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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 121, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/121\hfil A minimax inequality for a class of functionals]
{A minimax inequality for a class of functionals and
applications to the existence of solutions for
two-point boundary-value problems}

\author[G. A. Afrouzi, S. Heidarkhani\hfil EJDE-2006/121\hfilneg]
{Ghasem Alizadeh Afrouzi, Shapour Heidarkhani}  % in alphabetical order


\address{Ghasem Alizadeh Afrouzi \newline
Department of Mathematics\\
Faculty of Basic Sciences \\
Mazandaran University, Babolsar, Iran}
\email{afrouzi@umz.ac.ir}

\address{Shapour Heidarkhani\newline
Department of Mathematics\\
Faculty of Basic Sciences \\
Mazandaran University, Babolsar, Iran}
\email{s.heidarkhani@umz.ac.ir}

\date{}
\thanks{Submitted August 22, 2006. Published October 2, 2006.}
\subjclass[2000]{35J65}
\keywords{Minimax inequality; critical point; three solutions; \hfill\break\indent
 multiplicity results; Dirichlet problem}

\begin{abstract}
In this paper, we establish an equivalent statement to minimax
inequality for a special class of functionals. As an application,
we prove the existence of three solutions to the Dirichlet
problem
\begin{gather*}
-u''(x)+m(x)u(x) =\lambda f(x,u(x)),\quad x\in (a,b),\\
u(a)=u(b)=0,
\end{gather*}
where $\lambda>0$, $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous
function which changes sign on $[a,b]\times \mathbb{R}$ and $m(x) \in C([a,b])$
is a positive function.
\end{abstract}

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

\section{Introduction}

 Given two G\^ateaux differentiable
functionals $\Phi$ and $T$ on a real Banach space $X$, the minimax
inequality
 \begin{equation} \label{e1}
 \sup_{\lambda\geq 0}\inf_{u\in X}(\Phi(u)+\lambda(\rho-T(u)))
<\inf_{u\in X}\sup_{\lambda\geq 0}(\Phi(u)+\lambda(\rho-T(u))), \quad
 \rho \in \mathbb{R},
 \end{equation}
 plays a fundamental role for establishing the existence of at least
three critical points for the functional $\Phi(u)-\lambda T(u)$.

 In this work some conditions that imply the minimax
 inequality \eqref{e1} are pointed out and equivalent formulations are
 proved.

 In this paper, our approach is based on a three
critical-point theorem proved in \cite{r1} (Theorem \ref{thmA})
which stated below for the reader's
convenience. Also we state a technical lemma that enables us to apply
the theorem.

 Lemma \ref{lem2.1} below establishes an equivalent statement of minimax inequality
\eqref{e1}  for a special class of functionals, while its consequences
(Lemmas \ref{lem2.4} and \ref{lem2.6}) guarantee some conditions so that minimax
inequality holds.

 Finally, we apply Theorem \ref{thmA} to elliptic equations, by using an immediate
consequence of Lemma \ref{lem2.1}: We consider the boundary-value problem
 \begin{equation} \label{e2}
 \begin{gathered}
-u''(x)+m(x)u(x) =\lambda f(x,u(x)),\quad x\in (a,b),\\
u(a)=u(b)=0,
\end{gathered}
\end{equation}
where $\lambda>0$, $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous
function which changes sign on $[a,b]\times \mathbb{R}$, $m$ is a
continuous, positive function and we establish some conditions on
$f$ so that problem \eqref{e2} admits at least three weak solutions.

 We say that $u$ is a weak solution to \eqref{e2} if
$u\in W^{1,2}_{0}([a,b])$ and
$$
\int_{a}^{b} u'(x)
v'(x)dx+\int_{a}^{b}m(x)u(x)v(x)dx-\lambda\int_{a}^{b}f(x,u(x))v(x)dx=0
$$
for every $v\in W^{1,2}_{0}([a,b])$.

By arguments similar to those in problem \eqref{e2}, we will have
the existence of at least three weak solutions for the problem
\begin{equation} \label{e3}
\begin{gathered}
-u''(x)+m(x)u(x)=\lambda h_{1}(x)h_{2}(u(x)),\quad x\in (a,b)\\
u(a)=u(b)=0,
\end{gathered}
\end{equation}
where $h_{1}\in C([a,b])$ is a function which changes sign on $[a,b]$
and $h_{2}\in C(\mathbb{R})$ is a positive function.
The existence of at least three weak solutions is also proved for the problem
\begin{equation} \label{e4}
\begin{gathered}
-u''(x)+m(x)u(x)=\lambda f(u(x)),\quad x\in (a,b)\\
u(a)=u(b)=0,
\end{gathered}
\end{equation}
where $f:\mathbb{R}\to \mathbb{R}$ is a continuous function which
changes sign on $\mathbb{R}$.

