\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 36, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/36\hfil Positive solutions]
{Positive solutions for a $2n$th-order $p$-Laplacian  boundary
 value problem involving all derivatives}

\author[Y. Ding, J. Xu, X. Zhang\hfil EJDE-2013/36\hfilneg]
{Youzheng Ding, Jiafa Xu, Xiaoyan Zhang}  % in alphabetical order

\address{Youzheng Ding \newline
School of  Mathematics, Shandong University,
Jinan 250100,  Shandong,  China}
\email{dingyouzheng@139.com}

\address{Jiafa Xu \newline
School of  Mathematics, Shandong University,
Jinan 250100,  Shandong,  China}
\email{xujiafa292@sina.com}

\address{Xiaoyan Zhang \newline
School of  Mathematics, Shandong University,
Jinan 250100,  Shandong,  China}
\email{zxysd@sdu.edu.cn}

\thanks{Submitted September 10, 2012. Published January 30, 2013.}
\subjclass[2000]{34B18, 45J05, 47H11}
\keywords{Integro-ordinary differential equation; a priori estimate;
index; \hfill\break\indent fixed point; positive solution}

\begin{abstract}
 In this work, we  are mainly concerned  with the   positive
 solutions for the  $2n$th-order $p$-Laplacian  boundary-value
 problem
 \begin{gather*}
 -(((-1)^{n-1}x^{(2n-1)})^{p-1})'
 =f(t,x,x',\ldots,(-1)^{n-1}x^{(2n-2)},(-1)^{n-1}x^{(2n-1)}),\\
 x^{(2i)}(0)=x^{(2i+1)}(1)=0,\quad (i=0,1,\ldots,n-1),
 \end{gather*}
 where $n\ge 1$ and  $f\in C([0,1]\times \mathbb{R}_+^{2n},
 \mathbb{R}_+)(\mathbb{R}_+:=[0,\infty))$.
 To overcome the difficulty resulting from all derivatives,
 we first convert  the above problem into a boundary value problem
 for an associated second order integro-ordinary differential equation
 with $p$-Laplacian operator.  Then, by virtue of the classic fixed
 point index theory, combined with a priori estimates of positive solutions,
 we establish some results on the existence and multiplicity of positive
 solutions for the above  problem. Furthermore, our nonlinear term $f$ is
 allowed to grow superlinearly and sublinearly.
\end{abstract}

\maketitle 
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
%\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

In this paper, we investigate  the existence and multiplicity of
positive solutions for the following $2n$th-order $p$-Laplacian
boundary value problem involving all derivatives
\begin{equation}\label{Problem}
\begin{gathered}
 -(((-1)^{n-1}x^{(2n-1)})^{p-1})'
 =f(t,x,x',\ldots,(-1)^{n-1}x^{(2n-2)},(-1)^{n-1}x^{(2n-1)}),\\
 x^{(2i)}(0)=x^{(2i+1)}(1)=0,\quad (i=0,1,\ldots,n-1),
 \end{gathered}
\end{equation}
where $f\in C([0,1]\times \mathbb{R}_+^{2n},\mathbb{R}_+)$.  Here,
by a positive solution \eqref{Problem} we mean a function $u\in
C^{2n}[0,1]$ that solves \eqref{Problem} and satisfies $u(t)> 0$ for
all $t\in (0,1]$.

 We are here interested in the case where $f$
depends explicitly on all derivatives. When $f$  involves all even
derivatives explicitly,  many researchers
\cite{AOS,Bai-Ge,Davis,YuhongMa,Wei3} study the
 Lidstone boundary value problem
\begin{equation}\label{Lidstone-1}
\begin{gathered}
(-1)^nu^{(2n)}=f(t,u,-u'',\ldots,(-1)^{n-1}u^{(2n-2)}), \quad n\geq 2,\\
u^{(2i)}(0)=u^{(2i)}(1)=0,\quad i=0,1,2,\ldots,n-1.
\end{gathered}
\end{equation}

Yang \cite{Yang1} considered the existence and uniqueness of
positive solutions for the following generalized Lidstone boundary
value problem
\begin{equation}\label{Lidstone-2}
\begin{gathered}
(-1)^nu^{(2n)}=f(t,u,-u'',\ldots,(-1)^{n-1}u^{(2n-2)}),\\
\alpha_0u^{(2i)}(0)-\beta_0u^{(2i+1)}(0)=0,
\alpha_1u^{(2i)}(1)-\beta_1u^{(2i+1)}(1)=0, \; i=0,1,2,\ldots,n-1,
\end{gathered}
\end{equation}
where $\alpha_j\ge 0$, $\beta_j\ge 0$ $(j=0,1)$ and
$\alpha_0\alpha_1+\alpha_0\beta_1+\alpha_1\beta_0>0$.  In view of
the symmetry, the results in
\cite{AOS,Bai-Ge,Davis,YuhongMa,Wei3,Yang1} demonstrate that 
problems \eqref{Lidstone-1} and \eqref{Lidstone-2} are essentially
identical with  second-order Dirichlet problem and Sturm-Liouville
problem (the case $n=1$), respectively.

Yang, O'Regan and Agarwal \cite{Yang4} studied  the
existence and multiplicity of positive solutions for the
second-order boundary value problem depending on the first-order
derivative $u'$
\begin{equation}\label{yang4}
\begin{gathered}
u''+f(t,u,u')=0,\\
u(0)=u'(1)=0.
\end{gathered}\end{equation}
In order to overcome the difficulty resulting from the first-order
derivative, they imposed  the Bernstein-Nagumo condition
\cite{SB,MN} on the nonlinear term $f$ to establish several
existence theorems for \eqref{yang4}.

Yang and O'Regan \cite{Yang2} studied  the
existence, multiplicity and uniqueness of positive solutions for the
$2n$th-order boundary value problem involving all derivatives of odd
orders
\begin{equation}\label{yang}
\begin{gathered}
(-1)^nu^{(2n)}=f(t,u,u',-u''',\ldots,(-1)^{n-1}u^{(2n-1)}),\\
u^{(2i)}(0)=u^{(2i+1)}(1)=0,\quad i=0,1,2,\ldots,n-1,
\end{gathered}
\end{equation}
where $n\ge 2$ and  $f\in C([0,1]\times \mathbb{R}_+^{n+1},
\mathbb{R}_+)$ depends on $u$ and all derivatives of odd orders. As
application, they utilized   their  results to discuss the positive
symmetric solutions for a Lidostone problem involving an open
question posed by Eloe \cite{Eloe}.  Yang \cite{Yang5} discussed a
 $2n$th-order ordinary differential equation
involving all derivatives, and the results improved and extended the
corresponding ones in \cite{Yang1,Yang4,Yang2}.

