\documentclass[twoside]{article}
\usepackage{amssymb} % font used for R in Real numbers
\pagestyle{myheadings}

\markboth{\hfil Periodic solutions \hfil EJDE--2001/38}
{EJDE--2001/38\hfil Morched Boughariou \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 38, pp. 1--17. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Periodic solutions for a class of non-coercive Hamiltonian systems
 %
\thanks{ {\em Mathematics Subject Classifications:} 34C25, 37J45.
\hfil\break\indent
{\em Key words:} 
Hamiltonian systems, non-coercive, periodic solutions, minimax argument.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted January 3, 2001. Published May 28, 2001.} }
\date{}
%
\author{ Morched Boughariou }
\maketitle

\begin{abstract}
 We prove the existence of non-constant $T$-periodic
 orbits of the Hamiltonian  system
 $$\displaylines{
 \dot q =H_p (t, p(t), q(t))\cr
 \dot p =-H_q (t, p(t), q(t)),
 }$$
 where $H$ is a $T$-periodic function in $t$, non-convex and
 non-coercive in $(p,q)$, and has the form
 $H(t,p,q)\sim |q|^{\alpha}(|p|^{\beta}-1)$ with  $\alpha>\beta>1$.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{corollary}{Corollary}[section]

\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11
\@addtoreset{equation}{section}
\catcode`@=12

\section{Introduction}

We study the existence of $T$-periodic solutions of the Hamiltonian
system
\begin{eqnarray}
&\dot q =H_p (t, p(t), q(t)) &\label{HS}\\
&\dot p =-H_q (t, p(t), q(t)).&\nonumber
\end{eqnarray}
 Here, $H(t,p,q):\mathbb{R}\times \mathbb{R}^N \times \mathbb{R}^N \to
\mathbb{R}$ $(N\geq 3)$ is $T$-periodic in $t$ and differentiable in
$(p,q)$. We also assume that $H,H_p,H_q$ are continuous.

Most of the existence results use coercivity
(i.e., $H(t,p,q)\to \infty $ as $|(p,q)|\to \infty)$ or
convexity assumptions in $H(t,.)$;
see \cite{R,BR,E,CE,B} and references therein. The purpose of this paper is to study non-coercive
 and non-convex Hamiltonians. Typically,
 $$ H(t,p,q)\sim |q|^{\alpha}(|p|^{\beta}-1);
\quad\alpha>\beta>1.$$
To state our existence result, we introduce the following
hypotheses.
For constants $\alpha>\beta>1$, $r>0$, $a_1,\dots ,a_{8}>0$
and functions $A$, $K_i \in C(\mathbb{R}^N,\mathbb{R})$ with
$K_i(0)= 0$ $(i=1,2,3)$, we assume:
\begin{enumerate}
\item[(H1)] $H(t+{T\over 2},p,q)=H(t,-p,-q)$ for all $t,p,q$;

\item[(H2)] (i) $H(t,p,q)\leq a_1|q|^\alpha |p|^\beta$ for all $t,p,q$;
\\ \quad (ii) $H(t,p,q) \geq a_2|q|^\alpha |p|^\beta - K_1(q)$
for all  $t,p,q$;

\item[(H3)] $-H(t,p,q)+H_p(t,p,q)p \geq a_3|q|^\alpha (|p|^\beta +1)
-a_4$ for all $t,p,q$;

\item[(H4)] $|H_p(t,p,q)|\leq a_5|q|^\alpha (|p|^{\beta -1}+1)+a_6|q|$
for all $t,p,q$;

\item[(H5)] $|H_q(t,p,q)|\leq A(q)(|p|^\beta +1)$ for all $t,p,q$;

\item[(H6)] (i) $H_q(t,p,q)q-H_p(t,p,q)p \geq a_7 H(t,p,q)+K_2(q)$
for all $t,p,|q|\leq r$;
\\ \quad (ii) $|H_p (t,p,q)|^{\beta \over {\beta -1} }
\leq a_8 |q|^{\alpha \over {\beta -1}}
\big( |q|^\alpha |p|^\beta +K_3(q)\big)$ for all $t,p,|q|\leq r$.
\end{enumerate}
Our main result is as follows.

\begin{theorem} Under assumptions (H1)-(H6), System (\ref{HS})
has at least one non-constant $T$-periodic solution $(p(t),q(t))$ with
$q(t) \neq 0$ for all $t$.
 \end{theorem}

\paragraph{Remark.} If $H(t,p,q)=a(t)|q|^\alpha(|p|^\beta -1)$ with
$\alpha>\beta>1$ and $a(t) \in C(\mathbb{R},\mathbb{R})$ is a
 $T\over 2$-periodic and positive function , then (H1)-(H6) hold.


\paragraph{Remark.} The condition $\alpha >\beta$ is necessarily for
the existence of non-constant $T$-periodic solution. More precisely,
in case
$$H(t,p,q)=|q|^\alpha(|p|^\beta -1),$$
if $(p(t),q(t))$ is a non-constant $T$-periodic solution of (\ref{HS}),
then \\
(i) $\alpha >\beta$;
\\
(ii) there exists a constant $C>0$ such that
$$|q(t)|^\alpha (|p(t)|^\beta -1)=C>0 \hbox{ for all }t \in \mathbb{R}.$$
In particular, $q(t) \neq 0$ for all $ t \in \mathbb{R}$.

