\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 126, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/126\hfil $p$-Laplacian boundary value problems] {Positive solutions of boundary value problems with $p$-Laplacian} \author[Q. Kong, M. Wang\hfil EJDE-2010/126\hfilneg] {Qingkai Kong, Min Wang} % in alphabetical order \address{Qingkai Kong \newline Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA} \email{kong@math.niu.edu} \address{Min Wang \newline Department of Mathematics, Northern Illinois University, Dekalb, IL 60115, USA} \email{mwang@math.niu.edu} \thanks{Submitted June 17, 2010. Published September 7, 2010.} \subjclass[2000]{34B15, 34B18} \keywords{Boundary value problem with $p$-Laplacian; positive solution; \hfill\break\indent existence and nonexistence; eigenvalue criteria; fixed point; index theory} \begin{abstract} In this article, we study a class of boundary value problems with $p$-Laplacian. By using a Green-like'' functional and applying the fixed point index theory, we obtain eigenvalue criteria for the existence of positive solutions. Several explicit conditions are derived as consequences, and further results are established for the multiplicity and nonexistence of positive solutions. Extensions are also given to partial differential BVPs with $p$-Laplacian defined on annular domains. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} In this article, we study the following boundary value problem (BVP) that includes the equation with $p$-Laplacian \label{e1.1} -(\phi(q(t)u'))'=w(t)f(t,u),\quad 00$,$a>0$,$f:[0,1]\times\mathbb{R}_+\to\mathbb{R}_+$is continuous,$q\in L[0,1]$with$q(t)\ge \delta>0$on$[0,1]$, and$w\in L[0,1]$with$w(t)\ge0$a.e. on$[0,1]$, and$\int_{0}^{1}w(t)dt>0$. BVPs with$p$-Laplacian have been investigated for decades. Results are obtained for the existence of positive solutions for different BCs. To name a few, see \cite{ALO, FZG} for Dirichlet BCs, \cite{DGP,PP} for periodic BCs, and \cite{Y} for the general separated BCs. For the work on$m$-point$p$-Laplacian BVPs, see \cite{FG, FGJ, GP, LZ} and the references therein. As a special case with$p=1$, the BVPs consisting of \eqref{e1.1} and various BCs have been extensively studied. We refer to the reader \cite{AO, BBCW, E1, JCOA, KK4, KK5, KK6, KK7, KW1, KW2, KW3, OW} and references therein. Among various criteria for the existence of positive solutions, some were established using the first eigenvalue of an associated Sturm-Liouville problem (SLP), see, for example \cite{E1, KK1, K, KW1, KW3, NT}. Such eigenvalue criteria are usually sharper than criteria obtained in some other ways especially when they involve the behavior of$f$as$u$near$0$and$\infty$. Therefore, a natural question arises: Are there parallel eigenvalue criteria for the$p$-Laplacian BVP \eqref{e1.1}, \eqref{e1.2} using the first eigenvalue of an associated half-linear SLP? To the best knowledge of the authors, no answers can be found in the literature although the spectral theory for half-linear SLPs has been well developed, see \cite{BD, E, KK2, KK3, KNT}. The main difficulties for the extension