\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 38, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/38\hfil Multiple positive solutions] {Multiple positive solutions for singular multi-point boundary-value problems with a positive parameter} \author[C.-G. Kim, E. K. Lee \hfil EJDE-2014/38\hfilneg] {Chan-Gyun Kim, Eun Kyoung Lee } % in alphabetical order \address{Chan-Gyun Kim \newline Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795, USA} \email{cgkim75@gmail.com} \address{Eun Kyoung Lee (corresponding author)\newline Department of Mathematics and Education, Pusan National University, Busan 609-735, Korea} \email{eunkyoung165@gmail.com} \thanks{Submitted July 3, 2013. Published February 5, 2014.} \subjclass[2000]{34B10, 34B16} \keywords{Singular problem; multi-point boundary value problems; \hfill\break\indent positive solution; $p$-Laplacian; multiplicity} \begin{abstract} In this article we study the existence, nonexistence, and multiplicity of positive solutions for a singular multi-point boundary value problem with positive parameter. We use the fixed point index theory on a cone and a well-known theorem for the existence of a global continuum of solutions to establish our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Consider the singular multi-point boundary-value problem \begin{gather} \label{ePl} (\varphi_{p}(u'(t)))'+\lambda f(t,u(t))=0,\quad t \in (0,1), \\ \label{eBC} u(0)= \sum_{i=1}^{m-2} a_i u(\xi_i),\quad u(1)=\sum_{i=1}^{m-2} b_i u(\xi_i), \end{gather} where $\varphi_{p}(s)=|s|^{p-2}s,~p>1,~\lambda$ a nonnegative real parameter, $\xi_i \in (0,1)$ with $0<\xi_1<\xi_2 < \dots < \xi_{m-2}<1$, $a_i,b_i \in [0,1)$ with $0 \le \sum_{i=1}^{m-2} a_i<1$, $0 \le \sum_{i=1}^{m-2} b_i<1$, and $f \in C((0,1) \times [0,\infty),(0,\infty))$. Here, $f(t,u)$ may be singular at $t=0$ and/or $1$ and satisfies the following conditions: \begin{itemize} \item[(F1)] for all $M>0$, there exists $h_M \in \mathcal{A}$ such that $f(t,u) \le h_M(t)$, for all $u \in [0,M]$ and all $t \in (0,1)$, where $$ \mathcal{A}=\{h :\int_{0}^{1/2} \varphi_{p}^{-1} \Big( \int_{s}^{1/2} h(\tau) d \tau \Big) ds + \int_{1/2}^{1} \varphi_{p}^{-1} \Big( \int_{1/2}^{s} h(\tau) d \tau \Big) ds < \infty\}: $$ \item[(F2)] there exists $[\alpha,\beta] \subset (0,1)$ such that $\lim_{u \to \infty}f(t,u)/u^{p-1}=\infty$ uniformly in $[\alpha,\beta]$. \end{itemize} By a positive solution of problem \eqref{ePl}-\eqref{eBC}, we mean a function $u \in C[0,1] \cap C^1(0,1)$ with $\varphi_p(u') \in C^1(0,1)$ that satisfies \eqref{ePl}-\eqref{eBC} and $u>0$ in $(0,1)$. Here $\|\cdot\|$ denotes the usual maximum norm in $C[0,1]$. Motivated by the work of Bitsadze \cite{Bitsadze84,Bitsadze85}, the study of multi-point boundary value problem for linear second-order ordinary differential equations was initially done by Il'in and Moiseev \cite{Il'in1,Il'in2}. Gupta \cite{Gupta92} studied three-point boundary value problems for nonlinear ordinary differential equations. Since then, many researchers have studied nonlinear second-order multi-point boundary value problems under various conditions on the nonlinear term. We refer the reader to \cite{Bai,Feng,Hu,Infante,Kim09A,Liu03,Ma05,Sun,wang,Wong08,Xu04} and references therein. Problem \eqref{ePl}-\eqref{eBC} is a singular boundary value problem since $f$ is allowed to have singularity at $t=0$ and/or $1$. Singular problems have been extensively studied in the literature. For the case of two-point boundary value problems, the results were proved in \cite{agarwal:eop02,Choi,Dal,Ha,Kim09,Lee5,Wong93,Xu,Yang} and for multi-point boundary value problems, the results were proved in \cite{Feng,Hu,Kim09A,Liu03,Sun,Xu04}. However, there are few results for multi-point boundary value problems having nonlinear term which does not satisfy $L^1$-Carath\'eodory condition. Recently, in semi-linear case, Sun et al. \cite{Sun} studied the following singular three-point boundary-value problem \begin{equation}\label{0.1} \begin{gathered} y''+\mu a(t)g_1(t,y)=0,\quad t\in(0,1)\\ y(0)-\beta y'(0)=0, \quad y(1)=\alpha y(\eta), \end{gathered} \end{equation} where $\mu>0$ is a parameter, $\beta>0$, $0<\eta<1$, $0<\alpha\eta<1$, $(1-\alpha\eta)+\beta(1-\alpha)>0$, $a \in C((0,1),(0,\infty))$ satisfies $0<\int_0^1(\beta+s)(1-s)a(s)ds<\infty$, and $g_1 \in C([0,1]\times(0,\infty),(0,\infty))$ may be singular at $y=0$. Without any monotone or growth conditions imposed on the nonlinearity $g_1$, using fixed point index theorem, they obtained not only the existence results of positive solutions to the problem \eqref{0.1}, but also the explicit interval about positive parameter $\mu$. Kim \cite{Kim09A}, in $p$-Laplacian case, presented some sufficient conditions for one or multiple positive solutions to the problem \eqref{ePl}-\eqref{eBC}, where $f(t,u)=h(t)g_2(t,u)$, $h \in \mathcal{A}, g_2 \in C([0,1]\times [0,\infty),[0,\infty))$. To the authors' knowledge, in the case of $p$-Laplacian, there is no result about the global structure of positive solutions for parameter $\lambda \in (0,\infty)$ to multi-point boundary-value problems with the nonlinear term admitting stronger singularity than $L^1(0,1)$ at $t=0$ and/or $1$. The following is the main result in this paper. \begin{theorem}\label{Thm1.2} Assume that {\rm (F1)} and {\rm (F2)} hold. Assume in addition that $f(t,u)=h(t)g(t,u)$, where $h \in \mathcal{A}$ and $g \in C((0,1) \times [0,\infty),(0,\infty))$ satisfies \begin{itemize} \item[(A1)] for all $N>0$ and all $\epsilon>0$, there exists $\delta=\delta(N,\epsilon)>0$ such that if $u,v \in [0,N]$ and $|u-v|<\delta$, then $|g(t,u)-g(t,v)|<\epsilon$, for all $t \in (0,1)$, \item[(A2)] $\inf\{g(t,u)~|~t \in (0,1),~u \in [0,\infty)\}>0$. \end{itemize} Then there exists $\lambda^*>0$ such that problem \eqref{ePl}-\eqref{eBC} has at least two positive solutions for $\lambda \in (0,\lambda^*)$, at least one positive solution for $\lambda=\lambda^*$ and no positive solution for $\lambda > \lambda^*$. \end{theorem} The above result is an extension of previous works for two-point boundary-value problems by Choi \cite{Choi}, Wong \cite{Wong93}, Dalmasso \cite{Dal}, Ha and Lee \cite{Ha}, Lee \cite{Lee5}, Xu and Ma \cite{Xu}, and Kim \cite{Kim09}. The rest of this article is organized as follows. In Section 2, the operator for problem \eqref{ePl}-\eqref{eBC} is introduced, and well-known facts such as Picone-type identity and Global continuation theorem are presented. In Section 3, the proofs of our results (Theorem \ref{thm3.4} and Theorem \ref{Thm1.2}) and examples for nonlinear term to illustrate our results are given. \section{Preliminaries} First we introduce the operator corresponding to problem \eqref{ePl}-\eqref{eBC}. Throughout this section we assume that (F1) holds. Set $$ \mathcal{K}=\{u \in C[0,1] : u\text{i s a nonnegative concave function on $[0,1]$, $u$ satisfies \eqref{eBC}}\}. $$ Then $\mathcal{K}$ is an ordered cone in $C[0,1]$. For $(\lambda,u) \in [0,\infty) \times \mathcal{K}$, we define $x_{\lambda,u} : [0,1] \to \mathbb{R}$ as $x_{\lambda,u}(t)=x_{\lambda,u}^1(t)-x_{\lambda,u}^2(t)$, where \begin{equation*} x_{\lambda,u}^1(t)=A^{-1}{\sum_{i=1}^{m-2}a_i} \int_0^{\xi_i}\varphi_p^{-1}\Big[\int_s^t \lambda f(\tau,u(\tau))d\tau\Big]ds +\int_0^t\varphi_p^{-1}\Big[\int_s^t \lambda f(\tau,u(\tau))d\tau\Big]ds \end{equation*} and \begin{equation*} x_{\lambda,u}^2(t)=B^{-1}{\sum_{i=1}^{m-2}b_i}\int_{\xi_i}^1\varphi_p^{-1} \Big[\int_t^s \lambda f(\tau,u(\tau))d\tau\Big]ds +\int_t^1\varphi_p^{-1}\Big[\int_t^s \lambda f(\tau,u(\tau))d\tau\Big]ds. \end{equation*} Here $$ A=1-\sum_{i=1}^{m-2}a_i, \quad