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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 115, pp. 1--13.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/115\hfil Periodic solutions]
{Periodic solutions for a class of second-order Hamiltonian systems}
\author[G. Bonanno, R. Livrea\hfil EJDE-2005/115\hfilneg]
{Gabriele Bonanno, Roberto Livrea}  % in alphabetical order

\address{Gabriele Bonanno \hfill\break
Dipartimento di Informatica, Matematica, Elettronica e Trasporti\\
Facolt\`a di Ingegneria, Universit\`a di Reggio Calabria\\
Via Graziella (Feo di Vito), 89100 Reggio Calabria, Italy}
\email{bonanno@ing.unirc.it}

\address{Roberto Livrea \hfill\break
Dipartimento di Patrimonio Architettonico e Urbanistico\\
Facolt\`{a} di Architettura, Universit\`{a} di Reggio Calabria\\
Salita Melissari, 89100 Reggio Calabria, Italy}
\email{roberto.livrea@unirc.it}

\date{}
\thanks{Submitted August 4, 2005. Published October 21, 2005.}
\thanks{Reasearch supported by RdB (ex 60\% MIUR) of
Reggio Calabria University}
\subjclass[2000]{34B15, 34C25}
\keywords{Second order Hamiltonian systems; eigenvalue problem;
\hfill\break\indent
 periodic solutions; critical points; multiple solutions}

\begin{abstract}
 Multiplicity results for an eigenvalue second-order Hamiltonian
 system are investigated. Using suitable critical points arguments,
 the existence of an exactly determined open interval of positive
 eigenvalues for which the system admits at least three distinct
 periodic solutions is established. Moreover, when the energy
 functional related to the Hamiltonian system is not coercive,
 an existence result of two distinct periodic solutions is given.
\end{abstract}

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


\section{Introduction}

Recently, several authors studied problems of the type
\begin{equation} \label{eP}
\begin{gathered}
   \ddot u=\nabla_u F(t,u)\quad \text{a.e.  in  } [0,T]\\
   u(T)-u(0)=\dot u(T)-\dot u(0)=0
    \end{gathered}
\end{equation}
 establishing, under suitable assumptions, existence or
multiplicity of periodic solutions. We refer the reader
 to the book of Mawhin and Willem \cite{M-W} for basic results.
and to \cite{B-L,B-N,C,F,T,T-W}
for more recent results. In particular, in \cite{B-N}  Brezis
and  Nirenberg assumed that:
\begin{itemize}
\item [(a)] $F(t,0)=0$, $\nabla_u F(t,0)=0$.
\item [(b)] $ \lim_{|u| \to +\infty}F(t,u)=+\infty$ uniformly in
$t$.
\item [(c)] For some constant vector $u_0$,
$$\int_0^T F(t,u_0)\,dt<\int_0^T F(t,0)\,dt.
$$
\item [(d)] There exists $r>0$ and an integer $k\ge 0$ such that
$$
-{1 \over 2}(k+1)^2 w^2|u|^2\le F(t,u)- F(t,0)
\le -{1 \over 2}k^2 w^2|u|^2
$$
for all $|u|\le r$, a.e. $t \in [0,T]$, where $w=2\pi/T$.
\end{itemize}

Under the previous assumptions, they proved that problem \eqref{eP}
admits three periodic solutions (see \cite[Theorem 7]{B-N}). In
\cite{T} and \cite{T-W}, relaxing the coercivity of the potential
and exploiting assumption (d), three periodic solutions to \eqref{eP}
are still ensured (see  \cite[Theorems 2 and 4]{T-W} and
\cite[Theorem 2]{T}). Further, the existence of one periodic
solution to \eqref{eP} is guaranteed when (d) is not required and a
weaker type of coercivity is assumed (see  \cite[Theorems 1 and
3]{T-W} and  \cite[Theorem 1]{T}). Very recently, in \cite{F}, if
$F(t,u)={1 \over 2}A(t)u\cdot u-b(t)G(u)$, the existence of three
periodic solutions to \eqref{eP} is ensured without assuming (d), but
still requiring a condition that implies the coercivity of the
energy functional related to the Hamiltonian system, in addition
to the following:
\begin{itemize}
\item [(e)] There exist $\sigma>0$ and $u_0\in \mathbb{R}^N$ such
that
$$
|u_0|<\sqrt{\frac{\sigma}{\sum_{i,j=1}^N\|a_{ij}\|_\infty\
T}}\quad\text{and}\quad
G(u_0)=\sup_{|u|\le \overline k\sqrt \sigma}G(u),
$$
that is, $G$ achieves its maximum in the interior of the ball of radius
$\overline k \sqrt \sigma$,
where $\overline k$ is the constant of the Sobolev embedding and
$a_{ij}$ are the entries of the matrix $A$
(see \cite[assumptions 1 and 3 of Theorem 2.1]{F}).
\end{itemize}

The aim of this paper is twofold: on the one hand we prove the
existence of three periodic solutions to \eqref{eP} (see Theorem
\ref{main}) when neither condition (d) nor condition (e) are
required, as Remarks \ref{oss1} and \ref{oss2} show; moreover, in
our context, condition (c) together with a limit condition on %neighbourhood of
$G$ at zero imply the key assumption of Theorem \ref{main} (see
Remark \ref{onB-N}). On the other hand we establish the existence
of two periodic solutions (see Theorem \ref{main3}) when, in
addition, condition (b) can be removed, that is the energy
functional related to the differential problem need not be
coercive (see Remark \ref{oss3}). In our approach condition (a) is
not required, as Example \ref{es1} and \ref{es2} show.
To be precise, we study the following problem
\begin{equation} \label{ePl}
\begin{gathered}
      \ddot u=A(t)u-\lambda b(t)\nabla G(u)\quad \text{a.e. in  } [0,T]\\
      u(T)-u(0)=\dot u(T)-\dot u(0)=0
    \end{gathered}
\end{equation}
and establish the existence of an explicit open interval of
positive parameters $\lambda$ for which \eqref{ePl} admits three
or two distinct periodic solutions. We also observe that problems
of type \eqref{ePl} were studied in \cite{B-L} and \cite{C}, but
there only an upper bound of the interval of positive parameters
$\lambda$ for which \eqref{ePl} admits three distinct periodic
solutions was established.