lie in the facts that no Green's functions can be found for equations with$p$-Laplacian since the solutions of half-linear equations do not form a linear space and the important Lagrange bracket property for linear SLPs is not satisfied by the half-linear SLPs. In this paper, by constructing a Green-like'' functional and applying a different fixed point index theory, we obtain eigenvalue criteria for the$p$-Laplacian BVP \eqref{e1.1}, \eqref{e1.2}. More specifically, we show that BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution if the first eigenvalue of an associated half-linear SLP satisfies certain relations with the behavior of the function$f$as$u$near$0$and$\infty$. Some explicit conditions are derived as consequences, and further results are also given for the multiplicity and nonexistence of positive solutions. Our work is new and improves most existing results on BVPs with$p$-Laplacian when restricted to problem \eqref{e1.1}, \eqref{e1.2}. Finally, we extend our results to partial differential BVPs with$p$-Laplacian on annular domains and hence obtain criteria for the existence, multiplicity, and nonexistence of positive radial solutions. This paper is organized as follows: after this introduction, we state our main results in Section 2. The proofs are given in Section 3. Extensions to$p$-Laplacian partial differential equations are given in Section 4. Several examples are presented in Section 5 as applications. \section{Main Results} For the function$f$given in \eqref{e1.1}, define $$\label{e2.1} \begin{gathered} f_0=\liminf_{u\to0^+}\min_{t\in[0,1]}f(t,u)/u^p,\quad f^0=\limsup_{u\to0^+}\max_{t\in[0,1]}f(t,u)/u^p,\\ f_\infty=\liminf_{u\to\infty}\min_{t\in[0,1]}f(t,u)/u^p,\quad f^\infty=\limsup_{u\to\infty}\max_{t\in[0,1]}f(t,u)/u^p. \end{gathered}$$ Consider the half-linear SLP consisting of the equation \begin{eqnarray}\label{e2.2} -(\phi(q(t) u'))'=\lambda w(t)\phi(u),\ 0(a\alpha\beta)^{-p}$; \item[(b)] $f_0>(a\alpha\beta)^{-p}$ and $f^\infty<((a+q^*)\beta)^{-p}$. \end{itemize} \end{corollary} Next, we derive criteria for the existence of positive solutions based on the behavior of $f(t,u)$ for $u$ in two disjoint closed intervals. Below we use the notation $\|u\|=\max_{t\in[0,1]}|u(t)|$. \begin{theorem}\label{t2.2} Assume there exist $00$ such that \eqref{e2.4} holds. Then \begin{itemize} \item[(a)] BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution $u$ with $\|u\|\leq l_1$ if $f_0>\lambda_0$; \item[(b)] BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution $u$ with $\|u\|\ge l_1$ if $f_{\infty}>\lambda_0$. \end{itemize} \end{theorem} \begin{theorem} \label{t2.4} Assume there exists $l_2>0$ such that \eqref{e2.5} holds. Then \begin{itemize} \item[(a)] BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution $u$ with $\|u\|\leq l_2$ if $f^0<\lambda_0$; \item[(b)] BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution $u$ with $\|u\|\ge l_2$ if $f^{\infty}<\lambda_0$. \end{itemize} \end{theorem} Combining Theorems \ref{t2.3} and \ref{t2.4} we obtain a result on the existence of at least two positive solutions. \begin{theorem} \label{t2.5} Assume either \begin{itemize} \item[(a)] $f_0>\lambda_0$ and $f_{\infty}>\lambda_0$, and there exists $l>0$ such that \label{e2.6} f(t,u)0$such that $$\label{e2.7} f(t,u)>l^p(a\beta)^{-p}\text{ for all } (t, u)\in[0,1]\times[\alpha l,l].