Equations of the $p$-Laplacian form occur in the study of
non-Newtonian fluid theory and the turbulent flow of a gas in a
porous medium. Since 1980s, there exist a very large number of
papers devoted to the existence of solutions for differential
equations with $p$-Laplacian,  for instance, see
\cite{Graef,Ke,Li,Lakmeche,Wang,Jiafa,Wei,Yang3,Yang6,Zhang} and the
references therein.

Yang and his coauthors \cite{Jiafa,Yang3,Yang6}  studied some
boundary value problems with the $p$-Laplacian operator.
Yang  and  O'Regan  \cite{Yang3} studied the existence and multiplicity 
of positive solutions for the focal problem involving both the $p$-Laplacian
and the first order derivative
\begin{equation}\label{JiafaXu}
\begin{gathered}
((u')^{p-1})'+f(t,u,u')=0,\quad t\in (0,1),\\
 u(0)=u'(1)=0,
\end{gathered}
\end{equation}
where $p>1$ and $f\in C([0,1]\times \mathbb{R}_+^2,\mathbb{R}_+)$.
Moreover, they applied  their  main results obtained here  to
establish the existence of positive symmetric solutions to the
Dirichlet problem
\begin{equation}
\begin{gathered}
(|u'|^{p-2}u')'+f(u,u')=0,\quad t\in (-1,0)\cup (0,1),\\
 u(-1)=u(1)=0.
\end{gathered}
\end{equation}

However, the existence  of positive solutions for $p$-Laplacian
equation with the nonlinear term involving the derivatives, such as
Lidstone problem,  has not been extensively studied yet. To the best
of our knowledge, only \cite{Guo-Ge,Su-Wei,Zhao-Ge} is devoted to
this direction. Guo and Ge \cite{Guo-Ge} considered the
following boundary-value problem
\begin{equation}\label{GG}
\begin{gathered}
(\Phi(y^{(2n-1)}))'=f(t,y,y'',\ldots,y^{(2n-2)}),\quad 0\le t\le 1,\\
y^{(2i)}(0)=y^{(2i)}(1)=0,\quad i=0,1,2,\ldots,n-1,
\end{gathered}
\end{equation}
where $f\in C([0,1]\times \mathbb{R}^n, \mathbb{R})
(\mathbb{R}:=(-\infty,+\infty))$. Some growth conditions are imposed
on $f$ which yield the existence of at least two symmetric positive
solutions by using a fixed point theorem in cones. An interesting
feature in  \cite{Guo-Ge} is that  the nonlinearity  $f$ may  be
sign-changing.



Motivated by the works mentioned above, in particular
\cite{Jiafa,Yang1,Yang4,Yang2,Yang5,Yang3,Yang6}, in this work, we
discuss the existence and multiplicity of positive solutions for
\eqref{Problem}. To overcome the difficulty resulting from all
derivatives, we first transform \eqref{Problem} into a boundary
value problem for an associated second order integro-ordinary
differential equation. Then, we will use    fixed point index theory
to establish our main results based on a priori estimates achieved
by utilizing some properties of concave functions, properties
including Jensen's inequalities and our inequality \eqref{le4}
below. The results obtained here improve some existing results in
the literature.

\section{Preliminaries}

Let $E:=C^1[0,1]$, $\|u\|:=\max\{\|u\|_0,\|u'\|_0\}$, where
$\|u\|_0:=\max_{t\in [0,1]}|u(t)|$. Furthermore, let
 $P:=\{u\in E: u(t)\geq 0,u^\prime(t)\geq 0, \forall t\in [0,1]\}$. 
Then $E$ is a real Banach space and $P$  a cone on $E$. 
For any positive integer $i\ge 2$,  we denote
\[
k_1(t,s):=\min\{t,s\}, \quad
k_i(t,s):=\int_0^1k_{i-1}(t,\tau)k_1(\tau,s)\,{\rm d}  \tau, \forall
t,s\in [0,1].
\] 
Define
\[
(B_iu)(t):=\int_0^1 k_i(t,s)u(s)\,{\rm d}  s,\quad
 h_i(t,s):=\partial k_i(t,s)/\partial t, \quad i=1,2,\ldots,
\]
Then 
\[
 ((B_iu)(t))':=\int_0^1 h_i(t,s)u(s)\,{\rm d}  s,\quad i=1,2,\ldots,
\]
and  $B_i,B_i': E\to E$ are completely continuous linear
operators  and $B_i,B_i'$ are  also positive operators.

Let $(-1)^{n-1}x^{(2n-2)}:=u$, it is easy to see that \eqref{Problem} is
equivalent to the following system of integro-ordinary differential
equations
\begin{equation}\label{integro-diff}
\begin{gathered}
-((u')^{p-1})'=f(t,(B_{n-1}u)(t),((B_{n-1}u)(t))',\ldots,u,u'),\\
u(0)=u'(1)=0.
\end{gathered}
\end{equation}
 Furthermore, the above system can be written in the form
\begin{equation}\label{integral}
u(t)=\int_0^t\Big(\int_s^1
f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))
\,{\rm d} \tau\Big)^{\frac{1}{p-1}}\,{\rm d}  s.
\end{equation} 
Denote by
\[
(Au)(t):=\int_0^t\Big(\int_s^1
f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))
 \,{\rm d} \tau\Big)^{\frac{1}{p-1}}\,{\rm d}  s.
\]
 Hence, $f\in C([0,1]\times \mathbb{R}_+^{2n},\mathbb{R}_+)$
implies that
 $A:P\to P$ is a completely continuous operator, and the existence of
positive solutions for \eqref{integro-diff} is equivalent to that of
positive fixed points of $A$.


