\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2021 (2021), No. 40, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2021 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2021/40\hfil Radially symmetric $p$-Laplacian equations] {Existence of sign-changing solutions for radially symmetric $p$-Laplacian equations with various potentials} \author[W.-C. Wang \hfil EJDE-2021/40\hfilneg] {Wei-Chuan Wang} \address{Wei-Chuan Wang \newline Department of Civil Engineering and Engineering Management, Center for General Education, National Quemoy University, Kinmen, Taiwan 892, ROC} \email{wangwc@nqu.edu.tw, wangwc72@gmail.com} \thanks{Submitted September 8, 2020. Published May 7, 2021.} \subjclass{34A12, 34B15, 34A55} \keywords{Nonlinear $p$-Laplacian equation; sign-changing solution; \hfill\break\indent blow-up solution} \begin{abstract} In this article, we study the nonlinear equation $$\big(r^{n-1}|u'(r)|^{p-2}u'(r)\big)'+r^{n-1}w(r)|u(r)|^{q-2}u(r)=0,$$ where $q>p>1$. For positive potentials ($w>0$), we investigate the existence of sign-changing solutions with prescribed number of zeros depending on the increasing initial parameters. For negative potentials, we deduce a finite interval in which the positive solution will tend to infinity. The main methods using in this work are the scaling argument, Pr\"{u}fer-type substitutions, and some integrals involving the $p$-Laplacian. \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 purpose of this article is to investigate some properties related to the radially symmetric problem for \begin{equation} \label{eq1.20} -\Delta_p u=g(|x|,u),\quad\text{on }\Omega\subseteq \mathbb{R}^n, \end{equation} where $\Delta_p u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$ and $n\geq 1$. The $p$-Laplacian operator $\Delta_p u$ itself has the originally physical meaning, and also can be treated as a generalization of the Laplacian operator. The quantity $p$ is a characteristic of the medium of non-Newtonian fluids or nonlinear diffusion problems. 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, we refer the readers to \cite{D85,D01,la,li,PR74,SFT20,SW91,Takeuchi12} and their references. Also, some results for radial solutions related to \eqref{eq1.20} have been obtained in \cite{BR04,cgy13,gmy06,JYW,KYY96,KYY98,RW97,MRW97,WC14,WC17}. In \cite{BR04}, the authors extended the eigenvalue theory to the radially symmetric $p$-Laplcian in $\mathbb{R}^n$ corresponding to Weyl's limit point and Weyl's limit circle theories in the case $p=2$. In \cite{gmy06}, the authors determined the structure of positive radial solutions related to \eqref{eq1.20}. In particular, Kabeya et al.\ \cite{KYY96} deduced the existence of radially {\em fast-decay} solutions of \eqref{eq1.20} with prescribed number of zeros in $(0,\infty)$. They also considered further boundary problems with similar results. More recently, the authors \cite{cgy13} derived the existence of radial solutions having prescribed number of sign changes on $(0,\infty)$ for $n\geq p>1$. Another direction is to deduce the existence of {\sl blow-up} solutions. A solution $u$ of \eqref{eq1.20} is called {\sl boundary blow-up} if $\lim_{k\to \infty}u(x_k)=\infty$ for each sequence in $\Omega$ which converges to a point on $\partial \Omega$. For $p=2$ this type of problem has a long history. Bieberbach \cite{B16} and Rademacher \cite{Ra43} started to study this theme. Bieberbach was motivated by problems in geometry, Rademacher by a problem in mathematical physics. Later, Keller \cite{K57} derived a well-known result. The author gave necessary and sufficient conditions on the the growth of $g(u)$ at infinity to guarantee that such solutions exist. Under more restrictive assumptions, but for general $N$-dimensional domains, the blow-up problem has been studied by Bandle and Marcus \cite{BM92,BM95}, Lazer and McKenna \cite{LM941,LM942} for $p=2$ and by Diaz and Letelier \cite{DL93} for general $p>1$. For a nonlinear radial $p$-Laplacian, Reichel and Walter \cite{RW97} employed a strong comparison principle to develop some properties of boundary blow-up solutions. Also, McKenna {\sl et al.