Conditions that guarantee the existence of multiple solutions to
differential equations are of interest because physical processes
described by differential equations can exhibit more that one
solution. For example, certain chemical reactions in tubular
reactors can be mathematically described by a nonlinear,
two-point boundary value problem with the interest in seeing if
multiple steady-states to the problem exist. For a recent
treatment of chemical reactor theory and multiple solutions see
\cite[section 7]{a2} and references therein.

 In recent years, many authors have studied multiple solutions
from several points of view and with different approaches and we
refer to \cite{a1,a3,b1,k1} and the references therein for
more details, for instance, in their interesting paper \cite{a3}, the
authors studied problem
\begin{equation} \label{e5}
\begin{gathered}
  u''+\lambda f(u)=0,\\
u(0)=u(1)=0,
\end{gathered}
\end{equation}
(in the case independent of $\lambda$)
by using a multiple fixed-point theorem to obtain three
symmetric positive solutions under growth conditions on $f$.

 Also, in \cite{b1}, the author proves multiplicity results for the problem
\eqref{e5} which for each $\lambda\in[0,+\infty[$, admits at least
three solutions in $W^{1,2}_{0}([0,1])$ when  $f$ is a continuous function.

 In particular, in \cite{a1} we obtained the existence of an
 interval $\Lambda\subseteq[0,+\infty[$ and a positive real number $q$ such
that, such that for each
 $\lambda\in\Lambda$ problem
\begin{equation}
\begin{gathered}
\Delta_{p}u+\lambda f(x,u)=a(x)|u|^{p-2}u\quad \textrm{in } \Omega,\\
u=0\quad \textrm{on }\partial\Omega,
\end{gathered}
\end{equation}
where $\Delta_{p}u=\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is the
$p$-Laplacian operator,  $\Omega\subset \mathbb{R}^{N}(N\geq2)$ is
non-empty bounded open set with smooth boundary $\partial\Omega$,
$p>N$, $\lambda>0$, $f:\Omega\times \mathbb{R}\to \mathbb{R}$ is a continuous
function and positive weight function
$a(x) \in C(\overline{\Omega})$, admits at least three weak solutions whose
norms in $W^{1,p}_{0}(\Omega)$ are less than $q$.

 For additional approaches to the existence of multiple solutions
 to boundary-value problems, see \cite{a2,h1,h2} and
 references therein.

\section{Main results}

  First, we recall the three critical point theorem by Ricceri \cite{r1}
when choosing $h(\lambda)=\lambda\rho$.

\begin{theorem} \label{thmA}
Let $X$ be a separable and reflexive real
Banach space; $ \Phi:X \to \mathbb{R}$ a continuously
G\^ateaux differentiable and sequentially weakly lower
semicontinuous functional whose G\^ateaux derivative
admits a continuous inverse on $X^{*}$; $\Psi:X\to \mathbb{R}$
a continuously
 G\^ateaux differentiable functional whose
 G\^ateaux derivative is compact. Assume that
$$
\lim_{  \|u\| \to +\infty}(\Phi(u)+\lambda \Psi(u))=+\infty
$$
for all $\lambda\in[0,+\infty[$, and that there exists
$\rho\in \mathbb{R}$ such that
 $$
\sup_{\lambda\geq 0}\inf_{u\in X}(\Phi(u)+\lambda \Psi(u)
+\lambda\rho)<\inf_{u\in X}\sup_{\lambda\geq 0}(\Phi(u)+\lambda \Psi(u)
+\lambda\rho).
$$
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that, for each
$\lambda\in\Lambda$, the equation
$$
\Phi^{\prime}(u)+\lambda \Psi^{\prime}(u)=0
$$
has at least three solutions in $X$ whose norms
are less than $q$.
\end{theorem}