Indeed,  by (\ref{HS}) we have
$$\int _0^T p\dot q dt =\beta \int_0^T |q|^\alpha |p|^\beta dt
 =\alpha \int _0^T |q|^\alpha( |p|^\beta -1)dt.$$
Then
$$(\alpha -\beta) \int_0^T |q|^\alpha |p|^\beta dt
= \alpha \int_0^T |q|^\alpha dt.$$
Since $(p,q)$ is non-constant, one can see that $q\neq 0$ and
$\alpha>\beta$.
Also note that (ii) follows from the conservation of the energy.
\medskip

To show the existence of a $T$-periodic solution of (\ref{HS}), we use a
variational method; we introduce the functional
 $$I(p,q)=\int_0^T[p\dot q-H(t,p,q)]dt  $$
defined on the function space
$$E=\{(p,q)\in L^\gamma (0,T;\mathbb{R}^N)\times
W^{1,{\gamma \over {(\gamma -1)}}}(0,T;\mathbb{R}^N);\ q(0)=q(T)\}$$
where $\gamma=\alpha +\beta$.
Critical points of $I(p,q)$ on $E$ correspond to $T$-periodic solutions of
(\ref{HS}). We remark that the correspondence is one-to-one.


Since it is difficult to verify the Palais-Smale compactness condition
for $I(p,q)$, we introduce in  the following section, modified
functionals and a finite dimensional  approximation. We will use a
minimax argument.

\section{Modified functionals and other preliminaries}

As stated in the introduction, we will find
a critical point of the functional $I(p,q)$ on $E =P \times Q$ where
$$P=L^\gamma (0,T;\mathbb{R}^N), \quad
 Q=\{q \in W^{1,{\gamma \over{(\gamma -1)}}}(0,T;\mathbb{R}^N); q(0)=q(T)\}.$$
We set
$$\Lambda =\{q \in Q; \; q(t)\not = 0 \; \hbox {for all } t\}$$
and introduce the modified functionals
$$\displaylines{
I_\delta(p,q)=\int_0^T[p\dot q -H(t,p,q)+{\delta \over {|q|^\gamma}} ]dt,\cr
I_{\delta,\varepsilon}(p,q)=\int_0^T[p\dot q -H(t,p,q)+{\delta\over {|q|^\gamma}}+\varepsilon (|q|^\gamma-
 |p|^\gamma)]dt
}$$
for $\delta, \; \varepsilon \in [0,1]$.
Since $\gamma \geq \beta>1 $, by (H2), (H4), and (H5),
 we can see that $I_{\delta,\varepsilon} \in C^1(P \times
\Lambda ;\mathbb{R})$.

To get the existence of a $T$-periodic solution for a symmetric Hamiltonians, we have to restrict
our functionals to a subsets of $E$. We set
$$E_0=\{(p,q)\in E;\; (p,q)(t+{T\over 2})=-(p,q)(t)\}$$
with norm
$$\|(p,q)\|_{E_0}=\|p\|_\gamma + \|\dot q\|_{\gamma \over {\gamma -1}}$$
where
$$\|u\|_s = (\int_0^T |u(t)|^s dt )^{1/s}\hbox{  for all } s \geq 1.$$
For $m \in\mathbb{N}$, we define
$$\displaylines{
P_m=Q_m\cr
=\big\{p(t)=\sum_{|j|\leq m} \theta_je^{2i\pi j t\over T}; \;p(t+{T\over 2})=-p(t),  \theta_j\in {
\mathbb C}^N,\theta_{-j}
 =\bar{\theta_j} ,|j|\leq m\big\},\cr
E_m=P_m\times Q_m,\cr
\Lambda _m=\{q\in  Q_m; \; q(t) \not = 0 \; {\hbox {for all}}\;t\},\cr
 \partial \Lambda_m =\{q \in Q_m ;\; q(t_0)=0 \; \hbox {for some}\; t_0\}
}$$
and we consider the restriction of $I_{\delta,\varepsilon} (p,q):$
$$I_{\delta,\varepsilon,m}=I_{\delta,\varepsilon}/_{P_m \times{\Lambda}_m} :P_m \times{\Lambda}_m \to
\mathbb{R}.$$
The main reason for introducing such subspaces are the following Lemmas.

\begin{lemma} For any $u\in Q$ such that $u(t+{T\over 2})=-u(t)$, we have
$$\|u\|_\infty \leq \int_0^T |\dot u |dt.$$
\end{lemma}

\paragraph{Proof.} Let $u\in Q$ such that $u(t+{T\over 2})=-u(t)$. Then for
all $t\in [0,T]$, we have
$$
|u(t)|={1\over 2}|u(t+{T\over 2})-u(t)|
={1\over 2}|\int_t^{t+{T\over 2}}\dot u \;ds|
\leq \int_0^T |\dot u |ds.
$$
Thus we obtain the desired result. \hfill$\diamondsuit$

\begin{lemma} Suppose $(p,q) \in P_m \times \Lambda _m$ is such that
\begin{equation}
I'_{\delta,\varepsilon,m}(p,q)(h,k)=0 \quad \hbox{for all } (h,k)\in E_m.
\label{eq:2.1}
\end{equation}
Then $(p,q)$ is a critical point for $I_{\delta,\varepsilon,m}$.
\end{lemma}

\paragraph{Proof.} It is sufficient to remark that, by (H1),
$I'_{\delta,\varepsilon,m}(p,q)\in E_m$. Since
$I'_{\delta,\varepsilon,m}(p,q)$ belongs also to $E_m^\perp$
from \ref{eq:2.1}, we have the
conclusion. \hfill$\diamondsuit$\medskip

