\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
\begin{equation} \label{e1.1}
 -(\phi(q(t)u'))'=w(t)f(t,u),\quad  0<t<1,
\end{equation}
and the boundary condition (BC)
\begin{equation}\label{e1.2}
(q u')(0)=0,\quad u(1)+a(q u')(1)=0,
\end{equation}
where $\phi(u)=|u|^{p-1}u$ with $p>0$, $a>0$,
$f:[0,1]\times\mathbb{R}_+\to\mathbb{R}_+$ 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
\begin{equation}\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}
\end{equation}
Consider the half-linear SLP
consisting of the equation
\begin{eqnarray}\label{e2.2}
 -(\phi(q(t) u'))'=\lambda w(t)\phi(u),\ 0<t<1,
\end{eqnarray}
and BC \eqref{e1.2}.
SLP \eqref{e2.2}, \eqref{e1.2} is called the SLP associated with BVP
\eqref{e1.1}, \eqref{e1.2}.
It is well known that SLP \eqref{e2.2}, \eqref{e1.2} has infinite number
of real eigenvalues $\{\lambda_n\}_{n=0}^{\infty}$ satisfying
$$
-\infty<\lambda_0<\lambda_1<\dots<\lambda_n<\cdots, \ 
\text{ and } \lambda_n\to\infty;
$$
and the eigenfunction $v_n$ associated with $\lambda_n$ has exactly $n$
zeros in $(0,1)$.
We refer to the reader
Binding and Dr\'abek \cite{BD} and Kong and Kong \cite{KK2}
for the details. Moreover, we have the following result.

\begin{lemma}\label{l2.1}
The first eigenvalue $\lambda_0$ of  SLP \eqref{e2.2}, \eqref{e1.2} is
positive.
\end{lemma}

Our major result below is on the existence of positive solutions of BVP
\eqref{e1.1}, \eqref{e1.2} using the relationships among $\lambda_0$, $f_0$,
and $f_{\infty}$.

\begin{theorem}\label{t2.1}
Let $\lambda_0$ be the first eigenvalue of SLP \eqref{e2.2}, \eqref{e1.2}.
Then BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution
if either $f^0<\lambda_0<f_\infty$ or $f^\infty<\lambda_0<f_0$.
\end{theorem}

Let
\begin{equation}\label{e2.3}
q^*=\sup_{t\in[0,1]}1/q(t),\quad
\alpha=a/(a+q^*),\quad
\beta=\phi^{-1}\Big(\int_{0}^{1}w(\tau)d\tau\Big),
\end{equation}
where $\phi^{-1}$ is the inverse function of $\phi$.
It is easy to see that $0<\alpha<1$. The following corollary, which
gives explicit conditions without using $\lambda_0$,
follows directly from Theorem \ref{t2.1}.

\begin{corollary}\label{c2.1}
 BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution if
 either of the following holds:
\begin{itemize}
\item[(a)] $f^0<((a+q^*)\beta)^{-p}$  and $f_\infty>(a\alpha\beta)^{-p}$;
\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 $0<l_1<l_2$ (respectively, $0<l_2<l_1$), such that
\begin{gather}\label{e2.4}
 f(t,u)\le l_1^p((a+q^*)\beta)^{-p}\quad \text{for all
 }(t,u)\in[0,1]\times[\alpha l_1,l_1] ,\\
\label{e2.5}
 f(t,u)\ge l_2^p(a\beta)^{-p}\quad\text{for all }(t,u)\in[0,1]\times[\alpha
 l_2,l_2].
\end{gather}
Then BVP \eqref{e1.1}, \eqref{e1.2} has at least one positive solution $u$
 with $l_1\le\|u\|\le l_2$ (respectively, $l_2\le\|u\|\le l_1$).
\end{theorem}

As extensions of Theorems \ref{t2.1} and \ref{t2.2}, we have the
following results.

