\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 106, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2012/106\hfil Global continuum and multiple positive solutions] {Global continuum and multiple positive solutions to a p-Laplacian boundary-value problem} \author[C.-G. Kim, J. Shi \hfil EJDE-2012/106\hfilneg] {Chan-Gyun Kim, Junping Shi} % 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{Junping Shi \newline Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795, USA.\newline School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China} \email{shij@math.wm.edu} \thanks{Submitted May 25, 2012. Published June 25, 2012.} \subjclass[2000]{34B18, 34C23, 35J25} \keywords{ Upper and lower solution; positive solution; p-Laplacian;\hfill\break\indent uniqueness; multiplicity} \begin{abstract} A $p$-Laplacian boundary-value problem with positive nonlinearity is considered. The existence of a continuum of positive solutions emanating from $(\lambda,u)=(0,0)$ is shown, and it can be extended to $\lambda=\infty$. Under an additional condition on the nonlinearity, it is shown that the positive solution is unique for any $\lambda>0$; thus the continuum $\mathcal{C}$ is indeed a continuous curve globally defined for all $\lambda>0$. In addition, by the upper and lower solutions method, existence of three positive solutions is established under some conditions on the nonlinearity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Consider a boundary-value problem $$\label{Plambda} \begin{gathered} (w(t)\varphi_{p}(u'(t)))' + \lambda h(t)f(u(t))=0, \quad t \in (0,1), \\ u(0) =u(1)=0, \end{gathered}$$ where $\varphi_p(x) := |x|^{p-2}x$, $p>1$ and $\lambda$ is a nonnegative parameter. For the functions in \eqref{Plambda}, throughout the paper, we assume that the following hypotheses hold: \begin{itemize} \item[(H1)] $h \in C([0,1],(0,\infty))$; \item[(W1)] $w\in C([0,1],[0,\infty))$, $w(t)>0$ for $t \in (0,1)$ and $\varphi_p^{-1}(1/w) \in L^1(0,1)$; \item[(F1)] $f \in C([0,\infty),(0,\infty))$, and $\lim_{u \to \infty} \frac{f(u)}{\varphi_p(u)}=0$. \end{itemize} Our main result is that under the above conditions for the weight functions $w$, $h$, and the nonlinearity $f$, a continuum $\mathcal{C}$ of positive solutions of \eqref{Plambda} emanates from $(\lambda,u)=(0,0)$, and $\mathcal{C}$ can be extended to $\lambda=\infty$. Thus we establish the existence of at least one positive solution of \eqref{Plambda} for any $\lambda>0$. Under an additional condition \begin{itemize} \item[(F2)] $\frac{f(u)}{\varphi_p(u)}$ is strictly decreasing on $(0,\infty)$, \end{itemize} it is shown that the positive solution of \eqref{Plambda} is unique for any $\lambda>0$, thus the continuum $\mathcal{C}$ is indeed a continuous curve globally defined for all $\lambda>0$ in this case. On the other hand, we show that under some different conditions for $f$, problem \eqref{Plambda} has at least three positive solutions, which indicates that the bifurcation diagram for \eqref{Plambda} cannot be a monotone curve in this case. These results demonstrate the rich structure of the solution set of \eqref{Plambda}. Problem \eqref{Plambda} or similar equations arises from mathematical models of chemical catalysis \cite{MR590000,MR824204,MR1305768}, combustion theory \cite{MR1631537,Williams}, and ecological modeling \cite{jiang2009bistability,MR2802109}. Common to this type of models is that the equation may possess multiple positive steady state solutions, and the bifurcation diagram of the positive steady states may have multiple turning points (saddle-node bifurcation points). For the reaction-diffusion case $(p=2)$, many results on multiple positive solutions and $S$-shaped bifurcation diagrams have been obtained, see for example \cite{MR613056,MR590000,MR2228186,MR824204,MR1781263,MR1709753}. Moreover