B=1-\sum_{i=1}^{m-2}b_i. $$ For $\lambda>0$, $\lim_{t \to 0^+}x_{\lambda,u}(t)<0$ and $\lim_{t \to 1^-}x_{\lambda,u}(t)>0$. Indeed we can rewrite $x_{\lambda,u}^1(t)$ as \begin{align*} &x_{\lambda,u}^1(t) \\ &=A^{-1}\Big(-{\sum_{i=1}^{m-2}a_i} \int_t^{\xi_i}\varphi_p^{-1}\Big[\int_t^s \lambda f(\tau,u(\tau))d\tau\Big]ds +\int_0^t\varphi_p^{-1}\Big[\int_s^t \lambda f(\tau,u(\tau))d\tau\Big]ds\Big). \end{align*} By (F1), there exists $h_2 \in \mathcal{A}$ such that $$ 0 \le \int_0^t\varphi_p^{-1}\Big[\int_s^t \lambda f(\tau,u(\tau))d\tau\Big]ds \le \int_0^t\varphi_p^{-1}\Big[\int_s^t h_2(\tau)d\tau\Big]ds, $$ and $$ \lim_{t \to 0^+} \int_0^t\varphi_p^{-1} \Big[\int_s^t \lambda f(\tau,u(\tau))d\tau\Big]ds=0. $$ Clearly $\lim_{t \to 0^+}x_{\lambda,u}^2(t)>0$, and thus $\lim_{t \to 0^+}x_{\lambda,u}(t)<0$. In a similar manner we can show $\lim_{t \to 1^-}x_{\lambda,u}(t)>0$. Since $x_{\lambda,u}$ is continuous and strictly increasing in $(0,1)$, there exists a unique zero $A_{\lambda,u} \in (0,1)$ such that $x_{\lambda,u}(A_{\lambda,u})=0$. For $\lambda=0$, we may take $A_{0,u}=0$ since $x_{0,u}\equiv 0$. Then, for $(\lambda,u)\in [0,\infty) \times \mathcal{K}$, \begin{align*} &A^{-1}{\sum_{i=1}^{m-2}a_i}\int_0^{\xi_i}\varphi_p^{-1} \Big[\int_s^{A_{\lambda,u}} \lambda f(\tau,u(\tau))d\tau\Big]ds +\int_0^{A_{\lambda,u}}\varphi_p^{-1} \Big[\int_s^{A_{\lambda,u}} \lambda f(\tau,u(\tau))d\tau\Big]ds\\ &=B^{-1}{\sum_{i=1}^{m-2}b_i}\int_{\xi_i}^1\varphi_p^{-1} \Big[\int_{A_{\lambda,u}}^s \lambda f(\tau,u(\tau))d\tau\Big]ds +\int_{A_{\lambda,u}}^1\varphi_p^{-1} \Big[\int_{A_{\lambda,u}}^s \lambda f(\tau,u(\tau))d\tau\Big]ds. \end{align*} Define $H:[0,\infty)\times \mathcal{K} \to C[0,1]$ as \begin{equation*} H(\lambda,u)(t)= \begin{cases} A^{-1}{\sum_{i=1}^{m-2}a_i}\int_0^{\xi_i}\varphi_p^{-1} \big[\int_s^{A_{\lambda,u}} \lambda f(\tau,u(\tau))d\tau\big]ds\\ +\int_0^t\varphi_p^{-1} \big[\int_s^{A_{\lambda,u}} \lambda f(\tau,u(\tau))d\tau\big]ds, &0 \le t \le A_{\lambda,u},\\[4pt] B^{-1}{\sum_{i=1}^{m-2}b_i}\int_{\xi_i}^1\varphi_p^{-1} \big[\int_{A_{\lambda,u}}^s \lambda f(\tau,u(\tau))d\tau\big]ds\\ +\int_{A_{\lambda,u}}^1\varphi_p^{-1} \big[\int_{A_{\lambda,u}}^s \lambda f(\tau,u(\tau))d\tau\big]ds, &A_{\lambda,u} \le t \le 1. \end{cases} \end{equation*} In view of the definition of $A_{\lambda,u}$, $H(\lambda,u)$ is well-defined, $\|H(\lambda,u)\|$ $=H(\lambda,u)(A_{\lambda,u})$, and $H(\lambda,u) \in \mathcal{K}$ for all $(\lambda,u) \in [0,\infty)\times \mathcal{K}$ (see, e.g., \cite[Lemma 2.2]{Feng}). \begin{lemma}\label{Lem3.1} Problem \eqref{ePl}-\eqref{eBC} has a positive solution $u$ if and only if $H(\lambda,\cdot)$ has a fixed point $u$ in $\mathcal{K}$ for $\lambda>0$. \end{lemma} \begin{proof} We assume that $u$ is a positive solution of problem \eqref{ePl}-\eqref{eBC}. If $\lambda=0$, $u\equiv0$ by the facts that $0\le\sum_{i=1}^{m-2}a_i<1$ and $0\le\sum_{i=1}^{m-2}b_i<1$. Thus $\lambda>0$. Since $u'$ is strictly decreasing in $(0,1)$, $u \in \mathcal K$. From the fact that $u$ satisfies $(BC)$, $\max\{u(0),u(1)\}0$, then one can easily see that $u$ is a positive solution of problem \eqref{ePl}-\eqref{eBC}. \end{proof} \begin{lemma}\label{Lem3.2} Let $M>0$ be given and let $\{(\lambda_n,u_n)\}$ be a sequence in $[0,\infty)\times \mathcal{K}$ with $|\lambda_n|+\|u_n\| \le M$. If $A_{\lambda_n,u_n} \to 0$ (or 1) as $n \to \infty$, then $\lambda_n \to 0$ and $\|H(\lambda_n,u_n)\| \to 0$ as $n \to \infty$. \end{lemma} \begin{proof} We only prove the case $A_{\lambda_n,u_n} \to 0$ as $n \to \infty$ since the other case can be showed in a similar manner. By the definition of $A_{\lambda,u}$, we can easily know $\lambda_n \to 0$ as $n \to \infty$. By (F1), there exists $h_M \in \mathcal{A}$ such that $f(t,u) \le h_M(t)$, $t \in (0,1)$, $u \in [0,M]$. For sufficiently