\begin{theorem}\label{t2.3}
Assume there exists $l_1>0$ 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
\begin{equation}\label{e2.6}
f(t,u)<l^p((a+q^*)\beta)^{-p} \text{ for all }(t, u)\in[0,1]\times
[\alpha l,l];  or
\end{equation}
\item[(b)] $f^0<\lambda_0$ and
 $f^{\infty}<\lambda_0$, and
there exists $l>0$ such that
\begin{equation}\label{e2.7}
f(t,u)>l^p(a\beta)^{-p}\text{ for all } (t, u)\in[0,1]\times[\alpha
l,l].
\end{equation}
\end{itemize}
Then BVP \eqref{e1.1}, \eqref{e1.2} has at least two positive
solutions $u_1$ and $u_2$ with $\|u_1\|<l<\|u_2\|$.
\end{theorem}


Note that in Theorem \ref{t2.5}, the inequalities in
\eqref{e2.6} and \eqref{e2.7} are strict and hence are different from
those in \eqref{e2.4} and \eqref{e2.5}
in Theorem \ref{t2.2}. This is to guarantee that the two solutions
$u_1$ and $u_2$ are different.


By applying Theorem \ref{t2.2} repeatedly, we obtain criteria for the
existence of multiple positive solutions.

\begin{theorem} \label{t2.6}
Let $\{l_i\}_{i=1}^{N}\subset \mathbb{R}$ such that
$0<l_1<l_2<\dots<l_N$. Assume either
\begin{itemize}
\item[(a)] $f$ satisfies \eqref{e2.6} with $r=l_i$  when $i$ is odd,
and satisfies
\eqref{e2.7} with $r=l_i$ when $i$ is even; or
\item[(b)] $f$ satisfies \eqref{e2.6} with $r=l_i$  when $i$ is even,
and satisfies
\eqref{e2.7} with $r=l_i$  when $i$ is odd.
\end{itemize}
Then BVP \eqref{e1.1}, \eqref{e1.2} has at least $N-1$ positive
solutions $u_i$ with $l_i<\|u_i\|<l_{i+1}$, $i=1,2,\dots,N-1$.
\end{theorem}


\begin{theorem} \label{t2.7}
Let $\{l_i\}_{i=1}^\infty\subset \mathbb{R}$ such that
$0<l_1<l_2<\dots$ and $\lim_{i\to \infty }l_{i}=\infty $. Assume either
\begin{itemize}
\item[(a)] $f$ satisfies \eqref{e2.4} with $l_1=l_i$  when $i$ is odd,
and satisfies
\eqref{e2.5} with $l_2=l_i$  when $i$ is even; or
\item[(b)] $f$ satisfies \eqref{e2.4} with $l_1=l_i$  when $i$ is even,
and satisfies
\eqref{e2.5} with $l_2=l_i$  when $i$ is odd.
\end{itemize}
Then BVP \eqref{e1.1}, \eqref{e1.2} has an infinite number of
positive solutions.
\end{theorem}

The following is an immediate consequence of Theorem \ref{t2.7}.

\begin{corollary} \label{c2.2}
Let $\{l_i\}_{i=1}^\infty\subset \mathbb{R}$ such that
$0<l_1<l_2<\dots$ and $\lim_{i\to \infty }l_{i}=\infty $.
Let $E_1=\cup_{i=1}^{\infty}[\alpha
l_{2i-1},l_{2i-1}]$ and $E_2=\cup_{i=1}^{\infty}[\alpha l_{2i},l_{2i}]$.
Assume
$$
\limsup_{E_1\ni
u\to\infty}\max_{t\in[0,1]}\frac{f(t,u)}{u^p}<((a+q^*)\beta)^{-p}, \quad
\liminf_{E_2\ni u\to\infty}\min_{t\in[0,1]}\frac{f(t,u)}{u^p}>(a\alpha\beta)^{-p}.
$$
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
\begin{equation} \label{e3.2}
 (qu')(0)=0,\quad (qu')(1)=0.
\end{equation}
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
\begin{equation}\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}
\end{equation}


\begin{remark}\label{r3.1} \rm
We observe that the operator $\Gamma$ defined by \eqref{e3.3} is the
same as
\begin{equation}
\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{equation}
\end{remark}