for some cases, careful analysis can lead to exact multiplicity of solutions and exact $S$-shaped bifurcation diagrams, see for example, \cite{MR1834115,MR1610804,MR1799061,MR1285087,MR1631537,MR1305768}. The $p$-Laplacian equation \eqref{Plambda} is considerably harder than the corresponding reaction-diffusion equation (which is the special case of $p=2$). The mathematical difficulty comes from the nonlinearity and degeneracy of the differential operator and the failure of linearization and continuation technique in many places. While in general fewer existence and multiplicity results have been proved for $p$-Laplacian equations, we mention the nice work in \cite{MR904816,MR2592712,MR1492111,MR1064016} as examples, and a nice survey of earlier work was given in \cite{MR1695379}. For one-dimensional case, more precise results can be established as several exact multiplicity of solutions and exact $S$-shaped bifurcation diagrams have been shown in \cite{MR2800152,MR2445267} recently for $f$ satisfying (F1), and similar uniqueness result when $f$ also satisfies (F2) has been also proved in \cite{MR2651739}. But there are very few results for the existence of three positive solutions when the problem is spatially non-homogenous as the one we consider here, and the nature of global continuum of the positive solutions has not been exploited before either. In a recent work \cite{Kim2012} by the first author, an existence result for the three positive solutions was proved for nondecreasing nonlinearity $f$ with $p$-sublinear at infinity, and in this paper we consider non-monotone nonlinearity $f$ with a positive falling zero as well as $p$-sublinear at infinity. The rest of this article is organized as follows. In Section 2, we introduce some preliminary results such as definitions and lemmas. In Section 3, our main results are given and also an example to illustrate our result is presented. \section{Preliminaries} First we recall the general framework for the solutions of \eqref{Plambda}. Consider the Banach space $$C_w[0,1]=\{u \in C[0,1]\cap C^1(0,1) : w\varphi_p(u') \in C[0,1]\}$$ with norm $$\|u\|_w=\|u\|_\infty+\|w^{1/(p-1)}u'\|_\infty,$$ where $\|v\|_\infty=\max_{t\in[0,1]}|v(t)|$. Note that if $w(t) >0$ for $t \in [0,1]$, then $(C_w[0,1],\|\cdot\|_w )$ is equivalent to $(C^1[0,1], \|\cdot\|_1 )$. Here $\|u\|_1=\|u\|_\infty+\|u'\|_\infty$. We set $\mathcal{K}_w=\{u \in C_w[0,1] : u(t) \ge 0,~t \in [0,1]\}$. Then $\mathcal{K}_w$ is an ordered cone in $C_w[0,1]$, and let $L^1_+(0,1)$ denote the set of nonnegative Lebesgue-integrable functions on $[0,1]$. To define the solution operator to \eqref{Plambda}, let us consider the $p$-Laplacian problem $$\label{EK} \begin{gathered} (w(t)\varphi_{p}(u'(t)))' + K(t)=0,~a.e.~ in~ (0,1), \\ u(0) =u(1)=0, \end{gathered}$$ where $K \in L^1_+(0,1)$. By a solution $u$ of problem \eqref{EK}, we understand a function $u \in C_w[0,1]$ with $w\varphi_{p}(u')\in AC[0,1]$ which satisfies \eqref{EK}. We recall the following lemmas from \cite{Kim2012}. \begin{lemma}[\cite{Kim2012}] \label{Lem2.1} Assume {\rm (W1)} is satisfied, and let $u$ be the solution of \eqref{EK} with $K>0$, a.e. in $[0,1]$. Then there exists a unique point $A_u \in (0,1)$ such that $(w^{1/(p-1)}u')(t)>0$, $t\in[0,A_u)$, $u'(A_u)=0$ and $(w^{1/(p-1)}u')(t)<0$, $t \in(A_u,1]$. \end{lemma} \begin{lemma}[\cite{Kim2012}] \label{Lem2.2} Assume {\rm (W1)} is satisfied, and let $B$ and $C$ be positive constants such that $B < C$. Then there exists $\epsilon=\epsilon(C/B)>0$ such that $A_u \in (2\epsilon,1-2\epsilon)$ for all possible solutions $u$ of \eqref{EK} with $K \in K_B^C$, where $$K_B^C=\{ K \in L^1(0,1):K \ge B,\,\int_0^1 K(s)ds\le C\}$$ and $A_u$ is the unique point in Lemma \ref{Lem2.1} such that $u'(A_u)=0$. \end{lemma} \begin{remark}\label{Rem2.3} \rm