\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 $$\label{eq1.1} \begin{gathered} (u'(x)^{(p-1)})'+(p-1)\lambda u(x)^{(p-1)}-(p-1)q(x)v(x)^{(p-1)}=0,\\ (v'(x)^{(p-1)})'+(p-1)\lambda v(x)^{(p-1)}+(p-1)q(x)u(x)^{(p-1)}=0, \end{gathered}$$ with the initial conditions $$\label{eq1.2} u(0)=v(0)=0,~u'(0)=v'(0)=\lambda^{1/p},$$ 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 $$\label{eq1.3} \begin{gathered} u''(x)+\lambda u(x)-q(x)v(x)=0,\\ v''(x)+\lambda v(x)+q(x)u(x)=0, \end{gathered}$$ 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 $$\label{eq1.6} q(x)-\int_0^1q(t)dt=\lim_{m\to \infty}[p(m\pi_p)^p(m\ell_{i_m(x)}^{(m)}-1)],$$ 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 $$\label{eq2.1} W=0\quad\text{for }|x-x_0|\leq \frac{1}{K},\quad x\in I.$$ \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 $$\label{eq2.2} \begin{gathered} u'=U^{(p^*-1)},\\ U'=(p-1)[qv^{(p-1)}-\lambda u^{(p-1)}],\\ v'=V^{(p^*-1)},\\ V'=-(p-1)[qu^{(p-1)}-\lambda v^{(p-1)}], \end{gathered}$$ 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 $$\label{eq2.3} \frac{\lambda^{1/p}}{2}|x-0|<|u(x)|,~|v(x)|<2\lambda^{1/p}|x-0|\quad \text{for }x\in I.$$ 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{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} It follows from the mean value theorem, that for $a_1$ and $a_2$ of the same sign, $$\label{eq2.5} a_1^{(p-1)}-a_2^{(p-1)}=(p-1)(a_1-a_2)|\bar{a}|^{p-2}\,,$$ 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 $$\label{eq2.6} |S_p(x)|^{p}+|S'_p(x)|^{p}=1,$$ and $$\label{eq2.7} (S_p)''= \frac{-S_p^{(p-1)} S'_p}{(S'_p)^{(p-1)}} =\frac{-S_p^{(p-1)}}{|S'_p|^{p-2}}.$$ 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 $$\label{eq2.10} \lambda R(x,\lambda)^p=\lambda|u(x,\lambda)|^p+|u'(x,\lambda)|^p,\quad \lambda r(x,\lambda)^p=\lambda|v(x,\lambda)|^p+|v'(x,\lambda)|^p,$$ 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 $$\label{eq2.15} \theta(0,\lambda)=\phi(0,\lambda)=0,~~and~~R(0,\lambda)=r(0,\lambda)=1.$$ 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 $$\label{eq2.16} 2\exp[-c_1\lambda^{\frac{1-p}{p}}x]\leq R(x,\lambda)^{p-1}+r(x,\lambda)^{p-1}\leq 2\exp[c_2\lambda^{\frac{1-p}{p}}x],$$ where $c_1,~c_2$ are some positive constants. (ii) For fixed $x>0$ and sufficiently large $\lambda$, we have $$\label{eq2.17} \frac{r(x,\lambda)}{R(x,\lambda)}=1+o(1).$$ 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 $$\label{eq2.18} y(x)^{p-1}\leq 1+o(1)$$ 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)] $$\label{eq3.2} \lim_{\lambda \to \infty}\lambda^{1/p}\theta(1,\lambda)=\infty.$$ \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 $$\label{eq3.1} \theta'(x_n,\lambda)=1>0.$$ This proves (ii). For (iii), integrating \eqref{eq2.11} over $[0,1]$ and applying (i), one obtains $$\label{eq3.3} \lambda^{1/p}\theta(1,\lambda) =\lambda^{1/p}-\lambda^{\frac{1-p}{p}}\int_0^1q(t) (\frac{r(t,\lambda)}{R(t,\lambda)})^{p-1}S_p (\lambda^{1/p}\theta(t,\lambda))S_p(\lambda^{1/p}\phi(t,\lambda))^{(p-1)}dt.