\begin{remark}\label{r3.2} \rm
 It is easy to see that for any $u\in C_+[0,1]$
\begin{equation}
 a\le G_u(t,s)\le a+q^*\text{ on }[0,1]\times[0,1],\label{e3.5}
\end{equation}
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*}
\begin{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.
\end{equation}
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
\begin{equation}\label{e3.8}
     K=\{u\in C[0,1]\ |\ \min_{t\in [0,1]}u(t)\ge\alpha\|u\|\},
\end{equation}
 where $\alpha$ is defined by \eqref{e2.3}. Clearly, $K$ is a cone
 contained in $C_+[0,1]$.
For $l>0$, define
\begin{equation} \label{e3.9}
K_l=\{u\in K|\|u\|< l\}, \quad \partial K_l=\{u\in K|\|u\|=l\},
\end{equation}
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}
\begin{equation} \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}
\end{equation}
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 $0<l_1<l_2$ satisfy
$$
\mathfrak{i}(\Gamma,K_{l_1},K)=0 \quad \text{and} \quad
\mathfrak{i}(\Gamma,K_{l_2},K)=1;
$$
or
$$
\mathfrak{i}(\Gamma,K_{l_1},K)=1 \quad \text{and} \quad
\mathfrak{i}(\Gamma,K_{l_2},K)=0.
$$
 Then $\Gamma$ has a fixed point in
$K_{l_2}\setminus \overline K_{l_1}$.
\end{lemma}

To prove Theorem \ref{t2.1}, we also need the lemma below,
see \cite[Lemma 2.3.1 and Corollary 2.3.1]{GL} for details.

\begin{lemma}\label{l3.3}
Let $l>0$. 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 $f^0<\lambda_0<f_{\infty}$.
Let $\lambda_0$ be the first eigenvalue of SLP \eqref{e2.2}, \eqref{e1.2}
with an associated positive eigenfunction $v_0$.
Define $\Gamma_1:C_+[0,1]\to C_+[0,1]$ as
\begin{equation}
\begin{aligned}
&(\Gamma_1 u)(t) \\
&=\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(s)\phi(u(s))ds\Big).
\end{aligned} \label{e3.10}
\end{equation}
It is easy to verify that
$\lambda_0^{-1/p}$ is  an  eigenvalue  of  $\Gamma_1$
with $v_0$ as an associated eigenfunction, i.e., $\Gamma_1
v_0=\lambda_0^{-1/p}v_0$. Hence $v_0=\lambda_0^{1/p}\Gamma_1v_0$.


Since $f^0<\lambda_0$, there exists $\underline l>0$ 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{equation}
\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}
\end{equation}
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 $t\in[0,1]$
\begin{equation} \label{e3.11a}
 u_0(t)=\mu_0(\Gamma u_0)(t)<\mu_0\lambda_0^{1/p}(\Gamma_1u_0)(t).
\end{equation}
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{equation}
\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}
\end{equation}
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 $f^{\infty}<\lambda_0<f_0$ can be proved similarly.
We omit the details.
\end{proof}