} \cite{MRW97} have treated the radial case for $g(u)=|u|^q$ and general $p>1$. We mention a part of work in \cite{MRW97}, where the authors developed the existence of blow-up solutions for the one-dimensional case ($n=1$) first and applied a crucial inequality to obtain the existence of blow-up solutions to the general case ($n\geq 1$). By considering radially symmetric solutions to \eqref{eq1.20}, we are led to study the nonlinear problem \begin{gather} \left(r^{n-1}|u'(r)|^{p-2}u'(r)\right)'+r^{n-1}w(r)|u(r)|^{q-2}u(r)=0,\label{eq1.1}\\ u(0)=\alpha>0,\quad u'(0)=0,\label{eq1.2} \end{gather} where $r=|x|$ and $'=\frac{d}{dr}$. Motivated by the previous results \cite{cgy13,KYY96,KK09,MRW97,W19,WC17}, we study two issues related to \eqref{eq1.1}-\eqref{eq1.2}. When $w>0$, we investigate the existence of sign-changing solutions with the prescribed number of zeros in a finite interval. For this issue, we consider a right endpoint condition \begin{equation}\label{eq1.3} u(1)=0 \end{equation} for the sake of simplicity. We denote the solution of \eqref{eq1.1}-\eqref{eq1.2} by $u(r;\alpha)$. The following results (Theorem \ref{th1.1} and \ref{th1.2}) can be treated as the {\sl Sturmian theory} which is related to the existence of solutions having prescribed number of zeros. Some results closely relevant to this issue can be referred to \cite{cgy13,KYY96,KYY98,W19,WC14,WC17} and their bibliographies. In this article we to consider a wide class of potential functions and employ the interesting methods, scaling arguments and Pr\"{u}fer-type substitutions, to achieve the goal. The method seems to be classical and can be found in \cite{KYY96,W19,Y94}. However, this extension is not trivial and need more subtle arguments in the analysis of generalized polar coordinates. Throughout this paper we assume the following conditions: \begin{itemize} \item[(A1)] $q>p$; \item[(A2)] $w\in C^1(\mathbb{R})$, and $|w|\geq \delta_1$ on $[0,\infty)$ for some $\delta_1>0$; \item[(A3)] $\kappa:=\max \big\{\frac{|w'(r)|}{w(r)}:r\in [0,1]\big\}$; \item[(A4)] $p>n$, $w_1:=\max\{|w(r)|:r\in [0,1]\}$ and $\mathbb{K}:=w_1\big(\frac{w(0)}{\delta_1}e^{\kappa}\big)^{\frac{q-p}{q}}$. \end{itemize} Note that (A1)--(A3) hold for the existence of solutions with prescribed number of zeros to the two-endpoint boundary condition case (Theorem \ref{th1.1}). (A4) is added to the case of multi-point boundary conditions (Theorem \ref{th1.2}). Here is our first result. \begin{theorem}\label{th1.1} Assume that $w>0$ in \eqref{eq1.1} and {\rm (A1)--(A3)} hold. Then there exists a strictly increasing sequence of positive numbers $\{\alpha_n\}_{n=1}^{\infty}$, such that the solution $u(r;\alpha_n)$ is a solution to the BVP \eqref{eq1.1}-\eqref{eq1.3}. Moreover, $u(r;\alpha_n)$ has exactly $n-1$ zeros in $(0,1)$ for $n\in \mathbb{N}$. \end{theorem} \begin{remark} \label{th1.28} \rm The initial parameter corresponding to the solution with the prescribed number of zeros is not unique usually. For the case of $n=1$, the author \cite{T09} showed that such a sequence is unique, provided $w\in C^2(\mathbb{R})$, $([w(r)]^{-1/p})''\leq 0$ on $\mathbb{R}$ and $10$ in \eqref{eq1.1} and {\rm (A1)--(A4)} hold. Also assume that \begin{equation} \label{eq1.5} 1-\sum_{i=1}^{d}|\tau_i|>0. \end{equation} Then there exists a strictly increasing sequence of positive initial values $\{\alpha_n\}_{n=1}^{\infty}$, such that the solution $u(r;\alpha_n)$ is a solution to the multi-point boundary value problem \eqref{eq1.1}-\eqref{eq1.2} and \eqref{eq1.4}. Moreover, $u(r;\alpha_n)$ has exactly $n$ or $n+1$ zeros in $(0,1)$ for $n\in \mathbb{N}$. \end{theorem} Next, the counterpart of this paper is to investigate the negative potential case ($w<0$). We intend to discuss the blow-up solutions of \eqref{eq1.1}-\eqref{eq1.2}. Motivated