The proofs of the above-mentioned results are all based on
critical point theorems. In particular, the results in
\cite{B-N},
\cite{T} and \cite{T-W} are obtained exploiting the critical
points theorem of  Brezis and  Nirenberg (\cite[Theorem 4]{B-N}).
 In \cite{B-L} and \cite{C} the main tool is the three
critical points theorem of  Bonanno \cite[Theorem 2.1]{Bonanno4}
(which is a consequence of the three critical points theorem of
Ricceri \cite[Theorem 3]{Ricceri2}). While in \cite{F} the scope
is achieved putting together the variational principle of
Ricceri \cite[Theorem 2.5]{Ricceri1} and the classical mountain
pass theorem of  Pucci and  Serrin \cite[Corollary 1]{P-S}.
Here, our results are based on multiple critical points theorems
established by  Averna and  Bonanno \cite[Theorem B]{A-B} and
by  Bonanno \cite[Theorem 2.1]{Bonanno} (where the variational
principle of  Ricceri \cite[Theorem 2.1]{Ricceri1} was applied),
that we recall in Section 2 (see Theorems \ref{thmA} and \ref{thmB}).

The present paper is organized as follows. Section 2 is devoted to
preliminaries and basic results; while in Section 3 we establish
the multiplicity results for Problem \eqref{ePl}.


\section{Preliminaries}
Let $T$ be a positive real number, $N$ a positive integer and
consider a matrix-valued function $A:[0,T]\to\mathbb{R}^{N\times N}$.
We assume that $A$ satisfies
\begin{itemize}

\item[(A1)] $A:[0,T]\to\mathbb{R}^{N\times N}$ is a map
 into the space
of $N\times N$ symmetric matrices with $A\in L^\infty([0,T])$ and
there exists a positive constant $\mu$ such that
\[
A(t)w\cdot w\geq \mu |w|^2
\]
for every $w\in\mathbb{R}^N$ and a.e. in $[0,T]$.
\end{itemize}

Recall that $H^1_T$ is the Sobolev space of all functions
$u\in L^2([0,T],\mathbb{R}^N)$ that admit a weak derivative $\dot{u}\in
L^2([0,T],\mathbb{R}^N)$. We emphasize that, in defining this kind of
weak derivative, the {\it test} functions belong to the space
$C^\infty_T$ of functions that are infinitely differentiable
and $T-$periodic from $\mathbb{R}$ into $\mathbb{R}^N$. Moreover, for each
$u\in H^1_T$ one has that $\int_0^T \dot u(t)dt=0$ and $u$ is
absolutely continuous (for more details we refer the reader to
\cite[pp. 6-7]{M-W}).

For each $u, v\in H^1_T$, we define
\begin{equation}\label{inner}
\langle u, v\rangle=\int_0^T \dot u(t)\cdot\dot v(t)dt+\int_0^T
A(t)u(t)\cdot v(t)dt.
\end{equation}
Since $A(t)$ is symmetric, (\ref{inner}) defines an inner product in $H^1_T$.\\
Then we define a norm in $H^1_T$ by putting $\|u\|=\langle u,
u\rangle^{1\over 2}$ for each $u\in H^1_T$.

Observe that
\begin{eqnarray}\label{elliptic2}
A(t)\xi\cdot \xi=\sum_{i,j=1}^N
a_{ij}(t)\xi_i\xi_j\leq\sum_{i,j=1}^N
|a_{ij}(t)||\xi_i||\xi_j|\leq\sum_{i,j=1}^N\|a_{ij}\|_\infty|\xi|^2.
\end{eqnarray}
Hence, if we put
$$
m=\min\{1,\mu\},\quad
M=\max\big\{1,\sum_{i,j=1}^N\|a_{ij}\|_\infty\big\},
$$
using (A1) and (\ref{elliptic2}), we  see that our
norm $\|\cdot\|$ is equivalent to the usual norm. Indeed one has
\begin{equation}\label{equivalent}
\sqrt{m}\|u\|_*\leq\|u\|\leq\sqrt{M}\|u\|_*,
\end{equation}
where, for each $u\in H^1_T$,
$$
\|u\|_*=\Big(\int_0^T |\dot
u(t)|^2dt+\int_0^T |u(t)|^2 dt\Big)^{1/2}\,.
$$
It is well known that $(H^1_T,\|\cdot\|_*)$ is compactly embedded in
$C^0([0,T],\mathbb{R}^N)$ (see for instance \cite{A}), hence, from
(\ref{equivalent}), we conclude that
\begin{equation}\label{ksegnato}
\bar k=\sup_{u\in H^1_T,\; u\neq 0} {\|u\|_{C^0}\over{\|u\|}}
\end{equation}
is finite. We are able to give an upper estimate of $\bar k$ in
the following manner. Fix $u\in H^1_T$ and consider $t_0\in[0,T]$ such
that $|u(t_0)|=\min_{\tau\in[0,T]}|u(\tau)|$. We can write
\begin{equation} \label{embed}
\begin{aligned}
|u(t)|&=\big|\int_{t_0}^t \dot
u(\tau)d\tau+u(t_0)\big|\\
& \leq \int_0^T|\dot
u(\tau)|d\tau+{1\over T}\int_0^T|u(t_0)|d\tau\\
&\leq \int_0^T|\dot u(\tau)|d\tau+{1\over
T}\int_0^T|u(\tau)|d\tau \\
&\leq\sqrt{T}\Big(\int_0^T|\dot u(\tau)|^2 d\tau\Big)^{1/2}
+{1\over\sqrt{T}}\Big(\int_0^T|u(\tau)|^2 d\tau\Big)^{1/2}\\
&\leq\sqrt 2 \max\big\{\sqrt T, {1\over\sqrt T
}\big\}\|u\|_*
\end{aligned}
\end{equation}
for each $t\in[0,T]$. Hence, from (\ref{embed}) and
(\ref{equivalent}), if we put
\begin{equation}\label{k}
 k=\sqrt{2\over m}\max\big\{\sqrt T, {1\over\sqrt T }\big\}
\end{equation}
one has
\begin{equation}\label{estimate}
\bar k\leq  k.
\end{equation}
In the sequel we shall make use of the constants
\begin{equation}\label{costanti}
L=\frac{1}{ k^2 T\sum_{i,j=1}^N\|a_{ij}\|_\infty},\quad
R=\frac{L}{1+L}.
\end{equation}
Now, let $b\in L^1([0,T])\setminus\{0\}$ which is a.e. nonnegative
and $G\in C^1(\mathbb{R}^N)$.\\
Put
$$
\Phi(u)={1\over 2}\|u\|^2
\quad\mbox{and}\quad
\Psi(u)=-\int_0^T b(t)G(u(t))dt
$$
for each $u\in H^1_T$. There are no difficulties in verifying that
$\Phi$ is a continuously G\^{a}teaux differentiable functional
whose G\^{a}teaux derivative admits a continuous inverse. In
addition, $\Phi$ is a continuous and convex functional, so that it
is sequentially lower semicontinuous too. Thanks to the
Rellich-Kondrachov theorem, $\Psi$ is a well-defined continuously
G\^{a}teaux differentiable functional whose G\^{a}teaux derivative
is a compact operator. In particular, for  $u, v\in H^1_T$, one has
\begin{gather*}
\Phi'(u)(v)=\int_0^T \dot u(t)\cdot\dot v(t)dt+\int_0^T
A(t)u(t)\cdot v(t)dt, \\
\Psi'(u)(v)=-\int_0^T b(t)\nabla G(u(t))\cdot v(t)dt\,.
\end{gather*}