\begin{proof}[Proof of Corollary \ref{c2.1}]
It suffices to show that
$$
((a+q^*)\beta)^{-p}\le\lambda_0\le (a\alpha\beta)^{-p},
$$
and then the conclusion follows from Theorem \ref{t2.1}.
Let $\lambda_0$ be the first eigenvalue of SLP \eqref{e2.2}, \eqref{e1.2}
with an associated positive eigenfunction $v_0$. Let $\Gamma_1$ be
defined by \eqref{e3.10}.
Then as shown in the proof of Theorem \ref{t2.1}, we have
$v_0=\lambda_0^{1/p}\Gamma_1v_0$.
Moreover, for $t\in [0,1]$,
\begin{equation}
(\Gamma_1
v_0)(t)=\int_0^1G_1(t,s)\phi^{-1}
\Big(\int_0^1w(\tau)\phi(v_0(\tau))d\tau\Big)ds,
\label{e3.12}
\end{equation}
where
\[
 G_1(t,s)=\begin{cases}
a, &0\le s\le t,\\
a+\frac{1}{q(s)}\phi^{-1}
\Big(\frac{\int_{0}^sw(\tau)\phi(v_0(\tau))d\tau}
{\int_{0}^{1}w(\tau)\phi(v_0(\tau))d\tau}\Big), &t\le s\le 1.
\end{cases}
\end{equation*}
Clearly, $a\le G_1(t,s)\le a+q^*$.
By \eqref{e2.3}
\begin{align*}
\|v_0\|
&=\max_{t\in[0,1]} v_0(t)=\max_{t\in[0,1]}\lambda_0^{1/p}(\Gamma_1v_0)(t)\\
&= \max_{t\in[0,1]}\lambda_0^{1/p}\int_0^1G_1(t,s)\phi^{-1}
\Big(\int_0^1w(\tau)\phi(v_0(\tau))d\tau\Big)ds\\
&\le \lambda_0^{1/p}\int_0^1(a+q^*)\phi^{-1}
\Big(\int_0^sw(\tau)d\tau\Big)\|v_0\|ds\\
&= \lambda_0^{1/p}(a+q^*)\beta\|v_0\|.
\end{align*}
Therefore,  $\lambda_0\ge ((a+q^*)\beta)^{-p}$.


Similar to \eqref{e3.9a} we have that $v_0(t)\ge \alpha\|v_0\|$ for $t\in
[0,1]$. Thus
\begin{align*}
 \|v_0\|
&\ge \lambda_0^{1/p}(\Gamma_1 v_0)( t)
 =\lambda_0^{1/p}\int_0^1G_1(t,s)\phi^{-1}
 \Big(\int_0^1w(\tau)\phi(v_0(\tau))d\tau\Big)ds\\
&\ge \lambda_0^{1/p}\int_0^1a\phi^{-1}
\Big(\int_0^1w(\tau)d\tau\Big)\alpha\|v_0\|ds\\
&= \lambda_0^{1/p}a\alpha \beta\|v_0\|;
\end{align*}
i.e., $\lambda_0\le (a\alpha \beta)^{-p}$.
This completes the proof.
\end{proof}

To prove Theorem \ref{t2.2} we need
the following well-known lemma on fixed point indices. See \cite{D,GL}
for details.


\begin{lemma}\label{l3.5}
Let $l>0$ 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 $l_1<l_2$, and
$\Gamma$ has a fixed point $u\in K_{l_1}\setminus \overline K_{l_2}$
if $l_1>l_2$.
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 $0<l_2<l_1$ with
$\mathfrak{i}(\Gamma,K_{l_2},K)=0 $. Then the conclusion follows
from Lemma \ref{l3.4}.
Part (b) can be proved similarly.
\end{proof}


\begin{proof}[Proof of Theorem \ref{t2.5}]
(a) Assume there exists $l>0$ such that \eqref{e2.6} holds. Then there
exist $l_1$ and $l_2$
such that $l_1<l<l_2$ and $f(t,u)<(l_i^p((a+q^*)\beta)^{-p}$ on
$[0,1]\times[\alpha l_i,l_i]$, $i=1,2$.
By Theorem \ref{t2.3} (a) and (b), BVP \eqref{e1.1}, \eqref{e1.2}
 has two positive solutions $u_1$ and $u_2$ satisfying
$\|u_1\|\le l_1$  and $\|u_2\|\ge l_2$.

Part (b) can be proved similarly.
\end{proof}

Theorems \ref{t2.6} and \ref{t2.7} can be obtained by applying Theorem
\ref{t2.2} repeatedly. We omit the details.