 Here and in the sequel, $X$ will denote the Sobolev space
$W^{1,2}_{0}([a,b])$ with the  norm
$$\| u\|\
:=\Big(\int_{a}^{b}|u'(x)|^{2}dx \Big)^{1/2},
$$
$f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function and
$g:[a,b]\times \mathbb{R}\to \mathbb{R}$ is defined by
$$
g(x,t)=\int^{t}_{0}f(x,\xi)d\xi
$$
for   each $(x,t)\in[a,b]\times \mathbb{R}$.
Now, we define
$$
\|u\|_{*}:=\Big(\int_{a}^{b}(|u'(x)|^2+m(x)|u(x)|^2)dx\Big)^{1/2}.
$$
So the Poincar\'e's inequality and the positivity of the function
$m(x)\in C([a,b])$, there exist positive suitable constants
$c_{1}$ and $c_{2}$ such that
 \begin{equation} \label{e7}
  c_{1}\|u\|\leq\|u\| _{*}\leq c_{2}\|u\|
\end{equation}
(i.e., the above norms are equivalent).
We now introduce two positive special functionals on the Sobolev space
$X$: For $u\in X$, let
\begin{gather*}
\Phi(u):=\frac{\|u\|_{*}^2}{2},\\
T(u):=\int_{a}^{b}g(x,u(x))dx\,
\end{gather*}
 Let $\rho, r \in \mathbb{R}$, $w\in X$ be such that $0<\rho<T(w)$ and
$0<r<\Phi(w)$. We put
\begin{gather}
\beta_{1}(\rho,w):=\rho\frac{\Phi(w)}{T(w)}, \label{e8}\\
\beta_{2}(r,w):=r\frac{T(w)}{\Phi(w)}, \label{e9}\\
\beta_{3}(\rho,w):=\frac{1}{c_{1}}\big( \frac{b-a}{2}\beta_{1}(\rho,w)
\big)^{1/2}, \label{e10}
\end{gather}
Clearly, $\beta_{1}(\rho,w)$, $\beta_{2}(r,w)$ and  $\beta_{3}(\rho,w)$ are
positive. Now, we put
$$
\delta_{1}:=\inf\{\frac{(b-a)^{1/2}}{2c_{1}}\| u
\|_{*} \in \mathbb{R}^{+};  T(u)\geq
 \rho \},
$$
\begin{align*}
\delta_{2}:=\inf\big\{&
 \frac{(b-a)^{1/2}}{2c_{1}} \| u \|_{*} \in \mathbb{R}^{+}, \text{ such that}\\
&(b-a)\max_{(x,t)\in[a,b]\times[-\frac{(b-a)^{1/2}}{2c_{1}}\|u\|_{*},\
\frac{(b-a)^{1/2}}{2c_{1}}\|u\|_{*}] }g(x,t)\geq \rho \big\}
\end{align*}
and
 \begin{equation} \label{e11}
 \delta_{\rho}:=\delta_{1}-\delta_{2}.
 \end{equation}
Clearly, $\delta_{1} \geq \delta_{2}$.
Taking into account that for every $u\in X$,
$$
\max_{x\in [a,b]}|u(x)|\leq \frac{(b-a)^{1/2}}{2}\|u\|
$$
 and \eqref{e7}, we have
$$
\max_{x\in [a,b]}|u(x)|\leq
\frac{(b-a)^{1/2}}{2c_{1}}\|u\|_{*}
$$
for each $u\in X$. So that
$$
T(u)=\int_{a}^{b}g(x,u(x))dx \leq (b-a)\max g(x,t)
$$
where $(x,t)\in[a,b]\times[-\frac{(b-a)^{1/2}}{2c_{1}}\|u\|_{*}\ ,\
\frac{(b-a)^{1/2}}{2c_{1}}\|u\|_{*}]$.
Namely
$$
T(u)\leq (b-a)\max g(x,t),
$$
where $(x,t)\in[a,b]\times[-\frac{(b-a)^{1/2}}{2c_{1}} \|
u \|_{*}\ ,\  \frac{(b-a)^{1/2}}{2c_{1}} \| u
\|_{*}\ ]$;
therefore,
$\{\frac{(b-a)^{1/2}}{2c_{1}}\|u\|_{*} \in \mathbb{R}^{+}; T(u)\geq \rho
\} $ is a subset of
\begin{align*}
\big\{& \frac{(b-a)^{1/2}}{2c_{1}} \|
u \|_{*} \in \mathbb{R}^{+} \text{ such that}\\
&(b-a)\max_{(x,t)\in[a,b]\times[-\frac{(b-a)^{1/2}}{2c_{1}}
\| u \|_{*}\ ,\ \frac{(b-a)^{1/2}}{2c_{1}}
 \| u \|_{*} \ ]} g(x,t)\geq \rho \big\}.
\end{align*}
So, we have $\delta_1\geq \delta_2$ and
$\delta_{\rho}\geq0$.