\begin{lemma} \label{lem2.1} 
Let $\kappa:=1-2/e$ and $\psi(t):=te^t, t\in [0,1]$. Then $\psi(t)$ 
is nonnegative on $[0,1]$ and
\begin{equation}\label{Gineq}
\kappa \psi(s)\le \int_0^1 k_1(t,s)\psi(t)\,{\rm d}  t\le \psi(s).
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2.2} 
Let  $u$ is concave, increasing and nonnegative on $[0,1]$. Then
\begin{equation}\label{le4}
\int_0^1u(t)\psi(t)\,{\rm d}  t\ge \kappa e\|u\|.
\end{equation}
\end{lemma}

\begin{proof}  
The concavity of $u$ and $\max_{t\in [0,1]}u(t)=u(1)=\|u\|$ imply
\[ 
\int_0^1 u(t)\psi(t)\,{\rm d}  t =\int_0^1
u(t\cdot 1+(1-t)\cdot 0)\psi(t)\,{\rm d}  t\ge u(1)\int_0^1t\psi(t)\,{\rm d}  t=
\kappa e\|u\|.
\] 
This completes the proof.
\end{proof}

 \begin{lemma}[\cite{Yang2}] \label{lem2.3}
Let $u\in P$ and $q>0$. Then
\begin{equation}\label{hkeq}
\int_0^1 \Big[(B_{n-1}u^q)(t)+2\sum_{i=0}^{n-2}((B_{n-1-i}u^q)(t))'\Big]
\psi(t)\,{\rm d}  t=\int_0^1 u^q(t)\psi(t)\,{\rm d}  t.
\end{equation} 
\end{lemma}

\begin{lemma}[\cite{GuoDajun}] \label{lem2.4} 
 Let $\Omega \subset E$ be a bounded open set and  
$A:\overline{\Omega}\cap P\to P$ is a
completely continuous operator. If there exists
 $v_0\in P\setminus\{0\}$ such that $v-Av\neq \lambda v_0$  for all 
$ v\in \partial\Omega\cap P$ and $\lambda \geq 0$, then 
$i(A,\Omega\cap P,P)=0$,  where $i$ is the fixed point index on $P$. 
 \end{lemma}

 \begin{lemma}[\cite{GuoDajun}] \label{lem2.5}  
Let $\Omega \subset E$ be a bounded open set with $0\in \Omega$. Suppose
$A:\overline{\Omega}\cap P\to P$ is a completely continuous
operator. If $v\neq \lambda Av $ for all $v\in \partial\Omega \cap P$  and  
$  0\leq \lambda\leq 1 $, then $i(A,\Omega\cap P,P)=1$.  
\end{lemma}

 \begin{lemma}[Jensen's inequalities] \label{lem2.6}
Let $\theta>0$ and $\varphi\in C([0,1],\mathbb R^+)$. Then
\begin{gather*}
\Big(\int_0^1 \varphi (t)\,{\rm d}  t\Big)^\theta
\leq \int_0^1 (\varphi(t))^\theta\,{\rm d}  t, \quad
\text{if }\theta\geq 1,\\
\Big(\int_0^1 \varphi (t)\,{\rm d}  t\Big)^\theta
\geq \int_0^1 (\varphi(t))^\theta\,{\rm d}  t,\quad \text{if }
0<\theta\leq 1.
\end{gather*}
 \end{lemma}

\section{Main results}

For brevity, we define $y=(y_1,y_2,\ldots,y_{2n-1},y_{2n})\in \mathbb  R^{2n}_+$,
$\gamma_p:=\max\{1,2^{p-2}\}$, $p_*=\min\{1,p-1\}$, $p^*=\max\{1,p-1\}$,
$\mathscr{K}_i:=\max_{t,s\in [0,1]}k_i(t,s)>0$,
$\mathscr{H}_i:=\max_{t,s\in [0,1]}h_i(t,s)>0$,
\begin{gather*}
\beta_p:=\Big\{2^{p_*-1}\kappa\Big[(n-1)
\Big(\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p_*-1}+1\Big]\Big\}
^{1-p/p_*},
\\
\alpha_p:=\Big\{2^{p^*-1}\Big[(n-1)
\Big(\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p^*-1}+1\Big]\Big\}
^{1-p/p^*}.
\end{gather*}
\begin{itemize}
\item[(H1)]  $f\in C([0,1]\times \mathbb{R}_+^{2n},\mathbb{R}_+)$.

\item[(H2)] There exist $a_1>\beta_p$ and $c>0$ such that
\[
f(t,y)\ge a_1\Big(\sum_{i=1}^{n-1}(y_{2i-1}+2(n-i)y_{2i})+y_{2n-1}\Big)^{p-1}-c,
\quad\text{for all }y\in \mathbb R^{2n}_+\text{ and }t\in [0,1].
\]


 \item[(H3)] For any $M_0>0$ there is a function 
$\Phi_{M_0}\in C(\mathbb{R}_+,\mathbb{R_+})$ such that
\begin{gather*}
f(t,y)\le \Phi_{M_0}(y_{2n}^{p-1}), \forall 
 (t,y)\in [0,1]\times [0,{M_0}]^{2n-1}\times \mathbb{R}_+,\\
\int_\delta^\infty\frac{\,{\rm d} \xi}{\Phi_{M_0}(\xi)}=\infty\quad
\text{for any }\delta>0.
\end{gather*}

\item[(H4)] There exist $b_1\in (0,\alpha_p)$ and $r>0$ such
that
\[
f(t,y)\le b_1\Big(\sum_{i=1}^{n-1}(y_{2i-1}+2(n-i)y_{2i})+y_{2n-1}\Big)^{p-1}
\quad \text{for all $y\in [0,r]^{2n}$  and }t\in [0,1].
\]

\item[(H5)] There exist $a_2>\beta_p$ and $r>0$ such that
\[
f(t,y)\ge a_2\Big(\sum_{i=1}^{n-1}(y_{2i-1}+2(n-i)y_{2i})+y_{2n-1}\Big)^{p-1}
\text{ for all $y\in [0,r]^{2n}$ and }t\in [0,1].
\]


\item[(H6)] There exist $b_2\in (0,\alpha_p)$ and $c>0$ such
that
\[
f(t,y)\le b_2\Big(\sum_{i=1}^{n-1}(y_{2i-1}+2(n-i)y_{2i})+y_{2n-1}\Big)^{p-1}+c
\text{ for all $y\in \mathbb R^{2n}_+$  and }t\in [0,1].
\]