\begin{proof}[Proof of Corollary \ref{c2.2}]
 From the assumption we see that for sufficiently large $i$
$$
\frac{f(t,u)}{u^p}<((a+q^*)\beta)^{-p}\quad \text{for all
}(t,u)\in[0,1]\times[\alpha l_{2i-1},l_{2i-1}]
$$
and
$$
\frac{f(t,u)}{u^p}>(a\alpha\beta)^{-p}\quad \text{for all
}(t,u)\in[0,1]\times[\alpha l_{2i},l_{2i}].
$$
This shows that for sufficiently large $i$,
$$
f(t,u)<u^p((a+q^*)\beta)^{-p}\le l_{2i-1}^p((a+q^*)\beta)^{-p}\quad
\text{on }[0,1]\times[\alpha l_{2i-1},l_{2i-1}]
$$
and
$$
f(t,u)>u^p(a\alpha\beta)^{-p}\ge(\alpha
l_{2i})^p(a\alpha\beta)^{-p}=l_{2i}^p(a\beta)^{-p}\quad
\text{on }[0,1]\times[\alpha l_{2i},l_{2i}].
$$
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 $t\in[0,1]$, $f(t,u(t))<u^p(t)((a+q^*)\beta)^{-p}\le
l^p((a+q^*)\beta)^{-p}$. 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+q^*)\beta)\int_{0}^{1}G_{u}(t,s)\phi^{-1}
\Big(\int_{0}^{1}w(\tau)d\tau\Big)ds
\le l,
\end{align*}
which contradicts $\|u\|=l$. Therefore, BVP \eqref{e1.1}, \eqref{e1.2}
has no positive solutions.

(b) Assume BVP \eqref{e1.1}, \eqref{e1.2} has a positive solution $u$
with $\|u\|=l$ for some $l>0$.
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 $0<r_1<r_2$, $n\in\mathbb{N}$, and denote
$\Omega=B(0,r_2)\setminus\overline{B(0,r_1)}$, where $B(0,r)$
is the ball in $\mathbb{R}^n$ centered at $0$ with radius $r$.
Consider the scalar BVP
\begin{gather}
-\operatorname{div}(|\nabla v|^{p-1}\nabla v)=h(|x|)f(v)\quad
\text{in }\Omega,  \label{e4.1}\\
\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),\label{e4.2}
\end{gather}
where  $x=(x_1,\dots,x_n)\in\mathbb{R}^n$, $\operatorname{div}(y)$
is the divergence of $y:\mathbb{R}^n\to\mathbb{R}$, $b\in\mathbb{R}$,
$\nabla v$ is the gradient of $v$, and $\partial v/{\partial\nu}$
is the outward normal derivative of $v$ along $\partial B(0,r_i)$,
 $i=1,2$.
We assume that
 $h\in L[r_1,r_2]$, $h\ge0$ a.e. on $(r_1,r_2)$, and
 $\int_{r_1}^{r_2}h(s)ds>0$.

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 $q\equiv1$,
\begin{equation}\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}
\end{equation}
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<r<r_2,
\end{eqnarray}
and the BC
\begin{equation}\label{e4.5}
 \frac{d\tilde v}{dr}(r_1)=0,\quad
 \tilde v(r_2)+b\frac{d\tilde v}{dr}(r_2)=0,
\end{equation}
where $\tilde v(r)=v(|x|)$.
In fact, the proof for the case when $p=1$ is given in many books such
as  \cite{E2} which can be easily extended to the general case.

Let $\tilde v(r)$ be a positive solution of
BVP \eqref{e4.4}, \eqref{e4.5}
and $u(t)=\tilde v(r(t))$.  Then
$\frac{d\tilde v}{dr}=\frac{du}{dt}\frac{dt}{dr}$.
We note from the definition of $t(r)$ that
$$
\frac{dt}{dr}=\frac{r^{\frac{1-n}{p}}}{\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds}.
$$
By \eqref{e4.4},
\begin{align*}
r^{n-1}h(r)f(\tilde v)
&=-\frac{d}{dr}\Big(r^{n-1}\phi(\frac{d\tilde v}{dr})\Big)
=-\frac{d}{dt}\Big(r^{n-1}\phi(\frac{du}{dt}\frac{dt}{dr})\Big)
 \frac{dt}{dr}\\
&= -\frac{d}{dt}\Big(\Big(\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds
 \Big)^{-p}\phi(\frac{du}{dt})\Big)\frac{dt}{dr}\\
&=-\Big(\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds\Big)^{-p}\frac{d}{dt}
 \Big(\phi(\frac{du}{dt})\Big)\frac{dt}{dr}\\
&= -r^{\frac{1-n}{p}}\Big(\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds
 \Big)^{-p-1}\frac{d}{dt}\left(\phi(\frac{du}{dt})\right).
\end{align*}
Therefore,
\[
-\frac{d}{dt}\left(\phi(\frac{du}{dt})\right)
=h(r(t))r^{\frac{(p+1)(n-1)}{p}}(t)
\Big(\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds\Big)^{p+1}f(u)=w(t)f(u),
\]
which means that $u(t)$ is a positive solution of \eqref{e1.1}
under \eqref{e4.3}. It is also easy to see that $u(t)$ satisfies BC
\eqref{e1.2}.

