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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 45, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/45\hfil
Existence of multiple positive solutions]
{Existence of multiple positive solutions for discrete problems
 with p-Laplacian via variational methods}

\author[Y. Tian, W. Ge\hfil EJDE-2011/45\hfilneg]
{Yu Tian, Weigao Ge}

\address{Yu Tian \newline
 School of Science, Beijing University of Posts and Telecommunications, 
 Beijing 100876, China}
\email{tianyu2992@163.com}

\address{Weigao Ge \newline
Department of Applied Mathematics, 
Beijing Institute of Technology, Beijing 100081, China}
\email{gew@bit.edu.cn}

\thanks{Submitted February 22, 2011. Published April 4, 2011.}
\thanks{Supported by grants 11001028 from the National Science
Foundation for  Young Scholars, \hfill\break\indent
and BUPT2009RC0704 from the Chinese Universities Scientific Fund}
\subjclass[2000]{39A10, 34B18, 58E30}
\keywords{Discrete boundary value problem;  variational methods;
\hfill\break\indent  mountain pass theorem}

\begin{abstract}
 Using critical point theory, we prove the existence of multiple
 positive solutions for second-order discrete boundary-value
 problems with  p-Laplacian.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

  In recent years, a great deal of work has been
done in the study of the existence of multiple positive
solutions for discrete boundary value problems
describing  physical and biological phenomena.
For the background and summary of results, we refer the reader to
the monograph by Agarwal et al \cite{a2}, and for some recent
contributions to \cite{a1,a3}.
Various fixed point theorems have been applied for obtaining
solutions, among them, Krasnosel'skii fixed  point theorem,
Leggett-Williams fixed point theorem, fixed point theorem
in cones; see \cite{a4,a5,h1,m1,t1} and the
 references therein.

There is also a trend to study difference equation  using
variational methods which lead to many interesting results;
see for example \cite{a3,g1,l1,y1}.
Li \cite{l1} studied the existence
of solutions for the problem
\begin{equation}
\begin{gathered}
\Delta(p(k)\Delta x(k-1))+f(k, x(k))=g(k)\\
x(0)=x(T+1)=0,
\end{gathered}
\end{equation}
where $f\in C(\mathbb{R}^2, \mathbb{R})$, $p, g\in
C(\mathbb{R},\mathbb{R})$. Using  variational methods, the existence
of at least one non-trivial solution was obtained. Agarwal et al
\cite{a3} show  the existence of multiple positive solutions for the
discrete boundary-value problem
\begin{equation}
\begin{gathered}
\Delta^2 y(k-1)+f(k, y(k))=0, \quad k\in [1, T],\\
y(0)=0=y(T+1),
\end{gathered} \label{discrete}
\end{equation}
where $[1, T]$ is the discrete interval $\{1, 2, \dots, T\}$,
$\Delta y(k)=y(k+1)-y(k)$,
$f\in C([1, T]\times [0, \infty), \mathbb{R})$
satisfies $f(k, 0)\ge0$, for all $k\in [1, T]$. They applied
critical point theory under the following conditions:
\begin{itemize}
\item[(a)] $\min_{k\in[1, T]}\liminf_{u\to\infty}
\frac{f(k,u)}{u}>\lambda_1$, where
$ \lambda_1$ is the smallest eigenvalue of
$\Delta^2y(k-1)+\lambda y(k)=0$, $y\in H$;

\item[(b)] there is a positive constant $M$, independent of $\lambda$,
such that $\|y\|\neq M$ for every solution $y\ge 0$ of the
equation
\[
\Delta^2 y(k-1)+\lambda f(k, y(k))=0, \quad y\in H,\; \lambda\in( 0, 1].
\]
\end{itemize}
We remark that is not easy to verify Condition (b) in applications.