\item[(H7)] $f$ is increasing in $y$ and there is a constant
$\omega > 0$ such that
\[
\int_0^1 f^{p^*/(p-1)}(s,\omega,\ldots,\omega)\,{\rm d}  s<\omega.
\]
\end{itemize}

\begin{remark} \label{rmk3.1} \rm
A function $f$ is said to be increasing in $y$ if $f(t, x) \le  f(t, y)$
holds for every pair $x, y \in\mathbb{R}^{2n}_+ $ with $x \le y$,
where the partial ordering $\le $ in $\mathbb{R}^{2n}_+$ is
understood componentwise.\end{remark}


\begin{theorem} \label{thm3.2} 
If {\rm (H1)--(H4)} hold, then
\eqref{Problem} has at least one positive solution.
\end{theorem}

\begin{proof} 
 Let 
\[
\mathscr{M}_1: =\{u\in P: u=Au+\lambda
\varphi, \text{ for some } \lambda\ge 0\},
\] 
where
$\varphi(t):=te^{-t}$. Clearly, $\varphi(t)$ is nonnegative and
concave on $[0, 1]$. We claim $\mathscr{M}_1$ is bounded. We first
establish the a priori bound of $\|u\|_0$ for $\mathscr{M}_1$.
Indeed, $u\in\mathscr{M}_1$ implies $u$ is concave (by the concavity
of $A$ and $\varphi$) and $u(t) \ge (Au)(t)$. By definition we
obtain
\begin{equation}
u(t)\ge \int_0^t
\Big(\int_s^1 f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',
\ldots,u(\tau),u'(\tau))\,{\rm d} \tau\Big)^{\frac{1}{p-1}}\,{\rm d}  s
\end{equation}
for all $u\in\mathscr{M}_1$. Note that $p_*, p_*/p-1\in [0,1]$.
 Now, by  (H2), we find
\begin{equation}
\Big[a_1\Big(\sum_{i=1}^{n-1}(y_{2i-1}+2(n-i)y_{2i})+y_{2n-1}
\Big)^{p-1}\Big]^{\frac{p_*}{p-1}}
\le (f(t,y)+c)^{\frac{p_*}{p-1}}\le
f^{\frac{p_*}{p-1}}(t,y)+c^{\frac{p_*}{p-1}}.
\end{equation}
Combining this and Jensen's  inequality, we obtain
\begin{equation}\label{A-1}
\begin{aligned}
& u^{p_*}(t)\\
& \ge \Big[\int_0^t\Big(\int_s^1
f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))\,{\rm d}
 \tau\Big)^{\frac{1}{p-1}}\,{\rm d}  s\Big]^{p_*}\\
& \ge \int_0^t\int_s^1
f^{\frac{p_*}{p-1}}(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',\ldots,u(\tau),
 u'(\tau))\,{\rm d} \tau\,{\rm d}  s\\
& =\int_0^1 k_1(t,s)f^{\frac{p_*}{p-1}}(s,(B_{n-1}u)(s),((B_{n-1}u)(s))',
 \ldots,u(s),u'(s))\,{\rm d}  s\\
&\ge a_1^{\frac{p_*}{p-1}} \int_0^1
 k_1(t,s)\Big[\sum_{i=1}^{n-1}\left((B_iu)(s)+2(n-i)((B_iu)(s))'\right)+u(s)
 \Big]^{p_*}\,{\rm d}  s-\frac{c^{\frac{p_*}{p-1}}}{2}\\
&\ge 2^{p_*-1}a_1^{\frac{p_*}{p-1}} \int_0^1 k_1(t,s)\Big[
 \sum_{i=1}^{n-1}\Big((B_iu)(s)+2(n-i)((B_iu)(s))'\Big)\Big]^{p_*}\,{\rm d}  s\\
&\quad +2^{p_*-1}a_1^{\frac{p_*}{p-1}} \int_0^1 k_1(t,s)u^{p_*}(s)\,{\rm d}
  s-\frac{c^{\frac{p_*}{p-1}}}{2}\\
&= 2^{p_*-1}a_1^{\frac{p_*}{p-1}} \int_0^1
 k_1(t,s)\Big[\sum_{i=1}^{n-1}\int_0^1
 (k_i(s,\tau)+2(n-i)h_i(s,\tau))u(\tau)\,{\rm d} \tau\Big]^{p_*}\,{\rm d}  s\\
&\quad +2^{p_*-1}a_1^{\frac{p_*}{p-1}} \int_0^1 k_1(t,s)u^{p_*}(s)\,{\rm d}
  s-\frac{c^{\frac{p_*}{p-1}}}{2}\\
& = 2^{p_*-1}a_1^{\frac{p_*}{p-1}} \int_0^1 k_1(t,s)\Big[\int_0^1
 \frac{\sum_{i=1}^{n-1}(k_i(s,\tau)+2(n-i)h_i(s,\tau))}{\sum_{i=1}^{n-1}
 (\mathscr{K}_i+\mathscr{H}_i)}\sum_{i=1}^{n-1}(\mathscr{K}_i\\
&\quad +\mathscr{H}_i)
 u(\tau)\,{\rm d} \tau\Big]^{p_*}\,{\rm d}  s
 +2^{p_*-1}a_1^{\frac{p_*}{p-1}}
 \int_0^1 k_1(t,s)u^{p_*}(s)\,{\rm d}  s-\frac{c^{\frac{p_*}{p-1}}}{2}\\
& \ge \Big(2\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p_*-1}
 a_1^{\frac{p_*}{p-1}}
 \int_0^1 k_1(t,s)\\
&\quad\times \Big[\sum_{i=1}^{n-1}\int_0^1
 (k_i(s,\tau)+2(n-i)h_i(s,\tau)) u^{p_*}(\tau)\,{\rm d} \tau\Big]\,{\rm d}  s\\
&\quad +2^{p_*-1}a_1^{\frac{p_*}{p-1}} \int_0^1 k_1(t,s)u^{p_*}(s)\,{\rm d}
 s-\frac{c^{\frac{p_*}{p-1}}}{2}\\
&=\Big(2\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p_*-1}
 a_1^{\frac{p_*}{p-1}}
 \int_0^1 k_1(t,s)\\
&\quad\times \Big[\sum_{i=1}^{n-1}((B_iu^{p_*})(s)+2(n-i)((B_iu^{p_*})(s))')
 \Big]\,{\rm d}  s\\
&\quad +2^{p_*-1}a_1^{\frac{p_*}{p-1}} \int_0^1
k_1(t,s)u^{p_*}(s)\,{\rm d}  s-\frac{c^{\frac{p_*}{p-1}}}{2}.
\end{aligned}
\end{equation}
Multiply both sides of the above expression by  $\psi(t)$ and integrate
over [0,1] and use \eqref{Gineq} and \eqref{hkeq} to obtain
\begin{equation}
\begin{aligned} 
& \int_0^1 \psi(t)u^{p_*}(t)\,{\rm d}  t\\
&\ge \Big(2\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p_*-1}
 a_1^{\frac{p_*}{p-1}}\kappa \int_0^1
 \psi(t)\Big[\sum_{i=1}^{n-1}\big((B_iu^{p_*})(t)\\
&\quad +2(n-i)((B_iu^{p_*})(t))'\big) \Big]\,{\rm d}  t
 +2^{p_*-1}a_1^{\frac{p_*}{p-1}}\kappa \int_0^1
 \psi(t)u^{p_*}(t)\,{\rm d}  t-\frac{c^{\frac{p_*}{p-1}}}{2}
\\
& =2^{p_*-1}a_1^{\frac{p_*}{p-1}}\kappa\Big[(n-1)
\Big(\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p_*-1}+1\Big]
\int_0^1 \psi(t)u^{p_*}(t)\,{\rm d}  t-\frac{c^{\frac{p_*}{p-1}}}{2}.
\end{aligned}
\end{equation}
Therefore,
\[
\int_0^1 \psi(t)u^{p_*}(t)\,{\rm d}  t
\le \frac{c^{\frac{p_*}{p-1}}}{2^{p_*}a_1^{\frac{p_*}{p-1}}\kappa
\Big[(n-1) \Big(\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p_*-1}+1
\Big]-2}:=\mathscr{N}_1.
\]
Recall that every  $u \in \mathscr{M}_1$ is concave and increasing on
$[0,1]$. So is $u^{p_*}$ with $p_*\in (0,1]$. Now Lemma \ref{lem2.2} yields
\begin{equation}
\|u\|_0\le  (\kappa e)^{-1/p_*}\mathscr{N}_1^{1/p_*}
\end{equation}
for all $u\in\mathscr{M}_1$,