 Our main results depend on the following lemma:

\begin{lemma} \label{lem2.1}
Assume that there exist $\rho\in \mathbb{R}$, $w\in X$
such that
\begin{itemize}
\item[(i)] $0<\rho<T(w)$,
\item[(ii)] $(b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)
+\delta_{\rho}, \ \beta_{3}(\rho,w)-\delta_{\rho}] }g(x,t)<\rho$,
where $\beta_{3}(\rho,w)$ is given by \eqref{e10} and
$\delta_{\rho}$ by \eqref{e11}.
\end{itemize}
Then, there exists $\rho\in \mathbb{R}$ such that
$$
\sup_{\lambda\geq 0}\inf_{u\in X}(\Phi(u)+\lambda(\rho-T(u)))<\inf_{u\in
X}\sup_{\lambda\geq 0}(\Phi(u)+\lambda(\rho-T(u))).
$$
\end{lemma}

\begin{proof} From (ii), we obtain
$$
\beta_{3}(\rho,w)-\delta_{\rho}\notin \{ l \in \mathbb{R}^{+}:
(b-a)\max_{(x,t)\in [a,b]\times[-l,l]}g(x,t)\geq \rho \}.
$$
Moreover
$$
\inf\{l\in \mathbb{R}^{+};(b-a)\max_{(x,t)\in
[a,b]\times[-l,l]}g(x,t)\geq \rho \}\geq
\beta_{3}(\rho,w)-\delta_{\rho};
$$
in fact, arguing by contradiction, we assume that there is
$l_{1}\in \mathbb{R}^{+}$ such that
$$
(b-a)\max_{(x,t)\in [a,b]\times[-l_{1},l_{1}]}g(x,t)\geq \rho
$$
and
$$
l_{1}<\beta_{3}(\rho,w)-\delta_{\rho},
$$
so
$$(b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)+\delta_{\rho}\
 ,\ \beta_{3}(\rho,w)-\delta_{\rho}]
}g(x,t)\geq (b-a)\max_{(x,t)\in
[a,b]\times[-l_{1},l_{1}]}g(x,t)\geq \rho\,.
$$
This is a
contradiction. So
$$
\inf\{l\in \mathbb{R}^{+};(b-a)\max_{(x,t)\in
[a,b]\times[-l,l]}g(x,t)\geq \rho \}>\beta_{3}(\rho,w)-\delta_{\rho}.
$$
Therefore,
\begin{align*}
&\inf\{\frac{(b-a)^{1/2}}{2c_{1}} \| u \|_{*} \in
\mathbb{R}^{+}:\\
&(b-a)\max_{(x,t)\in
[a,b]\times[-\frac{(b-a)^{1/2}}{2c_{1}}\| u \|_{*}\ ,
\ \frac{(b-a)^{1/2}}{2c_{1}} \| u \|_{*}]}g(x,t)\geq \rho \}\\
&>\beta_{3}(\rho,w)-\delta_{\rho};
\end{align*}
namely $\beta_{3}(\rho,w)<\delta_1$.
So, we have
$$
\inf \{ \frac{\| u \|_{*}^2}{2} \in
\mathbb{R}^{+};T(u)\geq \rho \}>\beta_{1}(\rho,w),
$$
or equivalently
$$
\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}>
 \rho\frac{\Phi(w)}{T(w)},
$$
and, taking in to account that (i) holds, one has
$$
 \frac{\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}}{\rho}
> \frac{\Phi(w)-\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}}{T(w)-\rho}.
$$
 Now, there exists $\lambda \in \mathbb{R}$ such that
$$
\lambda>\frac{\Phi(w)-\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}}{T(w)-\rho}
$$
and
$$
\lambda<\frac{\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}}{\rho}.
$$
or equivalently
 $$
\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}>\Phi(w)+\lambda(\rho-T(w))
$$
and
$$
\lambda\rho<\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}.
$$
Therefore, thanks to the $0<\rho<T(w)$, we obtain
 \begin{equation} \label{e12}
\inf_{u\in X}(\Phi(u)+\lambda(\rho- T(u)))
<\inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}.
\end{equation}
On other hand,
\begin{equation} \label{e13}
\inf_{u\in X}(\Phi(u)+\lambda(\rho- T(u)))\leq
  (\Phi(0)+\lambda(\rho-T(0)))=\lambda \rho.
\end{equation}
So, with \eqref{e12} and \eqref{e13}, one has
$$
\sup_{\lambda\geq 0}\inf_{u\in X}(\Phi(u)+\lambda(\rho-T(u)))<
 \inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\}.
$$
Therefore, thanks to the
$$
\inf_{u\in X} \sup_{\lambda\geq 0}(\Phi(u)+\lambda(\rho-T(u)))=
 \inf\{ \Phi(u);\ u\in T^{-1}([\rho,+\infty[)\},
$$
we have
$$
\sup_{\lambda\geq 0}\inf_{u\in X}(\Phi(u)+\lambda(\rho-T(u)))
<\inf_{u\in X}\sup_{\lambda\geq 0}(\Phi(u)+\lambda(\rho-T(u))).
$$
\end{proof}