To the best of our knowledge, very few authors have
studied the existence of multiple positive solutions for discrete
boundary value problem with a p-Laplacian by using variational
methods. As a result the goal of this paper is to fill the gap in
this area.
It is well known that positive solutions are very
important in applications. Motivated by the above results,  in
this paper, we study the existence of multiple positive solutions for
the  second-order discrete boundary-value problem (BVP)
\begin{equation}\label{000}
\begin{gathered}
\Delta(\Phi_p(\Delta y(k-1)))+f(k, y(k))=0, \quad k\in[1, T],\\
y(0)=0=y(T+1),
\end{gathered}
\end{equation}
where $T$ is a positive integer, $[1, T]$ is the discrete
interval $\{1, \dots, T\}$ and $\Delta y(k)=y(k+1)-y(k)$
is the forward difference operator, $p>1$,
$ \Phi_p(y):=|y|^{p-2}y$,
$f\in C([1, T]\times [0, +\infty), [0, +\infty))$,
$f(k, 0)\not\equiv 0$ for $k\in [1, T]$,
$F(k, x)=\int_{0}^{x}f(k, s)ds$.
For a review of variational methods, we refer the reader
to \cite{m2,r1}.

 Our aim of this paper is to apply critical point theory
to  \eqref{000} and prove the existence of two positive solutions.
We impose some conditions on the nonlinearity $f$ that
are different from those in \cite{a2} for $p=2$, and
are easy to verify.


 In this article, we assume the following conditions:
\begin{itemize}

\item[(C1)] there exist $\mu>p$,
$h\in C([1, T]\times[0, +\infty), [0,+\infty))$,
$l:[1, T]\to (0, +\infty)$, $\min_{k\in[1, T]}l(k)>0$ such that
\[
f(k, y)=l(k)\Phi_{\mu}(y)+h(k, y);\]

\item[(C2)] there exist functions $c, d: [1, T]\to [0, +\infty)$ such that
\[
h(k, y)\le c(k)+d(k)\Phi_p(y).
\]
\end{itemize}

\section{Related Lemmas}

 Here, and in the sequel, we denote
 \[
Y=W_{0}^{1, p}[0,T+1]=\{y:[0, T+1]\to R:y(0)=y(T+1)=0\}
\]
whihc is a $T$-dimensional Banach space with the norm
\[
\|y\|=\Big(\sum_{k=1}^{T+1}|\Delta y(k-1)|^p\Big)^{1/p}\,.
\]


\begin{lemma}\label{lem0}
 Let $y^{\pm}=\max\{\pm y, 0\}$, then the following five
properties hold:
\begin{itemize}
\item[(i)] $y=y^+ -y^-$;
\item[(ii)] $\|y^+\|\le \|y\|$;
\item[(iii)] $y^+(t)y^-(t)=0, (y^+)'(t)(y^-)'(t)=0$ for
$t\in[0, T+1]$;
\item[(iv)] $\Phi_p(y)y^+=|y^{+}|^p$, $\Phi_p(y)y^-=-|y^{-}|^p$.
\end{itemize}
\end{lemma}


 \begin{lemma}\label{lem1}
 If $y$ is a solution of the equation
\begin{equation}
\Delta(\Phi_p(\Delta y(k-1)))+f(k, y^+(k))=0, \quad y\in Y, \label{eq1+}
\end{equation}
then $y\ge 0$, $y(k)\not\equiv 0$, $k\in[0, T+1]$ and hence it
 is a solution of  \eqref{000}.
 \end{lemma}

 \begin{proof}
If $y$ is a solution of  \eqref{eq1+}, then
 \begin{equation}\label{12}
\begin{aligned}
 0&=\sum_{k=1}^{T}\left[\Delta(\Phi_p(\Delta y(k-1)))
 +f(k, y^+(k))\right]y^-(k)\\
 &=\Phi_p(\Delta y(k-1))y^-(k)|_{k=1}^{T+1}
 -\sum_{i=1}^{T}\Phi_p(\Delta y(k))\Delta y^-(k)
 +\sum_{k=1}^{T}f(k, y^+(k))y^-(k)\\
&\geq -\Phi_p(y(1))y^-(1)+\sum_{k=1}^{T}|\Delta y^-(k)|^p\\
&=|y^-(1)|^p+\sum_{k=2}^{T+1}|\Delta y^-(k-1)|^p,
 \end{aligned}
\end{equation}
 so $\Delta y^-(k)=0$, $k\in[1, T]$ and $y^-(1)=0$,
which yield that $y^-(k)=0, k\in[1, T+1]$; that is, $y\ge0$.
If  $y(k)= 0$ for every  $k\in[0, T+1]$, the fact $f(k, 0)\not\equiv0$
for every $k\in [1, T]$ gives a contradiction.
\end{proof}


\begin{remark}\label{rem1}\rm
By Lemma \ref{lem1}, to find  positive solutions of
 \eqref{000} it suffices to obtain solutions of \eqref{eq1+}.
\end{remark}