Let us recall that a critical point for the functional
$\Phi+\lambda\Psi$ is any $u\in H^1_T$ such that
\begin{equation}\label{critical}
\Phi'(u)(v)+\lambda\Psi'(u)(v)=0
\end{equation}
for each $v\in H^1_T$.
Moreover, a solution for problem \eqref{ePl} is any $u\in
C^1([0,T],\mathbb{R}^k)$ such that $\dot u$ is absolutely continuous
and
\begin{gather*}
   \ddot u=A(t)u-\lambda b(t)\nabla G(u)\quad \text{a.e.  in } [0,T]\\
      u(T)-u(0)=\dot u(T)-\dot u(0)=0.
 \end{gather*}
We claim that each critical point for the functional
$\Phi+\lambda\Psi$ is a solution for problem \eqref{ePl}. In
fact, since $C^\infty_T$ is a subset of $H^1_T$, we can observe that
if $u$ is a critical point for the functional $\Phi+\lambda\Psi$,
then $\dot u\in H^1_T$ and, in particular,
\[
\ddot u=A(t)u-\lambda b(t)\nabla G(u) \quad \text{a.e.  in }
[0,T].
\]
Hence,
\[
\int_0^T \dot u(t)dt=\int_0^T\ddot u(t)dt=0
\]
and $u(T)-u(0)=\dot u(T)-\dot u(0)=0$;
that is, $u$ is a solution for problem \eqref{ePl}.

Let us recall a recent result, due to Averna and Bonanno
\cite[Theorem B]{A-B}, which is the main tool to reach our
goal.

\begin{theorem}[{\cite[Theorem B]{A-B}}]  \label{thmA}
 Let $X$ be a reflexive Banach Space, $\Phi:X\to\mathbb{R}$
a continuously G\^{a}teaux differentiable, coercive and
sequentially weakly lower semicontinuous functional whose
G\^{a}teaux derivative admits a continuous inverse on $X^*$,
$\Psi:X\to\mathbb{R}$ a continuously G\^{a}teaux differentiable
functional whose G\^{a}teaux derivative is compact. Put, for each
$r>\inf_X \Phi$,
\begin{gather*}
\varphi_1(r)=\inf_{x\in\Phi^{-1}(]-\infty,r[)}\frac{\Psi(x)
-\inf_{\overline{\Phi^{-1}(]-\infty,r[)}^w}\Psi}{r-\Phi(x)},
\\
\varphi_2(r)=\inf_{x\in\Phi^{-1}(]-\infty,r[)}\sup_{y\in
\Phi^{-1}([r,+\infty[)}\frac{\Psi(x)-\Psi(y)}{\Phi(y)-\Phi(x)},
\end{gather*}
where $\overline{\Phi^{-1}(]-\infty,r[)}^w$ is the closure of
$\Phi^{-1}(]-\infty,r[)$ in the weak topology, and assume that
\begin{itemize}
\item[(i)] There is $r\in\mathbb{R}$ such that
$\inf_X\Phi<r$ and
$\varphi_1(r)<\varphi_2(r)$.
\\
Further, assume that:
\item[(ii)]
$\lim_{\|x\|\to+\infty}(\Phi(x)+\lambda\Psi(x))=+\infty$\quad
for all $\lambda\in]{1\over
\varphi_2(r)},{1\over\varphi_1(r)}[$.
\end{itemize}
Then, for each $\lambda\in]{1\over
\varphi_2(r)},{1\over\varphi_1(r)}[$ the equation
(\ref{critical}) has at least three solutions in $X$.
\end{theorem}

We also use the following theorem concerning two critical
points.