\begin{remark} \label{rmk2.2} \rm
Note that  $\sup_{\lambda\geq0}\inf_{u\in
X}(\Phi(u)+\lambda (\rho-T(u)))$ is well defined, because
$\lambda\to\inf_{u\in X}(\Phi(u)+\lambda (\rho-T(u)))$ is upper
semicontinuous in $[0,+\infty[$ and tends to $-\infty$ as
$\lambda\to +\infty$.
\end{remark}

\begin{remark} \label{rmk2.3} \rm
 If  $\beta_{3}(\rho,w)-\delta_{\rho}\leq 0$ in Lemma \ref{lem2.1},; then
then the lemma still holds.
Because, $\beta_{3}(\rho,w)\leq\delta_{1}-\delta_{2}\leq\delta_{1}$,
and by arguing  as in the proof of Lemma \ref{lem2.1}, the results holds.
\end{remark}

If instead of condition (ii) in Lemma \ref{lem2.1}, we put
$$
(b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)\ ,\ \beta_{3}(\rho,w)]
}g(x,t)<\rho,
$$
then the result holds, because
\begin{align*}
&(b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)+\delta_{\rho}\ ,\
\beta_{3}(\rho,w)-\delta_{\rho}]}g(x,t)\\
&\leq (b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)\ ,\
\beta_{3}(\rho,w)] }g(x,t)<\rho.
\end{align*}
So, we have the following result.

\begin{lemma} \label{lem2.4}
Assume that there exist $\rho\in \mathbb{R}, \ w\in X$
such that
\begin{itemize}
\item[(i)] $0<\rho<T(w)$,
\item[(ii)] $(b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)\ ,\
\beta_{3}(\rho,w)] }g(x,t)<\rho$,
where $\beta_{3}(\rho,w)$ is given by \eqref{e10}
\end{itemize}
 Then, there exists $\rho\in \mathbb{R}$ such that
$$
\sup_{\lambda\geq 0}\inf_{u\in X}(\Phi(u)+\lambda(\rho-T(u)))
<\inf_{u\in X}\sup_{\lambda\geq 0}(\Phi(u)+\lambda(\rho-T(u))).
$$
\end{lemma}


\begin{proposition} \label{prop2.5}
The following assertions are equivalent:
\begin{itemize}
\item[(a)] There are $\rho\in \mathbb{R}$, $w\in X$ such that\\
(i) $0<\rho<T(w)$, \\
(ii) $(b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)\ ,\
\beta_{3}(\rho,w)] }g(x,t)<\rho$,
where $\beta_{3}(\rho,w)$ is
given by \eqref{e10}.
\item[(b)] There are $r\in \mathbb{R}$, $w\in X$ such that\\
(iii) $0<r<\Phi(w)$,\\
(iv)  $(b-a)\max_{(x,t)\in[a,b]\times[-\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}\
,\ \frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}]
}g(x,t)<\beta_{2}(r,w)$,
where $\beta_{2}(r,w)$ is given by
\eqref{e9}.
\end{itemize}
\end{proposition}

\begin{proof} (a) $\Rightarrow$ (b). First we note that
$0<\Phi(w)$, because if $\Phi(w)=0$, one has
$\frac{(b-a)^{1/2}}{2c_{1}}\|w\|_{*}=0$. Hence, taking into
account (ii), one has
$$
T(w)\leq (b-a)\max_{(x,t)\in[a,b]\times[-\frac{(b-a)^{1/2}}{2c_{1}}
\| w \|_{*}\ ,\ \frac{(b-a)^{1/2} }{2c_{1}} \| w
\|_{*}\ ]} g(x,t)  = 0 ,
$$
and that is in contradiction to (i).
 We now put $\beta_{1}(\rho,w)=r$. We
obtain $\rho=\beta_{2}(r,w)$ and
$\beta_{3}(\rho,w)=\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}$.
Therefore, from (i) and (ii), one has
$0<r<\Phi(w)$ and
$$
(b-a)\max_{(x,t)\in[a,b]\times[-\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r} \ ,\
\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}] }g(x,t)<\beta_{2}(r,w).
$$

(b) $\Rightarrow$ (a) First we note that $0<T(w)$, because if
$0\geq T(w)$, from (iii) one has $ r\frac{T(w)}{\Phi(w)}\leq0$;
namely, $\beta_{2}(r,w)\leq0$. Hence, from (iv) one has
$$
0=T(0)\leq
(b-a)\max_{(x,t)\in[a,b]\times[-\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}\
,\ \frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}] }g(x,t)<0,
$$
 and this is a contradiction. We now put $\beta_{2}(r,w)=\rho$.
We obtain $r=\beta_{1}(\rho,w)$ and
$\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}=\beta_{3}(\rho,w)$.
Therefore, from (iii) and (iv), we have the conclusion.
\end{proof}