 For $y\in Y$, put
\begin{equation}\varphi(y)
:=\sum_{k=1}^{T+1}\big[\frac{1}{p}|\Delta y(k-1)|^p-F(k, y^+(k))
+f(k, 0)y^-(k)\big].\label{phi}
\end{equation}
Clearly, the functional $\varphi$ is $C^1$ with
\begin{equation}
\langle\varphi'(y), z\rangle
=\sum_{k=1}^{T+1}\left[\Phi_p(\Delta y(k-1))\Delta
z(k-1)-f(k,y^+(k))z(k)\right]\label{phi'}
\end{equation}
for every $z\in Y$. So the solutions of \eqref{eq1+} are
precisely the critical points of the functional $\varphi$.

\begin{lemma}\label{inequality}
For $y\in Y$, we have  $\|y\|_{\infty}\le (T+1)^{1/q}\|y\|$,
where
\[
\|y\|_{\infty}=\max_{i\in[0, T+1]}|y(i)|.
\]
\end{lemma}
\begin{proof}
For $y\in Y$, it follows from H\"older's inequality, that
\begin{align*}
|y(k)|&=\big|y(0)+\sum_{i=0}^{k-1}\Delta y(i)\big|
\le\sum_{i=0}^{T}|\Delta y(i)|\\
&\le(T+1)^{1/q}\Big(\sum_{i=0}^{T}|\Delta
y(i)|^p\Big)^{1/p}=(T+1)^{1/q}\|y\|,
\end{align*}
 which completes the proof.
\end{proof}

\begin{lemma}[{\cite[Theorem 38.A]{z1}}] \label{th38.a}
For the functional $F: M\subseteq X\to[-\infty, +\infty]$
with $M\neq \emptyset$, $\min_{u\in M}F(u)=\alpha$ has a
solution when the following conditions hold:
\begin{itemize}
\item[(i)] $X$ is a real reflexive Banach space;
\item[(ii)] $M$ is bounded and weak sequentially closed; i.e., by
definition, for each sequence $(u_n)$ in $M$ such that
$u_n\rightharpoonup u$ as $n\to\infty$, we always have
$u\in M$;
\item[(iii)] $F$ is weak sequentially lower semi-continuous on $M$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{g1}]\label{guo}
Let $E$ be a Banach space and $\varphi\in C^1(E, R)$ satisfy
Palais-Smale condition. Assume there exist $x_0, x_1\in E$, and a
bounded open neighborhood $\Omega$ of $x_0$ such that
$x_1\not\in \overline{\Omega}$ and
\[
\max\{\varphi(x_0), \varphi(x_1)\}
<\inf_{x\in \partial\Omega}\varphi(x).
\]
Let
$\Gamma=\{h:\, h:[0, 1]\to E \text{ is continuous}, h(0)=x_0,
h(1)=x_1\}$
and
\[c=\inf_{h\in \Gamma}\max_{s\in[0, 1]}\varphi(h(s)).
\]
Then $c$ is a critical value of $\varphi$; that is,
there exists  $x^*\in E$ such that $\varphi'(x^*)=\Theta$
and $\varphi(x^*)=c$, where
$c>\max\{\varphi(x_0), \varphi(x_1)\}$.
\end{lemma}

\begin{lemma}\label{ps}
Suppose that {\rm (C1), (C2)} hold. Furthermore, we
assume
\begin{itemize}
\item[(C3)] $(T+1)^{p/q}\sum_{k=1}^{T+1}d(k)<\frac{\mu}{p}-1$.
\end{itemize}
Then the functional $\varphi$ satisfies Palais-Smale condition;
i.e., every sequence $\{y_n\}$ in $Y$ satisfying $\varphi(y_n)$ is
bounded and $\varphi'(y_n)\to 0$ has a convergent
subsequence.
\end{lemma}

\begin{proof}
Since $Y$ is a finite dimensional Banach space, we only need to
show that  $(y_n)$ is a bounded sequence in $Y$.

