m)-\theta(0,\lambda_m)\\
&=x-\frac{1}{\lambda_m}\int_0^xq(t)
(\frac{r(t,\lambda)}{R(t,\lambda)})^{p-1}S_p
(\lambda^{1/p}\theta(t,\lambda))S_p(\lambda^{1/p}\phi(t,\lambda))^{(p-1)}dt.
\end{aligned}
\end{equation}
Letting $x=1$, by Theorem \ref{thm1.2}, Lemma \ref{lem2.4} and the
initial condition \eqref{eq1.2}, one obtains
\begin{equation}\label{eq4.3}
\frac{m\pi_p}{\lambda_m^{1/p}}=1+O(\frac{1}{\lambda_m^{\frac{p-1}{p}}});
\end{equation}
i.e.,
\begin{equation}\label{eq4.4}
\lambda_m^{1/p}=m\pi_p+O(\frac{1}{m^{p-1}}).
\end{equation}
Again, integrating \eqref{eq2.11} over $[0,x]$, with the associated
$\lambda_m$, and applying \eqref{eq4.4}, one gets
\begin{equation}\label{eq4.5}
\lambda_m^{1/p}\theta(x,\lambda_m)=m\pi_px+O(\frac{1}{m^{p-1}}).
\end{equation}
Similarly, from \eqref{eq2.13}, we  obtain
\begin{equation}\label{eq4.6}
\lambda_m^{1/p}\phi(x,\lambda_m)=m\pi_px+O(\frac{1}{m^{p-1}}).
\end{equation}
Hence, by \eqref{eq4.5},
\begin{equation} \label{eq4.7}
S_p(\lambda_m^{1/p}\theta(x,\lambda_m))
=S_p(m\pi_px)+S_p'(m\pi_px)O(\frac{1}{m^{p-1}})+o(\frac{1}{m^{p-1}});
\end{equation}
i.e.,
\begin{equation} \label{eq4.8}
S_p(\lambda_m^{1/p}\theta(x,\lambda_m))=S_p(m\pi_px)+o(1).
\end{equation}
And the same asymptotic formula is true for 
$S_p(\lambda_m^{1/p}\phi(x,\lambda_m))$. 
Now substituting \eqref{eq4.8} into \eqref{eq4.2} and taking $x=1$, one obtains
\begin{equation}
\begin{aligned}
\frac{m\pi_p}{\lambda_m^{1/p}}
&= 1-\frac{1}{\lambda_m}\int_0^1q(t)|S_p(m\pi_pt)|^pdt+o(\frac{1}{\lambda_m}) \\
&= 1-\frac{1}{p\lambda_m}\int_0^1q(t)dt-\frac{1}{\lambda_m}
\int_0^1q(t)[|S_p(m\pi_pt)|^p-\frac{1}{p}]dt+o(\frac{1}{\lambda_m}),
\end{aligned}\label{eq4.9}
\end{equation}
for sufficiently large $m$.
By a generalized Riemann-Lebesgue lemma, the asymptotic estimate \eqref{eq4.1} 
is valid.
\end{proof}

Hence, the asymptotic formula for $\lambda_m$ is
\begin{equation}
\label{eq4.10}
\lambda_m=(m\pi_p)^p+\int_0^1q(t)dt+o(1).
\end{equation}
Next we derive the asymptotic formula of the nodal length.