 The following lemma is another consequence of Lemma \ref{lem2.1}.


\begin{lemma} \label{lem2.6}
Assume that there exist $r \in \mathbb{R},\ w\in X$ such that
\begin{itemize}
\item[(i)] $0<r<\Phi(w)$,
\item[(ii)] $(b-a)\max_{(x,t)\in[a,b]\times[-\frac{1}{c_{1}}
\sqrt{\frac{b-a}{2}r}\ ,\ \frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}]
}g(x,t)<\beta_{2}(r,w)$,
where $\beta_{2}(r,w)$ is given by \eqref{e9}.
\end{itemize}
Then, there exists $\rho\in \mathbb{R}$ such that
$$
\sup_{\lambda\geq
0}\inf_{u\in X}(\Phi(u)+\lambda(\rho-T(u)))<\inf_{u\in
X}\sup_{\lambda\geq 0}(\Phi(u)+\lambda(\rho-T(u))).
$$
\end{lemma}

The above lemma follows from Lemma \ref{lem2.4} and Proposition \ref{prop2.5}.


 Finally, we are interested in ensuring the existence of at least
three weak solutions for the Dirichlet problem \eqref{e2}.
Now, we have the following result.

\begin{theorem} \label{thm2.7}
 Assume that there exist $\rho \in \mathbb{R}$, $a_{1}\in L^1([a,b])$,
$w\in X$ and a positive constant $\gamma$ with $\gamma<2$ such that
\begin{itemize}
\item[(i)] $ 0<\rho<\int_{a}^{b}g(x,w(x))dx$,

\item[(ii)] $(b-a)\max_{(x,t)\in[a,b]\times[-\beta_{3}(\rho,w)\ ,\
\beta_{3}(\rho,w)] }g(x,t)<\rho$

\item[(iii)] $g(x,t)\leq a_{1}(x)(1+|t|^\gamma)$ almost everywhere
in $[a,b]$ and for each $t\in \mathbb{R}$,
where $\beta_{3}(\rho,w)$ is given by \eqref{e10}.

\end{itemize}
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that,  for each
$\lambda\in\Lambda$, problem \eqref{e2}
 admits at least three
solutions in X whose norms are less than $q$.
\end{theorem}

\begin{proof} For each $u\in $X, we put
\begin{gather*}
\Phi(u)=\frac{\|u\|_{*}^2}{2},\\
\Psi(u)=-\int_{a}^{b}g(x,u(x))dx.\\
J(u)=\Phi(u)+\lambda \Psi(u).
\end{gather*}
 In particular, for each $u,v\in X$ one has
\begin{gather*}
\Phi'(u)(v)=\int_{a}^{b}( u'(x) v'(x)+m(x)u(x)v(x))dx,
\Psi'(u)(v)=-\int_{a}^{b}f(x,u(x))v(x)dx.
\end{gather*}
It is well known that the critical points of $J$ are the weak
solutions of \eqref{e2}, our goal is to prove that $\Phi$ and $\Psi$
satisfy the assumptions of Theorem \ref{thmA}. Clearly, $\Phi$ is a
continuously G\^ateaux  differentiable and sequentially
weakly lower semi continuous functional whose G\^ateaux
derivative admits a continuous inverse on $X^{*}$ and $\Psi$ is a
continuously G\^ateaux differentiable functional whose
G\^ateaux derivative is compact.

Thanks to (iii), for each $\lambda>0$ one has
$$\lim_{\| u\|\rightarrow+\infty}
(\Phi(u)+\lambda \Psi(u))=+\infty\,.
$$
Furthermore, thanks to Lemma \ref{lem2.4}, from (i) and (ii), we have
$$
\sup_{\lambda\geq 0}\inf_{u\in X}(\Phi(u)+\lambda \Psi(u)
+\lambda\rho)<\inf_{u\in X}\sup_{\lambda\geq 0}(\Phi(u)+\lambda \Psi(u)
+\lambda\rho).
$$
 Therefore, we can apply Theorem \ref{thmA}. It follows that there exists an open
interval $\Lambda\subseteq[0,+\infty[$ and a positive real number
$q$ such that, for each $\lambda\in\Lambda$, problem \eqref{e2}
 admits at least three solutions in X whose norms are less than $q$.
\end{proof}