For this, by Lemma \ref{lem0} (iv) and \eqref{phi'} we have
\begin{equation}\label{23}
\begin{aligned}
\langle \varphi'(y_n),y_{n}^{-}\rangle
&= \sum_{k=1}^{T+1}\left[\Phi_p(\Delta
y_n(k-1))\Delta y_{n}^{-}(k-1)-f(k,
y_{n}^{+}(k))y_{n}^{-}(k)\right]\\
&\leq -\sum_{k=1}^{T+1}|\Delta y_{n}^{-}(k-1)|^p=-\|y_{n}^{-}\|^p.
\end{aligned}
\end{equation}
Set $w_{n}^{-}=\frac{y_{n}^{-}}{\|y_{n}^{-}\|}$. Dividing by
$\|y_{n}^{-}\|$ on the both sides of the above inequality, we have
\[
\|y_{n}^{-}\|^{p-1}\le-\langle\varphi'(y_n), w_{n}^{-}\rangle\to 0\quad
\text{as }n\to\infty.
\]
So $y_{n}^{-}\to 0$ in $Y$.

  Now we show that $(y_{n}^{+})$ is bounded. By \eqref{phi}
\eqref{phi'} we have
\begin{equation}\label{34}
\begin{aligned}
\frac{\mu}{p}\|y_n\|^p-\|y_{n}^{+}\|^p
&=\mu\varphi(y_n)-\langle\varphi'(y_n),
y_{n}^{+}\rangle-\sum_{k=1}^{T+1}\mu f(k,0)y_{n}^{+}(k)\\
&\quad +\sum_{k=1}^{T+1}\left[\mu F(k, y_{n}^{+}(k))- f(k,
y_{n}^{+}(k))y_{n}^{+}(k)\right].\end{aligned}\end{equation} By
(C1) (C2) Lemma \ref{inequality} one has
\begin{equation}\label{67}
\begin{aligned}
&\sum_{k=1}^{T+1}\left[\mu F(k, y_{n}^{+}(k))- f(k,
y_{n}^{+}(k))y_{n}^{-}(k)\right]\\
&\le \sum_{k=1}^{T+1}\left[c(k)y_{n}^{+}(k)+d(k)|y_{n}^{+}(k)|^p\right]\\
&\le (T+1)^{1/q}\|y_{n}^{+}\|\sum_{k=1}^{T+1}c(k)+
(T+1)^{p/q}\|y_{n}^{+}\|^p\sum_{k=1}^{T+1}d(k).
\end{aligned}
\end{equation}
Substituting \eqref{67} into \eqref{34}, in view of Lemma
\ref{lem0} (ii), one has
\begin{equation}\label{67890}
\begin{aligned}
\big(\frac{\mu}{p}-1\big)\|y_{n}^{+}\|^p
&\le\mu\varphi(y_n)-\langle\varphi'(y_n), y_{n}^{+}\rangle
+(T+1)^{1/q}\|y_{n}^{+}\|\sum_{k=1}^{T+1}c(k)\\
&\quad + (T+1)^{p/q}\|y_{n}^{+}\|^p\sum_{k=1}^{T+1}d(k).
\end{aligned}
\end{equation}
Suppose that $(y_{n}^{+})$ is unbounded. Passing to a subsequence,
we may assume if necessary, that $\|y_{n}^{+}\|\to\infty$
as $n\to\infty$. Dividing the both sides of \eqref{67890}
by $\|y_{n}^{+}\|^p$, denoting
$w_{n}^{+}=\frac{y_{n}^{+}}{\|y_{n}^{+}\|}$, we have
\begin{equation} \label{e.29}
\begin{aligned}
\frac{\mu}{p}-1
&\le\frac{\mu\varphi(y_n)}{\|y_{n}^{+}\|^p}-\frac{\langle\varphi'(y_n),
w_{n}^{+}\rangle}{\|y_{n}^{+}\|^{p-1}}
+(T+1)^{1/q}\|y_{n}^{+}\|^{1-p}\sum_{k=1}^{T+1}c(k)\\
&\quad + (T+1)^{p/q}\sum_{k=1}^{T+1}d(k).
\end{aligned}
\end{equation}
Since $\varphi(y_n)$ is bounded and $\varphi'(y_n)\to 0,
y_{n}^{-}\to0$ in $Y$, let $n\to\infty$, we have
\[
\frac{\mu}{p}-1\le(T+1)^{p/q}\sum_{k=1}^{T+1}d(k),
\]
which contradicts to (C3).  Therefore, $(y_n)$ is bounded in $Y$.
\end{proof}