which implies the a priori bound of $\|u\|_0$ for $\mathscr{M}_1$,
as claimed. It follows, from the boundedness of $\|u\|_0$ for
$\mathscr{M}_1$, that there is $\lambda_0 > 0$ such that
$\lambda\le \lambda_0$ for all $\lambda\in \Lambda$,
where
\[
\Lambda:=\{\lambda\ge 0: \text{ there exists  } u\in \mathscr{M}_1 
\text{ such that } u=Au+\lambda \varphi\}.
\]
If $u\in \mathscr{M}_1$, then
\[
u'(t)=\Big(\int_t^1 f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',
\ldots,u(\tau),u'(\tau))\,{\rm d} \tau\Big)^{\frac{1}{p-1}}+\lambda(1-t)e^{-t}
\]
for some $\lambda\ge 0$, and by (H3),
\begin{align*} 
(u')^{p-1}(t)
& \le \gamma_p\int_t^1 f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',
 \ldots,u(\tau),u'(\tau))\,{\rm d} \tau+\gamma_p\lambda_0^{p-1}\\
& \le \gamma_p\int_t^1
\Phi_{M_0}((u')^{p-1}(\tau))\,{\rm d} \tau+\gamma_p\lambda_0^{p-1}.
\end{align*}
Let $v(t):= (u')^{p-1}(t)$. Then $v(t)\in C([0,1], \mathbb{R}_+)$
and $v(1)=0$. Moreover,
\[
v(t)\le \gamma_p\int_t^1
\Phi_{M_0}(v(\tau))\,{\rm d} \tau+\gamma_p\lambda_0^{p-1}.
\]
Let $F(t):= \int_t^1 \Phi_{M_0}(v(\tau))\,{\rm d} \tau$.
Then
\[
-F'(t)=\Phi_{M_0}(v(t))\le
\Phi_{M_0}(\gamma_pF(t)+\gamma_p\lambda_0^{p-1}). 
\] 
Therefore,
\[
\int_{\gamma_p\lambda_0^{p-1}}^{v(t)}\frac{\,{\rm d} \xi}{\Phi_{M_0}(\xi)}\le
\int_{\gamma_p\lambda_0^{p-1}}^{\gamma_pF(t)
+\gamma_p\lambda_0^{p-1}}\frac{\,{\rm d} \xi}{\Phi_{M_0}(\xi)}\le
\gamma_p(1-t).
\]
Hence there is $\mathscr{N}_2 > 0$ such
that\[\|(u')^{p-1}\|_0=\|v\|_0=v(0)\le \mathscr{N}_2.\] Let
$\mathscr{N}_3:=\max\{(\kappa
e)^{-1/p_*}\mathscr{N}_1^{1/p_*},\mathscr{N}_2^{1/p-1}\}$.
Then
\[
\|u\|\le\mathscr{N}_3, \quad \forall u\in \mathscr{M}_1.
\] 
This proves the boundedness of $\mathscr{M}_1$. As a result of this, for
every $R>\mathscr{N}_3$, we have
\[
u-Au\not=\lambda \psi,\quad \forall u\in\partial B_R\cap P, \;
\lambda\geq 0.
\] 
Now by  Lemma \ref{lem2.4}, we obtain
\begin{equation}\label{fi-1}
i(A,B_R\cap P,P)=0.
\end{equation} 
Let 
\[
\mathscr{M}_2:=\{u\in \overline{B}_r\cap P:u=\lambda Au \text{ for some }
 \lambda\in [0,1]\}.
\] 
We shall prove $\mathscr{M}_2= \{0\}$. Indeed, if 
$u\in \mathscr{M}_2$,  we have for any $ u\in \overline{B}_r\cap P$
\begin{equation}
u(t)\le \int_0^t\Big(\int_s^1
f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))\,{\rm d} \tau
\Big)^{\frac{1}{p-1}}\,{\rm d}  s.
\end{equation} 
 Notice  that $p^*, p^*/p-1\ge 1$. Now, similar to
\eqref{A-1}, by Jensen's inequality and (H4), we obtain
\begin{equation}\label{A-2}
\begin{aligned} 
&u^{p^*}(t)\\
& \le \Big[\int_0^t\Big(\int_s^1
f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))
\,{\rm d} \tau\Big)^{\frac{1}{p-1}}\,{\rm d}  s\Big]^{p^*}\\
& \le  \int_0^1 k_1(t,s)f^{p^*/(p-1)}(s,(B_{n-1}u)(s),((B_{n-1}u)(s))',
 \ldots,u(s),u'(s))\,{\rm d}  s\\
& \le  b_1^{p^*/(p-1)} \int_0^1
k_1(t,s)\Big[\sum_{i=1}^{n-1}\Big((B_iu)(s)+2(n-i)((B_iu)(s))'\Big)+u(s)
\Big]^{p^*}\,{\rm d}  s\\
&\le  2^{p^*-1}b_1^{p^*/(p-1)} \int_0^1
k_1(t,s)\Big[\sum_{i=1}^{n-1}\int_0^1
(k_i(s,\tau)+2(n-i)h_i(s,\tau))u(\tau)\,{\rm d} \tau\Big]^{p^*}\,{\rm d}  s\\
&\quad +2^{p^*-1}b_1^{p^*/(p-1)} \int_0^1 k_1(t,s)u^{p^*}(s)\,{\rm d}  s\\
& = 2^{p^*-1}b_1^{p^*/(p-1)} \int_0^1 k_1(t,s)
 \Big[\int_0^1 \frac{\sum_{i=1}^{n-1}(k_i(s,\tau)+2(n-i)h_i(s,\tau))}
 {\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)}\\
&\quad\times \sum_{i=1}^{n-1}
 (\mathscr{K}_i+\mathscr{H}_i) u(\tau)\,{\rm d} \tau\Big]^{p^*}\,{\rm d}  s
 +2^{p^*-1}b_1^{p^*/(p-1)} \int_0^1 k_1(t,s)u^{p^*}(s)\,{\rm d}  s\\
& \le \Big(2\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p^*-1}
 b_1^{p^*/(p-1)} \int_0^1 k_1(t,s)
 \Big[\sum_{i=1}^{n-1}((B_iu^{p^*})(s)\\
&\quad +2(n-i)((B_iu^{p^*})(s))') \Big]\,{\rm d}  s
 +2^{p^*-1}b_1^{p^*/(p-1)}
 \int_0^1 k_1(t,s)u^{p^*}(s)\,{\rm d}  s.
\end{aligned}
 \end{equation}
 Multiply both sides of the above expression by  $\psi(t)$ and integrate
 over $[0,1]$ and
use \eqref{Gineq} and \eqref{hkeq} to obtain
\begin{equation}\begin{aligned}  
& \int_0^1 \psi(t)u^{p^*}(t)\,{\rm d}  t\\
&\le \Big(2\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p^*-1}
b_1^{p^*/(p-1)}
\int_0^1 \psi(t)\Big[\sum_{i=1}^{n-1}
\big((B_iu^{p^*})(t)\\
&\quad +2(n-i)((B_iu^{p^*})(t))'\big)\Big]\,{\rm d}  t
 +2^{p^*-1}b_1^{p^*/(p-1)} \int_0^1 \psi(t)u^{p^*}(t)\,{\rm d}  t\\
& =2^{p^*-1}b_1^{p^*/(p-1)}\Big[(n-1)
\Big(\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p^*-1}+1\Big]
\int_0^1 \psi(t)u^{p^*}(t)\,{\rm d}  t.
\end{aligned}
\end{equation}
Therefore, $\int_0^1 \psi(t)u^{p^*}(t)\,{\rm d}  t=0$, whence
$u(t)\equiv 0, \forall u\in\mathscr{M}_2$. As a result,
$\mathscr{M}_2= \{0\}$,
as claimed. Consequently,
\[
u\neq \lambda Au, \quad \forall u\in \partial B_r\cap P, \;\lambda\in [0,1].
\]
Now  Lemma \ref{lem2.5} yields 
\begin{equation}\label{index11}
i(A,B_r\cap P,P)=1.
\end{equation} 
Combining this with \eqref{fi-1} gives
\[
i(A,(B_R\backslash \overline{B}_r)\cap P,P)=0-1=-1.
\] 
Hence the operator $A$ has at least one fixed point on 
$(B_R\setminus \overline{B}_r)\cap P$ and therefore \eqref{Problem} 
has at least one positive solution. This completes the proof.
\end{proof}