The opposite direction can be verified by reversing the argument. We
omit the details.
\end{proof}


Clearly, all the assumptions of BVP \eqref{e1.1}, \eqref{e1.2} are
guaranteed by \eqref{e4.3}. In this case, since
 $q\equiv1$, from \eqref{e2.3} we have
\begin{equation}\label{e4.6}
\alpha=\frac{br_2^{\frac{1-n}{p}}}{br_2^{\frac{1-n}{p}}
+\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds},
\end{equation}
\begin{equation}
\begin{aligned}
\beta&= \phi^{-1}\Big(\int_0^1h(r(\tau))r^{\frac{(p+1)(n-1)}{p}}(\tau)
\Big(\int_{r_1}^{r_2}s^{\frac{1-n}{p}}ds\Big)^{p+1}d\tau\Big)\\
&= \phi^{-1}\Big(\int_{r_1}^{r_2}h(r)r^{n-1}\Big(\int_{r_1}^{r_2}
s^{\frac{1-n}{p}}ds\Big)^pdr\Big).
\end{aligned}\label{e4.7}
\end{equation}

Let
$f_0, f^0, f_{\infty}, f^{\infty}$ be defined by \eqref{e2.1},
$\lambda_0$ the first eigenvalue of SLP \eqref{e2.2}, \eqref{e1.2}
associated with BVP \eqref{e1.1}, \eqref{e1.2} with $q\equiv1$, and $a$
and $w$ defined by \eqref{e4.3}.
Denote $\|v\|=\max_{x\in\Omega}|v(x)|$ for $v\in C(\Omega,\mathbb{R})$.

Now we apply the results in Section 2 to derive criteria for the
existence, multiplicity, and nonexistence of positive radial solutions
of BVP \eqref{e4.1}, \eqref{e4.2}.
In the theorems below, $a$,  $\alpha$, and $\beta$ are defined by
\eqref{e4.3}, \eqref{e4.6}, and \eqref{e4.7}, respectively.


\begin{theorem}\label{t4.1}
BVP \eqref{e4.1}, \eqref{e4.2} has at least one positive radial
solution if
 either $f^0<\lambda_0<f_\infty$  or
$f^\infty<\lambda_0<f_0$.
\end{theorem}

\begin{corollary}\label{c4.1}
 BVP \eqref{e4.1}, \eqref{e4.2} has at least one positive radial
solution  if either
\begin{itemize}
\item[(a)] $f^0<((a+1)\beta)^{-p}$  and
 $f_\infty>(a\alpha\beta)^{-p}$; 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 $0<l_1<l_2$ (respectively, $0<l_2<l_1$) such that
\begin{gather}\label{e4.8}
 f(v)\le l_1^p((a+1)\beta)^{-p}\text{ for all }v\in[\alpha l_1,l_1],\\
\label{e4.9}
 f(v)\ge l_2^p(a\beta)^{-p}\text{ for all }v\in [\alpha l_2,l_2].
\end{gather}
Then  BVP \eqref{e4.1}, \eqref{e4.2} has at least one positive radial solution
 $v$ with $l_1\le\|v\|\le l_2$ (respectively, $l_2\le\|v\|\le l_1$).
\end{theorem}