The proof of Theorem 1.1 will be done as follows:
In section 3, we introduce a minimax method to $I_{\delta,\varepsilon,m}$. For $\delta,
\varepsilon \in ]0,1]$ and $m \in \mathbb{N}$, we establish the existence
of a sequence $(p_{\delta,\varepsilon,m},q_{\delta,\varepsilon,m})
\in P_m \times \Lambda_m$ such that
\begin{equation}
I'_{\delta,\varepsilon,m}(p_{\delta,\varepsilon,m},q_{\delta, \varepsilon,m})=0,\label{eq:2.2}
\end{equation}
\begin{equation}I_{\delta,\varepsilon,m}(p_{\delta,\varepsilon,m},q_{\delta,\varepsilon,m})\leq \bar c
 \label{eq:2.3}\end{equation}
where $\bar c>0$ is a constant independent of $\delta,\varepsilon$ and
$m$. From  \ref{eq:2.2}-\ref{eq:2.3}, we can find uniform estimates for
$(p_{\delta,\varepsilon,m},q_{\delta,\varepsilon,m})$ and we can extract,
 in section 4, a subsequence converging to $ (p_{\delta,\varepsilon},
q_{\delta,\varepsilon})\in (P\times \Lambda )\cap E_0$.
Next in Section 5, we pass to the limit as $\varepsilon \to 0$ and obtain
a  critical points $(p_\delta,q_\delta) \in (P \times \Lambda)\cap E_0 $
of $I_\delta$ such that
\begin{equation}
I_\delta (p_\delta,q_\delta)\leq \bar c.
\label{eq:2.4}
\end{equation}
Finally in Section 6, we pass to the limit as $\delta \to 0$. Lemma 2.1 plays a essential role to
obtain a non-constant $T$-periodic solution
$(p,q)=\lim (p_\delta,q_\delta)$ of (\ref{HS}).

In the sequel, we use the projection operator
$$\displaylines{
{\mathop{\rm proj}}_m : L^s (0, T; \mathbb{R}^N) \to \hbox{span}
\{e^{2i\pi jt \over T }; |j| \leq m\}\,,\cr
({\mathop{\rm proj}}_m u) (t) = \sum_{|j| \leq m}\theta_j e^{2i\pi j t\over T}
\quad \hbox {for} \quad u(t) = \sum_{j\in {\mathbb Z}}\theta_j
e^{2i\pi j t\over T}.
}$$

\begin{lemma} For any $s\in ]1,+\infty[$, there exists a constant $K_s >0$ independent of $m\in \mathbb{N}
$ such that
$$\| {\mathop{\rm proj}}_m u\|_s \leq K_s \|u\|_s \quad \hbox{for all}\quad u\in L^s (0, T; \mathbb{R}^N).$$
\end{lemma}
This lemma is a special case of Steckin's theorem \cite[Theorem 6.3.5]{EG}.
In  sections 3, 4, 5, and 6, we will assume (H1)-(H6).

\section{A minimax method for $I_{\delta,\varepsilon,m}$}

In this part, we study the existence of critical points in
$P_m \times \Lambda_m$ of $I_{\delta,\varepsilon,m}$
for  $\delta,\varepsilon \in ]0,1]$ and $m\in \mathbb{N}$.
First, we give some a priori estimates and verify the
Palais-Smale condition (PS)  for
$I_{\delta,\varepsilon,m}$.

\begin{lemma} (i) For any $M_1 >0$, there exists a constant
$C_0 = C_0 (M_1)> 0$ independent of $\delta,\varepsilon \in ] 0,1]$
and $m\in \mathbb{N}$ such that:
If $(p, q) \in P_m \times \Lambda_m$ satisfies
\begin{eqnarray}
&I_{\delta,\varepsilon, m} (p, q)  \leq M_1,& \label{eq:3.1}\\
&I'_{\delta,\varepsilon, m}(p,q)=0,& \label{eq:3.2}
\end{eqnarray}
then
$$\displaylines{
\int _0^T |q|^\alpha |p|^\beta dt+\int _0^T |q|^\alpha dt \leq C_0,\cr
\varepsilon \int_0^T\big(|q|^\gamma + |p|^\gamma\big) dt +
 \delta \int_0^T {1\over {|q|^\gamma}}dt \leq C_0.
}$$
(ii) For any $\delta,\varepsilon \in ]0, 1]$ and $m\in \mathbb{N}$, if $(p_j, q_j)_{j=1}^\infty
\subset P_m \times
\Lambda_m$ satisfies
$$(p_j, q_j) \to (p_0, q_0) \in P_m \times \partial \Lambda_m,$$
then $I_{\delta,\varepsilon, m} (p_j, q_j) \to +\infty$.\\
(iii) For any $\delta,\varepsilon \in ]0, 1]$ and $m\in \mathbb{N}$, $I_{\delta,\varepsilon, m}
$ satisfies
the condition (PS) on $P_m \times \Lambda_m$; i.e.,
if $(p_j, q_j)_{j \in \mathbb{N}} \subset P_m \times \Lambda_m$ satisfies $
I_{\delta,\varepsilon, m}(p_j, q_j) \to c>0$ and
$(I_{\delta,\varepsilon, m})' (p_j, q_j) \to 0$, then
$(p_j,q_j)$ possesses a subsequence converging in $E_{m}$ to some
$(p, q)\in P_m \times \Lambda_m$.
\end{lemma}

\paragraph{Proof.} (i) Let $\delta,\varepsilon \in ]0, 1]$ and $m \in \mathbb{N}$. We assume $
(p,q) \in P_m\times \Lambda_m$ satisfies \ref{eq:3.1} and \ref{eq:3.2}
for $M_1 >0$. We have
$$I'_{\delta,\varepsilon,m}(p,q)(p,0)=\int_0^T[p\dot q-H_p(t,p,q)p
-\varepsilon \gamma |p|^\gamma]dt.$$
Hence,
\begin{eqnarray}
\lefteqn{I_{\delta,\varepsilon ,m}(p,q)-I'_{\delta,\varepsilon,m}
(p,q)(p,0)}\label{eq:3.3}\\
&=&\int_0^T[-H(t,p,q) +H_p(t,p,q)p
+ {\delta  \over {|q|^\gamma}} +\varepsilon |q|^\gamma
+\varepsilon(\gamma -1)|p|^\gamma ]dt. \nonumber
\end{eqnarray}
By the assumptions \ref{eq:3.1} and \ref{eq:3.2}, we get
$$\int_0^T[ -H(t,p,q)+H_p(t,p,q)p+{\delta \over{|q|^\gamma}} +\varepsilon|q|^\gamma+\varepsilon (\gamma
-1) |p|^\gamma]dt  \leq M_1.$$
From (H3), it follows that
$$\int _0^T[a_3|q|^\alpha (|p|^\beta +1) -a_4+{\delta\over {|q|^\gamma}} +
\varepsilon |q|^\gamma+\varepsilon (\gamma-1)|p|^\gamma] dt \leq M_1.$$
Thus we obtained (i).