$$ \end{itemize} Then BVP \eqref{e1.1}, \eqref{e1.2} has at least two positive solutions$u_1$and$u_2$with$\|u_1\|(a\alpha\beta)^{-p}. Then BVP \eqref{e1.1}, \eqref{e1.2} has an infinite number of positive solutions. \end{corollary} Finally, we present a result on the nonexistence of positive solutions of BVP \eqref{e1.1}, \eqref{e1.2}. \begin{theorem} \label{t2.8} BVP \eqref{e1.1}, \eqref{e1.2} has no positive solutions if \begin{itemize} \item[(a)] f(t,u)/u^p<((a+q^*)\beta)^{-p} for all (t,u)\in [0,1]\times(0,\infty), or \item[(b)] f(t,u)/u^p>(a\alpha\beta)^{-p} for all (t,u)\in [0,1]\times(0,\infty). \end{itemize} \end{theorem} \section{Proofs} \begin{proof}[Proof of Lemma \ref{l2.1}] To prove this lemma, we need to normalize BC \eqref{e1.2} using the generalized sine and cosine functions established by Elbert, see \cite{E} for the detail. It can be shown that \eqref{e1.2} is equivalent to the BC \begin{eqnarray}\label{e3.1} (qu')(0)=0,\quad C(\theta^*)u(1)-S(\theta^*)(qu')(1)=0, \end{eqnarray} where C(\theta) and S(\theta) are the generalized sine and cosine functions, \theta^*\in(\pi_p/2,\pi_p) with \pi_p=2\pi((p+1)\sin(\pi/(p+1)))^{-1} such that S(\theta^*)/C(\theta^*)=-a. Now we treat \eqref{e3.1} as a function of \theta and let \lambda_0(\theta) be the first eigenvalue of SLP \eqref{e2.2}, \eqref{e3.1} for \theta\in [\pi_p/2,\pi_p). By \cite[Corollary 3.9]{KK2}, \lambda_0 is strictly increasing. Note that \eqref{e3.1} with \theta=\pi_p/2 becomes $$\label{e3.2} (qu')(0)=0,\quad (qu')(1)=0.$$ In this case, \lambda_0(\pi_p/2)=0 is the first eigenvalue of SLP \eqref{e2.2}, \eqref{e3.2} with an associated eigenfunction v_0\equiv 1. As a result, \lambda_0(\theta)>0 for \theta\in(\pi_p/2,\pi_p). In particular, \lambda_0(\theta^*)>0, i.e., the first eigenvalue of SLP \eqref{e2.2}, \eqref{e1.2} is positive. \end{proof} With \|u\|=\max_{t\in[0,1]}|u(t)|, it is clear that (C[0,1],\|\cdot\|) is a Banach space. Let C_+[0,1]=\{u\in C[0,1]\ |\ u\ge0\text{ on }[0,1]\}. Define \Gamma:\ C_+[0,1]\to C[0,1] by \begin{eqnarray} (\Gamma u)(t)=\int_0^1G_u(t,s)\phi^{-1}\Big(\int_0^1w(\tau)f(\tau, u(\tau))d\tau\Big)ds,\quad t\in(0,1) , \label{e3.3} \end{eqnarray} where \phi^{-1} is the inverse function of \phi, and $$\label{e3.4} G_{u}(t,s)=\begin{cases} a, &0\le s\le t,\\ a+\frac{1}{q(s)}\phi^{-1} \Big(\frac{\int_{0}^sw(\tau)f(\tau,u(\tau))d\tau}{\int_{0}^{1}w(\tau) f(\tau,u(\tau))d\tau}\Big) , &t\le s\le 1. \end{cases}$$ \begin{remark}\label{r3.1} \rm We observe that the operator \Gamma defined by \eqref{e3.3} is the same as \begin{aligned} (\Gamma u)(t) &=\int_t^1\frac 1{q(s)}\phi^{-1} \Big({\int_0^sw(\tau)f(\tau,u(\tau))d\tau}\Big)ds\\ &\quad +a\phi^{-1}\Big(\int_0^1w(\tau)f(\tau,u(\tau))d\tau\Big). \end{aligned} \label{e3.6} \end{remark} \begin{remark}\label{r3.2} \rm It is easy to see that for any u\in C_+[0,1] $$a\le G_u(t,s)\le