\begin{theorem} \label{thm3.3} 
If {\rm (H1), (H5), (H6)} are satisfied,
then \eqref{Problem} has at least one positive
solution.
\end{theorem}

\begin{proof} Let
\[
\mathscr{M}_3:=\{u\in \overline{B}_r\cap P: u=Au+\lambda \psi\
\text{for some}\ \lambda\ge 0\}.
\] 
We claim $\mathscr{M}_3\subset\{0\}$. Indeed, if $u\in\mathscr{M}_3$,
then we have $u\geq Au$ by definition. That is,
\begin{equation}
u(t)\ge \int_0^t\Big(\int_s^1 f(\tau,(B_{n-1}u)(\tau),
((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))\,{\rm d} \tau\Big)^{\frac{1}{p-1}}
 \,{\rm d}  s.
\end{equation}
 Similar to $\mathscr{M}_2= \{0\}$, we can also
obtain $\mathscr{M}_3\subset\{0\}$.  As a result of this, we have
\[
u-Au\not=\lambda \psi, \forall u\in\partial B_r\cap P, \lambda\geq 0.
\] 
Now Lemma \ref{lem2.4} gives
\begin{equation}\label{theorem2index0}
i(A,B_r\cap P,P)=0.
\end{equation} 
Let 
\[
\mathscr{M}_4:=\{u\in  P:u=\lambda Au \text{ for some } \lambda\in [0,1]\}.
\] 
We assert  $\mathscr{M}_4$ is bounded. We first establish the a priori bound 
of $\|u\|_0$ for $\mathscr{M}_4$. Indeed, if $u\in \mathscr{M}_4$, 
then  $u$ is concave and $u\leq Au$, which can be written in the form
\begin{equation}
u(t)\le \int_0^t\Big(\int_s^1
f(\tau,(B_{n-1}u)(\tau),((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))
\,{\rm d} \tau\Big)^{\frac{1}{p-1}}\,{\rm d}  s,
\end{equation} 
for all $ u\in \mathscr{M}_4$. Note that $p^*, p^*/p-1\ge 1$. Now
by (H6) and  Jensen's inequality, we obtain
\begin{equation}\label{A-4}
\begin{aligned}  u^{p^*}(t)
& \le \Big[\int_0^t\Big(\int_s^1 f(\tau,(B_{n-1}u)(\tau),
((B_{n-1}u)(\tau))',\ldots,u(\tau),u'(\tau))\,{\rm d} \tau\Big)^{\frac{1}{p-1}}
 \,{\rm d}  s\Big]^{p^*}\\
& \le \int_0^1 k_1(t,s)f^{p^*/(p-1)}(s,(B_{n-1}u)(s),
 ((B_{n-1}u)(s))',\ldots,u(s),u'(s))\,{\rm d}  s\\
& \le \int_0^1 k_1(t,s)\Big\{b_2\Big[\sum_{i=1}^{n-1}
 \big((B_iu)(s)+2(n-i)((B_iu)(s))'\big)
 +u(s)\Big]^{p-1}\\
&\quad +c\Big\}^{p^*/(p-1)}\,{\rm d}  s\\
&\le b_3^{p^*/(p-1)} \int_0^1
 k_1(t,s)\Big[\sum_{i=1}^{n-1}\left((B_iu)(s)+2(n-i)((B_iu)(s))'\right)+u(s)
 \Big]^{p^*}\,{\rm d}  s\\
&\quad +\frac{c_1^{p^*/(p-1)}}{2}\\
&\le 2^{p^*-1}b_3^{p^*/(p-1)} \int_0^1 k_1(t,s)
 \Big[\sum_{i=1}^{n-1}\left((B_iu)(s)+2(n-i)((B_iu)(s))'\right)\Big]^{p^*}
 \,{\rm d}  s\\
&\quad +2^{p^*-1}b_3^{p^*/(p-1)} \int_0^1 k_1(t,s)u^{p^*}(s)\,{\rm d}
  s+\frac{c_1^{p^*/(p-1)}}{2}\\
&=  2^{p^*-1}b_3^{p^*/(p-1)} \int_0^1 k_1(t,s)
 \Big[\int_0^1 \frac{\sum_{i=1}^{n-1}(k_i(s,\tau)+2(n-i)h_i(s,\tau))}
 {\sum_{i=1}^{n-1} (\mathscr{K}_i+\mathscr{H}_i)}\\
&\quad\times 
 \sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)
 u(\tau)\,{\rm d} \tau\Big]^{p^*}\,{\rm d}  s
 +2^{p^*-1}b_3^{p^*/(p-1)}
 \int_0^1 k_1(t,s)u^{p^*}(s)\,{\rm d}  s\\
&\quad +\frac{c_1^{p^*/(p-1)}}{2}\\
& \le \Big(2\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)
 \Big)^{p^*-1}b_3^{p^*/(p-1)} \int_0^1 k_1(t,s)
 \Big[\sum_{i=1}^{n-1}((B_iu^{p^*})(s)\\
&\quad +2(n-i)((B_iu^{p^*})(s))') \Big]\,{\rm d}  s
 +2^{p^*-1}b_3^{p^*/(p-1)} \int_0^1
k_1(t,s)u^{p_*}(s)\,{\rm d}  s\\
&\quad +\frac{c_1^{p^*/(p-1)}}{2},
\end{aligned}
\end{equation}
for all $u\in \mathscr{M}_4$, $b_3\in (b_2,\alpha_p)$ and $c_1>0$
being chosen so that
\[
(b_2z+c)^{p^*/(p-1)}\le b_3^{p^*/(p-1)}