\noindent
(ii) By (H2)(i), we have for all $(p,q) \in P_m\times \Lambda _m$
\begin{equation}
I_{\delta,\varepsilon,m}(p,q) \geq \int_0^T [ p\dot q-a_1 |q|^\alpha |p|^\beta
+\varepsilon  (|q|^\alpha-|p|^\gamma)] dt+\delta  \int _0^T{1\over{|q|^\gamma}}dt. \label{eq3.3'}
\end{equation}
Since $\delta \int _0^T {1\over |q_j|^\gamma } dt \to \infty$, we
get the conclusion easily.

\noindent
(iii) Let $(p_j,q_j)_{(j \in \mathbb{N})} \subset P_m \times \Lambda _m$ be a sequence satisfying
the assumptions of the condition (PS).We may assume that
\begin{equation}
I_{\delta,\varepsilon,m}(p_j,q_j) \to c, \label{eq:3.4}\end{equation}
\begin{equation}
\|I'_{\delta,\varepsilon,m}(p_j,q_j)\|_{E_m^\star}\to 0. \label{eq:3.5}\end{equation}We prove that $(p_j,q_j)
$ possesses a convergent subsequence to some
$(p,q) \in P_m \times \Lambda _m$.
By (H3) and \ref{eq:3.3}-\ref{eq:3.5}, for large $j$,
\begin{eqnarray*}
\int _0^T [a_3|q_j|^\alpha (|p_j|^\beta +1) -a_4]dt
+ \delta \int _0^T {1\over {|q_j|^\gamma}}dt &&\\
+\varepsilon \int_0^T |q_j|^\gamma dt
+ \varepsilon (\gamma -1)\int _0^T |p_j|^\gamma dt
&\leq& 2c+\|p_j\|_{\gamma}\,.
\end{eqnarray*}
Thus, for some  constant $C_1>0$ independent of $j$,
$$\int _0^T |q_j|^\alpha dt,\; \int _0^T |p_j|^\gamma dt \leq C_1 \quad  \hbox
 {for all }j \in \mathbb{N}.$$
Since $\dim E_m<\infty $, we can extract a subsequence -
still indexed by $(p_j,q_j)$ -, such that  $(p_j,
q_j)\to (p,q)\in E_m$.
By (ii), we necessarily have $q \in \Lambda_m$.

Next, we apply to $I_{\delta,\varepsilon ,m}$ a minimax
 argument related to the one in \cite {T1}.
This argument will play an important role in obtaining a critical points
$(p_{\delta, \varepsilon,m},q_{\delta,
\varepsilon,m})\in P_m \times\Lambda _m$ with uniform upper
bound of critical values.
We define
$$\Gamma_{m }=\{ A(p,\xi) \in C \big(P_m  \times  S^{N-2},
P_m\times \Lambda _m \big) ;\;
 A(p,\xi)=\big(p,\sigma_0(\xi)\big)  \hbox {for large }\|p\|_\beta \}$$
where
$$\sigma_0  :S^{N-2}= \{ \xi =({\xi}_1,\dots ,{\xi}_{N-1})
\in \mathbb{R}^{N-1} :\sum _{j=1} ^{N-1} |{\xi}_j|^2 =1 \}
 \to Q_m$$
is given by
$$ \sigma_0(\xi)(t) =\hbox{cos} {2\pi t\over T}({\xi}_1,\dots ,{\xi}_{N-1},0)
+\hbox{sin} {2\pi t\over T }(0,\dots 0,1).$$
We remark that $A_0(p,\xi)=(p,\sigma_0(\xi))\in \Gamma _{m} $ and
 $\Gamma_{m }\not =  \emptyset $.
Then we define the minimax values of
$I_{\delta,\varepsilon ,m}$ as follows
$$c_{\delta,\varepsilon ,m} =\inf_{A \in \Gamma_{m} }\sup_{(p,\xi)\in P_m \times S^{N-2}}I_{\delta,
\varepsilon ,m}(A (p,\xi)).$$
 \begin{proposition} For any $\delta,\varepsilon \in ]0,1]$ and
$m \in \mathbb{N}$, there exists a constant
$\underline c (\delta,\varepsilon)>0$  such that
$$ c_{\delta,\varepsilon,m} \geq \underline c(\delta,\varepsilon)>0. $$
\end{proposition}

To prove this proposition, we need the following result.

\begin{lemma} For any $A \in \Gamma _{m }$ and $\lambda >0$, we have
$$A (P_m  \times S^{N-2} )\cap {\cal D}_{m,\lambda}\not = \emptyset$$
where
$${\cal D}_{m,\lambda}=\{(p,q) \in P_m \times \Lambda _m ;\; p= \lambda{{\mathop{\rm proj}}}_m
(|\dot q|^{{1\over {\gamma -1}}-1} \dot q) \}.$$
\end{lemma}
The proof of this lemma will be given in the appendix.