$$ 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 $$\label{eq4.1} \lambda_m^{1/p}=m\pi_p+\frac{1}{p(m\pi_p)^{p-1}} \int_0^1q(t)dt+o(\frac{1}{m^{p-1}})$$ as $m\to \infty$. \end{theorem} \begin{proof} First, integrating \eqref{eq2.11} over $[0,x]$, with the associated $\lambda_m$, \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} Letting $x=1$, by Theorem \ref{thm1.2}, Lemma \ref{lem2.4} and the initial condition \eqref{eq1.2}, one obtains $$\label{eq4.3} \frac{m\pi_p}{\lambda_m^{1/p}}=1+O(\frac{1}{\lambda_m^{\frac{p-1}{p}}});$$ i.e., $$\label{eq4.4} \lambda_m^{1/p}=m\pi_p+O(\frac{1}{m^{p-1}}).$$ Again, integrating \eqref{eq2.11} over $[0,x]$, with the associated $\lambda_m$, and applying \eqref{eq4.4}, one gets $$\label{eq4.5} \lambda_m^{1/p}\theta(x,\lambda_m)=m\pi_px+O(\frac{1}{m^{p-1}}).$$ Similarly, from \eqref{eq2.13}, we obtain $$\label{eq4.6} \lambda_m^{1/p}\phi(x,\lambda_m)=m\pi_px+O(\frac{1}{m^{p-1}}).$$ Hence, by \eqref{eq4.5}, $$\label{eq4.7} S_p(\lambda_m^{1/p}\theta(x,\lambda_m)) =S_p(m\pi_px)+S_p'(m\pi_px)O(\frac{1}{m^{p-1}})+o(\frac{1}{m^{p-1}});$$ i.e., $$\label{eq4.8} S_p(\lambda_m^{1/p}\theta(x,\lambda_m))=S_p(m\pi_px)+o(1).$$ 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{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} 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 $$\label{eq4.10} \lambda_m=(m\pi_p)^p+\int_0^1q(t)dt+o(1).$$ 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 $$\label{eq4.11} \ell_i^{(m)}=\frac{1}{m}-\frac{1}{pm^{p+1}\pi_p^p}\int_0^1q(t)dt +\frac{1}{(m\pi_p)^p}\int_{x_i^{(m)}}^{x_{i+1}^{(m)}}q(t)|S_p(m\pi_p t)|^pdt +o(\frac{1}{m^{p+1}}).$$ \end{lemma} \begin{proof} Letting $\lambda=\lambda_m$ and integrating \eqref{eq2.11} from $x_i^{(m)}$ to $x_{i+1}^{(m)}$, we obtain $$\label{eq4.12} \frac{\pi_p}{\lambda_m^{1/p}}=\ell_i^{(m)}-\int_{x_i^{(m)}}^{x_{i+1}^{(m)}} \frac{q(t)}{\lambda_m} (\frac{r(t,\lambda_m)}{R(t,\lambda_m)})^{p-1}S_p (\lambda_m^{1/p}\theta(t,\lambda_m)) S_p(\lambda_m^{1/p}\phi(t,\lambda_m))^{(p-1)}dt.$$ By the parameter estimates \eqref{eq4.1}, we have $$\label{eq4.13} \frac{1}{\lambda_m^{1/p}}=\frac{1}{m\pi_p}-\frac{1}{p(m\pi_p)^{p+1}} \int_0^1q(t)dt+o(\frac{1}{m^{p+1}}).$$ 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 $x_{i}^{(m)}$, $$\label{eq4.14} x_i^{(m)}=\frac{i}{m}-\frac{i}{pm^{p+1}\pi_p^p}\int_0^1q(t)dt +\frac{1}{(m\pi_p)^p}\int_0^{x_i^{(m)}}q(t)|S_p(m\pi_p t)|^pdt +o(\frac{1}{m^p}),$$ 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{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} Substituting \eqref{eq4.15} into \eqref{eq4.11}, one obtains $$\label{eq4.16} p(m\pi_p)^p(m\ell_{i_m}^{(m)}-1)=q(x)-\int_0^1q(t)dt+o(1).$$ 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. \begin{thebibliography}{99} \bibitem{BB} J. Benedikt, P. Girg; \emph{Pr\"ufer transformation for the $p$-Laplacian}, Electronic J.\ Differential Equations, Conference \textbf{18} (2010), 1-13. \bibitem{bd} P. Binding, P. 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