  We also have the following existence result.

\begin{theorem} \label{thm2.8}
Assume that there exist $r \in \mathbb{R}$, $a_{2}\in L^1([a,b])$,
 $w\in X$ and a positive constant $\gamma$ with $\gamma<2$ such that
\begin{itemize}
\item[(i)] $0<r<\frac{\|w\|_{*}^{2}}{2}$;
\item[(ii)] $(b-a)\max_{(x,t)\in[a,b]\times[-\frac{1}{c_{1}}
\sqrt{\frac{b-a}{2}r}\ ,\ \frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}]
}g(x,t)<\beta_{2}(r,w)$;
\item[(iii)] $g(x,t)\leq a_{2}(x)(1+|t|^\gamma)$ almost everywhere
in $[a,b]$ and for each $t\in \mathbb{R}$,
where $\beta_{2}(r,w)$ is given by \eqref{e9}.
\end{itemize}
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that, for each
$\lambda\in\Lambda$, problem \eqref{e2}
 admits at least three
solutions in $X$ whose norms are less than $q$.
\end{theorem}

The above theorem follows from Lemma \ref{lem2.6} and Theorem \ref{thm2.7}.

 Let $h_{1}\in C([a,b])$ be a function which changes sign on $[a,b]$
 and $h_{2}\in C(\mathbb{R})$ be a positive function.
For for $(x,t)\in [a,b]\times \mathbb{R}$, put
$$
f(x,t)=h_{1}(x)h_{2}(t).
$$
For for $t\in \mathbb{R}$, put
 $$
\alpha(t)=\int^{t}_{0}h_{2}(\xi)d\xi\,.
$$
For  almost every $x\in [a,b]$, put
$$
a_{3}(x)=\frac{a_{1}(x)}{h_{1}(x)}
$$
Then, using Theorem \ref{thm2.7}, we
 have the following result.

\begin{theorem} \label{thm2.9}
Assume that there exist $\rho \in \mathbb{R}$, $a_{3}\in L^1([a,b])$,
$w\in X$ and a positive constant $\gamma$ with $\gamma<2$ such that
\begin{itemize}
\item[(i)] $0<\rho<\int_{a}^{b}(h_{1}(x)\alpha(w(x)))dx$;
\item[(ii)] $(b-a)\max_{x\in[a,b]}h_{1}(x)<\frac{\rho}{\alpha
(\beta_{3}(\rho,w))}$;
\item[(iii)] $\alpha(t)\leq a_{3}(x)(1+|t|^\gamma)$ almost everywhere in
$[a,b]$ and for each $t\in \mathbb{R}$,
where $\beta_{3}(\rho,w)$ is given by \eqref{e10}.
\end{itemize}
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that,  for each
$\lambda\in\Lambda$, problem \eqref{e3}
 admits at least three
solutions in X whose norms are less than $q$.
\end{theorem}

Put
$$
a_{4}(x)=\frac{a_{2}(x)}{h_{1}(x)}$$for almost every $x\in
[a,b]$.
Then, by Theorem \ref{thm2.8}, we have the following existence result.

\begin{theorem} \label{thm2.10}
Assume that there exist $r \in \mathbb{R}$, $a_{4}\in L^1([a,b])$, $w\in X$
 and a positive constant $\gamma$ with $\gamma<2$ such that
\begin{itemize}
\item[(i)] $0<r<\frac{\|w\|_{*}^{2}}{2}$;
\item[(ii)] $(b-a)\max_{x\in[a,b]
}h_{1}(x)<\frac{\beta_{2}(r,w)}{\alpha(\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r})}$;
\item[(iii)] $\alpha(t)\leq a_{4}(x)(1+|t|^\gamma)$ almost everywhere in
$[a,b]$ and for each
$t\in \mathbb{R}$,
where $\beta_{2}(r,w)$ is given by \eqref{e9}.
\end{itemize}
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that, for each
$\lambda\in\Lambda$, problem \eqref{e3} admits at least three
solutions in $X$ whose norms are less than $q$.
\end{theorem}

 We now want to point out two simple consequences of Theorems
  \ref{thm2.7} and \ref{thm2.8}. Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous
function which changes sign on $\mathbb{R}$. For  $t\in \mathbb{R}$, put
$g(t)=\int^{t}_{0}f(\xi)d\xi$.
 So we have the following results.

