\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\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[)\}.
$$
Therefore, thanks to the $0<\rho0$ one has
$$\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