a+q^*\text{ on }[0,1]\times[0,1],\label{e3.5}$$ where q^* is defined by \eqref{e2.3}. \end{remark} \begin{lemma}\label{l3.1} A function u(t) is a solution of \eqref{e1.1}, \eqref{e1.2} if and only if u is a fixed point of \Gamma. \end{lemma} \begin{proof} Assume u(t) is a solution of BVP \eqref{e1.1}, \eqref{e1.2}. From \eqref{e1.1} and the first BC in \eqref{e1.2} we see that for any t\in(0,1) \begin{equation*} (qu')(t)=-\phi^{-1}\Big({\int_0^tw(\tau)f(\tau,u(\tau))d\tau}\Big), \end{equation*} $$\label{e3.7} u(t)=u(0)-\int_0^t\frac 1{q(s)}\phi^{-1}\Big({\int_0^s w(\tau)f(\tau,u(\tau))d\tau}\Big)ds.$$ Then by the second BC in \eqref{e1.2}, we have u(0)=\int_0^1\frac 1{q(s)} \phi^{-1}\Big(\int_0^s w(\tau)f(\tau,u(\tau))d\tau \Big)ds +a\phi^{-1}\Big(\int_0^1w(\tau)f(\tau,u(\tau))d\tau\Big). By \eqref{e3.7} and \eqref{e3.6}, \begin{align*} u(t)&=\int_t^1\frac 1{q(s)}\phi^{-1}\Big(\int_0^s w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &\quad +a\phi^{-1}\Big(\int_0^1w(\tau)f(\tau,u(\tau))d\tau\Big)=(\Gamma u)(t). \end{align*} Thus, u is a fixed point of the operator \Gamma. The opposite direction can be verified by reversing the argument. We omit the details. \end{proof} Let $$\label{e3.8} K=\{u\in C[0,1]\ |\ \min_{t\in [0,1]}u(t)\ge\alpha\|u\|\},$$ where \alpha is defined by \eqref{e2.3}. Clearly, K is a cone contained in C_+[0,1]. For l>0, define $$\label{e3.9} K_l=\{u\in K|\|u\|< l\}, \quad \partial K_l=\{u\in K|\|u\|=l\},$$ and let \mathfrak{i}(\Gamma,K_l,K) be the fixed point index of \Gamma on K_l with respect to K. \begin{lemma}\label{l3.2} \Gamma is completely continuous and \Gamma K\subset K. \end{lemma} \begin{proof} By \eqref{e3.6}, it is easy to see that \Gamma  is completely continuous on C_+[0,1]. For any u\in K, by \eqref{e2.3}, \eqref{e3.3}, and \eqref{e3.5} \label{e3.9a} \begin{aligned} \min_{t\in [0,1]}(\Gamma u)(t) &= \min_{t\in [0,1]}\int_{0}^{1}G_{u}(t,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &\ge \int_{0}^{1}a\phi^{-1}\Big(\int_{0}^{1}w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &= \alpha\int_{0}^{1}(a+q^*)\phi^{-1} \Big(\int_{0}^{1}w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &\ge \alpha\max_{t\in [0,1]}\int_{0}^{1}G_{u}(t,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &= \alpha\|\Gamma u\|. \end{aligned} Therefore, \Gamma K\subset K. \end{proof} Our proofs for the existence of positive solutions are based on the following fixed point index theorem, see \cite[page 529, A2, A3]{Z} for the detail. \begin{lemma} \label{l3.4} Let 00. Then \begin{itemize} \item[(a)] \mathfrak{i}(\Gamma,K_l,K)=1 if u\neq \mu\Gamma u for all u\in \partial K_l and \mu\in[0,1]; \item[(b)] \mathfrak{i}(\Gamma,K_l,K)=0 if there exists v_0\in K\setminus\{0\} such that u-\Gamma u\neq \mu v_0 for all u\in \partial K_l and \mu\ge0. \end{itemize} \end{lemma} \begin{proof}[Proof of Theorem \ref{t2.1}] Assume f^0<\lambda_00 such that f(t,u)<\lambda_0 u^p=\lambda_0\phi(u) for any (t,u)\in[0,1]\times[0,\underline l]. For any u\in\partial K_{\underline l}, \alpha \underline l\le u(t)\le \underline l on [0,1]. By \eqref{e3.6} and \eqref{e3.10}, for