\begin{theorem} \label{thm2.11} Assume that
there exist $\rho \in \mathbb{R}$, $w\in X$ and two positive constants
 $\gamma$ and $\eta$ with $\gamma<2$ such that
\begin{itemize}
\item[(i)] $0<\rho<\int_{a}^{b}g(w(x))dx$;
\item[(ii)] $(b-a)\max_{t\in[-\beta_{3}(\rho,w)\ ,\ \beta_{3}(\rho,w)]
}g(t)<\rho$;
\item[(iii)] $g(t)\leq \eta(1+|t|^\gamma)$ for each $t\in \mathbb{R}$,
where $\beta_{3}(\rho,w)$ is given by \eqref{e10}.
\end{itemize}
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that,  for each
$\lambda\in\Lambda$, problem \eqref{e4} admits at least three
solutions in $X$ whose norms are less than $q$.
\end{theorem}

\begin{theorem} \label{thm2.12}
 Assume that there exist $r \in \mathbb{R}$, $w\in X$ and two positive
constants $\gamma$ and $\mu$ with $\gamma<2$ such that
\begin{itemize}
\item[(i)] $0<r<\frac{\|w\|_{*}^{2}}{2}$;
\item[((ii)] $(b-a)\max_{t\in[-\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r} ,\
\frac{1}{c_{1}}\sqrt{\frac{b-a}{2}r}]
}g(t)<\beta_{2}(r,w)$;
\item[(iii)] $g(t)\leq \mu(1+|t|^\gamma)$ for
each $t\in \mathbb{R}$,
where $\beta_{2}(r,w)$ is given by \eqref{e9}.
\end{itemize}
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that, for each
$\lambda\in\Lambda$, problem \eqref{e4}
 admits at least three
solutions in $X$ whose norms are less than $q$.
\end{theorem}

\begin{example} \label{exa2.13} \rm
 Let $\Omega=(0,1)$ and consider the problem
\begin{equation} \label{e14}
\begin{gathered}
-u''+{e^x}u=\lambda (e^{u}u^2(3+u)),\quad x\in(0,1)\\
u(0)=u(1)=0.
\end{gathered}
\end{equation}
Then, there exists an open interval $\Lambda\subseteq[0,+\infty[$
and a positive real number $q$ such that, for each
$\lambda\in\Lambda$, problem \eqref{e14}
 admits at least three
solutions in $W^{1,2}_{0}([0,1])$ whose norms are less than $q$.
In fact, by choosing $\rho=\frac{1}{4}$ and
$$
w(x)=\begin{cases}
x, & x\in(0,1)\\
0, &\text{otherwise}
\end{cases}
$$
so that $\beta_{3}(\rho,w)=\frac{1}{c_{1}} (\frac{e-1}{96-32e})^{1/2}$,
all assumptions of Theorem \ref{thm2.11}, are satisfied with $\gamma=1$,
$c_{1}$ is positive constant such that the inequality \eqref{e7} hold
for $m(x)=e^x$ and $\eta$ sufficiently large, also with choose
$r=\frac{1}{2}$ so that $\beta_{2}(r,w)=\frac{6-2e}{e-1}$, all
assumptions of Theorem \ref{thm2.12}, are satisfied with $\mu$
sufficiently large.
\end{example}

\begin{thebibliography}{99}

\bibitem{a1} G. a. Afrouzi, S. Heidarkhani;
{\em Three solutions for a Dirichlet boundary value problem involving
the p-Laplacian,} Nonlinear Anal. (to appear)

\bibitem{a2} R. P. Agarwal, H. B. Thompson, C. C. Tisdell;
  {\em On the existence of multiple solutions to boundary value problems
for second order, ordinary differentntial equations}.
Dynam. Systems Appl. (in press)

\bibitem{a3} R. I. Avery, J. Henderson; {\em Three symmetric positive
solutions for a second-order boundary value problem,} Appl. Math.
Lett. 13 (2000) 1-7.

\bibitem{b1}G. Bonanno, {\em Existence of three solutions for a two point boundary value
problem,} Appl. Math. Lett. 13 (2000) 53-57.

\bibitem{h1} Johnny Henderson, H. b. Thompson;
{\em Existence of multiple solutions for second order boundary
value problems.} J. Differential Equations 166 (2000), no. 2, 443-454.

\bibitem{h2}  Johnny Henderson, H. B. Thompson; {\em Multiple symmetric
positive solutions for a second order boundary value problem.}
 Proc. Amer. Math. Soc. 128 (2000), no. 8, 2373-2379.

\bibitem{k1} P. Korman, T. Ouyang; {\em Exact multiplicity results
for two classes of boundary value problem,} Diff. Integral
Equations 6 (1993) 1507-1517.

\bibitem{r1} B. Ricceri; {\em On a three critical points theorem,}
Arch. Math. (Basel) 75 (2000) 220-226.

\end{thebibliography}

\end{document}