by the interesting idea as in \cite{MRW97} and under some minor assumption of $w$, we plan to discuss the existence of blow-up solutions in a finite interval. We also derive such an interval associated with the initial parameter $\alpha$ precisely by analyzing some integrals involving the $p$-Laplcian. \begin{theorem} \label{th1.3} For nonincreasing $w$ with $w<0$, assume that {\rm (A1)} and {\rm (A2)} hold. The nonlinear problem \eqref{eq1.1}-\eqref{eq1.2} has at least one positive blow-up solution $u(r;\alpha)$ in $(0,R_\alpha)$, where \begin{equation} \label{eq1.6} R_\alpha:=\sqrt[p]{\frac{nq(p-1)}{p\delta_1\alpha^{q-p}}} \Big(\frac{1}{p-1}(2^p-1)^{\frac{p-1}{p}}+\frac{2p}{q-p}\Big). \end{equation} That is, the positive solution $u(r;\alpha)$ tends to infinity as $r$ tends to $R\leq R_\alpha$. Moreover, such a positive blow-up solution can not occur when the problem is considered in a finite interval as $q\leq p$. \end{theorem} The article is organized as follows. Some elementary properties related to \eqref{eq1.1}-\eqref{eq1.2} and the proofs of Theorem \ref{th1.1} and \ref{th1.2} will be given in Section 2. The existence of blow-up solutions (Theorem \ref{th1.3}) will be represented in Section 3. \section{Preliminaries and Sturmian theory} First, the existence of solutions to \eqref{eq1.1}-\eqref{eq1.2} is valid and can be found in \cite{gmy06,KYY96,KYY98,MRW97,RW97,WC14,WC17}. Here we quote the following to coincide with our setting. Also the regularity requirements for a solution $u$ are $u\in C^1$ and $r^{n-1}|u'|^{p-2}u'\in C^1$. \begin{theorem}[{\cite [Theorem EUCD]{MRW97},\cite[Theorems 1 and 4]{RW97},\cite[Corollary 2.3]{WC14}}] \label{th2.1} \hfil Assume conditions {\rm (A1)} and {\rm (A2)} hold. For the positive potential function ($w>0$), there exists a unique local solution $u(r;\alpha)$ of \eqref{eq1.1}-\eqref{eq1.2}. Moreover, the solution $u(r;\alpha)$ can be extended to the whole real axis. \end{theorem} Now, we introduce a Pr\"{u}fer-type substitution for the solution $u(r;\alpha)$ of \eqref{eq1.1}-\eqref{eq1.2} by using the generalized sine function $S_p(r)$. 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 reader's convenience. The function $S_p$ satisfies \begin{gather} \label{eq2.1} |S_{p}'(r)|^{p}+\frac{|S_{p}(r)|^{p}}{p-1}=1, \\ \label{eq2.2} (|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 $S_{p}(\frac{\pi_{p}}{2}+n\pi_p)=(-1)^{n}$, $S'_{p}(n\pi_p)=(-1)^{n}$, $S_{p}(n\pi_p)=0$ and $S'_{p}(\frac{\pi_{p}}{2}+n\pi_p)=0$ for $n\in \mathbb{Z}$. With the help of the generalized sine function, we introduce the phase-plane coordinates $\rho>0$ and $\theta$ for the solution $u(r;\alpha)$ of \eqref{eq1.1}-\eqref{eq1.2} as follows: \begin{equation} \label{eq2.3} \begin{gathered} |u(r;\alpha)|^{p-2}u(r;\alpha) =\rho(r;\alpha)|S_p(m\theta(r;\alpha))|^{p-2}S_p(m\theta(r;\alpha)),\\ r^{n-1}|u'(r;\alpha)|^{p-2}u'(r;\alpha) =g \rho(r;\alpha)|S_p'(m\theta(r;\alpha))|^{p-2}S_p'(m\theta(r;\alpha)), \end{gathered} \end{equation} with \begin{equation} \label{eq2.4} m\theta(0;\alpha)=\frac{\pi_p}{2}\quad\text{and}\quad \rho(0;\alpha)=\Big(\frac{\alpha}{p-1}\Big)^{p-1}, \end{equation} where $m$ and $g$ are some positive constants that will be specified later. Then \begin{gather} \label{eq2.5} \left(g\rho(r;\alpha)\right)^{\frac{p}{p-1}} =\frac{g^{\frac{p}{p-1}}}{p-1}|u(r;\alpha)|^p+r^{\frac{p(n-1)}{p-1}}|u'(r;\alpha)|^p,\\ \label{eq2.20} \frac{r^{n-1}|u'|^{p-2}u'}{|u|^{p-2}u}=\frac{g|S'_p|^{p-2}S'_p}{|S_p|^{p-2}S_p}. \end{gather} Differentiating both sides of \eqref{eq2.20} with respect to $r$ and employing \eqref{eq1.1} and \eqref{eq2.1}-\eqref{eq2.3}, one can obtain \begin{equation} \begin{aligned} &mg\theta'(r;\alpha) \\ &=\frac{r^{n-1}}{p-1}w(r)|u(r;\alpha)|^{q-p}|S_p(m\theta(r;\alpha))|^p+ g^{\frac{p}{p-1}}\left(r^{\frac{1-n}{p-1}}|S_p'(m\theta(r;\alpha))|^p\right), \end{aligned} \label{eq2.6} \end{equation} and \begin{equation}\label{eq2.7} \begin{aligned} \frac{\rho'(r;\alpha)}{\rho(r;\alpha)} &=\big[r^{\frac{1-n}{p-1}}g^{\frac{1}{p-1}}-r^{n-1}g^{-1}w(r)|u(r;\alpha)|^{q-p}\big] \\ &\quad\times |S_p(m\theta(r;\alpha))|^{p-2}S_p(m\theta(r;\alpha))S_p'(m\theta(r;\alpha)). \end{aligned} \end{equation} Employing the above, one can conclude that $u(r;\alpha)$ is the solution of \eqref{eq1.1}-\eqref{eq1.2} if and only if $\{\theta(r;\alpha),\rho(r;\alpha)\}$ satisfies \eqref{eq2.6}-\eqref{eq2.7} and \eqref{eq2.4}. Motivated by a similar idea in \cite{W19} (or \cite{KYY96}), we introduce the scaling argument. Assume that $\{\alpha_i\}$ is a positively and strictly increasing sequence which tends to infinity, and define the sequence $\{\mu_i\}$ to satisfy the following relation: \begin{equation} \label{eq2.8} \mu_i=\max \{x>0: x^pw(x)=\alpha_i^{p-q}\} \end{equation} for $i\in \mathbb{N}$. Note that $t^pw(t)=O(t^p)$ and $q>p$. Hence, if $\{\alpha_i\}$ is a positively increasing sequence which tends to infinity, then the corresponding sequence $\{\mu_i\}$ satisfying \eqref{eq2.8} decreases to zero. Then, the scaled function $v_i$ is defined by \begin{equation} \label{eq2.9} v_i(r)=\frac{u(\mu_i r;\alpha_i)}{\alpha_i}. \end{equation} By \eqref{eq1.1} and \eqref{eq2.8}-\eqref{eq2.9}, a direct calculation yields that $v_i$ satisfies \begin{gather} \left(r^{n-1}|v_i'(r)|^{p-2}v_i'(r)\right)'+r^{n-1}\frac{w(\mu_ir)}{w(\mu_i)}|v_i(r)|^{q-2}v_i(r) =0,\label{eq2.10}\\ v_i(0)=1,\quad v_i'(0)=0. \label{eq2.11} \end{gather} From Theorem \ref{th2.1} and for each fixed $i$, the function $v_i$ which solves \eqref{eq2.10}-\eqref{eq2.11} exists on $[0,\mu_i^{-1}]$. By the assumption on $w$, $\frac{w(\mu_ir)}{w(\mu_i)}\to 1$ as $\mu_i \to 0$ uniformly on any bounded interval in $[0,\infty)$. Thus $v_i$ converges to a function $V$ uniformly on any bounded interval in $[0,\infty)$, where $V$ solves \begin{equation}\label{eq2.12} \begin{gathered} \big(r^{n-1}|V'|^{p-2}V'\big)'+r^{n-1}|V|^{q-2}V=0,\\ V(0)=1,\quad V'(0)=0. \end{gathered} \end{equation} Next we define an energy functional for the scaled function $v_i(r)$ and deduce an a priori estimate for this energy. Let a functional $E[v_i](r,\alpha)$ be defined by \begin{equation} \label{eq2.17} E[v_i](r,\alpha)\equiv \frac{|v_i'(r)|^p}{p}+\frac{w(\mu_i r)}{q(p-1)w(\mu_i)}|v_i(r)|^q \end{equation} with \begin{equation} \label{eq2.30} E[v_i](0,r)=\frac{w(0)}{q(p-1)w(\mu_i)}. \end{equation} Note that from \eqref{eq2.10}, one can obtain the following equation by multiplying $v_i'(r)$, \begin{align*} &(n-1)r^{n-2}|v_i'(r)|^{p}+(p-1)r^{n-1}|v_i'(r)|^{p-2}v_i'(r)v_i''(r) \\ &+r^{n-1}\frac{w(\mu_ir)}{w(\mu_i)}|v_i|^{q-2}v_i(r)v_i'(r)=0. \end{align*} i.e., for $r\neq 0$, \begin{equation}\label{eq2.18} \frac{-|v_i'(r)|^p}{r}=\frac{(p-1)}{(n-1)}|v_i'(r)|^{p-2}v_i'(r)v_i''(r) +\frac{w(\mu_ir)}{(n-1)w(\mu_i)}|v_i(r)|^{q-2}v_i(r)v_i'(r). \end{equation} Since there exists a unique solution in $[0,\mu_i^{-1}]$, by Theorem \ref{th2.1} and \eqref{eq2.9}, all the terms on the right-hand side of \eqref{eq2.18} are bounded in $[0,\mu_i^{-1}]$ and $v_i'$ tends to zero as $r$ vanishes by the initial condition. This implies that \begin{equation}\label{eq2.19} \lim_{r\to 0^{+}}\frac{|v_i'(r)|^p}{r}=0 \end{equation} and the term $\frac{|v_i'(r)|^p}{r}$ is bounded in $[0,\mu_i^{-1}]$. Then, it follows from \eqref{eq2.10} and \eqref{eq2.18} that for $r\in (0,\mu_i^{-1}]$, \begin{equation} \begin{aligned} \frac{d}{dr}E[v_i](r,\alpha) &=E[v_i]'(r,\alpha) \\ &= v_i'(r)\Big[|v_i'(r)|^{p-2}v_i''(r)+\frac{w(\mu_ir)}{(p-1)w(\mu_i)}|v_i(r)|^{q-2}v_i(r)\Big]\\ &\quad + \frac{\mu_iw'(\mu_i r)}{q(p-1)w(\mu_i)}|v_i(r)|^q \\ &=-\frac{(n-1)}{(p-1)}\cdot \frac{|v_i'(r)|^p}{r} +\frac{\mu_iw'(\mu_i r)}{q(p-1)w(\mu_i)}|v_i(r)|^q \\ &\leq \mu_i\kappa \frac{|u'(r)|^p}{p}+\mu_i\kappa\frac{w(\mu_i r)}{q(p-1)w(\mu_i)}|v_i(r)|^q \\ &= \mu_i\kappa E[v_i](r,\alpha), \end{aligned} \label{eq2.99} \end{equation} where $\kappa=\max \big\{\frac{|w'(\mu_ir)|}{w(\mu_ir)}:r\in [0,\mu_i^{-1}]\big\} =\max \big\{\frac{|w'(r)|}{w(r)}:r\in [0,1]\big\}$. In particular, the above inequality holds for the whole interval $[0,\mu_i^{-1}]$ by \eqref{eq2.19}. Hence, for any $r\in [0,\mu_i^{-1}]$ \begin{equation}\label{eq2.31} E[v_i](r,\alpha)\leq E[v_i](0,\alpha)e^{\mu_i\kappa r} =\frac{w(0)e^{\mu_i\kappa r}}{q(p-1)w(\mu_i)} \end{equation} by \eqref{eq2.99} and \eqref{eq2.30}. This means that both $v_i(r;\alpha)$ and $v_i'(r;\alpha)$ are bounded as long as the solution exists. \begin{proposition} \label{th2.2} Assume the conditions {\rm (A1)} and {\rm (A2)} hold. Let $\mu_i$ be defined as in \eqref{eq2.8}. Then, $v_i$ satisfies \begin{equation} \label{eq2.32} |v_i(r)|\leq \Big(\frac{w(0)}{w(\mu_ir)}e^{\kappa}\Big)^{1/q} \leq \Big(\frac{w(0)}{\delta_1}e^{\kappa}\Big)^{1/q} \end{equation} for $r\in [0,\mu_i^{-1}]$, where $\kappa=\max \big\{\frac{|w'(r)|}{w(r)}:r\in [0,1]\big\}$. Moreover, the function $V$ solving \eqref{eq2.12} satisfies the uniform boundedness, \begin{equation}\label{eq2.33} |V(r)|\leq e^{\kappa/q} \end{equation} on any bounded interval in $[0,\infty)$. \end{proposition} Now we prove the result related to the Sturmian theory. \begin{proof}[Proof of Theorem \ref{th1.1}] From the Pr\"{u}fer angular equation \eqref{eq2.6}, one has \begin{equation} \begin{aligned} & mg\theta'(r;\alpha) \\ &=\frac{r^{n-1}}{p-1}w(r)|u(r;\alpha)|^{q-p}|S_p(m\theta(r;\alpha))|^p+g^{\frac{p}{p-1}} r^{\frac{1-n}{p-1}}|S_p'(m\theta(r;\alpha))|^p. \end{aligned} \label{eq2.13} \end{equation} Applying the scaling argument \eqref{eq2.9} and choosing $m=g^{\frac{1}{p-1}}=\alpha^{\frac{q-p}{p}}$, the phase equation \eqref{eq2.13} can be rewritten as \begin{equation}\label{eq2.14} \begin{aligned} \theta'(r;\alpha) &=\frac{r^{n-1}}{p-1}w(r)|v(\mu^{-1}r)|^{q-p}|S_p(\alpha^{\frac{q-p}{p}}\theta(r;\alpha))|^p \\ &\quad +r^{\frac{1-n}{p-1}}|S_p'(\alpha^{\frac{q-p}{p}}\theta(r;\alpha))|^p. \end{aligned} \end{equation} Note that $|S_p(m\theta(r;\alpha))|^p$ and $|S_p'(m\theta(r;\alpha))|^p$ will not vanish at the same point by \eqref{eq2.1}. And if $S_p(m\theta(r;\alpha))$ tends to zero, $|S_p'(m\theta(r;\alpha))|$ will approach to one. Furthermore, $v(\mu^{-1}r)$ and $S_p(m\theta(r;\alpha))$ vanish at the same point by \eqref{eq2.3} and \eqref{eq2.9}. Integrating the phase equation \eqref{eq2.14} over $[0,r]$ for $r\in (0,1]$, one can obtain \begin{equation}\label{eq2.15} \begin{aligned} m\theta(r;\alpha) &=\frac{\pi_p}{2}+m\int_0^r\Big(\frac{s^{n-1}}{p-1}w(s)|v(\mu^{-1}s)|^{q-p} |S_p(m\theta(s;\alpha))|^p \\ &\quad +s^{\frac{1-n}{p-1}}|S_p'(m\theta(s;\alpha))|^p\Big)ds, \end{aligned} \end{equation} where $m=\alpha^{\frac{q-p}{p}}$. A detailed analysis similar as in \cite[Lemma 3]{RW99} (or \cite{BR04}) shows that $m\theta(r;\alpha)-\frac{\pi_p}{2}=O(r^n)$ as $r\to 0^+$. Hence, we can observe that for any $\alpha>0$ the integral term in \eqref{eq2.15} is bounded and never vanishes by the above explanation. And the Pr\"{u}fer phase $\theta$ is continuous dependence on $\alpha$ obviously. Then, one can conclude that for $r\in (0,1]$, \begin{equation} \label{eq2.16} \lim_{\alpha\to 0}\alpha^{\frac{q-p}{p}}\theta(r;\alpha)=\frac{\pi_p}{2},\quad \lim_{\alpha\to \infty}\alpha^{\frac{q-p}{p}}\theta(r;\alpha)=\infty. \end{equation} Now by \eqref{eq2.4} and \eqref{eq2.16}, there exists an increasing sequence of of positive numbers $\{\alpha_n\}_{n=1}^{\infty}$ such that $$\alpha_n^{\frac{q-p}{p}}\theta(0;\alpha_n)=\frac{\pi_p}{2}\quad \text{and}\quad \alpha_n^{\frac{q-p}{p}}\theta(1;\alpha_n)=n\pi_p.