t\in[0,1] \begin{aligned} &(\Gamma u)(t)\\ &= \int_t^1\frac 1{q(s)}\phi^{-1}\Big({\int_0^sw(\tau)f(\tau,u(\tau))d\tau}\Big) ds +a\phi^{-1}\Big(\int_0^1w(\tau)f(\tau,u(\tau))d\tau\Big)\\ &< \lambda_0^{1/p}\Big[\int_t^1\frac 1{q(s)}\phi^{-1}\Big({\int_0^sw(\tau)\phi(u(\tau))d\tau}\Big) ds +a\phi^{-1}\Big(\int_0^1w(\tau)\phi(u(\tau))d\tau\Big)\Big]\\ &=\lambda_0^{1/p}(\Gamma_1u)(t). \end{aligned}\label{e3.11} Without loss of generality, we assume that \Gamma u has no fixed point on \partial K_{\underline l}. For otherwise, the proof is done. We show that u\neq\mu\Gamma u for all u\in\partial K_{\underline l} and \mu\in[0,1]. Obviously, it is true for \mu=0, 1. So we only consider \mu\in(0,1). Assume the contrary, i.e., there exist u_0\in\partial K_{\underline l} and \mu_0\in(0,1) such that u_0(t)=\mu_0(\Gamma u_0)(t). By \eqref{e3.11}, for t\in[0,1] $$\label{e3.11a} u_0(t)=\mu_0(\Gamma u_0)(t)<\mu_0\lambda_0^{1/p}(\Gamma_1u_0)(t).$$ In view of the fact that u_0(t)>0 and v_0(t)>0 on [0,1], the set \{\mu\ |\ u_0(t)\le\mu v_0(t)\text{ for }t\in[0,1]\} is not empty. Define \mu_1=\min\{\mu\ |\ u_0(t)\le\mu v_0(t)\text{ for }t\in[0,1]\}. Then \mu_1>0, and from \eqref{e3.10} and by the nondecreasing property of \Gamma_1 we have that for t\in[0,1] $\lambda_0^{1/p}(\Gamma_1 u_0)(t)\le\lambda_0^{1/p}(\Gamma_1(\mu_1 v_0))(t) =\mu_1\lambda_0^{1/p}(\Gamma_1v_0)(t)=\mu_1v_0(t).$ Thus by \eqref{e3.11a} u_0(t)<\mu_0\mu_1v_0(t)<\mu_1v_0(t) on [0,1], which contradicts the definition of \mu_1. Therefore, u\neq\mu\Gamma u for all u\in\partial K_{\underline l} and \mu\in[0,1]. By Lemma \ref{l3.3} (a), \mathfrak{i}(\Gamma, K_{\underline l},K)=1. Since f_\infty>\lambda_0, there exists \tilde l >\underline l such that f(t,u)>\lambda_0u^p=\lambda_0\phi(u) for all (t,u)\in[0,1]\times(\tilde l,\infty) . Choose \bar l\ge\alpha^{-1}\tilde l . Then for any u\in\partial K_{\bar l}, u(t)\ge \alpha \bar l=\tilde l on [0,1]. By \eqref{e3.6} and \eqref{e3.10}, for t\in[0,1] \begin{aligned} &(\Gamma u)(t)\\ &= \int_t^1\frac 1{q(s)}\phi^{-1}\Big({\int_0^sw(\tau)f(\tau,u(\tau))d\tau}\Big)ds +a\phi^{-1}\Big(\int_0^1w(\tau)f(\tau,u(\tau))d\tau\Big)\\ &>\lambda_0^{1/p}\Big[\int_t^1\frac 1{q(s)}\phi^{-1}\Big({\int_0^sw(\tau)\phi(u(\tau))d\tau}\Big)ds +a\phi^{-1}\Big(\int_0^1w(\tau)\phi(u(\tau))d\tau\Big)\Big]\\ &=\lambda_0^{1/p}(\Gamma_1u)(t). \end{aligned} \label{e3.11b} Without loss of generality, we assume that \Gamma u has no fixed point on \partial K_{\bar l}. For otherwise, the proof is done. We show that u-\Gamma u\neq \mu v_0 for any u\in\partial K_{\bar l} and \mu\ge0. Obviously, it is true for \mu=0. so we only consider \mu>0. Assume the contrary, i.e., there exist u^0\in\partial K_{\bar l} and \mu^0>0 such that u^0-\Gamma u^0=\mu^0 v_0. Then on [0,1] \, we have u^0(t)=(\Gamma u^0)(t)+\mu^0 v_0(t)>\mu^0v_0(t). Define \mu_2=\max\{\mu\ |\ u^0(t)\ge\mu v_0(t)\text{ for }t\in[0,1]\}. Then \mu_2\ge\mu^0 and u^0(t)\ge\mu_2v_0(t) on [0,1]. From \eqref{e3.11b} we see that for t\in[0,1] \begin{align*} u^0(t) &=\Gamma u^0(t)+\mu^0v_0(t)> \lambda_0^{1/p}(\Gamma_1u^0)(t)+\mu^0v_0(t)\\ &\ge \lambda_0^{1/p}(\Gamma_1\mu_2v_0)(t)+\mu^0v_0(t) =\mu_2\lambda_0^{1/p}(\Gamma_1v_0)(t)+\mu^0v_0(t)\\ &= \mu_2v_0(t)+\mu^0v_0(t)=(\mu_2+\mu^0)v_0(t), \end{align*} which contradicts the definition of \mu_2. Therefore, u-\Gamma u\neq \mu v_0 for any u\in\partial K_{\bar l} and \mu\ge0. By Lemma \ref{l3.3} (b), \mathfrak{i}(\Gamma, K_{\bar l}, K)=0. By Lemma \ref{l3.4}, BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution. The case for f^{\infty}<\lambda_00 and assume \Gamma u\neq u for u\in\partial K_l. Then \begin{itemize} \item[(a)] \mathfrak{i}(\Gamma,K_l,K)=1 if \|\Gamma u\|\le\|u\| for u\in\partial K_l. \item[(b)] \mathfrak{i}(\Gamma,K_l,K)=0 if \|\Gamma u\|\ge\|u\| for u\in\partial K_l. \end{itemize} \end{lemma} \begin{proof}[Proof of Theorem \ref{t2.2}] Without loss of generality, we assume \Gamma u\neq u on \partial K_{l_1}\cup\partial K_{l_2}. For otherwise, \Gamma has a positive fixed point. For any u\in \partial K_{l_1}, \alpha l_1\le u(t)\le l_1 on [0,1]. From \eqref{e2.4}, f(t,u(t))\le l_1^p((a+q^*)\beta)^{-p} on [0,1]. Then by \eqref{e2.3} and \eqref{e3.5}, \begin{align*} \|\Gamma u\|&=\max_{t\in[0,1]} \int_{0}^{1}G_{u}(t,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &\le \max_{t\in[0,1]}\int_{0}^{1}G_u(t,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)d\tau\Big)l_1((a+q^*)\beta)^{-1}ds\le l_1. \end{align*} Thus \|\Gamma u\|\le\|u\|. By Lemma \ref{l3.5} (a), \mathfrak{i}(\Gamma,K_{l_1},K)=1 . For any u\in K_{l_2}, \alpha l_2\le u(t)\le l_2 on [0,1]. From \eqref{e2.5}, f(t,u(t))\ge l_2^p(a\beta)^{-p} on [0,1]. Then by \eqref{e2.3} and \eqref{e3.5} \begin{align*} \|\Gamma u\| &\ge \int_{0}^{1}G_u(t,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &\ge \int_{0}^{1}G_u(t,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)d\tau\Big)l_2(a\beta)^{-1}ds \ge l_2. \end{align*} Thus \|\Gamma u\|\ge\|u\|. By Lemma \ref{l3.5} (b), \mathfrak{i}(\Gamma,K_{l_2},K)=0 . By Lemma \ref{l3.4}, \Gamma has a fixed point u\in K_{l_2}\setminus \overline K_{l_1} if l_1l_2. In each case, u is a positive function with \min\{l_1,l_2\}\le\|u\|\le\max\{l_1,l_2\}. \end{proof} The proofs of Theorems \ref{t2.3} and \ref{t2.4} are in the same way and hence we only give the proof of Theorem \ref{t2.3}. \begin{proof}[Proof of Theorem \ref{t2.3}] (a) If there exists l_1>0 such that \eqref{e2.4} holds, then by the proof of Theorem \ref{t2.2}, \mathfrak{i}(\Gamma,K_{l_1},K)=1 . By the proof of Theorem \ref{t2.1}, f_0>\lambda_0 implies there exists 00 such that \eqref{e2.6} holds. Then there exist l_1 and l_2 such that l_1(a\alpha\beta)^{-p}\quad \text{for all }(t,u)\in[0,1]\times[\alpha l_{2i},l_{2i}]. This shows that for sufficiently large $i$, $$f(t,u)u^p(a\alpha\beta)^{-p}\ge(\alpha l_{2i})^p(a\alpha\beta)^{-p}=l_{2i}^p(a\beta)^{-p}\quad \text{on }[0,1]\times[\alpha l_{2i},l_{2i}].