\begin{theorem}[{\cite[Theorem 1.1]{Bonanno}}] \label{thmB}
Let $X$ be a reflexive real Banach
space, and let $\Phi,\Psi :X \to \mathbb{R}$ be two sequentially weakly
lower semicontinuous and G\^ateaux differentiable functionals.
Assume that $\Phi$ is (strongly) continuous and satisfies
$\lim_{\Vert x \Vert \to +\infty}\Phi (x)=+\infty$. Assume also
that there exist two constants $r_1$ and $r_2$ such that
\begin{itemize}
\item[(j)] $ \inf_X\Phi<r_1<r_2$;
\item[(jj)] $\varphi_1(r_1)<\varphi_2^*(r_1,r_2)$;
\item[(jjj)] $\varphi_1(r_2)<\varphi_2^*(r_1,r_2)$,
\end{itemize}
where $\varphi_1$ is defined as in Theorem \ref{thmA} and
\[
\varphi_2^*(r_1,r_2):=\inf_{x \in \Phi^{-1}(]-\infty,r_1[)}
\sup_{y \in \Phi^{-1}([r_1,r_2[)}
\frac{\Psi(x)-\Psi(y)}{\Phi(y)-\Phi(x)},
\]
Then, for each $\lambda \in ]{1 \over
\varphi_2^*(r_1,r_2)}, \min\{{1\over \varphi_1(r_1)},{1 \over
\varphi_1(r_2)}\}[,$ the functional $\Phi+\lambda\Psi$
admits at least two critical points which lie in
$\Phi^{-1}(]-\infty,r_1[)$ and $\Phi^{-1}([r_1,r_2[)$
respectively.
\end{theorem}

 We recall that Theorem \ref{thmA} and Theorem \ref{thmB} are based on
the variational principle stated by Ricceri \cite{Ricceri1}.

\section{Main results}

For the sake of simplicity, throughout this section we shall
assume that $G(0)=0$.
Our main result is the following.

\begin{theorem}\label{main}
Let $A$ be a matrix-valued function that satisfies assumption (A1).
and let $G\in C^1(\mathbb{R}^N,\mathbb{R})$. Assume that
there exist a positive constant $\gamma$ and a vector
$w_0\in\mathbb{R}^N$ with $\gamma<|w_0|$, such that
\begin{gather} \label{e1}
\frac{\max_{|w|\leq
\gamma}G(w)}{\gamma^2}<R\frac{G(w_0)}{|w_0|^2};\\
\limsup_{|w|\to\infty}\frac{G(w)}{|w|^2}<\frac{\max_{|w|\leq
\gamma}G(w)}{\gamma^2}. \label{e2}
\end{gather}
where $R$ is defined in \eqref{costanti}.
Then, for every function $b\in L^1([0,T])\setminus \{0\}$ that is
a.e. nonnegative and for every
$\lambda$ in $]\frac{1}{2\|b\|_1 k^2}\frac{1}{R}\frac{|w_0|^2}{
G(w_0)},\frac{1}{2 \|b\|_1 k^2} \frac{\gamma^2}{\max_{|w|\leq
\gamma}G(w)}[$, problem \eqref{ePl} admits at least three
solutions.
\end{theorem}

\begin{proof}
Fix $b\in L^1([0,T])\setminus \{0\}$ that is a.e. nonnegative.
Denote by $X$ the space $H^1_T$ and, for each $u\in X$, put
\[
\Phi(u)={1\over 2}\|u\|^2,\quad \Psi(u)=-\int_0^T b(t)G(u(t))dt.
\]
 As we saw
in Section 2, $\Phi$ and $\Psi$ are continuously G\^{a}teaux
differentiable and sequentially weakly lower semicontinuous
functionals. In particular $\Phi'$ admits a continuous inverse on
$X^*$ and $\Psi'$ is compact.

Since $G(0)=0$, $\max_{|w|\leq \gamma}G(w)\geq 0$. Hence,
we distinguish  two cases.
First, assume  $\max_{|w|\leq \gamma}G(w)> 0$ and fix
$\lambda$ in $\big]\frac{1}{2\|b\|_1
k^2}\frac{1}{R}\frac{|w_0|^2}{G(w_0)},\frac{1}{2 \|b\|_1 k^2}
\frac{\gamma^2}{\max_{|w|\leq \gamma}G(w)}\big[$.
By assumption \eqref{e2}, we can find two positive numbers
$\delta$ and $\delta'$, with
$$
\limsup_{|w|\to\infty}\frac{G(w)}{|w|^2}<\delta<\frac{\max_{|w|\leq
\gamma}G(w)}{\gamma^2}
$$
such that
$G(w)\leq\delta|w|^2+\delta'$
for each $w\in\mathbb{R}^N$. Fix  For each $u\in X$ one has
\begin{equation}\label{coecivity}
\begin{aligned}
\Phi(u)+\lambda\Psi(u)&\geq {1\over
2}\|u\|^2-\lambda\delta\int_0^T b(t)|u(t)|^2dt-\lambda\delta'\|b\|_1 \\
& \geq{1\over
2}\|u\|^2-\lambda\delta\|b\|_1\|u\|^2_{C^0}-\lambda\delta'\|b\|_1\\
&\geq\left({1\over 2}-\lambda\delta
k^2\|b\|_1\right)\|u\|^2-\lambda\delta'\|b\|_1\\
&>\frac{1}{2}\left(1-\delta\frac{\gamma^2}{\max_{|w|\leq
\gamma}G(w)} \right)\|u\|^2-\lambda\delta'\|b\|_1.
\end{aligned}
\end{equation}
Hence $\Phi+\lambda\Psi$ is coercive.