 z^{p^*/(p-1)}+c_1^{p^*/(p-1)}, \forall z\geq 0.
\]
Multiply  both sides of \eqref{A-4} by  $\psi(t)$ and
integrate over $[0,1]$ and use \eqref{Gineq} and \eqref{hkeq} to
obtain
\begin{equation}
\begin{aligned}  
& \int_0^1 \psi(t)u^{p^*}(t)\,{\rm d}  t\\
&\le \Big(2\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p^*-1}
 b_3^{p^*/(p-1)}
\int_0^1 \psi(t)\Big[\sum_{i=1}^{n-1}\big((B_iu^{p^*})(t)\\
&\quad +2(n-i) ((B_iu^{p^*})(t))'\big) \Big]\,{\rm d}  t
 +2^{p^*-1}b_3^{p^*/(p-1)} \int_0^1
\psi(t)u^{p^*}(t)\,{\rm d}  t+\frac{c_1^{p^*/(p-1)}}{2}\\
& =2^{p^*-1}b_3^{p^*/(p-1)}\Big[(n-1)
\Big(\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\Big)^{p^*-1}+1\Big]
\int_0^1 \psi(t)u^{p^*}(t)\,{\rm d}  t\\
&\quad +\frac{c_1^{p^*/(p-1)}}{2}.
\end{aligned}
\end{equation}
Therefore,
\[
\int_0^1 \psi(t)u^{p^*}(t)\,{\rm d}  t
 \le \frac{c_1^{p^*/(p-1)}}{2-2^{p^*}b_3^{p^*/(p-1)}
\big[(n-1) \big(\sum_{i=1}^{n-1}(\mathscr{K}_i+\mathscr{H}_i)\big)^{p^*-1}+1\big]}
:=\mathscr{N}_4.
\]
This, together with Jensen's inequality and $\psi(t)/e\in [0,1]$
(Note that $p^*\ge 1$), leads to
\begin{equation}
e\int_0^1 u(t)\frac{\psi (t)}{e} \,{\rm d}  t
\le e\Big(\int_0^1 u^{p^*}(t)\big(\frac{\psi(t)}{e}\big)^{p^*}  \,{\rm d}
t\Big)^{1/p^*} \le e^{\frac{p^*-1}{p^*}}
\mathscr{N}_4^{1/p^*}
\end{equation}
for all $u\in\mathscr{M}_4$. From Lemma \ref{lem2.2}, we find
\[
\|u\|_0\le \kappa^{-1} e^{-1/p^*}\mathscr{N}_4^{1/p^*}
:=\mathscr{N}_5, \forall u\in \mathscr{M}_4,
\]
which implies the a priori bound of $\|u\|_0$ for $\mathscr{M}_4$,
as claimed. Furthermore, for any  positive integer $i\ge 1$,  this
estimate leads to
\[
\|(B_iu)\|_0=(B_iu)(1)\le \mathscr{N}_5, \quad \forall u\in\mathscr{M}_4
\]
and for each  positive integer $i\ge 2$, we see
\[
\|(B_iu)'\|_0=(B_iu)'(0)=\int_0^1 (B_{i-1}u)(t)\,{\rm d}  t\le  \mathscr{N}_5,
 \forall u\in\mathscr{M}_4.
\]
Moreover, for $i=1$, we have
\[
\|(B_1u)'\|_0=(B_1u)'(0)=\int_0^1 u(t)\,{\rm d}  t
\le  \mathscr{N}_5,  \forall u\in\mathscr{M}_4.
\]
Combining these and (H6), we have
\begin{gather*}
-((u')^{p-1})'\le f(t,(B_{n-1}u)(t),((B_{n-1}u)(t))',\ldots,u,u'),\\
u(0)=u'(1)=0.
\end{gather*}
Let $(u')^{p-1}(t):=w'(t)$. Then 
$w\in C([0,1], \mathbb{R}_+)$ and $u'(1)=0$ implies $w'(1)=0$. Therefore,
\[
-w''(t)\le n^2 \mathscr N_5, \forall u\in\mathscr{M}_4,
\] 
so that
\[
\|w'\|_0=w'(0)\le n^2 \mathscr N_5. 
\]
Consequently,
\[
\|u'\|_0=\|w'\|_0^{1/p-1}\le (n^2 \mathscr N_5)^{1/p-1},\quad \forall
u\in\mathscr{M}_4.
\]
Let $\mathscr{N}_6:=\max\{\mathscr{N}_5,(n^2
\mathscr N_5)^{1/p-1}\}$. Then
\[
\|u\|\le\mathscr{N}_6, \quad \forall u\in \mathscr{M}_4.
\] 
This proves the boundedness of $\mathscr{M}_4$. As
a result of this, for every $R>\mathscr{N}_6$, we have
\[
u\neq \lambda Au, \quad \forall u\in
\partial B_R\cap P, \lambda\in [0,1].
\] 
Now Lemma \ref{lem2.5} yields
\begin{equation}\label{index10}
i(A,B_R\cap P,P)=1.
\end{equation}
Combining this with \eqref{theorem2index0}
gives 
\[
i(A,(B_R\backslash \overline{B}_r)\cap P,P)=1-0=1.
\] 
Hence
the operator $A$ has at least one fixed point on 
$(B_R\setminus \overline{B}_r)\cap P$ and therefore 
\eqref{Problem} has at least
one positive solution. This completes the proof.
\end{proof}