\begin{lemma}For sufficiently small $\lambda_\varepsilon >0$,
there exists a constant $c(\delta,\varepsilon)>0$
 such that
$$I_{\delta,\varepsilon,m}(p,q)\geq c(\delta,\varepsilon)>0
\quad {\hbox {for all}}\; (p,q)\in{\cal D}_{m,\lambda_\varepsilon}$$
where ${\cal D}_{m,\lambda_\varepsilon}$ is given in Lemma 3.2.
\end{lemma}

\paragraph{Proof.} Let $(p,q) \in {\cal D}_{m,\lambda}$.
We recall that $\gamma=\alpha+\beta$. By the Young's inequality,
$$a_1 \int_0^T |q|^\alpha  |p|^\beta dt\leq {\alpha \over \gamma}\varepsilon \int_0^T|q|^\gamma dt+{
\beta \over \gamma}({a_1 \over {\varepsilon^{\alpha \over \gamma}}})^{\gamma\over \beta}\int_0^T|p
|^\gamma dt.$$
Thus, from \ref{eq3.3'},
$$I_{\delta,\varepsilon,m}(p,q)\geq \int_0^T p \dot q dt-a(\varepsilon) \int_0^T  |p|^\gamma dt +\delta
\int_0^T {1\over{|q|^\gamma}}dt$$
where $a(\varepsilon)=\varepsilon +{\beta \over \gamma}({a_1 \over {\varepsilon^{\alpha \over \gamma}}})^{
\gamma\over \beta}>0$.
Since $(p,q) \in {\cal D}_{m,\lambda}$,
\begin{equation}
\int _0^T p \dot q dt = \lambda \int _0^T |\dot q|^{{\gamma\over {\gamma -1}}}dt.
\label{eq:3.6}\end{equation}
Moreover, by Lemma 2.1 and Lemma 2.3
\begin{equation}
T^{1\over \gamma} \|\dot q\|_{\gamma \over {\gamma -1}} \geq
\int_0^T |\dot q|dt \geq\|q\|_\infty,
\label{eq:3.7}\end{equation}
\begin{equation}
\int_0^T |p|^\gamma dt = \lambda ^\gamma
\|{{\mathop{\rm proj}}}_m(|\dot q|^{{1\over {\gamma -1}}-1} \dot q
)\|_\gamma^\gamma  \leq
 \lambda ^\gamma K_{\gamma} ^\gamma\|\dot q\|_{\gamma \over
{\gamma -1}}^{\gamma \over {\gamma-1}}.  \label{eq:3.8}
\end{equation}
By \ref{eq:3.6} and \ref{eq:3.8}, we get
$$ I_{\delta,\varepsilon,m}(p,q)\geq (\lambda -a(\varepsilon)K_{\gamma}^\gamma \lambda ^\gamma )\|
\dot q \|_{\gamma \over {\gamma-1}}^{\gamma \over {\gamma-1}}
+\delta \int_0^T {1\over{|q|^\gamma}}dt.
$$
Taking $\lambda _\varepsilon$ small enough so that
$A_\varepsilon =\lambda_\varepsilon -a(\varepsilon
)K_\gamma^\gamma \lambda_\varepsilon ^\gamma >0$,
 from \ref{eq:3.7}, for all $(p,q) \in {\cal
D}_{m,\lambda_\varepsilon}$, we have
$$
I_{\delta,\varepsilon,m}(p,q)\geq
\inf_{q\in \Lambda}\big({A_\varepsilon \over{T^{1\over {\gamma -1}}}}
\|q\|_\infty^{\gamma \over {\gamma-1}}+{\delta T\over {\|q\|_\infty
^\gamma}}\big)
=c(\delta,\varepsilon)>0\,.
$$

\paragraph{Proof of Proposition 3.1}
 Let $\lambda _\varepsilon>0$ be as in Lemma 3.3.
By Lemma 3.2, we have
$$A (P_m \times S^{N-2} )\cap {\cal D}_{m,\lambda_\varepsilon}\not = \emptyset \quad {\hbox {for all}}
\;  A \in \Gamma_{m}.$$
Thus, we find that
 \begin{eqnarray*} c_{\delta,\varepsilon ,m} &=& \inf_{A \in \Gamma_{m} }\sup_{(p,\xi)\in P_m \times
S^{N-2}}I_{\delta,\varepsilon ,m}(A (p,\xi)) \\
&\geq & \inf_{(p,q)\in{\cal  D}_{m,\lambda_\varepsilon}}I_{\delta,\varepsilon,m}(p,q) \\
& \geq & c(\delta,\varepsilon)>0.\end{eqnarray*}
We choose $\underline c(\delta,\varepsilon)=c(\delta,\varepsilon)$,
we get the desired result. \hfill$\diamondsuit$\medskip

Now, we prove an existence result

\begin{proposition} For any $\delta,\varepsilon \in ]0,1]$ and
$m \in \mathbb{N}$, we have
\\
(i) $$0<\underline c(\delta,\varepsilon) \leq c_{\delta,\varepsilon,m} \leq \bar c$$
where $\bar c$ is independent of $\delta,\varepsilon$ and $m$.