$$ Therefore, the conclusion follows from Theorem \ref{t2.7}. \end{proof} \begin{proof}[Proof of Theorem \ref{t2.8}] (a) Assume BVP \eqref{e1.1}, \eqref{e1.2} has a positive solution $u$ with $\|u\|=l$ for some $l>0$. Then $u$ is a fixed point of the operator $\Gamma$ defined by \eqref{e3.3}. From the assumption, for any $t\in[0,1]$, $f(t,u(t))0$. Then $\alpha l\le u(t)\le l$ on $[0,1]$. From the assumption, for any $t\in[0,1]$, $f(t,u(t))>u^p(t)(a\alpha\beta)^{-p}\ge l^p(a\beta)^{-p}$. Hence \begin{align*} u(t)&= (\Gamma u)(t)=\int_{0}^{1}G_{u}(t,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)f(\tau,u(\tau))d\tau\Big)ds\\ &> l/(a\beta)\int_{0}^{1}G_{u}(t_1,s)\phi^{-1} \Big(\int_{0}^{1}w(\tau)d\tau\Big)ds\ge l, \end{align*} which contradicts $\|u\|=l$. Therefore, BVP \eqref{e1.1}, \eqref{e1.2} has no positive solutions. \end{proof} \section{Partial BVPs with $p$-Laplacian} In this section, we extend our results in Section 2 to BVPs for partial differential equations with $p$-Laplacian defined on annular domains. Let $00$. The next lemma shows the relation between the partial BVP \eqref{e4.1}, \eqref{e4.2} and the ordinary BVP \eqref{e1.1}, \eqref{e1.2}. \begin{lemma} \label{l4.1} Let $r=|x|$, $t=t(r):=\int_{r_1}^{r}s^{(1-n)/p}ds/\int_{r_1}^{r_2}s^{(1-n)/p}ds$, and $r=r(t)$ be its inverse function. Then BVP \eqref{e4.1}, \eqref{e4.2} has a positive radial solution $v(|x|)$ if and only if BVP \eqref{e1.1}, \eqref{e1.2} with $q\equiv1$, $$\label{e4.3} a=\frac{br_2^{\frac{1-n}{p}}}{\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds}, \quad\text{and}\quad w(t)=h(r(t))r^{\frac{(p+1)(n-1)}{p}}(t) \Big(\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds\Big)^{p+1}$$ has a positive solution $u(t)$. \end{lemma} \begin{proof} We first claim that the existence of a positive radial solution of BVP \eqref{e4.1}, \eqref{e4.2} is equivalent to the existence of positive solution of BVP consisting of the equation \begin{eqnarray}\label{e4.4} -\frac{d}{dr}(r^{n-1}\phi(\frac{d\tilde v}{dr}))=r^{n-1}h(r)f(\tilde v),\quad r_1(a\alpha\beta)^{-p}$; or \item[(b)]$f_0>(a\alpha\beta)^{-p}$and$f^\infty<((a+1)\beta)^{-p}$. \end{itemize} \end{corollary} \begin{theorem}\label{t4.2} Assume there exist$00$such that \eqref{e4.8} holds. Then \begin{itemize} \item[(a)] BVP \eqref{e4.1}, \eqref{e4.2} has at least one positive radial solution$v$with$\|v\|\leq l_1$if$f_0>\lambda_0$; \item[(b)] BVP \eqref{e4.1}, \eqref{e4.2} has at least one positive radial solution$v$with$\|v\|\ge l_1$if$f_{\infty}>\lambda_0$. \end{itemize} \end{theorem} \begin{theorem} \label{t4.4} Assume there exists$ l_2>0$such that \eqref{e4.9} holds. Then \begin{itemize} \item[(a)] BVP \eqref{e4.1}, \eqref{e4.2} has at least one positive radial solution$v$with$\|v\|\leq l_2$if$f^0<\lambda_0$; \item[(b)]BVP \eqref{e4.1}, \eqref{e4.2} has at least one positive radial solution$v$with$\|v\|\ge l_2$if$f^{\infty}<\lambda_0$. \end{itemize} \end{theorem} \begin{theorem} \label{t4.5} Assume either \begin{itemize} \item[(a)]$f_0>\lambda_0$,$f_{\infty}>\lambda_0$, and there exists$ l>0$such that $$\label{e4.10} f(v)< l^p((a+1)\beta)^{-p}\text{ for all }v\in[\alpha l,l]; or$$ \item[(b)]$f^0<\lambda_0$,$f^{\infty}<\lambda_0$, and there exists$l>0$such that $$\label{e4.11} f(v)\ge l^p(a\beta)^{-p}\text{ for all }v\in [\alpha l,l].$$ \end{itemize} Then BVP \eqref{e4.1}, \eqref{e4.2} has at least two positive radial solutions$v_1$and$v_2$with$\|v_1\|(a\alpha\beta)^{-p}.  