$$ This means that $u(r;\alpha_n)$ is a solution of \eqref{eq1.1}-\eqref{eq1.3} which has exactly $n-1$ zeros in $(0,1)$. The proof is complete. \end{proof} Next, we deal with multi-point boundary conditions. \begin{proof}[Proof of Theorem \ref{th1.2}] Recall that $m=g^{\frac{1}{p-1}}=\alpha^{\frac{q-p}{p}}$ are as in the proof of Theorem \ref{th1.1}. From \eqref{eq2.16} and the continuity of $\theta(r;\alpha)$ in $\alpha$, there exist a maximal $\alpha_n$ and a minimal $\alpha_{n+1}$ such that \begin{gather} \label{eq2.21} \alpha_n^{\frac{q-p}{p}}\theta(1;\alpha_n)=(n+\frac{1}{2})\pi_p,\quad \alpha_n^{\frac{q-p}{p}}\theta(1;\alpha_{n+1})=(n+\frac{3}{2})\pi_p, \\ \label{eq2.22} (n+\frac{1}{2})\pi_p<\alpha^{\frac{q-p}{p}}\theta(1;\alpha)<(n+\frac{3}{2})\pi_p \quad\text{for }\alpha_n<\alpha<\alpha_{n+1}. \end{gather} Now by \eqref{eq2.7},\eqref{eq2.9}, (A3), (A4), and \eqref{eq2.32}, for $r\in (0,1]$ and $j=n,n+1$ one can obtain \begin{align*} \frac{\rho'(r;\alpha_j)}{\rho(r;\alpha_j)} &\geq -\alpha_j^{\frac{q-p}{p}}\big(r^{\frac{1-n}{p-1}}+r^{n-1}w(r)|v(\mu_j^{-1}r)|^{q-p}\big)\\ &\geq -\alpha_j^{\frac{q-p}{p}}\Big(r^{\frac{1-n}{p-1}}+w_1\Big(\frac{w(0)}{\delta_1} e^{\kappa}\Big)^{\frac{q-p}{q}}\Big)\\ &= -\alpha_j^{\frac{q-p}{p}}\Big(r^{\frac{1-n}{p-1}}+\mathbb{K}\Big). \end{align*} Integrating the above inequality over $[r_i,1]$ ($1\leq i\leq d$), by (A4) one can get \begin{align*} \ln \frac{\rho(1;\alpha_j)}{\rho(r_i;\alpha_j)} &\geq -\alpha_j^{\frac{q-p}{p}}\Big(\frac{p-1}{p-n}(1-r_i^{\frac{p-n}{p-1}})+\mathbb{K}(1-r_i) \Big)\\ &\geq -\alpha_j^{\frac{q-p}{p}}\Big(\frac{p-1}{p-n}+\mathbb{K}\Big) >-\alpha_j^{\frac{q-p}{p}}\mathbb{K} \end{align*} for $j=n,n+1$. Then, \begin{equation}\label{eq2.23} \rho(r_i;\alpha_j)< e^{\big(\alpha_j^{\frac{q-p}{p}}\mathbb{K}\big)}\rho(1;\alpha_j), \quad i=1,2,3,\dots,d\quad\text{and}\quad j=n,n+1. \end{equation} By the Pr\"{u}fer-type substitution \eqref{eq2.5} and \eqref{eq2.21}, one can observe that \begin{equation} \label{eq2.24} \begin{gathered} |u(r_i;\alpha_j)| \leq \sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(r_i;\alpha_j)},\\ |u(1;\alpha_j)|=\sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(1;\alpha_j)}. \end{gathered} \end{equation} for $1\leq i\leq d$ and $j=n,n+1$. Now define $$\Gamma(\alpha)=u(1;\alpha)-\sum_{i=1}^{d}\tau_i e^{-\frac{\big(\alpha^{\frac{q-p}{p}}\mathbb{K}\big)}{p-1}}u(r_i;\alpha).$$ Assume that $n=2k-1$ for $k\in \mathbb{N}$. Note that \begin{equation}\label{eq2.25} u(1;\alpha_{2k-1})=u(1;\alpha_n)<0\quad\text{and}\quad u(1;\alpha_{2k})=u(1;\alpha_{n+1})>0 \end{equation} from \eqref{eq2.21} and \eqref{eq2.3}. By applying \eqref{eq2.23}-\eqref{eq2.25} and \eqref{eq1.5}, one can obtain that \begin{align*} &\Gamma(\alpha_{2k-1})\\ &= u(1;\alpha_{2k-1})-\sum_{i=1}^{d}\tau_i e^{-\frac{\big(\alpha_{2k-1}^{\frac{q-p}{p}}\mathbb{K}\big)}{p-1}}u(r_i;\alpha_{2k-1})\\ &\leq -\sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(1;\alpha_{2k-1})} +\sum_{i=1}^{d}|\tau_i|e^{-\frac{\big(\alpha_{2k-1}^{\frac{q-p}{p}}\mathbb{K}\big)} {p-1}}\sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(r_i;\alpha_{2k-1})}\\ &<-\sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(1;\alpha_{2k-1})} \\ &\quad +\sum_{i=1}^{d}|\tau_i|e^{-\frac{\big(\alpha_{2k-1}^{\frac{q-p}{p}}\mathbb{K}\big)}{p-1}} \sqrt[p-1]{(p-1)^{\frac{p-1}{p}} e^{\big(\alpha_{2k-1}^{\frac{q-p}{p}}\mathbb{K}\big)}\rho(1;\alpha_{2k-1})}\\ &=\sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(1;\alpha_{2k-1})} \Big(-1+\sum_{i=1}^{d}|\tau_i|\Big)<0, \end{align*} and \begin{align*} &\Gamma(\alpha_{2k}) \\ &= u(1;\alpha_{2k})-\sum_{i=1}^{d}\tau_ie^{-\frac{\big(\alpha_{2k}^{\frac{q-p}{p}}\mathbb{K}\big)} {p-1}}u(r_i;\alpha_{2k})\\ &\geq \sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(1;\alpha_{2k})} -\sum_{i=1}^{d}|\tau_i| e^{-\frac{\big(\alpha_{2k}^{\frac{q-p}{p}}\mathbb{K}\big)}{p-1}} \sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(r_i;\alpha_{2k})}\\ &> \sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(1;\alpha_{2k})}\\ &\quad -\sum_{i=1}^{d}|\tau_i|e^{-\frac{\big(\alpha_{2k}^{\frac{q-p}{p}}\mathbb{K}\big)}{p-1}} \sqrt[p-1]{(p-1)^{\frac{p-1}{p}}e^{\big(\alpha_{2k}^{\frac{q-p}{p}}\mathbb{K}\big)} \rho(1;\alpha_{2k})}\\ &=\sqrt[p-1]{(p-1)^{\frac{p-1}{p}}\rho(1;\alpha_{2k})} \Big(1-\sum_{i=1}^{d}|\tau_i|\Big)>0. \end{align*} By the continuity of $\Gamma(\alpha)$, there exists $\bar{\alpha}\in (\alpha_{2k-1},\alpha_{2k})$ such that $\Gamma(\bar{\alpha})=0$. It is similar to the case of $n=2k$ with $k\in \mathbb{N}$. Now in both cases, from \eqref{eq2.22} one has $$(n+\frac{1}{2})\pi_p<\bar{\alpha}^{\frac{q-p}{p}}\theta(1;\bar{\alpha})<(n+\frac{3}{2})\pi_p.