\begin{theorem} \label{thm3.4} 
If {\rm (H1)--(H3), (H6), (H7)} are satisfied. Then
\eqref{Problem} has at least two positive solutions.
\end{theorem}

\begin{proof} 
By (H2), (H3), and (H6), we know that \eqref{fi-1} and
\eqref{theorem2index0} hold. Note we may choose 
$R >\omega > r$ in \eqref{fi-1} and \eqref{theorem2index0} (see the
proofs of Theorems \ref{thm3.2} and \ref{thm3.3}). By (H7) and Jensen's inequality, we
have that for all $u \in \partial B_\omega \cap P$,
\begin{equation}\begin{aligned}
&[(Au)(t)]^{p^*}\\
& \le  \int_0^1
k_1(t,s)f^{p^*/(p-1)}(s,(B_{n-1}u)(s),((B_{n-1}u)(s))',\ldots,u(s),u'(s))\,{\rm d}
s\\
& \le  \int_0^1
f^{p^*/(p-1)}(s,(B_{n-1}u)(s),((B_{n-1}u)(s))',\ldots,u(s),u'(s))\,{\rm d}
s <\omega
\end{aligned} 
\end{equation}
and
\begin{equation}
\begin{aligned}
&[((Au)(t))']^{p^*}\\
& =\left(\int_t^1
f(s,(B_{n-1}u)(s),((B_{n-1}u)(s))',\ldots,u(s),u'(s))\,{\rm d}
s\right)^{p^*/(p-1)}
\\
&\le \int_0^1
f^{p^*/(p-1)}(s,(B_{n-1}u)(s),((B_{n-1}u)(s))',\ldots,u(s),u'(s))\,{\rm d}
s<\omega.
\end{aligned} 
\end{equation} 
Thus we obtain
\[
\|Au\|<\omega=\|u\|,\quad  \forall u \in \partial B_\omega \cap P,
\]
This implies
\[
u\neq \lambda Au, \quad \forall u\in \partial B_\omega\cap P, \;\lambda\in [0,1].
\]
Now  Lemma \ref{lem2.5} yields 
\begin{equation}
i(A,B_\omega\cap P,P)=1.
\end{equation} 
Combining this with \eqref{fi-1} and
\eqref{theorem2index0} gives 
\[
i(A,(B_R\backslash \overline{B}_\omega)\cap P,P)=0-1=-1,\quad 
i(A,(B_\omega\backslash \overline{B}_r)\cap P,P)=1-0=1.
\] 
Hence the operator $A$ has at least two fixed points, with one 
on $(B_R\setminus \overline{B}_\omega)\cap P$ and the other on 
$(B_\omega\setminus \overline{B}_r)\cap P$. 
Therefore, \eqref{Problem} has at least two
positive solutions. This completes the proof.
\end{proof}

\subsection*{Acknowledgments}
 The authors would like to thank the anonymous
referees for their careful and thorough reading of the original
manuscript.

This research supported by grants: 10971046 from the NNSF-China, 
 ZR2012AQ\-007 from Shandong Provincial Natural Science Foundation, 
 yzc12063 from GIIFSDU, and  2012TS020 from IIFSDU.

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