\begin{theorem}\label{t4.3}
Assume there exists $l_1>0$ 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
\begin{equation}\label{e4.10}
 f(v)< l^p((a+1)\beta)^{-p}\text{ for all }v\in[\alpha l,l]; or
\end{equation}
\item[(b)] $f^0<\lambda_0$,  $f^{\infty}<\lambda_0$, and
there exists $l>0$ such that
\begin{equation}\label{e4.11}
 f(v)\ge l^p(a\beta)^{-p}\text{ for all }v\in [\alpha l,l].
\end{equation}
\end{itemize}
Then BVP \eqref{e4.1}, \eqref{e4.2} has at least two positive radial
solutions $v_1$ and $v_2$ with $\|v_1\|<l<\|v_2\|$.
\end{theorem}


\begin{theorem} \label{t4.6}
 Let $\{l_i\}_{i=1}^{N}\subset \mathbb{R}$ such that
$0<l_1<l_2<\dots<l_N$. Assume either
\begin{itemize}
\item[(a)] $f$ satisfies \eqref{e4.10} with $r=l_i$  when $i$ is odd,
and satisfies
\eqref{e4.11} with $r=l_i$ when $i$ is even; or
\item[(b)] $f$ satisfies \eqref{e4.10} with $r=l_i$  when $i$ is even,
and satisfies
\eqref{e4.11} with $r=l_i$  when $i$ is odd.
\end{itemize}
Then BVP \eqref{e4.1}, \eqref{e4.2} has at least $N-1$ positive radial
solutions $v_i$ with $l_i<\|v_i\|<l_{i+1}$, $i=1,2,\dots,N-1$.
\end{theorem}


\begin{theorem}\label{t4.7}
 Let $\{l_i\}_{i=1}^\infty\subset \mathbb{R}$ such that
$0<l_1<l_2<\dots$ and $\lim_{i\to \infty }l_{i}=\infty $. Assume either
\begin{itemize}
\item[(a)] $f$ satisfies \eqref{e4.8} with $l_1=l_i$  when $i$ is odd,
and satisfies
\eqref{e4.9} with $l_2=l_i$  when $i$ is even; or
\item[(b)] $f$ satisfies \eqref{e4.8} with $l_1=l_i$  when $i$ is even,
and satisfies
\eqref{e4.9} with $l_2=l_i$  when $i$ is odd.
\end{itemize}
Then BVP \eqref{e4.1}, \eqref{e4.2} has an infinite number of
positive radial solutions.
\end{theorem}

\begin{corollary} \label{c4.2}
Let $\{l_i\}_{i=1}^\infty\subset \mathbb{R}$ such that
$0<l_1<l_2<\dots$ and $\lim_{i\to \infty }l_{i}=\infty $.
Let $E_1=\cup_{i=1}^{\infty}[\alpha l_{2i-1},l_{2i-1}]$ and
$E_2=\cup_{i=1}^{\infty}[\alpha l_{2i},l_{2i}]$.
Assume
$$
\limsup_{E_1\ni v\to\infty}\frac{f(v)}{v^p}<((a+1)\beta)^{-p}
\quad \text{and}\quad
\liminf_{E_2\ni v\to\infty}\frac{f(v)}{v^p}>(a\alpha\beta)^{-p}.
$$
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
\begin{equation}\label{e5.1}
\begin{array}{l}
 -(\phi(u'))'=f(u),\ 0<t<1,\\
 u'(0)=0,\ u(1)+(\theta^*-\pi_p/2)^{-1}u'(1)=0,
\end{array}
\end{equation}
where  $f(u)=[p(\theta^*-\pi_p/2)^{p+1}+c(\tan^{-1}(u)-\pi/4)]u^p$
with $0<|c|<p(\theta^*-\pi_p/2)^{p+1}4/\pi$.
Then BVP \eqref{e5.1} has at least one positive solution.

In fact,
$S(\theta)$ is the unique solution of the initial value problem
\begin{align*}
-(\phi(u'))'=p\phi(u),\\
u(0)=0, \quad u'(0)=1.
\end{align*}
Note that $S'(\pi_p/2)=0$. Hence $p$ is the first eigenvalue
of the SLP
\begin{gather*}
 -(\phi(u'))'=\lambda\phi(u),\\
u'(\pi_p/2)=0,\ u(\theta^*)+u'(\theta^*)=0,
\end{gather*}
with the associated eigenfunction $S(\theta)$.