Let us consider $\varphi_1$ and $\varphi_2$ given in Theorem \ref{thmA}. We
can observe that $\inf_X \Phi=\Phi(0)=0$ and that, for each $r>0$,
$0\in\Phi^{-1}(]-\infty,r[)$ and
$\overline{\Phi^{-1}(]-\infty,r[)}^w=\Phi^{-1}(]-\infty,r])$.
Fix $r>0$. One has
\begin{equation} \label{phi1<1}
\begin{aligned}
\varphi_1(r)& \leq\frac{-G(0)\|b\|_1-\inf_{\|v\|^2\leq
2r}\big(-\int_0^T b(t)G(v(t))dt\big)}{r} \\
&\leq\sup_{\|v\|^2\leq 2r}\frac{\int_0^T b(t)G(v(t))dt}{r}.
\end{aligned}
\end{equation}
Thanks to (\ref{k}) and (\ref{estimate}), it is easy to check that
\[
\{v\in X: \|v\|^2\leq 2r\}\subseteq\{v\in C^0: \|v\|_{C^0}^2\leq 2
 k^2 r\}.
\]
Hence, from (\ref{phi1<1}), bearing in mind that $b\geq 0$ a.e.
and that $G$ is continuous, we can write
\begin{equation}\label{phi1<2}
\varphi_1(r)\leq \|b\|_1\frac{\max_{|w|\leq  k \sqrt{2r}}G(w)}{r}.
\end{equation}
Let now $r=\gamma^2/(2 k^2)$ and consider the function $v\in
X$ defined by putting $v(t)=w_0$ for each $t\in[0,T]$. A simple
computation shows that $k\sqrt{\mu T}\geq\sqrt 2$. Therefore, from
$\gamma<|w_0|$ one has $\gamma< k\sqrt{\mu T}|w_0|$ and, in view
of condition ($\mathcal{A}$), we obtain
\[
\|v\|^2=\int_0^T A(t) w_0\cdot w_0 dt\geq T\mu|w_0|^2> 2r.
\]
On the other hand, from (\ref{elliptic2}), one has
\begin{equation}\label{v>}
\|v\|^2\leq T\sum_{i,j=1}^N\|a_{ij}\|_\infty|w_0|^2.
\end{equation}
For each $u\in X$ such that $\|u\|^2<2r$ one has
\begin{equation}\label{-psi<}
\int_0^T b(t)G(u(t))dt\leq\|b\|_1\max_{|w|\leq  k
\sqrt{2r}}G(w)=\|b\|_1\max_{|w|\leq \gamma}G(w)
\end{equation}
and
\begin{equation}\label{no}
0<\|v\|^2-\|u\|^2\leq \|v\|^2.
\end{equation}
We claim that
\begin{equation}\label{primo<}
\frac{\max_{|w|\leq\gamma}G(w)}{\gamma^2}<L\frac{G(w_0)-\max_{|w|\leq\gamma}G(w)}{|w_0|^2},
\end{equation}
where $L$ is defined in (\ref{costanti}). In fact, since $G(0)=0$,
$\gamma<|w_0|$ and thanks to assumption \eqref{e1} one has
\begin{equation} \label{<}
\begin{aligned}
&\frac{\max_{|w|\leq\gamma}G(w)}{\gamma^2}
+L\frac{\max_{|w|\leq\gamma}G(w)}{|w_0|^2}\\
& <(1+L)\frac{\max_{|w|\leq\gamma}G(w)}{\gamma^2}\\
&<R(1+L)\frac{G(w_0)}{|w_0|^2}
 =L\frac{G(w_0)}{|w_0|^2}.
\end{aligned}
\end{equation}
Hence (\ref{primo<}) holds and, consequently,
\begin{equation}\label{no1}
G(w_0)>\max_{|w|\leq \gamma}G(w).
\end{equation}
At this point, putting together (\ref{-psi<}), (\ref{no1}),
(\ref{no}) and (\ref{v>}) we can obtain
\begin{equation} \label{>}
\begin{aligned}
\frac{\int_0^T b(t)G(v(t))dt-\int_0^T
b(t)G(u(t))dt}{\|v\|^2-\|u\|^2}
&\geq\|b\|_1\frac{G(w_0)-\max_{w\leq
\gamma}G(w)}{\|v\|^2-\|u\|^2} \\
&\geq\|b\|_1\frac{G(w_0)-\max_{|w|\leq
\gamma}G(w)}{\|v\|^2}\nonumber\\
&\geq\|b\|_1\frac{G(w_0)-\max_{|w|\leq
\gamma}G(w)}{T\sum_{i,j=1}^N\|a_{ij}\|_\infty|w_0|^2} \\
&= L k^2\|b\|_1\frac{G(w_0)-\max_{|w|\leq \gamma}G(w)}{|w_0|^2}
\end{aligned}
\end{equation}
for each $u\in X$ such that $\|u\|^2<2r$. Hence, one has
\begin{equation} \label{phi2>}
\begin{aligned}
\varphi_2(r)& \geq 2\inf_{\|u\|^2<2r}\frac{\int_0^T
b(t)G(v(t))dt-\int_0^T b(t)G(u(t))dt}{\|v\|^2-\|u\|^2} \\
& \geq 2L k^2\|b\|_1\frac{G(w_0)-\max_{|w|\leq
\gamma}G(w)}{|w_0|^2}.
\end{aligned}
\end{equation}
Making use of (\ref{phi1<2}), (\ref{primo<})  and (\ref{phi2>}),
we conclude that
\begin{equation} \label{phi1<phi2}
\begin{aligned}
\varphi_1(r) &\leq \|b\|_1\frac{\max_{|w|\leq  k
\sqrt{2r}}G(w)}{r} \\
&= 2 k^2\|b\|_1\frac{\max_{|w|\leq\gamma}G(w)}{\gamma^2}\\
&<2L k^2\|b\|_1
\frac{G(w_0)-\max_{|w|\leq\gamma}G(w)}{|w_0|^2}
 \leq \varphi_2(r).
\end{aligned}
\end{equation}
Moreover, in view of (\ref{primo<}) and (\ref{phi1<phi2}), since
$\gamma<|w_0|$ and assumption \eqref{e1} holds, we have
\begin{equation}
\begin{aligned}
\frac{1}{\varphi_2(r)}
& \leq \frac{1}{2L k^2\|b\|_1}\frac{|w_0|^2}{G(w_0)
-\max_{|w|\leq\gamma}G(w)}\\
& < \frac{1}{2L k^2\|b\|_1}\frac{|w_0|^2}{G(w_0)
-\gamma^2 R \frac{G(w_0)}{|w_0|^2}}\\
& < \frac{1}{2L k^2\|b\|_1}\frac{|w_0|^2}{G(w_0)}\frac{1}{1-R}\\
& =\frac{1}{2\|b\|_1 k^2}\frac{1}{R}\frac{|w_0|^2}{G(w_0)}
\end{aligned}
\end{equation}
and
\begin{equation}
\frac{1}{\varphi_1(r)}\geq \frac{1}{2 \|b\|_1
k^2}\frac{\gamma^2}{\max_{|w|\leq \gamma}G(w)}.
\end{equation}
Hence, all assumptions of Theorem \ref{thmA} are satisfied and the proof is
complete once observed that, as we saw in Section 2, the critical
points of the functional $\Phi+\lambda\Psi$ are solutions for our
problem \eqref{ePl}.