$$ Hence, the above implies that $u(r;\bar{\alpha})$ has $n$ or $n+1$ zeros in $(0,1)$ and satisfies the multi-point boundary condition \eqref{eq1.4}. The proof is complete. \end{proof} \section{Blow-up solutions in finite intervals} In this section, we consider the negative potential ($w<0$) and focus on the issue related to the existence of unbounded solutions in a finite interval. Motivated by the interesting idea raised in \cite{MRW97}, we first study the one-dimensional case. \begin{theorem} \label{th3.1} Let $w<0$ and assume that {\rm (A1)} and {\em (A2)} hold. Then, the one-dimensional problem $(|u'|^{p-2}u')'+w(r)|u|^{q-2}u=0$ has at least one blow-up solution in a finite interval. That is, for $u(0)=\alpha>0$ one positive solution will tend to infinity in the finite interval $(0,\sqrt[p]{n^{-1}} R_{\alpha})$, where $R_{\alpha}$ is defined as in \eqref{eq1.6}. For $q\leq p$, such a blow-up solution satisfying this problem can not exist in any finite interval. \end{theorem} \begin{proof} For $w<0$, a positive unique local solution $u(r)$ in $J$ with $u(0)=\alpha>0$ and $u'(0)=0$ satisfies $u'>0$, and then $$\int_0^r(|u'|^{p-2}u')'u'ds= -\int_0^r w(s)|u|^{q-2}uu'ds \geq \delta_1\int_0^r|u|^{q-2}uu'ds$$ by (A2). The above implies that $$u'^p-\int_0^r(|u'|^{p-2}u')u''ds=\frac{p-1}{p}u'^p \geq \frac{\delta_1}{q}\left(u^{q}-\alpha^{q}\right).$$ i.e., $$\frac{u'}{\sqrt[p]{u^{q}-\alpha^{q}}}\geq \sqrt[p]{\frac{p \delta_1}{q(p-1)}}.$$ Then, $$\int_\alpha^{u(r)}\frac{du}{\sqrt[p]{u^{q}-\alpha^{q}}} \geq \sqrt[p]{\frac{p \delta_1}{q(p-1)}}\; r.$$ Hence, if the solution becomes infinite at $r=\ell$, then \begin{equation} \label{eq3.1} \begin{aligned} \ell &\leq \sqrt[p]{\frac{q(p-1)}{p\delta_1}}\int_{\alpha}^{\infty} \frac{du}{\sqrt[p]{u^{q}-\alpha^{q}}} =\sqrt[p]{\frac{q(p-1)}{p\delta_1\alpha^{q-p}}}\int_1^{\infty} \frac{ds}{\sqrt[p]{s^{q}-1}}\\ &=C\Big(\int_1^2+\int_2^{\infty}\Big)\frac{ds}{\sqrt[p]{s^{q}-1}} \\ &< C\Big(\int_1^2\frac{ds}{\sqrt[p]{s^{p}-1}}+\int_2^{\infty} \frac{ds}{\sqrt[p]{s^{q}-\frac{1}{2}s^{q}}}\Big)\quad\text{(letting $s^p-1=t$)} \\ &= C\Big(\frac{1}{p}\int_{0}^{2^p-1}t^{\frac{-1}{p}}(1+t)^{\frac{1-p}{p}}dt +2^{\frac{q}{p}}\int_2^{\infty}s^{\frac{-q}{p}ds}\Big) \\ &\leq C\Big(\frac{1}{p}\int_{0}^{2^p-1}t^{\frac{-1}{p}}dt+\frac{2 p}{q-p}\Big) \\ &=C\Big(\frac{1}{p-1}(2^p-1)^{\frac{p-1}{p}}+\frac{2p}{q-p}\Big) = \sqrt[p]{n^{-1}} R_{\alpha}, \end{aligned} %\label{eq3.2} \end{equation} where $C=\sqrt[p]{\frac{q(p-1)}{p\delta_1\alpha^{q-p}}}$ and $R_{\alpha}$ is defined as in \eqref{eq1.6}. This shows that there is at least one positive blow-up solution in $(0,\sqrt[p]{n^{-1}} R_{\alpha})$. For $q\leq p$ and a positive solution $u(r)$ with $u(0)=\beta>0$ and $u'(0)=0$, assume this unique local solution exists in $J=[0,a)$ and let $|w|\leq \delta_a$ in $J$. Then $$\int_0^r(|u'|^{p-2}u')'u'ds= -\int_0^r w(s)|u|^{q-2}uu'ds\leq \delta_a\int_0^r|u|^{q-2}uu'ds.$$ Apply the similar argument as in the above case ($q>p$) and let the solution become infinite at $r=R(\beta)$, where $$R(\beta)\geq \sqrt[p]{q\frac{(p-1)}{p \delta_{a}}}\int_\beta^{\infty} \frac{du}{\sqrt[p]{u^{q}-\beta^{q}}} =\tilde{C}\int_1^{\infty}\frac{ds}{\sqrt[p]{s^{q}-1}}=\infty,$$ where $\tilde{C}=\sqrt[p]{\frac{q(p-1)}{p\delta_{a}\beta^{q-p}}}$. This shows that the blow-up solution can not occur when the problem is considered in a finite interval as $q\leq p$. \end{proof} The following is a technical and crucial lemma whose main concept is quoted from \cite [Lemma 1]{MRW97}. It represents some elementary properties for solutions of \eqref{eq1.1} and the significant relationship between \eqref{eq1.1} and its corresponding one-dimensional problem ($n=1$). Here we give the details for the reader's convenience and make