\begin{lemma} \label{lem4.3}
For $m\to \infty$, the nodal length of the solution
$u(x,\lambda_m)$ satisfies
\begin{equation}
\label{eq4.11}
\ell_i^{(m)}=\frac{1}{m}-\frac{1}{pm^{p+1}\pi_p^p}\int_0^1q(t)dt
+\frac{1}{(m\pi_p)^p}\int_{x_i^{(m)}}^{x_{i+1}^{(m)}}q(t)|S_p(m\pi_p t)|^pdt
+o(\frac{1}{m^{p+1}}).
\end{equation}
\end{lemma}

\begin{proof}
Letting $\lambda=\lambda_m$ and integrating \eqref{eq2.11} 
from $x_i^{(m)}$ to $x_{i+1}^{(m)}$, we obtain
\begin{equation} \label{eq4.12}
\frac{\pi_p}{\lambda_m^{1/p}}=\ell_i^{(m)}-\int_{x_i^{(m)}}^{x_{i+1}^{(m)}}
\frac{q(t)}{\lambda_m} (\frac{r(t,\lambda_m)}{R(t,\lambda_m)})^{p-1}S_p
(\lambda_m^{1/p}\theta(t,\lambda_m))
S_p(\lambda_m^{1/p}\phi(t,\lambda_m))^{(p-1)}dt.
\end{equation}
By the parameter estimates \eqref{eq4.1}, we have
\begin{equation}\label{eq4.13}
\frac{1}{\lambda_m^{1/p}}=\frac{1}{m\pi_p}-\frac{1}{p(m\pi_p)^{p+1}}
\int_0^1q(t)dt+o(\frac{1}{m^{p+1}}).
\end{equation}
Substituting \eqref{eq2.17}, \eqref{eq4.5}-\eqref{eq4.6}, \eqref{eq4.10} 
and \eqref{eq4.13} into \eqref{eq4.12}, one can obtain \eqref{eq4.11}.
\end{proof}

As in the proof of Lemma \ref{lem4.3}, one can obtain the
asymptotic estimate, for the nodal points $x_{i}^{(m)}$,
\begin{equation}
\label{eq4.14}
x_i^{(m)}=\frac{i}{m}-\frac{i}{pm^{p+1}\pi_p^p}\int_0^1q(t)dt
+\frac{1}{(m\pi_p)^p}\int_0^{x_i^{(m)}}q(t)|S_p(m\pi_p t)|^pdt
+o(\frac{1}{m^p}),
\end{equation}
which show the existence of a dense subset of nodal points in $[0,1]$.


\begin{proof}[Proof of Theorem \ref{thm1.3}]
For any $x\in (0,1)$, write $i_m(x)=i_m$ for the sake of
simplicity. Recall an easy identity,
$$
(S_p(t)S_p'(t)^{(p-1)})'=1-p|S_p(t)|^p.
$$ 
It follows from the mean value theorem for integrals, \eqref{eq4.14}, 
the above identity  and a change of variables.
Then,
\begin{equation}
\begin{aligned}
\int_{x_{i_m}^{(m)}}^{x_{i_m+1}^{(m)}}q(t)|S_p(m\pi_p t)|^pdt
&= \frac{q(x)}{m\pi_p}\int_{m\pi_p x_{i_m}^{(m)}}^{m\pi_p x_{i_m+1}^{(m)}}
|S_p(\sigma)|^pd\sigma  \\
&= \frac{q(x)}{m\pi_p}\int_0^{\pi_p}|S_p(\sigma)|^pd\sigma (1+o(1))  \\
&= \frac{q(x)}{m\pi_p}\int_0^{\pi_p}[\frac{1}{p}-\frac{1}{p}
 (S_p(\sigma)S_p'(\sigma)^{(p-1)})']d\sigma (1+o(1)) \\
&= \frac{q(x)}{pm}(1+o(1)). 
\end{aligned}\label{eq4.15}
\end{equation}
Substituting \eqref{eq4.15} into \eqref{eq4.11}, one obtains
\begin{equation}\label{eq4.16}
p(m\pi_p)^p(m\ell_{i_m}^{(m)}-1)=q(x)-\int_0^1q(t)dt+o(1).
\end{equation}
Therefore, the asymptotic formula is valid.
\end{proof}

\subsection*{Acknowledgements}
  This research was partially supported 
by the National Science Council of
Taiwan, under contract  NSC 102-2115-M-507 -001.

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\end{document}