\section{Main Results}

\begin{theorem}\label{thm1}
Suppose that {\rm (C1)--(C3)} hold. Furthermore, we assume
\begin{itemize}
\item[(C4)] $(T+1)^{\frac{\mu}{q}}\sum_{k=1}^{T}b(k)
+(T+1)^{1/q}\sum_{k=1}^{T}c(k)
+(T+1)^{p/q}\sum_{k=1}^{T}d(k)<1$.
\end{itemize}
Then  \eqref{000} has two positive solutions $x_0, x^*$.
\end{theorem}

\begin{proof}
By Lemma \ref{ps}  the functional $\varphi$ satisfies
 Palais-Smale condition.
Now we shall show that there exists $R>0$ such that the functional
$\varphi$ has a local minimum $x_0\in B_R:=\{x\in X: \|x\|<R\}$.

Let $R=1$. First we claim that the functional $\varphi$ has a
minimum on $\overline{B}_R$. Clearly $\overline{B}_R$ is a bounded
and weak sequentially closed. Now we claim that $\varphi$ has a
minimum $x_0\in \overline{B}_{R}$. We will show that $\varphi$ is
weak sequentially lower semi-continuous on $\overline{B}_R$. For
this, let
\[
\varphi^1(y)=\frac{1}{p}\sum_{k=1}^{T+1}|\Delta
y(k-1)|^p,\quad\varphi^2(y)=\sum_{k=1}^{T+1}\left[-F(k,
y^+(k)) +f(k, 0)y^-(k)\right],
 \]
 then $\varphi(y)=\varphi^1(y)+\varphi^2(y)$. By
$y_n\rightharpoonup y$ on $Y$ we have $(y_n)$ uniformly converges
to $y$ in $C([0, T+1])$. So $\varphi^2$ is weak sequentially
continuous. Clearly $\varphi^1$ is continuous, which together with
the convexity of $\varphi^1$ we have $\varphi^1$ is weak
sequentially lower semi-continuous. Therefore, $\varphi$ is weak
sequentially lower semi-continuous on $B_R$. Besides, $Y$ is a
reflexive Banach space, $\overline{B}_R$  is a bounded and weak
sequentially closed, so our claim follows from Lemma \ref{th38.a}.


If $y_0\in \partial B_R$ and $y_0$ is a local minimum of the
functional $\varphi$, then it is also a minimizer of
$\varphi|_{\partial B_R}$, so the
gradient of $\varphi$ at $y_0$ point is in the direction of the
inward normal to $\partial B_R$. Since $y_0\in \partial
B_R=\partial B_1$ is a local minimum of the  functional $\varphi$,
$\varphi(y)$ have a conditional minimum at the point $y_0$ about
the condition $\varphi(y)=\frac{1}{p}(\|y\|^p-1)$. By \cite{g1},
there exists $\gamma\in [0, \infty)$ such that
\[
\langle\varphi'(y_0), v\rangle=-\gamma \langle\psi'(y_0), v\rangle\quad
\text{for all }v\in Y.
\]
 That is,
\begin{equation}\label{main4}
\Delta(\Phi_p(\Delta y_0(k-1)))+\lambda f(k, y_{0}^{+}(k))=0,
\quad y_0\in Y
\end{equation}
with $\lambda=\frac{1}{1+\gamma}\in(0,1]$, $\|y_0\|=R=1$ holds.