Now, let $\max_{|w|\leq \gamma}G(w)=0$. By assumption \eqref{e2}, we can
find a positive number $\bar\delta$ such that $G(w)<0$ for every
$w\in\mathbb{R}^N$ with $|w|>\bar\delta$. At this point, if
$\lambda>0$, one has
\begin{equation} \label{coecivitybis}
\begin{aligned}
\Phi(u)+\lambda\Psi(u)
&\geq{1\over 2}\|u\|^2-\lambda\int_{\{t\in[0,T]:
 |u(t)|\leq\bar\delta\}} b(t)G(u(t))dt \\
&\geq{1\over 2}\|u\|^2-\lambda\|b\|_1\max_{|w|\leq\bar\delta}G(w)
\end{aligned}
\end{equation}
for every $u\in X$. Hence $\Phi+\lambda\Psi$ is coercive.

Due to (\ref{phi1<2}), for $r=\gamma^2/(2k^2)$, one has
$\varphi_1(r)=0$. As well as, since $G(w_0)>0$, reasoning as in
(\ref{phi2>}) we obtain
$$
\varphi_2(r)\geq 2L k^2\|b\|_1\frac{G(w_0)}{|w_0|^2}>2R
k^2\|b\|_1\frac{G(w_0)}{|w_0|^2}>0.
$$
At this point we have
$$
\frac{1}{\varphi_2(r)}<\frac{1}{2\|b\|_1
k^2}\frac{1}{R}\frac{|w_0|^2}{G(w_0)}
$$
and we can conclude as in the previous case, where we agree to
read $1\over 0$ as $+\infty$.
\end{proof}

\begin{remark} \label{rmk3.1}
{\rm  We explicitly observe that, from the proof of Theorem \ref{main} we
obtain that, when $\max_{|w|\leq\gamma}G(w)=0$, the interval of
parameters for which problem \eqref{ePl} admits at least three
solutions is
$]\frac{1}{2\|b\|_1k^2}\frac{1}{L}\frac{|w_0|^2}{G(w_0)},+\infty[$.
Moreover, in this particular case, the conclusion can be also obtained by
standard arguments.}
\end{remark}



\begin{example}\label{es1}
{\rm Let $G:\mathbb{R}^2\to\mathbb{R}$ be defined by
\[
G(x,y)=\frac{\left(x^2+y^2\right)^{6}}{e^{x^2+y^2}}+x
\]
for every $(x,y)\in\mathbb{R}^2$. By choosing $\gamma=1$ and
$w_0\equiv(\sqrt 6,0)$ all assumptions of Theorem \ref{main} are
satisfied and so, for every function $b\in L^1([0,1])\setminus
\{0\}$ that is a.e. nonnegative and for every
$\lambda\in\left]\frac{1}{\|b\|_1}\frac{7}{100},\frac{1}{\|b\|_1}\frac{18}{100}\right[$,
the problem
 \begin{gather*}
      \ddot u=u-\lambda b(t)\nabla G(u)\quad \text{ a.e.  in  } [0,1]\\
      u(1)-u(0)=\dot u(1)-\dot u(0)=0
\end{gather*}
admits at least three nonzero solutions. In fact, it is enough to
observe that
\[
\frac{\max_{|w|\leq\gamma}G(w)}{\gamma^2}=\frac{1}{e}+1,
\]
$R=1/5$, $G(w_0)=\left(\frac{6}{e}\right)^6+\sqrt 6$ and
\[
\lim_{|w|\to +\infty}\frac{G(w)}{|w|^2}=0.
\]
}\end{example}

\begin{remark}\label{oss1}
{\rm Let $G$ be as in Example \ref{es1}, fix $b\in
C^0([0,1],\mathbb{R}^+)$ and $\lambda>0$. It is easy to see that, if we
put $F(t,w)=\frac{1}{2}|w|^2-\lambda b(t)G(w)$ for every $(t,w)\in
[0,1]\times\mathbb{R}^2$, one has that
\[
\liminf_{|w|\to 0}\frac{F(t,w)}{|w|^2}=-\infty
\]
uniformly with respect to $t$. Therefore, assumption (d) in the
introduction does not hold and, hence by \cite[Theorem 7]{B-N},
\cite[Theorem 4]{T-W} and  \cite[Theorem 2]{T} cannot be
applied.}
\end{remark}

\begin{example}\label{es2}
{\rm Let $G:\mathbb{R}\to\mathbb{R}$ be defined by
\[
G(w)=\begin{cases}
            e^{e^w}-e           & \text{if } w<2\\
            e^{e^2}(e^2 w+1-2e^2)-e & \text{if } w\geq 2.
            \end{cases}
\]
By choosing $\gamma=1$ and $w_0=2$ we are able to apply
Theorem \ref{main} and affirm that for every function
$b\in L^1([0,1])\setminus\{0\}$ that is a.e. nonnegative and for every
$\lambda\in]\frac{1}{\|b\|_1}\frac{19}{1000},\frac{1}{\|b\|_1}\frac{17}{100}[$,
the problem
   \begin{gather*}
      \ddot u=u-\lambda b(t) \dot G(u)\quad \text{a.e.  in  } [0,1]\\
      u(1)-u(0)=\dot u(1)-\dot u(0)=0
    \end{gather*}
admits at least three nonzero solutions. In fact, a simple
computation shows that
\[
\frac{\max_{|w|\leq\gamma}G(w)}{\gamma^2}=e^e-e,
\]
$R=1/3$ and $G(w_0)=e^{e^2}-e$ so that assumption \eqref{e1}
holds. Moreover
\[
\lim_{|w|\to +\infty}\frac{G(w)}{|w|^2}=0
\]
and \eqref{e2} is also true.}
\end{example}