\noindent
(ii) If $\|p\|_\beta $ is sufficiently large, then for all $\xi \in S^{N-2}$,
$$I_{\delta,\varepsilon,m}(A_0(p,\xi))\leq 0\,.$$

\noindent
(iii) There exists a critical point $(p_{\delta,\varepsilon,m},q_{\delta,
\varepsilon,m})\in P_m \times \Lambda _m $ of $I_{\delta,\varepsilon,m}$ such that
$$I_{\delta,\varepsilon,m}(p_{\delta,\varepsilon,m},q_{\delta,\varepsilon,m})
=c_{\delta,\varepsilon,m}.$$
\end{proposition}

\paragraph{Proof.} (i) By (H2)(ii), we have
\begin{eqnarray}\nonumber
I_{\delta,\varepsilon ,m}(A _0 (p,\xi)) &\leq &\int _0^T |p| |{d \over {dt}}\sigma _0(\xi)|dt -
a_2\int _0^T |\sigma_0 (\xi)|^\alpha |p|^\beta dt \\
\nonumber
&  &+ \int _0^T K_1(\sigma_0(\xi))dt  + \int_0^T({1\over{|\sigma_0(\xi)|^\gamma}}+|\sigma_0(\xi)|^\gamma)
dt  \\
&\leq& k_1 \|p\|_\beta -k_2 \|p\|_\beta ^\beta + k_3 \label{eq:3.9}\end{eqnarray}
for some positive constants $k_1,k_2,k_3$ independent of
$\delta$, $\varepsilon$ and $m$.
Since $\beta >1$, there exists a constant $\bar c >0$ independent of
$\delta,\varepsilon$ and $m$ such that
$$
c_{\delta,\varepsilon,m} \leq  \sup_{(p,\xi)\in P_m
\times S^{N-2}}I_{\delta,\varepsilon,m}(A_0(p,\xi))  \leq  \bar c.
$$
(ii) follows clearly from \ref{eq:3.9}.
\\
(iii) Since $I_{\delta,\varepsilon,m}$ satisfies the (PS) condition
and property (ii) of Lemma 3.1, then by a standard argument using the
deformation theorem and (ii), we can see that $ c_{\delta,\varepsilon,m}>0$
is a critical value of $I_{\delta,\varepsilon,m}$.
By Lemma 2.2, we get (iii). \hfill$\diamondsuit$ \medskip

As a corollary to (i) of Lemma 3.1 and the uniform estimates of
$ c_{\delta,\varepsilon,m}$, we have the following statements.

\begin{corollary}
Let $(p_{\delta,\varepsilon, m},q_{\delta,\varepsilon, m})\in P_m\times \Lambda_m$ be a
critical point of $I_{\delta,\varepsilon, m} $ obtained by Proposition 3.2.Then, there exists a constant $C_2
>0$ independent of $\delta,\varepsilon $ and $m$, such that for all $\delta,\varepsilon \in ]0,1]$ and $m \in {
\mathbb N}$, we have
 $$\displaylines{
 \rlap{\rm (i)}\hfill
 \int _0^T |q_{\delta,\varepsilon,m}|^\alpha |p_{\delta,\varepsilon,m}|^\beta dt+\int _0^T |q_{\delta,
\varepsilon,m}|^\alpha dt \leq C_2, \hfill\cr
\rlap{\rm (ii)}\hfill
\varepsilon \int_0^T\big(|q_{\delta,\varepsilon,m}|^\gamma +|p_{\delta,
\varepsilon,m} |^\gamma\big) dt\leq C_2, \hfill\cr
\rlap{\rm (iii)}\hfill
 \delta \int_0^T {1\over{|q_{\delta,\varepsilon,m}|^\gamma}}dt \leq C_2. \hfill
 }$$
\end{corollary}

\section{Limiting process as $m \to \infty$}

\begin{proposition} For any $\delta, \varepsilon \in ]0,1] $,
$(p_{\delta,\varepsilon,m},q_{\delta,\varepsilon,
m})$ possesses a subsequence converging in $E$ to
$(p_{\delta,\varepsilon},q_{\delta,\varepsilon}) \in(P \times\Lambda)\cap E_0 $.
 Moreover,
\begin{eqnarray}
I_{\delta,\varepsilon}(p_{\delta,\varepsilon},q_{\delta,\varepsilon})
&\leq& \bar c, \label{eq:4.1} \\
I'_{\delta,\varepsilon}(p_{\delta,\varepsilon},q_{\delta,\varepsilon})&=&0\,.
\label{eq:4.2}
\end{eqnarray}
\end{proposition}