Then BVP \eqref{e4.1}, \eqref{e4.2} has an infinite number of positive radial solutions. \end{corollary} \begin{theorem} \label{t4.8} BVP \eqref{e4.1}, \eqref{e4.2} has no positive radial solutions if \begin{itemize} \item[(a)] $f(v)/v^p<((a+1)\beta)^{-p}$ for all $v\in (0,\infty)$, or \item[(b)] $f(v)/v^p>(a\alpha\beta)^{-p}$ for all $v\in (0,\infty)$. \end{itemize} \end{theorem} \begin{remark} \label{r4.1}{\rm Note that when $r_1\to0+$, the annulus $\Omega$ for the domain of \eqref{e4.1} approaches a disk centered at the origin with radius $r_2$, and the first BC in \eqref{e4.2} reduces to $\frac{\partial v}{\partial \nu}|_{x=0}=0$ which is automatically satisfied by radial solutions. Hence, the $p$-Laplacian partial BVP defined on the disk \begin{gather} -\operatorname{div}(\phi(\nabla v))=h(|x|)f(v)\quad \text{in }B(0,r_2),\label{e4.12}\\ v+b\frac{\partial v}{\partial\nu}=0\quad \text{on }\partial B(0,r_2).\label{e4.13} \end{gather} can be treated as the limiting problem of BVP \eqref{e4.1}, \eqref{e4.2} as $r\to 0+$. Therefore, the results for BVP \eqref{e4.1}, \eqref{e4.2} can be extended to BVP \eqref{e4.12}, \eqref{e4.13} with the modification $r_1=0$. The only problem in this extension is that the integral $\int_{r_1}^{r_2}s^{(1-n)/p}ds$ may become divergent as $r_1\to 0+$. However, this does not occur under the additional assumption that $p+1-n>0$.} \end{remark} \section{Examples} In this section, we give several examples as applications of our results. \begin{example} \label{exa1} \rm Let $S(\theta)$ denote the general sine function and let $\theta^*\in(\pi_p/2,\pi_p)$ be a solution of $S(\theta)+S'(\theta)=0$. Consider the BVP \label{e5.1} \begin{array}{l} -(\phi(u'))'=f(u),\ 0\gamma_2$. \end{itemize} In fact, the equation in \eqref{e5.2} is of the form of \eqref{e1.1} with$w(t)\equiv1$and$f(u)=k(u^{p/2}+u^{2p})$. Clearly,$f_0=f_\infty=\infty$,$f(u)/u^p$is decreasing on$(0,l]$, increasing on$[l,\infty)$, and hence reaches minimum value at$l$. By \eqref{e2.3},$\alpha=a/(a+1)$,$\beta=1$. When$k=\gamma_1$,$f(\alpha l)/(\alpha l)^p=(a+1)^{-p}$. Hence for$u\in [\alpha l,l]$,$f(u)/u^p\le f(\alpha l)/(\alpha l)^p=(a+1)^{-p}$, which follows that$f(u)\le u^p(a+1)^{-p}\le l^p(a+1)^{-p}$. Therefore, by Theorem \ref{t2.3} (a), BVP \eqref{e5.2} has a positive solution$u_1$with$\|u_1\|\le l$. Similarly, by Theorem \ref{t2.3} (b) we can also show that BVP \eqref{e5.2} has a positive solution$u_2$with$\|u_2\|\ge l$. However,$u_1$and$u_2$may be the same when$\|u_1\|=\|u_2\|=l$. When$0\gamma_2$,$f(u)/u^p>(a\alpha\beta)^{-p}=(a+1)^p(a^2)^{-p}$on$(0,\infty)$. Then the conclusion follows from Theorem \ref{t2.8} (b). \end{example} \begin{example} \label{exa3} \rm Consider the BVP $$\label{e5.3} \begin{gathered} -\operatorname{div}(|\nabla v|^{p-1}\nabla v) =k(v^{-p/2}+v^{-2p})^{-1}\quad \text{in }\Omega,\\ \frac{\partial v}{\partial\nu}=0\quad \text{on }\partial B(0,r_1),\quad v+b\frac{\partial v}{\partial\nu}=0\quad \text{on }\partial B(0,r_2), \end{gathered}$$ where$0\gamma_3$; \item[(c)] BVP \eqref{e5.3} has no positive solutions when$0