large $n$, we have $A_{\lambda_n,u_n}<\xi_1$, \begin{align*} 0 \le H(\lambda_n,u_n)(0) &= A^{-1}{\sum_{i=1}^{m-2}a_i}\int_0^{\xi_i}\varphi_p^{-1} \Big[\int_s^{A_{\lambda_n,u_n}} \lambda_n f(\tau,u_n(\tau))d\tau\Big]ds\\ &\le \lambda_n A^{-1}\int_0^{\xi_{m-2}}\varphi_p^{-1} \Big[\int_s^{\xi_{m-2}} h_M(\tau)d\tau\Big]ds, \end{align*} and \begin{align*} \|H(\lambda_n,u_n)\|=H(\lambda_n,u_n)(0) +\lambda_n \int_0^{A_{\lambda_n,u_n}}\varphi_p^{-1} \Big[\int_s^{A_{\lambda_n,u_n}} h_M(\tau)d\tau\Big]ds. \end{align*} Thus $\|H(\lambda_n,u_n)\| \to 0$ as $n \to \infty$ since $h_M \in \mathcal{A}$ and $\lambda_n \to 0$ as $n \to \infty$. \end{proof} \begin{lemma}\label{Lem3.3} $H:[0,\infty)\times \mathcal{K} \to \mathcal{K}$ is completely continuous. \end{lemma} \begin{proof} By Lemma \ref{Lem3.2}, Ascoli-Arzel\`{a} theorem, and Lebesgue dominated convergence theorem, one can easily show the completely continuity of $H$ (e.g., see \cite{agarwal:eop02,Kim09A}). Thus we omit the proof here. \end{proof} Next we introduce the generalized Picone identity due to Jaros and Kusano (\cite{kusano:pis09}). Let us consider the following operators: \begin{gather*} l_{p}[y] \equiv (\varphi_{p}(y'))' + q(t) \varphi_{p}(y), \\ L_{p}[z] \equiv (\varphi_{p}(z'))' + Q(t) \varphi_{p}(z). \end{gather*} \begin{theorem}[{\cite[p~382]{kusano:pis00}}] \label{Thm2.1} Let $q(t)$ and $Q(t)$ be measurable functions on an interval $I$. If $y$ and $z$ are any functions such that $y, z, \varphi_{p}(y')$, $\varphi_{p}(z')$ are differentiable a.e. on $I$ and $z(t) \neq 0$ for $t \in I$, then the following holds \begin{equation} \label{2.1} \begin{aligned} &\frac{d}{dt} \Big\{ \frac {|y|^{p} \varphi_{p}(z')} {\varphi_{p}(z)} - y \varphi_{p}(y') \Big\}\\ & = (q - Q) |y|^{p} - \big[ |y'|^{p} + (p-1) | \frac{yz'}{z}|^{p} - p\varphi_{p}(y) y' \varphi_{p} \big( \frac{z'}{z} \big) \big] - y l_{p}[y] + \frac{|y|^{p}}{\varphi_{p}(z)} L_{p}[z]. \end{aligned} \end{equation} \end{theorem} \begin{remark}\label{Rem2.2} \rm By Young's inequality, we have $$ |y'|^{p} + (p-1) | \frac{yz'}{z}|^{p} - p\varphi_{p}(y)y' \varphi_{p} \big( \frac{z'}{z} \big) \geq 0, $$ and the equality holds if and only if $y'=y z'/z$ in $(a,b)$. \end{remark} Finally we recall a well-known theorem for the existence of a global continuum of solutions by Leray and Schauder \cite{MR1509338}. \begin{theorem}[{\cite[Corollary 14.12]{zeidler:nfa85}}] \label{Thm2.5} Let $X$ be a Banach space with $X \neq \{0\}$ and let $\mathcal{K}$ be an ordered cone in $X$. Consider \begin{equation}\label{2.1b} x=H(\mu,x), \end{equation} where $\mu \in [0,\infty)$ and $x \in \mathcal{K}$. If $H:[0,\infty)\times \mathcal{K} \to \mathcal{K}$ is completely continuous and $H(0,x)=0$ for all $x \in \mathcal{K}$. Then the solution component $\mathcal{C}$ of \eqref{2.1} in $[0,\infty)\times \mathcal{K}$ which contains $(0,0)$ is unbounded. \end{theorem} \section{Main results} Since $H(0,u)=0$ and $H(\lambda,0) \neq 0$ if $\lambda \neq 0$, by Lemma \ref{Lem3.3}, Theorem \ref{Thm2.5}, we obtain the following proposition. \begin{proposition}\label{Pro4.1} Assume that (F1) holds. Then there exists an unbounded continuum $\mathcal{C}$ emanating from $(0,0)$ in the closure of the set of positive solutions of problem \eqref{ePl}-\eqref{eBC} in $[0,\infty) \times \mathcal{K}$. \end{proposition} To see the shape of $\mathcal{C}$, we need lemmas regarding $\lambda$-direction block and {\it a priori} estimate. Using the generalized Picone identity (Theorem \ref{Thm2.1}) and the properties of the $p$-sine function \cite{delpino:hdl89,zhang:nns00}, we obtain the following two lemmas. \begin{lemma}\label{Lem4.2} Assume that {\rm (F1)} and {\rm (F2)} hold. Then there exists $\bar \lambda >0 $ such that if problem \eqref{ePl}-\eqref{eBC} has a positive solution $u_\lambda$, then $\lambda \leq \bar \lambda$. \end{lemma} \begin{proof} Let $u_\lambda$ be a positive solution of problem \eqref{ePl}-\eqref{eBC}. Since $f(t,u)>0$ for all $(t,u) \in (0,1)\times [0,\infty)$, by (F2), there exists $C_1> 0$ such that \begin{equation}\label{4.1} f(t,u)> C_1 \varphi_p(u)\quad \text{for } u \in [0,\infty),\; t \in [\alpha,\beta]. \end{equation} It is easy to check that $w(t) = S_{q}\left(\pi_{q}(t-\alpha)/(\beta-\alpha)\right)$, where $S_q$ is the $q$-sine function and $\frac{1}{p} + \frac{1}{q} = 1$, is a solution of \begin{gather*} (\varphi_{p}(w'(t)))' + \big(\frac{\pi_{q}}{\beta-\alpha}\big)^{p} \varphi_{p}(w(t)) = 0,\quad t \in(\alpha,\beta),\\ w(\alpha)=w(\beta)=0. \end{gather*} Taking $y = w$, $z = u_\lambda$, $q(t)=\left(\pi_{q}/(\beta-\alpha)\right)^{p}$ and $Q(t)=\lambda f(t,u_\lambda)/\varphi_p(u_\lambda)$ in \eqref{2.1} and integrating \eqref{2.1} from $\alpha$ to $\beta$, by Remark \ref{Rem2.2}, $$ \int_{\alpha}^{\beta} \Big(\big(\frac{\pi_{q}}{\beta-\alpha}\big)^{p} - \lambda \frac{f(t,u_\lambda)}{\varphi_p(u_\lambda)} \Big) |w|^{p} dt \ge 0. $$ It follows from \eqref{4.1} that $$ \Big(\big(\frac{\pi_{q}}{\beta-\alpha}\big)^{p} - \lambda C_1 \Big)\int_{\alpha}^{\beta} |w|^{p} dt \ge 0, $$ and thus the proof is complete. \end{proof} \begin{lemma}\label{Lem4.3} Assume that {\rm (F1)} and {\rm (F2)} hold, and let $J=[D,E]$ be a compact subset of $(0,\infty)$. Then there exists $M_J >0$ such that if $u$ is a positive solution of problem \eqref{ePl}-\eqref{eBC} with $\lambda \in J$, then $\|u\| \leq M_J$. \end{lemma} \begin{proof} Suppose on the contrary that there exists a sequence $\{u_{n}\}$ of positive solutions of problem \eqref{ePl}-\eqref{eBC} with $\lambda_n$ instead of $\lambda$, and $\{\lambda_{n}\} \subset J = [D,E]$ and $\|u_{n}\| \to \infty$ as $n \to \infty$. It follows from the concavity of $u_{n}$ for all $n$ that \begin{equation}\label{4.3} u_{n}(t) \ge \min\{\alpha,1-\beta\} \|u_{n}\|, \quad t \in (\alpha,\beta). \end{equation} Take $C = 2 D^{-1}\left(\pi_{q}/(\beta-\alpha)\right)^{p}> 0$. By (F2), there exists $K > 0$ such that $f(t,u) > C\varphi_{p}(u)$, for $t \in (\alpha,\beta)$, $u > K$. From the assumption, we get $\|u_{N}\| > (\min\{\alpha,1-\beta\})^{-1} K$, for sufficiently large $N$. Therefore, by \eqref{4.3}, we have $$ f(t,u_{N}(t)) > C \varphi_{p}(u_{N}(t)), \quad t \in (\alpha,\beta). $$ As in the proof of Lemma \ref{Lem4.2}, if we take $y(t) = S_{q}\left(\pi_{q}(t-\alpha)/(\beta-\alpha)\right)$ and $z=u_N$, by Theorem \ref{Thm2.1} and Remark \ref{Rem2.2}, $$ C \le D^{-1}\big(\frac{\pi_{q}}{\beta-\alpha}\big)^{p}. $$ This contradicts the choice of $C$, and thus the proof is complete. \end{proof} Setting $\lambda^*=\sup \{\mu>0$: for all $\lambda \in (0,\mu)$, there exists at least two positive solutions of problem \eqref{ePl}-\eqref{eBC}, then $\lambda^*>0$ is well-defined. Indeed by Proposition \ref{Pro4.1}, $\mathcal{C}$ emanates from $(0, 0)$, and problem \eqref{ePl}-\eqref{eBC} has a small solution near $(0, 0)$ for $\lambda \in (0,s)$ with small $s>0$. On the other hand, for any $M > 0$, define $\mathcal{C}_M = \{(\lambda,u) \in \mathcal{C} : \|u\| \ge M\}$ and the projection of $\mathcal{C}_M$ to the $\lambda$-axis as $\Lambda_M$. Then, by Lemma \ref{Lem4.2} and Lemma \ref{Lem4.3}, for large $M$, $\Lambda_M=(0,a_M]$, where $a_M>0$ and it is decreasing in $M$. This implies that, for any interval $(0,s)$ with small $s>0$, problem \eqref{ePl}-\eqref{eBC} also has a large solution for $\lambda \in (0,s)$. Thus $\lambda^*>0$ is well-defined. Moreover it follows from an easy compactness argument that problem \eqref{ePl}-\eqref{eBC} has at least two positive solution for $\lambda \in (0,\lambda^*)$ and at least one positive solution for $\lambda=\lambda^*$. The following is the first result in this