If $w$ and $K$ are symmetric with respect to $1/2$, one can easily see that the unique solution $u_K$ of \eqref{EK} is symmetric with respect to $1/2$. Hence $A_{u_K}=1/2$ in this case. \end{remark} Next we show that \eqref{EK} and \eqref{Plambda} can be rewritten into some integral forms. Problem \eqref{EK} can be equivalently written as $$u(t)=G_p(K)(t):= \int_0^t\varphi_p^{-1} \Big[ \frac{1}{w(s)}\Big(c(K)+\int_s^1K(\tau)d\tau\Big)\Big]ds,\quad t \in [0,1],$$ where $c: L^1(0,1) \to \mathbb{R}$ is a mapping satisfying $$\int_0^1\varphi_p^{-1}\Big[ \frac{1}{w(s)}\Big(c(K)+\int_s^1K(\tau)d\tau\Big)\Big]ds=0.$$ It can be proved that the mapping $c$ is continuous and it maps bounded sets in $L^1(0,1)$ into bounded sets in $\mathbb{R}$, and the mapping $G_p:L^1_+(0,1)\to \mathcal{K}_w$ is continuous and it maps equi-integrable sets of $L^1_+(0,1)$ into relatively compact sets of $\mathcal{K}_w$ by similar arguments as in the previous result (e.g., see \cite{MR1302138,MR1296134,MR1621038}). Define $H:[0,\infty) \times \mathcal{K}_w \to L^1_+(0,1)$ by $H(\lambda,u)(t)=\lambda h(t)f(u(t))$. Then it is well known that $H$ is a continuous operator which maps bounded sets of $[0,\infty)\times \mathcal{K}_w$ into equi-integrable sets of $L^1_+(0,1)$. Thus $T=G_p \circ H:[0,\infty)\times \mathcal{K}_w \to \mathcal{K}_w$ is completely continuous. Furthermore, \eqref{Plambda} has a positive solution $u$ if and only if $T(\lambda,\cdot)$ has a fixed point $u$ in $\mathcal{K}_w\setminus \{0\}$ for $\lambda>0$. Now we recall a well-known theorem for the existence of a global continuum of solutions to $T(\lambda,u)=0$: \begin{theorem}[{\cite[Corollary 14.12]{MR816732}}] \label{Thm2.4} Let $X$ be a Banach space with $X \neq \{0\}$ and let $\mathcal{K}$ be an ordered cone in $X$. Consider $$\label{2.1} x=T(\mu,x),$$ where $\mu \in [0,\infty)$ and $x \in \mathcal{K}$. If $T:[0,\infty)\times \mathcal{K} \to \mathcal{K}$ is completely continuous and $T(0,x)=0$ 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} Since $T(0,u)=0$ and $T(\lambda,0) \neq 0$ if $\lambda > 0$, by Theorem \ref{Thm2.4}, we obtain the following proposition. \begin{proposition}\label{Pro2.5} Assume {\rm (H1), (W1), (F1)} are satisfied. Then there exists an unbounded continuum $\mathcal{C}$ of positive solutions for \eqref{Plambda} emanating from $(0,0)$ in $[0,\infty) \times \mathcal{K}_w$. \end{proposition} Next we review the notion of order in $C_w[0,1]$, and also the upper and lower solutions of \eqref{Plambda}, which was introduced in \cite{Kim2012}. \begin{definition}\label{Def2.6} \rm Given functions $u,v, w: [0,1] \to \mathbb{R}$, we say that \begin{itemize} \item[(1)] $u \le v$ if for all $t \in[0,1]$, $u(t)\le v(t)$; and \item[(2)] $u \in [v, w]$ if $v\le u\le w$. \end{itemize} \end{definition} \begin{definition}\label{Def2.7} \rm Let $u,v \in C_w[0,1]$. We say that $u\prec v$ if and only if the following three conditions hold: \begin{itemize} \item[(i)] for all $t \in (0,1)$, $u(t) < v(t)$, \item[(ii)] either $u(0) < v(0)$ or $(w^{1/(p-1)}u')(0)<(w^{1/(p-1)}v')(0)$, \item[(iii)] either $u(1) < v(1)$ or $(w^{1/(p-1)}u')(1)>(w^{1/(p-1)}v')(1)$. \end{itemize} \end{definition} \begin{definition}\label{Def2.8} \rm A function $\alpha \in C_w[0,1]$ with $w \varphi_p(\alpha')$ absolutely continuous is called a lower solution of \eqref{Plambda} if \begin{itemize} \item[(i)] for a.e. $t \in (0,1)$, $(w(t)\varphi_{p}(\alpha'(t)))' + \lambda h(t)f(\alpha(t))\ge 0$, \item[(ii)] $\alpha(0) \le 0$ and $\alpha(1) \le 0$. \end{itemize} In the same way we define an upper solution of \eqref{Plambda} by reversing the above inequalities. \end{definition} \begin{definition}\label{Def2.9} \rm A function $\alpha \in C_w[0,1]$ with $w \varphi_p(\alpha')$ absolutely continuous is called a strict lower solution of \eqref{Plambda} if it is not a solution, $\alpha(0) \le 0$, $\alpha(1)\le 0$ and \begin{itemize} \item[(i)] for any $t_0 \in (0,1)$, there exist an open interval $I_0 \subseteq (0,1)$ and $\epsilon_0>0$ such that $t_0 \in I_0$ and for a.e. $t \in I_0$, for all $u \in [\alpha(t),\alpha(t)+\epsilon_0]$, one has $(w(t)\varphi_{p}(\alpha'(t)))' + \lambda h(t)f(u)\ge 0$, \item[(ii)] either $\alpha(0) < 0$ (respectively, $\alpha(1)<0)$ or there exist $\delta_0>0$ and $c_0 \in C_w[0,1]$ with $c_0\succ 0$ such that if $I_0=[0,\delta_0)$ (respectively, $I_0=(1-\delta_0,1])$, then for a.e. $t \in I_0$, for all $u \in [\alpha(t),\alpha(t)+c_0(t)]$, one has $(w(t)\varphi_{p}(\alpha'(t)))' + \lambda h(t)f(u)\ge 0$. \end{itemize} In a similar way, a function $\beta \in C_w[0,1]$ with $w \varphi_p(\beta')$ absolutely continuous is a strict upper solution of \eqref{Plambda} if it is not a solution, $\beta(0) \ge 0$, $\beta(1)\ge 0$ and \begin{itemize} \item[(i)] for any $t_0 \in (0,1)$, there exist an open interval $I_0 \subseteq (0,1)$ and $\epsilon_0>0$ such that $t_0 \in I_0$ and, for a.e. $t \in I_0$, for all $u \in [\beta(t)-\epsilon_0,\beta(t)]$, one has $(w(t)\varphi_{p}(\beta'(t)))' + \lambda h(t)f(u)\le 0$, \item[(ii)] either $\beta(0) > 0$ (respectively, $\beta(1)>0)$ or there exist $\delta_0>0$ and $c_0 \in C_w[0,1]$ with $c_0\succ 0$ such that if $I_0=[0,\delta)$ (respectively, $I_0=(1-\delta_0,1])$, then for a.e. $t \in I_0$, for all $u \in [\beta(t)-c_0(t),\beta(t)]$, one has $(w(t)\varphi_{p}(\beta'(t)))' + \lambda h(t) f(u)\le 0$. \end{itemize} \end{definition} With these definition, the following existence results were proved in \cite{Kim2012}. \begin{theorem}[\cite{Kim2012}]\label{Thm2.10} Assume that $\alpha$ and $\beta$ are lower and upper solutions of \eqref{Plambda} respectively such that $\alpha \le \beta$. Then \eqref{Plambda} has at least one solution $u$ such that $\alpha \le u \le \beta$. Moreover, if $\alpha$ and $\beta$ are strict, then $\alpha \prec u \prec \beta$. \end{theorem} \begin{theorem}[\cite{Kim2012}]\label{Thm2.11} Assume that $\alpha_1$ and $\beta_2$ are lower and upper solutions of \eqref{Plambda} respectively and $\alpha_2$ and $\beta_1$ are strict lower and upper solutions of \eqref{Plambda} respectively such that $$\alpha_1 \le \beta_1\le \beta_2,~\alpha_1\le\alpha_2 \le \beta_2,$$ and there exists $t_0 \in [0,1]$ with $\beta_1(t_0) < \alpha_2(t_0)$. Then there exist three solutions of \eqref{Plambda} such that \begin{gather*} \alpha_1 \le u_1\prec \beta_1,\quad \alpha_2 \prec u_2 \le \beta_2, \\ u_3 \in [\alpha_1,\beta_2]\setminus ([\alpha_1,\beta_1]\cup[\alpha_2,\beta_2]). \end{gather*} \end{theorem} Finally we introduce the generalized Picone identity due to Jaro\v{s} and Kusano \cite{MR1711081,MR1768900}. Let us consider the following differential operators: \begin{equation*} l_{p}[y] := (r\varphi_{p}(y'))' + q(t) \varphi_{p}(y), \end{equation*} \begin{equation*} L_{p}[z]:= (R\varphi_{p}(z'))' + Q(t) \varphi_{p}(z). \end{equation*} \begin{lemma}[{\cite[p.~382]{MR1768900}}] \label{Lem2.12} Let $r,q,R$ and $Q$ be real-valued continuous functions on an interval $I$. If $y$ and $z$ are any functions such that $y, z, r\varphi_{p}(y')$ and $R\varphi_{p}(z')$ are differentiable on $I$ and $z(t) \neq 0$ for $t \in I$, then, for $t \in I$, we have \label{2.2} \begin{aligned} &\frac{d}{dt} \Big( y\Big[r\varphi_p(y')-\varphi_p\big(\frac{y}{z}\big) R\varphi_p(z')\Big] \Big) \\ & = (r-R)|y'|^p+(Q-q) |y|^p +R\Big[ |y'|^p + (p-1) \big| \frac{yz'}{z}\big|^p - p\varphi_{p}(y)y' \varphi_{p} \big( \frac{z'}{z} \Big) \Big] \\ &\quad + y l_{p}[y] - \frac{|y|^p}{\varphi_{p}(z)} L_{p}[z]. \end{aligned} \end{lemma} \begin{remark}\label{Rem2.13} \rm By Young's inequality, we obtain $$|y'|^p + (p-1) \big|\frac{yz'}{z} \big|^p - p\varphi_{p}(y)y' \varphi_{p} \big( \frac{z'}{z} \big) \geq 0,$$ and the equality holds if and only if $y'=y z'/z$ on $I$. \end{remark} \section{Main result} Let $\mathcal{S}$ be the set of positive solutions of \eqref{Plambda} in $(0,\infty)\times \mathcal{K}_w$. For the sake of convenience, we use the following notation \begin{gather*} h_0:=\min_{t \in [0,1]} h(t),\;\; h^0:=\max_{t \in [0,1]} h(t),\\ \overline{w} :=\int_0^1\varphi_{p}^{-1}\big(\frac{1}{w(s)}\big)ds. \end{gather*} \begin{lemma}\label{Lem3.1} Assume {\rm (H1), (W1), (F1)} are satisfied, and let $\{(\lambda_n,u_n)\}_{n=1}^\infty$ be a sequence in $\mathcal{S}$ such that $\lambda_n\to \infty$ as $n \to \infty$. Then $\|u_n\|_\infty \to \infty$ as $n \to \infty$. \end{lemma} \begin{proof} Assume on the contrary that $\lambda_n \to \infty$ as $n \to \infty$, but there exists $M>0$ such that $\|u_n\|_\infty \le M$ for all $n$. Then from (F1) there exists $\delta > 0$ such that $f(u_n(t)) \ge 4\delta,~t \in [0,1]$. For each $n$, the function $u_n$ attains its maximum at the unique point $x_n\in (0,1)$ and $u_n'(x_n)=0$. Suppose that $x_n \ge 1/2$ (the case $x_n < 1/2$ is similar). Then, for $t \in (0,1/4)$, one has \begin{align*} u_n'(t) &= \varphi_p^{-1}\Big(\frac{1}{w(t)} \int_t^{x_n}\lambda_n h(s)f(u_n(s))ds\Big)\\ &\geq \varphi_p^{-1}\Big(\frac{1}{w(t)} \int_{1/4}^{1/2} \lambda_nh(s) f(u_n(s))ds \Big)\\ &\geq \varphi_p^{-1}(\lambda_n \delta h_0)\varphi_p^{-1}\big(\frac{1}{w(t)}\big), \end{align*} and hence, $$u_n(\tfrac14) \ge \varphi_p^{-1}(\lambda_n \delta h_0)\int_0^{1/4}\varphi_p^{-1} \big(\frac{1}{w(s)}\big)ds .$$ Since $\lambda_n \to \infty$ as $n \to \infty$, $u_n(\tfrac14) \to \infty$ as $n \to \infty$. This contradicts the fact that $\|u_n\|_\infty\le M$ for all $n$. \end{proof} \begin{lemma}\label{Lem3.2} Assume {\rm (H1), (W1), (F1)} are satisfied, and let $\{(\lambda_n,u_n)\}_{n=1}^\infty$ be a sequence in $\mathcal{S}$ such that $\|u_n\|_w \to \infty$ as $n \to \infty$. Then $\lambda_n\to \infty$ as $n \to \infty$. \end{lemma} \begin{proof} Assume on the contrary that $\|u_n\|_w \to \infty$ as $n \to \infty$, but there exists $L>0$ such that $\lambda_n\le L$ for all $n$. Then one can easily see that $\|u_n\|_\infty \to \infty$ as $n \to \infty$. Put $$\alpha = \frac{1}{2L h^0\overline{w}^{p-1}}.$$ By (F1), there exists $N_\alpha>0$ such that for all $u >N_\alpha$, $f(u) < \alpha u^{p-1}$. Let $M_\alpha=\max_{0\le u\le N_\alpha} f(u)$, $A_n := \{t \in [0,1] : u_n(t) \le N_\alpha \}$ and $B_n := \{t \in [0,1] : u_n(t) > N_\alpha \}$. Then $f(u_n(t))\le M_\alpha$, $t \in A_n$ and $f(u_n(t))\le \alpha u_n(t)^{p-1}$, $t \in B_n$. Put $u_n(x_n) = \|u_n\|_\infty$. Then \begin{align*} u_n(x_n) &= \int_0^{x_n} \varphi_{p}^{-1} \Big( \frac{1}{w(s)} \int_{s}^{x_n} \lambda_n h(\tau)f(u_n(\tau))d \tau \Big) ds \\ &\leq \varphi_{p}^{-1}(\lambda_nh^0) \int_0^1 \varphi_{p}^{-1} \Big(\frac{1}{w(s)}\Big[ \int_{A_n} f(u_n(\tau))d \tau + \int_{B_n} f(u_n(\tau))d \tau\Big] \Big) ds \\ &\leq \varphi_{p}^{-1}(\lambda_nh^0) \int_0^1 \varphi_{p}^{-1} \Big(\frac{1}{w(s)}\big[M_\alpha + \alpha\int_{B_n} u_n(\tau)^{p-1}d \tau\big] \Big) ds. \end{align*} Thus \begin{align*} \frac{1}{\varphi_{p}^{-1}(\lambda_nh^0)} &\leq \int_0^1 \varphi_{p}^{-1} \Big(\frac{1}{w(s)}\Big[\frac{M_\alpha}{\|u_n\|_\infty^{p-1}} + \alpha\int_{B_n} \frac{u_n(\tau)^{p-1}}{\|u_n\|_\infty^{p-1}}d \tau \Big] \Big) ds\\ &\leq \int_0^1\varphi_{p}^{-1}\Big(\frac{1}{w(s)} \big[\frac{M_\alpha}{\|u_n\|_\infty^{p-1}} + \alpha\big]\Big)ds. \end{align*} Letting $n \to \infty$, by Lebesgue's dominated convergence theorem, $$\frac{1}{\varphi_{p}^{-1}(Lh^0)}\le \varphi_{p}^{-1}(\alpha)\overline{w},$$ which contradicts the choice of $\alpha$. \end{proof} The following theorem can be easily obtained in view of Proposition \ref{Pro2.5}, Lemma \ref{Lem3.1} and Lemma \ref{Lem3.2}. \begin{theorem}\label{Thm3.3} Assume {\rm (H1), (W1), (F1)} are satisfied. Then there exists an unbounded continuum $\mathcal{C}$ of positive solutions for \eqref{Plambda} emanating from $(0,0)$ in $[0,\infty) \times \mathcal{K}_w$ such that \begin{itemize} \item[(i)] for each $\lambda>0$, there exists a positive solution $u_\lambda$ of \eqref{Plambda} such that $(\lambda,u_\lambda)\in \mathcal{C}$ and \item[(ii)] for $(\lambda,u_\lambda) \in \mathcal{S}$, $\lambda \to \infty$ if and only if $\|u_\lambda\|_w \to \infty$. \end{itemize} \end{theorem} Let $v$ be the unique solution of \begin{gather*} (w(t)\varphi_{p}(v'(t)))' + h(t)=0,~ t \in (0,1), \\ v(0) =v(1)=0. \end{gather*} Then $v \in C_w[0,1]$ and $v(t) >0$, $t \in(0,1)$. Note that condition (F1) implies that $f$ also satisfies the following condition (e.g., see \cite[Lemma 4.1]{MR2381665}) \begin{itemize} \item[(F1$^*$)] Let $f^*(u)=\max_{0\le s\le u}f(s)$, then $\lim_{u \to \infty} \frac{f^*(u)}{\varphi_p(u)}=0$. \end{itemize} Now we give a uniqueness result under the additional condition (F2). \begin{theorem}\label{Thm3.4} Assume {\rm (H1), (W1),(F1), (F2)} are satisfied. Then $\mathcal{S}=\mathcal{C}$ and $\mathcal{C}$ is the solution curve of positive solutions for \eqref{Plambda} such that \begin{itemize} \item[(i)] for each $\lambda>0$, there exists the unique positive solution $u(\lambda)$ of \eqref{Plambda} such that $(\lambda,u(\lambda))\in \mathcal{C}$, \item[(ii)] $\lambda \to \infty$ if and only if $\|u(\lambda)\|_w \to \infty$ and \item[(iii)] for any $0<\lambda_a<\lambda_b$, $u(\lambda_a)\prec u(\lambda_b)$. \end{itemize} \end{theorem} \begin{proof} First, we prove that \eqref{Plambda} has at most one positive solution for each $\lambda>0$. Assume on the contrary that there exists $\lambda>0$ such that $(\lambda,u_1)$ and $(\lambda,u_2)$ are two distinct positive solutions of \eqref{Plambda}. Without loss of generality, we may assume that there exists an interval $(a,b)(\subseteq (0,1))$ such that $u_1(t)>u_2(t),~t\in (a,b),u_1(a)=u_2(a)$ and $u_1(b)=u_2(b)$. Then we have four cases: (1) $a>0$, $b=1$; (2) $a=0$, $b=1$; (3) $a=0$, $b<1$; (4) $a>0$, $b<1$. We only prove the case (1) since other cases are similar. In this case, we know that $u_1'(a)\ge u_2'(a)$ and $(w^{1/(p-1)}u_1')(1)\le (w^{1/(p-1)}u_2')(1)<0$. Then, by L'Hospital's rule, $$\label{3.1} \lim_{t\to 1^-}\frac{u_1(t)}{u_2(t)}=\frac{(w^{1/(p-1)}u_1')(1)} {(w^{1/(p-1)}u_2')(1)}.$$ Taking $r(t)=R(t)=w(t),y(t)=u_1(t),z(t)=u_2(t)$, $q(t)=\frac{\lambda h(t)f(u_1(t))}{\varphi_p(u_1(t))}$ and $Q(t)=\frac{\lambda h(t) f(u_2(t))}{\varphi_p(u_2(t))}$ in Lemma \ref{Lem2.12} and integrating \eqref{2.2} from $a$ to $1$, by \eqref{3.1} and Remark \ref{Rem2.13}, \begin{align*} &-u_1(a)[(w\varphi_p(u_1'))(a)-(w\varphi_p(u_2'))(a)]\\ &\geq \lambda \int_a^1 h(s) \Big[\frac{f(u_2(s))}{\varphi_p(u_2(s))} -\frac{f(u_1(s))}{\varphi_p(u_1(s))}\Big] |u_1(s)|^p ds. \end{align*} By the fact that $u_1'(a)>u_2'(a)$, \begin{equation*} \lambda \int_a^1 h(s)\Big[\frac{f(u_2(s))}{\varphi_p(u_2(s))} -\frac{f(u_1(s))}{\varphi_p(u_1(s))}\Big]|u_1(s)|^p ds\le 0. \end{equation*} On the other hand, since $u_1(t)>u_2(t),~t \in (a,1)$, by $(F_2)$, $$\lambda \int_a^1 h(s)\Big[\frac{f(u_2(s))}{\varphi_p(u_2(s))} -\frac{f(u_1(s))}{\varphi_p(u_1(s))}\Big]|u_1(s)|^p ds>0.