\begin{remark}\label{oss2}
{\rm By the fact that the function $\overline \lambda G$, where
$\overline\lambda
\in]\frac{1}{\|b\|_1}\frac{19}{1000},\frac{1}{\|b\|_1}\frac{17}{100}[$
and $G$ is as in Example \ref{es2}, is increasing, condition (e)
in the introduction does not hold. Hence, \cite[Theorem 2.1]{F}
cannot be applied. Moreover, for fixed $b\in C^0([0,1],\mathbb{R}^+)$,
if we consider $F(t,w)=\frac{1}{2}|w|^2- b(t)[\overline\lambda
G(w)]$ for every $(t,w)\in [0,1]\times\mathbb{R}^2$, it is easy to
verify that
\[
\liminf_{|w|\to 0}\frac{F(t,w)}{|w|^2}=-\infty
\]
uniformly with respect to $t$ and condition (d) in Introduction
does not hold.}
\end{remark}

\begin{remark}\label{onB-N}
{\rm In our context, from (c) in Introduction one has
\begin{itemize}
\item [(b1)] $G(w_0)>0$ for some constant vector $w_0$.
\end{itemize}
If, in addition, we assume
\begin{itemize}
\item [(b2)] $\lim_{w\to 0} {G(w)\over |w|^2} =0$,
\end{itemize}
then it is easy to verify that (b1) and (b2) imply \eqref{e1} of Theorem
\ref{main}.}
\end{remark}

 As an immediate consequence of Theorem \ref{main}, we can
obtain the following result.

\begin{theorem}\label{conse}
Let $A$, $G$, $\gamma$ and $w_0$ be like in Theorem \ref{main}.
Then, for every $b\in L^1([0,T])\setminus\{0\}$ that is a.e.
nonnegative and such that $\|b\|_1$ is in the interval
$]\frac{1}{2k^2}\frac{1}{R}\frac{|w_0|^2}{ G(w_0)},\frac{1}{2 k^2}
\frac{\gamma^2}{\max_{|w|\leq \gamma}G(w)}[$, the
problem
\begin{gather*}
      \ddot u=A(t)u-b(t)\nabla G(u)\quad \text{a.e.  in  } [0,T]\\
      u(T)-u(0)=\dot u(T)-\dot u(0)=0
    \end{gather*}
admits at least three solutions.
\end{theorem}
\begin{proof}
Fix any $b\in L^1([0,T])\setminus\{0\}$ that is a.e. nonnegative
and such that $\|b\|_1$ is in $]\frac{1}{2
k^2}\frac{1}{R}\frac{|w_0|^2}{ G(w_0)},\frac{1}{2 k^2}
\frac{\gamma^2}{\max_{|w|\leq \gamma}G(w)}[$. Obviously, one
has that
\[
1\in\big]\frac{1}{2\|b\|_1 k^2}\frac{1}{R}\frac{|w_0|^2}{
G(w_0)},\frac{1}{2\|b\|_1 k^2} \frac{\gamma^2}{\max_{|w|\leq
\gamma}G(w)}\big[
\]
and we apply Theorem \ref{main}.
\end{proof}

Here is another multiplicity result in which  assumption \eqref{e2} is
not required.

\begin{theorem}\label{main3}
Let $A$ be a matrix-valued function satisfying condition
(A1) and $G\in C^1(\mathbb{R}^N)$. Put
$l=\min\big\{1,\frac{1}{k(T\sum_{i,j=1}^k\|a_{ij}\|_\infty)^{1/2}}\big\}$,
where $k$ is defined in (\ref{k}) and assume that there exist two
positive constants $\gamma_1,\ \gamma_2$ and a vector
$w_0\in\mathbb{R}^N$ such that $\gamma_1<|w_0|<l\gamma_2$ and
\begin{equation} \label{e3}
\max\big\{\frac{\max_{|w|\leq
\gamma_1}G(w)}{\gamma_1^2},\frac{\max_{|w|\leq
\gamma_2}G(w)}{\gamma_2^2}\big\}<R \frac{G(w_0)}{|w_0|^2},
\end{equation}
where $R$ is defined in (\ref{costanti}).
 Then, for every $b\in L^1([0,T])\setminus\{0\}$ that is a.e.
nonnegative, and  every
$\lambda$ in $\big]\frac{1}{2\|b\|_1 k^2}\frac{1}{R}\frac{|w_0|^2}{
G(w_0)},\frac{1}{2\|b\|_1 k^2}\min\big\{
\frac{\gamma_1^2}{\max_{|w|\leq \gamma_1}G(w)},
\frac{\gamma_2^2}{\max_{|w|\leq \gamma_2}G(w)}\big\}\big[$,
problem \eqref{ePl} admits at least two solutions
$u_{1,\lambda}$ and $u_{2,\lambda}$ such that
$\|u_{1,\lambda}\|_{C^0}\leq\gamma_1$ and
$\|u_{2,\lambda}\|_{C^0}\leq\gamma_2$.
\end{theorem}