work. \begin{theorem}\label{thm3.4} Assume that {\rm (F1)} and {\rm (F2)} hold. Then there exists $\lambda_* \ge \lambda^*>0$ such that problem \eqref{ePl}-\eqref{eBC} has at least two positive solutions for $\lambda \in (0,\lambda^*)$, at least one positive solution for $\lambda \in [\lambda^*,\lambda_*]$, and no positive solution for $\lambda>\lambda_*$. \end{theorem} \begin{proof} Define $\lambda_*=\sup\{ \lambda:$ problem \eqref{ePl}-\eqref{eBC} has at least one positive solution$\}$. Then by Lemma \ref{Lem4.2}, $\lambda^* \leq \lambda_*<\infty$. We only consider the case $\lambda^* < \lambda_*$, since the proof is done for the case $\lambda^* = \lambda_*$. For $\lambda\in [\lambda^*, \lambda_*)$, there exists $\hat \lambda \in [ \lambda, \lambda_{*})$ such that \eqref{ePl}-\eqref{eBC} with $\hat{\lambda}$ instead of $\lambda$, has a positive solution, say $\hat u$. Consider the modified problem \begin{equation} \label{eMl} %\tag*{$(M_{\lambda})$} \begin{gathered} (\varphi_{p}(u'(t)))' + \lambda \bar f(t,u(t))=0, \quad t \in (0,1), \\ u(0)= {\sum_{i=1}^{m-2} a_i} u(\xi_i),\quad u(1)={\sum_{i=1}^{m-2} b_i} u(\xi_i), \end{gathered} \end{equation} where $\bar f(t,u)=f(t,\gamma(t,u))$ and $\gamma : (0,1) \times \mathbb{R} \to \mathbb{R}$ is defined as \begin{equation*} \gamma(t,u) =\begin{cases} \hat u(t) , & \text{if } u > \hat u(t) ,\\ u, & \text{if }0 \le u \le \hat u(t) ,\\ 0, & \text{if }u < 0. \end{cases} \end{equation*} Then all solutions $u$ of \eqref{eMl} are concave and non-trivial. Define $T_\lambda : C[0,1] \to C[0,1]$ as \begin{equation*} T_{\lambda}(u)(t) = \begin{cases} A^{-1}{\sum_{i=1}^{m-2}a_i}\int_0^{\xi_i}\varphi_p^{-1} \big[\int_s^{\hat{A}} \lambda \bar f(\tau,u(\tau))d\tau\big]ds\\ +\int_0^t\varphi_p^{-1} \big[\int_s^{\hat{A}} \lambda \bar f(\tau,u(\tau))d\tau\big]ds, &0 \le t \le \hat{A}\\[4pt] B^{-1}{\sum_{i=1}^{m-2}b_i}\int_{\xi_i}^1\varphi_p^{-1} \big[\int_{\hat{A}}^s \lambda \bar f(\tau,u(\tau))d\tau\big]ds\\ +\int_{\hat{A}}^1\varphi_p^{-1} \big[\int_{\hat{A}}^s \lambda \bar f(\tau,u(\tau))d\tau\big]ds, &\hat{A} \le t \le 1.\end{cases} \end{equation*} where $\hat{A}$ satisfies \begin{align*} &A^{-1}{\sum_{i=1}^{m-2}a_i}\int_0^{\xi_i}\varphi_p^{-1} \Big[\int_s^{\hat{A}} \lambda \bar f(\tau,u(\tau))d\tau\Big]ds +\int_0^{\hat{A}}\varphi_p^{-1} \Big[\int_s^{\hat{A}} \lambda \bar f(\tau,u(\tau))d\tau\Big]ds\\ &=B^{-1}{\sum_{i=1}^{m-2}b_i}\int_{\xi_i}^1\varphi_p^{-1} \Big[\int_{\hat{A}}^s \lambda \bar f(\tau,u(\tau))d\tau\Big]ds +\int_{\hat{A}}^1\varphi_p^{-1} \Big[\int_{\hat{A}}^s \lambda \bar f(\tau,u(\tau))d\tau\Big]ds. \end{align*} It is easy to check that $T_{\lambda}$ is completely continuous on $C[0,1]$, and $u$ is a solution of \eqref{eMl} if and only if $u=T_\lambda u$. It follows from the definition of $\gamma$ and the continuity of $f$ that there exists $R_{1} > 0$ such that $\|T_\lambda u\| < R_{1}$ for all $u \in C[0,1]$. Then by Schauder fixed point theorem, there exists $u_\lambda \in C[0,1]$ such that $T_\lambda u_\lambda = u_\lambda$, and $u_\lambda$ is a positive solution of $(M_{\lambda})$. We first claim that $u_\lambda(0) \leq \hat u(0)$. If the claim is not true, $u_\lambda(0) > \hat u(0)$. Put $x(t)=u_\lambda(t)-\hat u(t)$. Then $$ 0< x(0)=u_\lambda(0)-\hat u(0)={\sum_{i=1}^{m-2} a_i}x(\xi_i) \le {\sum_{i=1}^{m-2} a_i} x(\xi_j)0$, $x'(\sigma)=0$, $x(a)=0$, and $x(t)>0$ for $t\in (a,\sigma]$. Since $\lambda <\hat \lambda$, for $t\in (a,\sigma]$, $(\varphi_p(u_\lambda'(t)))'>(\varphi_p(\hat u'(t)))'$ and integrating this from $t$ to $\sigma$, $u_\lambda'(t)<\hat u'(t)$. Again integrating from $a$ to $\sigma$, we have $x(\sigma)=u_\lambda(\sigma)-\hat u(\sigma) \hat u(t)$ for all $t \in (t_{1},t_{2})$. Then \begin{equation}\label{1} (\varphi_p(u_\lambda'(t)))'>(\varphi_p(\hat u'(t)))', \quad t\in (t_{1},t_{2}) \end{equation} and we can choose an interval $[b,c] \subset [t_{1},t_{2}]$ such that $u_\lambda'(b)> \hat u'(b)$ and $ u_\lambda'(c) <\hat u'(c) $. Using \eqref{1}, we can get the contradiction \begin{align*} 0 & > [\varphi_{p}(u_\lambda'(c) ) - \varphi_{p}(u_\lambda'(b) ) ] -[\varphi_{p}(\hat u'(c) ) - \varphi_{p}(\hat u'(b) ) ] \\ & = \int_{b}^{c} \big\{ [\varphi_{p}(u_\lambda'(t) ) ]' - [\varphi_{p}(\hat u'(t) ) ]' \big\} dt > 0. \end{align*} Therefore, by the definition of $\gamma$, $u_\lambda$ turns out a positive solution of problem \eqref{ePl}-\eqref{eBC}. Furthermore, by Lemma \ref{Lem4.3} and the complete continuity of $H$, we can show that problem \eqref{ePl}-\eqref{eBC}, with $\lambda_*$ instead of $\lambda$, has a positive solution $u_*$, and thus the proof is complete. \end{proof} Now we consider $f(t,u)=h(t)g(t,u)$ and let $u_*$ be a positive solution of problem \eqref{ePl}-\eqref{eBC}, with $\lambda_*$ instead of $\lambda$. \begin{lemma}\label{Lem4.4} Assume that {\rm (F1)} and {\rm (F2)} hold. Assume in addition that $g$ satisfies the conditions {\rm (A1)} and {\rm (A2)}. Then, for all $\lambda \in (0,\lambda_*)$, there exists $\delta_\lambda >0$ such that $\alpha_\lambda(t)=u_*(t)+\delta_\lambda$ satisfies \begin{equation}\label{4.4} (\varphi_p(\alpha_\lambda'(t)))' + \lambda h(t)g(t,\alpha_\lambda(t)) <0, \quad t \in (0,1). \end{equation} \end{lemma} \begin{proof} Let $\lambda$ be fixed in $(0,\lambda_*)$. Put $$ \epsilon=\frac{1}{2} [\lambda_*/\lambda-1]\inf_{t\in(0,1)} g(t,u_*(t))>0. $$ By (A1), there exists $\delta_\lambda>0$ such that if $u,v \in [0,\|u_*\|+1]$ and $|u-v|<\delta_\lambda$, then $|g(t,u)-g(t,v)|<\epsilon,~t \in (0,1)$. Put $\alpha_\lambda(t)=u_*(t)+\delta_\lambda$. Then \begin{align*} (\varphi_p(\alpha_\lambda'(t)))'+ \lambda f(t,\alpha_\lambda(t)) &= (\varphi_p(u_*'(t)))'+ \lambda f(t,u_*(t)+\delta_\lambda)\\ &= h(t)[-\lambda_*g(t,u_*(t))+ \lambda g(t,u_*(t)+\delta_\lambda)]. \end{align*} From this, if $\alpha_\lambda$ does not satisfy \eqref{4.4}, there exists $t_0\in(0,1)$ such that $$ -\lambda_*g(t_0,u_*(t_0))+ \lambda g(t_0,u_*(t_0)+\delta_\lambda)\ge 0, $$ and then $$ g(t_0,u_*(t_0)+\delta_\lambda) \geq \frac{\lambda_*}{\lambda}g(t_0,u_*(t_0)). $$ By the choice of $\delta_\lambda$, $$ \epsilon \geq \big(\frac{\lambda_*}{\lambda}-1\big)g(t_0,u_*(t_0)), $$ which contradicts the choice of $\epsilon$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{Thm1.2}] Suppose on the contrary that $\lambda^* < \lambda_*$. Let $\lambda$ be fixed with $\lambda^* \le \lambda < \lambda_*$. Then by showing that \eqref{ePl}-\eqref{eBC} has at least two positive solutions for $\lambda \in [\lambda^*,\lambda_*)$, we get a contradiction to the definition of $\lambda^*$, which completes the proof. By Lemma \ref{Lem4.4}, there exists $\delta_\lambda>0$ such that $\alpha_\lambda(t)=u_*(t)+ \delta_\lambda$ satisfies \eqref{4.4}. Consider the modified problem \begin{equation}\label{ebarMl} %\tag{$\ol M_\lambda$} \begin{gathered} (\varphi_p(u'(t)))'+\lambda h(t)g(t,\gamma_1(t,u(t)))=0,\\ u(0)={\sum_{i=1}^{m-2} a_i}u(\xi_i),\quad u(1)={\sum_{i=1}^{m-2} b_i} u(\xi_i), \end{gathered} \end{equation} where $\gamma_1 : (0,1) \times \mathbb{R} \to [0,\infty)$ is defined as $$ \gamma_1 (t,u)= \begin{cases} \alpha_\lambda(t),& \text{if } u>\alpha_\lambda(t),\\ u, & \text{if } 0 \leq u \leq \alpha_\lambda(t),\\ 0, &\text{if } u<0. \end{cases} $$ Let $u$ be a positive solution of \eqref{ebarMl}. Set $$ \Omega=\{u\in C[0,1] |~-1u(1)$. If the claim is not true, then there exists $[t_0,t_1] \subset (0,1)$ with $t_0 \le t_1$ such that $00$. Since $\alpha_\lambda$ satisfies \eqref{4.4}, $$ \max_{t \in [t_0-\delta_1,t_1+\delta_1]} \{(\varphi_p(\alpha_\lambda'(t)))'+ \lambda h(t)g(t,\alpha_\lambda(t))\}=-\epsilon_1<0. $$ By condition (A1), there exists $\delta_2>0$ such that if $|u-v|< \delta_2$ and $u,v \in [0, \|\alpha_\lambda\|]$, then $$ |g(t,u)-g(t,v)|< \epsilon_2, $$ where $\epsilon_2=\epsilon_1[2\lambda \max_{t \in [t_0-\delta_1,t_1+\delta_1]}h(t)]^{-1} >0$, and then there exists an interval $[a,b] \subset (t_0-\delta_1, t_1 +\delta_1)$ such that $$ (u-\alpha_\lambda)'(a)>0, ~(u-\alpha_\lambda)'(b)<0 $$ and $$-\delta_2 < \gamma(t,u(t))-\alpha_\lambda(t) =u(t)-\alpha_\lambda(t) \leq 0, \quad t \in [a,b]. $$ Consequently, \begin{gather*} \varphi_p(u'(a))-\varphi_p(\alpha_\lambda'(a))>0,\quad \varphi_p(u'(b))-\varphi_p(\alpha_\lambda'(b))<0,\\ g(t,\gamma(t,u(t))) < g(t,\alpha_\lambda(t))+ \epsilon_2,\quad t \in [a,b]. \end{gather*} Then, by the choice of $\epsilon_2$, \begin{align*} 0 & > \varphi_{p}(u'(b)) - \varphi_{p}(\alpha_\lambda'(b)) - \varphi_{p}(u'(a)) + \varphi_{p}(\alpha_\lambda'(a)), \\ & = \left[\varphi_{p}(u'(b)) - \varphi_{p}(u'(a)) \right] - \left[\varphi_{p}(\alpha_\lambda'(b)) - \varphi_{p}(\alpha_\lambda'(a)) \right] \\ & = \int_{a}^{b} \left\{ (\varphi_{p}(u'(t)))' - (\varphi_{p}(\alpha_\lambda'(t)))' \right \} dt \\ & = \int_{a}^{b} \left\{-\lambda h(t) g(t,\gamma(t,u(t))) -(\varphi_{p}(\alpha_\lambda'(t)))' \right \} dt \\ & > \int_{a}^{b} \left\{-\lambda h(t) [g(t,\alpha_\lambda(t))+ \epsilon_2] -(\varphi_{p}(\alpha_\lambda'(t)))' \right \} dt\\ &> \int_{a}^{b} \left(-\lambda h(t)\epsilon_2 -\left[(\varphi_p(\alpha_\lambda'(t)))'+ \lambda h(t)g(t,\alpha_\lambda(t))\right]\right)dt\\ &\ge \int_{a}^{b}(-\lambda \epsilon_2 h(t) +\epsilon_1 )dt \ge 0. \end{align*} This is a contradiction. Thus the claim is proved. Define \begin{equation*} Mu(t)= \begin{cases} A^{-1}{\sum_{i=1}^{m-2}a_i}\int_0^{\xi_i}\varphi_p^{-1} \big[\int_s^{A_{u}} \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\big]ds\\ +\int_0^t\varphi_p^{-1} \big[\int_s^{A_{u}} \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\big]ds, & 0\le t \le A_{u},\\[4pt] B^{-1}{\sum_{i=1}^{m-2}b_i}\int_{\xi_i}^1\varphi_p^{-1} \big[\int_{A_{u}}^s \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\big]ds\\ +\int_{A_{u}}^1\varphi_p^{-1} \big[\int_{A_{u}}^s \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\big]ds, & A_{u} \le t \le 1, \end{cases} \end{equation*} where $A_u$ is defined as \begin{align*} & A^{-1}{\sum_{i=1}^{m-2}a_i}\int_0^{\xi_i}\varphi_p^{-1} \Big[\int_s^{A_{u}} \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\Big]ds\\ &\quad +\int_0^{A_{u}}\varphi_p^{-1} \Big[\int_s^{A_{u}} \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\Big]ds\\ &=B^{-1}{\sum_{i=1}^{m-2}b_i}\int_{\xi_i}^1\varphi_p^{-1} \Big[\int_{A_{u}}^s \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\Big]ds\\ &\quad +\int_{A_{u}}^1\varphi_p^{-1} \Big[\int_{A_{u}}^s \lambda f(\tau,\gamma_1(\tau, u(\tau)))d\tau\Big]ds. \end{align*} Then $M:\mathcal{K} \to \mathcal{K}$ is completely continuous, and $u$ is a positive solution of \eqref{ebarMl} if and only if $u=Mu$ on $\mathcal{K}$. By simple calculation, there exists $R_1>0$ such that $\|Mu\|\overline{\lambda}$. Thus, for any open subset $\mathcal{O}$ in $ X$, $$ i(H(\lambda_{N_0},\cdot) , \mathcal{O}\cap \mathcal{K},\mathcal{K})=0. $$ By {\it a priori} estimate (Lemma \ref{Lem4.3}) with $I=[\lambda,\lambda_{N_0}]$, there exists $R_2(>R_1)$ such that all possible positive solutions $u$ of \eqref{ePl}-\eqref{eBC} with $\mu$ instead of $\lambda$ for $\mu \in[\lambda,\lambda_{N_0}]$, satisfy $\|u\|0$, and $q>p-1$. \end{example} \subsection*{Acknowledgments} Chan-Gyun Kim was supported by National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology, NRF-2011-357-C00006) Eun Kyoung Lee was supported by a 2-Year Research Grant of Pusan National University. This work was done when the first author visited College of William and Mary. 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