Multiplying $y_0(t)$ on the both sides of equation in
\eqref{main4}, then summing on $[1, T]$, we have
\begin{align*}
0&=  \sum_{k=1}^{T} \left[\Delta
(\Phi_p(\Delta y_0(k-1)))+\lambda f(k,
y_{0}^{+}(k))\right]\times y_0(k)\\
&= \Phi_p(\Delta
y_0(k-1))y_0(k)|_{k=1}^{T+1}-\sum_{k=1}^{T}\Phi_p(\Delta
y_0(k))\Delta y_0(k)+\sum_{k=1}^{T}\lambda f(k,
y_{0}^{+}(k))y_0(k)\\
&= -\Phi_p(y_0(1))y_0(1)-\sum_{k=1}^{T}|\Delta
y_0(k)|^p+\sum_{k=1}^{T}\lambda f(k, y_{0}^{+}(k))y_0(k)\\
&\leq -\|y_0\|^p+\sum_{k=1}^{T}\lambda f(k,
y_{0}^{+}(k))y_0(k).
\end{align*}
Then
\begin{align*}
\|y_0\|^p
&\leq \sum_{k=1}^{T}\lambda f(k, y_0(k))y_0(k)\\
&\leq \sum_{k=1}^{T}b(k)|y_0(k)|^{\mu}+c(k)y_0(k)+d(k)|y_0(k)|^p\\
&\leq (T+1)^{\frac{\mu}{q}}\|y_0\|^{\mu}\sum_{k=1}^{T}b(k)
+(T+1)^{1/q}\|y_0\|^{1/p}\sum_{k=1}^{T}c(k)\\
&\quad +(T+1)^{p/q}\|y_0\|^{p}\sum_{k=1}^{T}d(k).
\end{align*}
Since $\|y_0\|=1$, we have
\[
1\le (T+1)^{\mu/q}\sum_{k=1}^{T}b(k)+(T+1)^{1/q}
\sum_{k=1}^{T}c(k) +(T+1)^{p/q}\sum_{k=1}^{T}d(k),
\]
which contradicts (C4). Therefore, for any $\lambda\in(0, 1]$,
the solution of \eqref{main4} is not on $\partial B_R$.
Therefore, $y_0\in B_R$ and hence it is a local minimizer
of $\varphi$, and $\varphi(y_0)<\min_{y\in \partial B_R}\varphi(y)$.

Next we  show that there exists $y_1$ with $\|y_1\|>R=1$
such that $\varphi(y_1)<\min_{y\in \partial B_R}\varphi(y)$.
Let $\widetilde{e}(k)=1\in Y$. Then
\begin{equation}\label{2323}
\begin{aligned}
\varphi(\overline{\lambda}\widetilde{
e})&\leq -\sum_{k=1}^{T}[F(k, \overline{\lambda})-f(k,
0)\overline{\lambda}] \\
&=-\sum_{k=1}^{T}\big[\frac{l(k)\overline{\lambda}^{\mu}}{\mu}+H(k,
\overline{\lambda})-f(k, 0)\overline{\lambda}\big]\\
&\leq -\sum_{k=1}^{T}\frac{l(k)\overline{\lambda}^{\mu}}{\mu}+\sum_{k=1}^{T}\left[
c(k)\overline{\lambda} +d(k)\overline{\lambda}^p+f(k,
0)\overline{\lambda}\right].
\end{aligned}
\end{equation}
Since $\mu>p$, we have
$\lim_{\overline{\lambda}\to+\infty}\varphi(\overline{\lambda}\widetilde{e})=-\infty$.
So there exists sufficiently large $\overline{\lambda}_0$ with
$\|\overline{\lambda}_0\widetilde{e}\|>R$ such that
$\varphi(\overline{\lambda}_0\widetilde{e})<\min_{y\in
\partial B_R}\varphi(y)$.

 Lemma \ref{guo} now gives the critical  value
\[
c=\inf_{h\in \Gamma}\max_{t\in[0, 1]}\varphi(h(t)),
\]
 where
$\Gamma=\{h:\,h:[0, 1]\to E \text{ is continuous, }
h(0)=y_0, h(1)=y_1\}$;
that is, there exists $y^*\in Y$
such that $\varphi'(y^*)=0$. Therefore, $y_0, y^*$ are two
critical points of $\varphi$, and hence they are classical
solutions of \eqref{eq1+}. Lemma \ref{lem1} means $y_0, y^*$ are
positive solutions of problem \eqref{000}.
\end{proof}

\begin{corollary}
Suppose that {\rm (C1) (C4)} hold. Moreover we assume
\begin{itemize}
\item[(C2')] there exists $0\le s<p$, $c\in L^1([a, b], [0, +\infty))$,
$d\in C([a, b], [0, +\infty))$ such that
\[
h(t, x)\le c(t)+d(t)\Phi_s(x).
\]
\end{itemize}
Then \eqref{000} has two positive solutions $x_0, x^*$.
\end{corollary}

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