\paragraph{Proof.} By (ii) of Corollary 3.1, we can extract a subsequence - still indexed by $m$-
such that
$$(p_{\delta,\varepsilon, m},q_{\delta,\varepsilon, m} ) \rightharpoonup (p_{\delta,\varepsilon},
q_{\delta,\varepsilon}) \quad \hbox{weakly in}\; L^\gamma (0, T; \mathbb{R}^N).$$
We remark that $I_{\delta,\varepsilon, m}' (p_{\delta,\varepsilon,m}, q_{\delta,\varepsilon, m}) =0$ is
equivalent to
\begin{equation}
\dot q_{\delta,\varepsilon, m} = {\mathop{\rm proj}}_m[H_p (t,p_{\delta,\varepsilon, m},q_{\delta,\varepsilon , m})+
\varepsilon \gamma |p_{\delta, \varepsilon, m}|^{\gamma-2}p_{\delta,\varepsilon, m}] ,  \label{eq:4.3}
\end{equation}
\begin{equation}
\dot p_{\delta,\varepsilon, m} = \hbox{-proj}_m [H_q (t,p_{\delta,\varepsilon, m}, q_{\delta,\varepsilon , m}) +
\delta \gamma  {q_{\delta,\varepsilon, m} \over|q_{\delta,\varepsilon, m}|^{\gamma+ 2} }-\varepsilon \gamma
 |q_{\delta,\varepsilon, m}|^{\gamma -2}q_{\delta,\varepsilon, m}].\label{eq:4.4}\end{equation}
By (H4) and Lemma 2.3, we have from \ref{eq:4.3}
\begin{eqnarray*}
\|\dot q_{\delta,\varepsilon, m}\|_{\gamma\over {\gamma -1}} & \leq  & K_{\gamma \over{
\gamma -1}}[a_5\|(|q_{\delta,\varepsilon, m}|^ \alpha|p_{\delta,\varepsilon, m}|^{ (\beta -1)} )\|_{\gamma \over
{\gamma -1}} + a_5 \| q_{\delta,\varepsilon,m}\|_{\alpha{\gamma  \over{\gamma -1}}} ^\alpha \\&& +a_6\| q_
{\delta,\varepsilon,m}\|_{\gamma \over{\gamma -1}}+  \varepsilon\gamma\|p_{\delta,\varepsilon,m}
\|_\gamma  ^{\gamma -1} ].
\end{eqnarray*}
Using a H\"{o}lder's inequality and (i)-(ii) of Corollary 3.1, we can find a constant $C_3>0$
independent of  $m \in \mathbb{N}$, such that
$$\| q_{\delta,\varepsilon,m}\|_{W^{1,{\gamma \over {\gamma-1}}}(0,T;\mathbb{R}^N)} \leq C_3.$$
Thus we can see from (iii) of Corollary 3.1 that
\begin{equation}
q_{\delta,\varepsilon,m}\to q_{\delta,\varepsilon}\in \Lambda \hbox{  uniformly in } [0,T].
\label{eq:4.5}
\end{equation}
On the other hand, by (H5) and Lemma 2.3, we have from \ref{eq:4.4}
\begin{eqnarray*}
\|\dot p_{\delta,\varepsilon, m}\|_{\gamma \over {\gamma-1}} &\leq &  K_{\gamma \over
{\gamma-1}} [ \|A(q_{\delta,\varepsilon,m})|p_{\delta,\varepsilon,m}|^\beta \|_{\gamma \over
 {\gamma-1}}+\|A(q_{\delta,\varepsilon,m})\|_{\gamma \over {\gamma-1}} \\ &&+  \gamma \|\delta
{ q_{\delta,\varepsilon, m} \over|q_{\delta,\varepsilon, m}|^{\gamma+ 2} }-\varepsilon |q_{\delta,\varepsilon,
 m}|^{\gamma -2}q_{\delta,\varepsilon, m}\|_{\gamma \over {\gamma-1}}].\end{eqnarray*}
Using \ref{eq:4.5}, we find
$$\|p_{\delta ,\varepsilon,m}\|_{W^{1,{\gamma \over {\gamma-1}}}(0,T;
\mathbb{R}^N)}\leq C_4
$$
where $C_4>0$ is a constant independent of $m$.
The injection ${W^{1,{\gamma \over {\gamma-1}}}(0,T;\mathbb{R}^N)}
\subset L^\gamma (0,T;\mathbb{R}^N)$ is compact, thus we have
\begin{equation}
p_{\delta,\varepsilon,m}\to p_{\delta,\varepsilon} \hbox{ strongly in } L^\gamma (0,T
;\mathbb{R}^N) \hbox{ and uniformly in } [0,T].\label{eq:4.6}
\end{equation}
By (i) and (iii) of Proposition 3.2, we deduce that
$$\displaylines{
I_{\delta,\varepsilon}(p_{\delta,\varepsilon},q_{\delta,\varepsilon})=\lim_{m \to \infty}
I_{\delta,\varepsilon,m}(p_{\delta,\varepsilon,m},q_{\delta,\varepsilon,m})
\leq \bar c,\cr
I_{\delta,\varepsilon}'(p_{\delta,\varepsilon,},q_{\delta,\varepsilon})(h,k) =
\lim_{m \to \infty}I_{\delta,\varepsilon,m}'(p_{\delta,\varepsilon,m},
q_{\delta,\varepsilon,m})(h,k)=0
}$$
for all sums
$$h=\sum_{|j|\leq n}\theta_je^{{2i\pi jt}\over T} \;,\; k=\sum_{|j|\leq n}\psi_je^{{2i \pi jt}\over T}
\quad(\theta_j,\psi_j \in  {\bf C}^N ).$$
Therefore,
$I_{\delta,\varepsilon}'(p_{\delta,\varepsilon},q_{\delta,\varepsilon})
(h,k)=0$ for all $ (h,k) \in E$.

\section{Limiting process as $\varepsilon \to 0$}

We take the limit as $\varepsilon \to 0$ to obtain a critical point $(p_\delta,q_\delta )\in( P \times \Lambda)
\cap E_0  $ of $I_\delta $ with uniform upper bound for critical values.
As a consequence to Corollary 3.1, and \ref{eq:4.5}, \ref{eq:4.6} we have the
following lemma.

\begin{lemma} For any $\delta, \varepsilon \in
 ]0,1],\; \big(p_{\delta,\varepsilon},q_{\delta,\varepsilon} )\in( P \times \Lambda)\cap E_0$
 satisfies
$$\displaylines{
\rlap{\rm (i)}\hfill
\int _0^T |q_{\delta,\varepsilon}|^\alpha |p_{\delta,\varepsilon}|^\beta dt+\int _0^T |q_{\delta,\varepsilon}
|^\alpha dt \leq C_2, \hfill\cr
\rlap{\rm (ii)}\hfill
\varepsilon \int_0^T \big(|q_{\delta,\varepsilon}|^\gamma+|p_{\delta,\varepsilon}
 |^\gamma \big)dt\leq C_2, \hfill\cr
 \rlap{\rm (iii)}\hfill
 \delta \int_0^T {1\over {|q_{\delta,\varepsilon}|^\gamma}}dt \leq C_2. \hfill
}$$
\end{lemma}

\begin{proposition}
For any $\delta \in ]0,1]$, $(p_{\delta,\varepsilon},q_{\delta,\varepsilon})$
possesses a  subsequence converging in $E$ to
$(p_\delta,q_\delta)\in (P \times \Lambda)\cap E_0  $. Moreover,
$$I'_\delta (p_\delta, q_\delta)=0,$$
$$I_\delta (p_\delta, q_\delta)\leq \bar c.$$
\end{proposition}