Make the transformation $t=(\theta-\pi_p/2)/(\theta^*-\pi_p/2)$ in the
above problem. Then similar to the proof of Lemma \ref{l4.1},
we find that $p(\theta^*-\pi_p/2)^{p+1} $ is the first
eigenvalue of the SLP
\begin{gather*}
 -\frac{d}{dt}(\phi(\frac{du}{dt}))=\lambda\phi(u),\\
\frac{du}{dt}(0)=0,\quad
 u(1)+(\theta^*-\pi_p/2)^{-1}\frac{du}{dt}(1)=0.
\end{gather*}
Note that
$$
f_0=f^0=p(\theta^*-\pi_p/2)^{p+1}-c\pi/4 ,\quad
f_{\infty}=f^{\infty}=p(\theta^*-\pi_p/2)^{p+1}+c\pi/4.
$$
Then the conclusion follows from Theorem \ref{t2.1}.
 Note that in this example, $c$ can be arbitrarily close to 0.
\end{example}

\begin{example} \label{example2} \rm
Consider the ordinary BVP
\begin{equation}\label{e5.2}
\begin{gathered}
 -(\phi(u'))'=k(u^{p/2}+u^{2p}),\quad 0<t<1,\\
 u'(0)=0,\quad u(1)+au'(1)=0.
\end{gathered}
\end{equation}
Let $l=4^{-1/(3p)}$,
$\gamma_1=(al)^{p/2}((a+1)^{3p/2}+(al)^{3p/2})^{-1}$,
and $\gamma_2=l^{p/2}(a+1)^{p}a^{-2p}(1+l^{3p/2})^{-1}$.
Then
\begin{itemize}
\item[(a)] BVP \eqref{e5.2} has at least one positive solution when
$k=\gamma_1$;
\item[(b)] BVP \eqref{e5.2} has at least two positive solutions $u_1$
and $u_2$ with $\|u_1\|<l<\|u_2\|$ when $0<k<\gamma_1$;
\item[(c)] BVP \eqref{e5.2} has no positive solutions when $k>\gamma_2$.
\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 $0<k<\gamma_1$, by the similar argument as above and applying
Theorem \ref{e2.5} (a), we obtain the conclusion.

When $k>\gamma_2$, $f(u)/u^p>(a\alpha\beta)^{-p}=(a+1)^p(a^2)^{-p}$
on $(0,\infty)$. Then the conclusion follows from Theorem \ref{t2.8} (b).
\end{example}

\begin{example} \label{exa3} \rm
Consider the  BVP
\begin{equation}\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}
\end{equation}
where $0<r_1<r_2$.
Let $l=4^{1/(3p)}$ and $a$, $\alpha$, $\beta$  be defined
 by \eqref{e4.3},
\eqref{e4.6}, \eqref{e4.7} with $h\equiv1$. Denote
\begin{gather*}
\gamma_3=(a+1)^{p/2}((al)^{3p/2}+(a+1)^{3p/2})a^{-3p} l^{-p}\beta^{-p},
\\
\gamma_4=(l^{3p/2}+1)l^{-p}(a+1)^{-p}\beta^{-p}.
\end{gather*}
Then
\begin{itemize}
\item[(a)] BVP \eqref{e5.3} has at least one positive solution when
$k=\gamma_3$;
\item[(b)] BVP \eqref{e5.3} has at least two positive solutions when
$k>\gamma_3$;
\item[(c)] BVP \eqref{e5.3} has no positive solutions when $0<k<\gamma_4$.
\end{itemize}

In fact, the equation in \eqref{e5.3} is of the form of \eqref{e4.1}
with $w(t)\equiv1$ and $f(v)=k(v^{-p/2}+v^{-2p})^{-1}$.
It is clear that
$f^0=f^\infty=0$ and $f(v)/v^p$ reaches maximum at $l$.
By a similar argument to Example \ref{example2} and applying Theorems
\ref{t4.4}, \ref{t4.5}, and \ref{t4.8}, we can prove the results. We
omit the details.
\end{example}

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\end{document}