$$ This is a contradiction and thus \eqref{Plambda} has a unique positive solution $u(\lambda)$ for each $\lambda>0$. For $0<\lambda_a<\lambda_b$, it can be easily see that $u(\lambda_a)$ is a strict lower solution of \eqref{Plambda} with $\lambda=\lambda_b$. Put $\beta_1=\lambda_b C_b \frac{v}{\|v\|_\infty}$, where $C_b$ is the constant satisfying $$\frac{f^*(\lambda_b C_b)}{(\lambda_b C_b)^{p-1}} <\frac{1}{\lambda_b \|v\|_\infty^{p-1}}$$ and $u(\lambda_a)\le \beta_1$. Note that it is possible to choose such a constant $C_b$ in view of $(F_1^*)$. Then, for $t \in (0,1)$, \begin{align*} -(w(t)\varphi_{p}(\beta_1'(t)))' &= \Big(\lambda_b C_b \frac{1}{\|v\|_\infty}\Big)^{p-1}h(t)\\ &> \lambda_b h(t) f^*(\lambda_b C_b)\\ &\geq \lambda_b h(t)f^*(\beta_1(t))\\ &\geq \lambda_b h(t)f(\beta_1(t)), \end{align*} which implies that $\beta_1$ is a strict upper solution of \eqref{Plambda} with $\lambda=\lambda_b$. By Theorem \ref{Thm2.10}, there exists a positive solution $u_b$ of \eqref{Plambda}, with $\lambda=\lambda_b$, such that $u(\lambda_a)\prec u_b$, and $u_b$ must be same as $u(\lambda_b)$ since \eqref{Plambda} has at most one positive solution for each $\lambda>0$. Thus the proof is complete by Theorem \ref{Thm3.3}. \end{proof} For the rest of this article, we assume that $f$ satisfies \begin{itemize} \item[(F3)] There exist $m, M>0$ such that $f$ is nondecreasing on $(m,M)$. \end{itemize} Define $\bar f$ as \bar f(u)=\begin{cases} \hat f(u), &u \lambda h(t) f^*(a)\\ &\geq \lambda h(t)f^*(\beta_1(t))\\ &\geq \lambda h(t)f(\beta_1(t)). \end{align*} Thus \beta_1 is a strict upper solution of \eqref{Plambda}. Thirdly, let \alpha_2 be the unique solution of $$\label{3.2} \begin{gathered} (w(t)\varphi_{p}(\alpha_2'(t)))' + \lambda^*h_0 \bar f(\rho(t))=0,~ t \in (0,1), \\ \alpha_2(0) =\alpha_2(1)=0, \end{gathered}$$ where \lambda^* \in (\lambda_1,\lambda) and \rho(t) is defined as \rho(t)= \begin{cases} b\big[\int_0^{\epsilon_b} \varphi_p^{-1} \big(\frac{1}{w(s)}\big)ds\big]^{-1} \int_0^{t} \varphi_p^{-1} (\frac{1}{w(s)})ds, & t \in [0,\epsilon_b],\\ b, & t \in (\epsilon_b,1-\epsilon_b),\\ b\big[\int_{1-\epsilon_b}^1\varphi_p^{-1} \big(\frac{1}{w(s)}\big)ds\big]^{-1} \int_{t}^1\varphi_p^{-1} (\frac{1}{w(s)})ds, &t \in [1-\epsilon_b,1]. \end{cases} Since \lambda^*< \lambda_2, one has \begin{align*} \|\alpha_2\|_\infty &= \alpha_2(A_{\alpha_2})\\ &= \int_0^{A_{\alpha_2}}\varphi_p^{-1}\Big(\frac{1}{w(s)}\int_s^{A_{\alpha_2}} \lambda^*h_0\bar f(\rho(\tau))d\tau\Big)ds\\ &\leq \varphi_p^{-1}(\lambda^* h_0 \bar f(b))\overline{w} = \varphi_p^{-1}(\lambda^* h_0 f(b))\overline{w} < M. \end{align*} Note that \alpha_2=\varphi_p^{-1}(\lambda^*h_0) l, where l is the unique solution of \begin{gather*} (w(t)\varphi_{p}(l'(t)))' + \bar f(\rho(t))=0,~t \in (0,1), \\ l(0) =l(1)=0. \end{gather*} Since \|\rho\|_\infty=b, by Lemma \ref{Lem2.2}, A_{\alpha_2} =A_l \in (2\epsilon_b,1-2\epsilon_b). For t \in [0,\epsilon_b], $$\label{3.3} w(t)\varphi_p(\rho'(t)) = \varphi_p\Big(b\Big[\int_0^{\epsilon_b} \varphi_p^{-1} \big(\frac{1}{w(s)}\big)ds\Big]^{-1}\Big) \leq \varphi_p(b/C_{\epsilon_b})$$ and for t \in [1-\epsilon_b,1], $$\label{3.4} -w(t)\varphi_p(\rho'(t)) = \varphi_p\Big( b\Big[\int_{1-\epsilon_b}^1\varphi_p^{-1} \big(\frac{1}{w(s)}\big)ds\Big]^{-1}\Big) \leq \varphi_p(b/C_{\epsilon_b}).$$ For t \in [0,\epsilon_b], integrating \eqref{3.2} from t to A_{\alpha_2}, one has \begin{align*} w(t)\varphi_{p}(\alpha_2'(t)) & = \lambda^*h_0 \int_t^{A_{\alpha_2}}\bar f(\rho(s))ds\\ &\geq \lambda^*h_0 \int_{\epsilon_b}^{2\epsilon_b}\bar f(\rho(s))ds\\ &= \lambda^*h_0 \int_{\epsilon_b}^{2\epsilon_b}\bar f(b)ds\\ &= \lambda^*h_0 \epsilon_b \bar f(b) = \lambda^*h_0 \epsilon_b f(b)\\ &> \varphi_p\big(\frac{b}{C_{\epsilon_b}}\big). \end{align*} Thus by \eqref{3.3}, for t \in [0,\epsilon_b], \begin{equation*} w(t)\varphi_{p}(\alpha_2'(t)) > w(t)\varphi_p(\rho'(t)), \end{equation*} which implies \alpha_2(t) \ge \rho(t), \quad t \in [0,\epsilon_b]. Similarly, by \eqref{3.4}, \begin{equation*} \alpha_2(t) \ge \rho(t),\quad t \in [1-\epsilon_b,1]. \end{equation*} By the facts that A_{\alpha_2}\in (2\epsilon_b,1-2\epsilon_b) and \rho(t)=b, t \in [\epsilon_b,1-\epsilon_b], we have \alpha_2\ge \rho, which implies that, for t \in (0,1), $-(w(t)\varphi_{p}(\alpha_2'(t)))' = \lambda^* h_0 \bar f(\rho(t)) \leq \lambda^* h_0 \bar f(\alpha_2(t)) < \lambda h(t)f(\alpha_2(t)),$ and thus \alpha_2 is a strict lower solution of \eqref{Plambda}. Since \|\alpha_2\|_\infty \ge \|\rho\|_\infty=b>a=\|\beta_1\|_\infty, there exists t_0 \in (0,1) such that \alpha_2(t_0) > \beta_1(t_0). Finally, put \beta_2=\lambda C_\lambda \frac{v}{\|v\|_\infty}, where C_\lambda is the constant satisfying \frac{f^*(\lambda C_\lambda)}{(\lambda C_\lambda)^{p-1}} <\frac{1}{\lambda \|v\|_\infty^{p-1}},\quad \beta_1 \le \beta_2, \quad \alpha_2 \le \beta_2. Then, for t \in (0,1), we have \begin{align*} -(w(t)\varphi_{p}(\beta_2'(t)))' &= \Big(\lambda C_\lambda \frac{1}{\|v\|_\infty}\Big)^{p-1}h(t)\\ &> \lambda h(t) f^*(\lambda C_\lambda)\\ &\geq \lambda h(t)f^*(\beta_2(t))\\ &\geq \lambda h(t)f(\beta_2(t)), \end{align*} and \beta_2 is an upper solution of \eqref{Plambda}. Thus, by Theorem \ref{Thm2.11}, \eqref{Plambda} has three positive solutions for all \lambda \in (\lambda_1,\lambda_2). \end{proof} In the results so far, we assume that f is positive for all u\ge 0 as it satisfies (F1). If we assume that f has a positive falling zero instead of (F1); i.e., f satisfies \begin{itemize} \item[(F1')] f \in C([0,\infty),\mathbb{R}) and there exists k >0 such that (k-u)f(u)>0 for u \neq k, \end{itemize} then we can obtain results similar to Theorem \ref{Thm3.3}, Theorem \ref{Thm3.4} and Theorem \ref{Thm3.5} as follows. \begin{theorem}\label{Thm3.6} Assume {\rm (H1), (W1), (F1')} are satisfied. Then there exists an unbounded continuum \mathcal{C} of positive solutions for \eqref{Plambda} emanating from (0,0) in [0,\infty) \times\mathcal{K}_w such that for each \lambda>0, there exists a positive solution u_\lambda of \eqref{Plambda} such that (\lambda,u_\lambda)\in \mathcal{C}, and for each \lambda>0, \|u_{\lambda}\|_{\infty}\le k. \end{theorem} \begin{theorem}\label{Thm3.7} Assume {\rm (H1), (W1), (F1')} and \begin{itemize} \item[(F2')] \frac{f(u)}{\varphi_p(u)} is strictly decreasing on (0,k) \end{itemize} are satisfied. Then \mathcal{S}=\mathcal{C} and \mathcal{C} is the solution curve of positive solutions for \eqref{Plambda} such that \begin{itemize} \item[(i)] for each \lambda>0, there exists the unique positive solution u(\lambda) of \eqref{Plambda} such that (\lambda,u(\lambda))\in \mathcal{C} and \item[(ii)] for any 0<\lambda_a<\lambda_b, u(\lambda_a)\prec u(\lambda_b). \end{itemize} \end{theorem} \begin{theorem}\label{Thm3.8} Assume {\rm (H1), (W1), (F1'), (F3)} are satisfied, and there exist positive constants a,b such that a0 and M>1 are constants which will be chosen later, and f_1 and f_2 are the functions such that f_1(u)\le \exp[\alpha/(\alpha+1)] for u \in [0,1] and f satisfies (F1) or (F_1'). Clearly, (W1) and (F3) are satisfied for m=1 and M>1 . Note that, in the proof of Theorem \ref{Thm3.5}, \epsilon_b is just needed for verifying that A_{\alpha_2} \in (2\epsilon_b,1-2\epsilon_b). Since w is symmetric with respect to 1/2, so \rho is also symmetric with respect to 1/2. Then A_{\alpha_2}=1/2 (see Remark \ref{Rem2.3}), and we can take \epsilon_b=1/5. For a=1 and b=\alpha, \frac{\varphi_p(b)f^*(a)}{\varphi_p(a)f(b)} =\frac{\varphi_p(\alpha)f(1)}{\varphi_p(1)f(\alpha)} =\alpha^{p-1}\exp\big[\frac{\alpha}{\alpha+1}-\frac{\alpha}{2}\big]. $$For any h \in C([0,1],(0,\infty)), there exists \alpha=\alpha(h)>0 such that$$ \alpha^{p-1}\exp\big[\frac{\alpha}{\alpha+1}-\frac{\alpha}{2}\big] <\frac{h_0\epsilon_b\varphi_p(C_{\epsilon_b})}{\varphi_p(\|v\|_\infty)}, $$and$$ \frac{\varphi_p(b)f^*(a)\varphi_p(\|v\|_\infty)}{\varphi_p(a)f(b)h_0\epsilon_b \varphi_p(C_{\epsilon_b})}<1. $$Furthermore, we can choose M=M(\alpha) such that$$ \varphi_p\big(\frac{b}{M}\big)=\varphi_p\big(\frac{\alpha}{M}\big) <\frac{\epsilon_b \varphi_p(C_{\epsilon_b})}{\varphi_p(\overline{w})}.  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