it coincide with our setting. \begin{lemma}\label{th3.2} For $\ell>0$, assume that $u\in C^1(0,\ell)$ with $|u'|^{p-2}u'\in C^1(0,\ell)$ is a solution of \begin{equation}\label{eq3.3} \left(r^{n-1}|u'|^{p-2}u'\right)'+r^{n-1}g(u)=0~~~in~~ (0,\ell), \end{equation} where $g\in C(\mathbb{R})$ and $u'$ is bounded near zero. Then $u(0):=\lim_{r\to 0}u(r)$ exists, and, with this definition, \begin{itemize} \item[(i)] $u'(0)=0$, $u\in C^1[0,\ell)$ and $|u'|^{p-2}u'\in C^1[0,\ell)$; \item[(ii)]if the function $g$ is negative and nonincreasing, then $u'(r)\geq 0$ for $r>0$ and $$(|u'|^{p-2}u')'\leq r^{-n+1}\left(r^{n-1}|u'|^{p-2}u'\right)'\leq n(|u'|^{p-2}u')';$$ \item[(iii)] the function $v(r)=v(r;c;\mu)=cu(\mu r)~(c,\mu>0)$ satisfies $$r^{-n+1}\left(r^{n-1}|v'|^{p-2}v'\right)'+c^{p-1}\mu^{p}g(v/c)=0\quad\text{in }(0,\ell/\mu).$$ \end{itemize} \end{lemma} \begin{proof} From \eqref{eq3.3} and the boundedness of $u'$, one has \begin{equation}\label{eq3.4} r^{n-1}|u'|^{p-2}u'=-\int_0^rs^{n-1}g(u(s))ds=-r^n\int_0^1t^{n-1}g(u(rt))dt. \end{equation} The boundedness of $u'$ near zero implies that $\lim_{r\longrightarrow o^+}u(r):=\alpha$ exists. Letting $\alpha=u(0)$, one can obtain that $\frac{|u'|^{p-2}u'}{r}$ is bounded near zero, hence $u'(0)=0$. The other properties of (i) are valid by \eqref{eq3.4}. For (ii), by the assumption of $g$ one has $u'(r)\geq 0$ from \eqref{eq3.4} obviously. Besides, \begin{equation} \label{eq3.5} r^{-n+1}\left(r^{n-1}|u'|^{p-2}u'\right)' =(n-1)\frac{|u'|^{p-2}u'}{r}+(|u'|^{p-2}u')'\geq (|u'|^{p-2}u')'. \end{equation} Employing \eqref{eq3.4} and $g'\leq 0$, one can obtain that \begin{equation} \begin{aligned} (|u'|^{p-2}u')' &=-\int_0^1t^{n-1}g(u(rt))dt-r\Big[\int_0^1t^{n}g'(u(rt)u'(rt)dt\Big] \\ &\geq -\int_0^1t^{n-1}g(u(rt)dt=\frac{|u'|^{p-2}u'}{r}. \end{aligned}\label{eq3.6} \end{equation} Hence, by \eqref{eq3.5}-\eqref{eq3.6} \begin{align*} (|u'|^{p-2}u')' &\leq r^{-n+1}\left(r^{n-1}|u'|^{p-2}u'\right)'\\ &\leq (n-1)(|u'|^{p-2}u')'+(|u'|^{p-2}u')' =n(|u'|^{p-2}u')'. \end{align*} This completes the proof of (ii). Finally, (iii) is valid by a direct substitution. \end{proof} The following is a version of the comparison lemma for the radial $p$-Laplacian and can be found as a consequence of \cite{MRW97,RW97}. \begin{lemma}[Comparison] \label{th3.3} Let $0\leq a0$ and $$R_1=\sqrt[p]{\frac{q(p-1)n}{p\delta_1}} \Big(\frac{1}{p-1}(2^p-1)^{\frac{p-1}{p}}+\frac{2p}{q-p}\Big).$$ That is, $v$ becomes infinite as $r$ tends to $\ell\leq R_1$. Now let $u_1$ be the solution of $$r^{-n+1}\left(r^{n-1}|u_1'(r)|^{p-2}u_1'(r)\right)' =w(\mu^{-1}r)|u_1(r)|^{q-2}u_1(r),~~~u_1(0)=1,~u_1'(0)=0.$$ By Lemma \ref{th3.2} (ii), $(|u_1'(r)|^{p-2}u_1'(r))'\geq n^{-1}w(\mu^{-1}r)|u_1(r)|^{q-2}u_1(r)$. Then the comparison lemma gives $u_1\geq v$. That is, $u_1$ tends to infinity as $r\to \ell_1$ with $\ell_1\leq R_1$. Now we define $u_{\alpha}(r)=\alpha u_1(\mu r)$ with $\mu=\alpha^{\frac{q-p}{p}}$. Applying Lemma \ref{th3.2} (iii), one can obtain that $u_{\alpha}(r)$ solves \eqref{eq1.1}-\eqref{eq1.2}. Hence, $u_{\alpha}$ has the asymptote $R(\alpha)\leq \alpha^{\frac{p-q}{p}}R_1=R_{\alpha}$ (as in \eqref{eq1.6}), which implies that there is at least one positive blow-up solution in $(0,R_{\alpha})$. For $q\leq p$, such a blow-up solution of \eqref{eq1.1}-\eqref{eq1.2} can not occur when the problem is considered in a finite interval by applying Theorem \ref{th3.1} and Lemma \ref{th3.2} (ii) directly. The proof is complete. \end{proof} \subsection*{Acknowledgments} The author was partially supported by MOST 108-2115-M-507-001. \begin{thebibliography}{00} \bibitem{BM92} C. Bandle, M. Marcus; Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, \emph{J. d'Analyse Math.}, \textbf{58} (1992), 9-24. \bibitem{BM95} C. Bandle, M. Marcus; Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary, \emph{Ann. Inst. Henri Poincar}, \textbf{12} (1995), 155-171. \bibitem{B16} L. 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