\begin{proof}
Let $b\in L^1([0,T])\setminus \{0\}$ be a function that is a.e.
nonnegative, put $X=H^1_T$ and consider $\Phi$ and $\Psi$ as usual.
Let us introduce the following two positive numbers
$r_1=\frac{\gamma^2_1}{2k^2}$, $r_2=\frac{\gamma^2_2}{2k^2}$ and
verify that all assumptions of Theorem \ref{thmB} hold. Obviously the
functionals $\Phi$ and $\Psi$ satisfy the regularity conditions
required. Moreover, $\inf_X\Phi<r_1<r_2$. Consider the function
$v\in X$ as follows
\[
v(t)=w_0
\]
for each $t\in [0,T]$. Arguing as in Theorem \ref{main}, since
$\gamma_1<|w_0|< l\gamma_2$ and taking into account (\ref{elliptic2})
we obtain
\begin{gather*}
r_1<\Phi(v)\leq \frac{T}{2}\sum_{i,j=1}^N\|a_{ij}\|_\infty
|w_0|^2<r_2,\\
G(w_0)>\max_{|w|\leq\gamma_1}G(w).
\end{gather*}
Hence, by assumption \eqref{e3} and noting that $\gamma_1<|w_0|$, one has
\begin{equation} \label{phi2*>}
\begin{aligned}
\varphi_2^*(r_1,r_2)
 &\geq\inf_{x \in \Phi^{-1}(]-\infty,r_1[)}\frac{\Psi(x)
-\Psi(v)}{\Phi(v)-\Phi(x)}\\
 &=2\inf_{\|u\|^2<2r_1}\frac{\int_0^T b(t)G(v(t))dt
-\int_0^T b(t)G(u(t))dt}{\|v\|^2-\|u\|^2}\\
 &\geq 2L k^2\|b\|_1\frac{G(w_0)-\max_{|w|\leq\gamma_1}G(w)}{|w_0|^2}\\
 &> 2L k^2\|b\|_1(1-R)\frac{G(w_0)}{|w_0|^2}\\
 &= 2R k^2\|b\|_1\frac{G(w_0)}{|w_0|^2}.
\end{aligned}
\end{equation}
Moreover, as we saw in Theorem \ref{main},
\begin{gather}\label{phi1r1<}
\varphi(r_1)\leq 2
k^2\|b\|_1\frac{\max_{|w|\leq\gamma_1}G(w)}{\gamma_1^2} \\
\label{phi1r2<}
\varphi(r_2)\leq 2
k^2\|b\|_1\frac{\max_{|w|\leq\gamma_2}G(w)}{\gamma_2^2}.
\end{gather}
At this point, combining (\ref{phi1r1<}), (\ref{phi1r2<}),
assumption \eqref{e3} and (\ref{phi2*>}) we obtain
\begin{equation} \label{phi1<phi2*}
\begin{aligned}
\max\{\varphi_1(r_1),\varphi_1(r_2)\}
&\leq 2 k^2\|b\|_1\max\big\{\frac{\max_{|w|\leq\gamma_1}G(w)}{\gamma_1^2},
 \frac{\max_{|w|\leq\gamma_2}G(w)}{\gamma_2^2} \big\} \\
&< 2R k^2\|b\|_1\frac{G(w_0)}{|w_0|^2} \\
&<\varphi_2^*(r_1,r_2).
\end{aligned}
\end{equation}
Therefore all assumptions of Theorem \ref{thmB} are satisfied. Hence,
since by (\ref{phi1<phi2*}) one has
\begin{align*}
\frac{1}{\varphi_2^*(r_1,r_2)}
&<\frac{1}{2\|b\|_1 k^2}\frac{1}{R}\frac{|w_0|^2}{ G(w_0)}\\
&<\frac{1}{2\|b\|_1 k^2}\min\big\{\frac{\gamma_1^2}{\max_{|w|\leq
 \gamma_1}G(w)},\frac{\gamma_2^2}{\max_{|w|\leq \gamma_2}G(w)}\big\}\\
&\leq\min\big\{{1\over\varphi_1(r_1)},{1\over\varphi_1(r_2)}\big\},
\end{align*}
for each $\lambda\in\big]\frac{1}{2\|b\|_1
k^2}\frac{1}{R}\frac{|w_0|^2}{ G(w_0)},\frac{1}{2\|b\|_1
k^2}\min\big\{ \frac{\gamma_1^2}{\max_{|w|\leq \gamma_1}G(w)},
\frac{\gamma_2^2}{\max_{|w|\leq \gamma_2}G(w)}\big\}\big[$,
problem \eqref{ePl} admits at least two solutions
$u_{1,\lambda}$ and $u_{2,\lambda}$ such that
$\|u_{1,\lambda}\|^2<2r_1\leq\|u_{2,\lambda}\|^2<2r_2$. Then
thanks to (\ref{ksegnato}) and (\ref{estimate}), we can complete
the proof.
\end{proof}

\begin{example}\label{es3}
{\rm Let $G:\mathbb{R}^2\to\mathbb{R}$ be defined as follows
\[
G(w)=\begin{cases}
\frac{|w|^6}{e^{|w|^2}} &  \text{if } |w|\leq\sqrt 3\\
(\frac{3}{e})^3 \cos(|w|^2-3) &  \text{if }
\sqrt 3<|w|\leq\sqrt{3+\frac{15}{2}\pi}\\
(\frac{3}{e})^3 [e^{|w|^2-3-\frac{15}{2}\pi}-1]
& \text{if } |w|>\sqrt{3+\frac{15}{2}\pi}.
\end{cases}
\]
Theorem \ref{main3} guarantees that for every
$b\in L^1([0,1])\setminus\{0\}$ that is a.e. nonnegative and for every
$\lambda\in]\frac{3}{\|b\|_1},\frac{4}{\|b\|_1}[$ the
 problem
\begin{equation} \label{ePIl}
\begin{gathered}
 \ddot u=u-\lambda b(t)\nabla G(u)\quad \text{a.e.  in  } [0,1]\\
      u(1)-u(0)=\dot u(1)-\dot u(0)=0
\end{gathered}
\end{equation}
admits at least one nonzero solution $u_\lambda$ such that
$\|u_\lambda\|_{C^0}\leq \sqrt{3+ {15\over 2}\pi}$. To see this,
we can observe that
$$
k=\sqrt 2,\quad \sum_{i,j=1}^2\|a_{ij}\|_\infty=2,\quad l={1 \over 2},\quad
R={1\over 5}.
$$
Hence, Theorem \ref{main3} applies with $\gamma_1=1/2$,
$\gamma_2=\sqrt{3+{15\over 2}\pi}$ and $w_0\in\mathbb{R}^2$ such that
$|w_0|=\sqrt 3$.}
\end{example}

\begin{remark}\label{oss3}
{\rm We observe that in Example \ref{es3}, for every
positive $\lambda$, the energy functional related to problem
($P^I_\lambda$) is not coercive, that is condition (b) in
Introduction fails. Hence, we cannot apply  \cite[Theorem 1]{T-W}
or \cite[Theorem 1]{T}}.
\end{remark}


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\end{document}