\paragraph{Proof.} Since $I'_{\delta,\varepsilon}(p_{\delta\varepsilon}, q_{\delta,\varepsilon})=0$, we have
\begin{equation}
\dot q_{\delta,\varepsilon} =  H_p(t,p_{\delta,\varepsilon},q_{\delta,\varepsilon })+\varepsilon \gamma |
p_{\delta, \varepsilon}|^{\gamma-2}p_{\delta,\varepsilon} ,  \label{eq:5.1}\end{equation}
\begin{equation}
\dot p_{\delta,\varepsilon}  =   -[H_q(t,p_{\delta,\varepsilon}, q_{\delta,\varepsilon }) +\delta \gamma  {
q_{\delta,\varepsilon} \over|q_{\delta,\varepsilon}|^{\gamma+ 2} }-\varepsilon \gamma |q_{\delta,\varepsilon}
|^{\gamma -2}q_{\delta,\varepsilon}].\label{eq:5.2}\end{equation}
By (H4) and \ref{eq:5.1}, we can see from (i)-(ii) of Lemma 5.1 that
\begin{eqnarray*}
\int_0^T |\dot q_{\delta,\varepsilon}|dt  &\leq &a_5[  \int_0^T|q_{\delta,\varepsilon}|^ \alpha |p_{\delta,
\varepsilon}|^{\beta -1}dt +\int_0^T |q_{\delta,\varepsilon}|^\alpha dt ]+a_6 \int_0^T |q_{\delta,\varepsilon}|dt\\
&&+\varepsilon \gamma \int_0^T |p_{\delta,\varepsilon}|^{\gamma -1}dt\\
&\leq& C_5
  \end{eqnarray*}
where $C_5>0$ is a constant independent of $\varepsilon$.
Thus, we deduce that $(q_{\delta,\varepsilon}
)_\varepsilon$ is bounded in
$L^\infty (0,T;\mathbb{R}^N)$.

\noindent By (H4) and (\ref{eq:5.1}) again, we have
\begin{eqnarray*}
||\dot q_{\delta,\varepsilon}||_{\gamma\over {\gamma
-1}} & \leq  & a_5||(|q_{\delta,\varepsilon}|^
\alpha|p_{\delta,\varepsilon }|^{ \beta -1} )||_{\gamma \over
{\gamma -1}} + a_5 || q_{\delta,\varepsilon}||_{\alpha{\gamma
\over{\gamma -1}}} ^\alpha
\\&& +a_6|| q_ {\delta,\varepsilon}||_{\gamma \over{\gamma -1}}+
\varepsilon\gamma||p_{\delta,\varepsilon} ||_\gamma  ^{\gamma -1}.
\end{eqnarray*}
Here we will apply the H\"{o}lder's inequality $$||fg||_s \leq
||f||_{s\mu} ||g||_{s\nu}$$ with $f(t)=|q_{\delta,\varepsilon
}|^{\alpha \over \beta},\; g(t)=(|q_{\delta,\varepsilon} |^\alpha
|p_{\delta,\varepsilon}|^{\beta})^{{\beta-1}\over \beta},\;
s={\gamma \over {\gamma-1}},\;\mu={{(\gamma-1)\beta}\over \alpha}$
and $\nu={{(\gamma-1)\beta}\over {(\beta -1)\gamma}}$.

\noindent We verify that ${1\over\mu}+{1\over \nu} =1$. Then we
have
\begin{eqnarray*}
||(|q_{\delta,\varepsilon}|^\alpha |p_{\delta,\varepsilon
}|^{\beta-1})||_{\gamma \over
 {\gamma-1}}&=& ||(|q_{\delta,\varepsilon}|^{\alpha \over \beta})
(|q_{\delta,\varepsilon}|^\alpha |p_{\delta,\varepsilon
}|^{\beta})^{{\beta-1}\over \beta} ||_{\gamma \over {\gamma-1}}\\
&\leq& ||(|q_{\delta,\varepsilon}|^{\alpha \over
\beta})||_{{\gamma \beta}\over \alpha} || (|q_{\delta,\varepsilon
}|^\alpha |p_{\delta,\varepsilon}|^{\beta}) ^{{\beta-1}\over
\beta}||_{\beta \over {\beta-1}}\\ &=&||q_{\delta,\varepsilon
}||_\gamma ^{\alpha \over \beta} ||(|q_{\delta,\varepsilon}
|^\alpha |p_{\delta,\varepsilon}|^{\beta})||_1^{{\beta-1}\over
\beta}\\ &\leq& C_6
\end{eqnarray*}
where $C_6>0$ is a constant independent of $\varepsilon$.

\noindent Finally $(q_{\delta,\varepsilon})_\varepsilon$ is
bounded in $W^{1,{\gamma \over {\gamma-1}}}(0,T;{\bf R}^N)$. That
is we can extract a subsequence -still indexed by $\varepsilon$-
such that
\begin{equation}q_{\delta,\varepsilon}\to q_\delta \in \Lambda
\quad {\hbox {uniformly in}}\; [0,T]. \label{eq:5.3}\end{equation}
Since $\int _0^T|q_{\delta,\varepsilon}|^\alpha |p_{\delta,\varepsilon}|^\beta dt\leq C_2$, we get
\begin{equation}
\int _0^T |p_{\delta,\varepsilon}|^\beta dt\leq C_7 \label{eq:5.4}
\end{equation}
for some constant  $C_7>0$ independent of $\varepsilon$.
By (H5) and \ref{eq:5.2}-\ref{eq:5.4}, there exists a constant $C_8>0$
independent of $\varepsilon$ such that
\begin{eqnarray*}
\int_0^T |\dot p_{\delta,\varepsilon}|dt
&\leq &
\int_0^T A(q_{\delta,\varepsilon})\big ( |p_{\delta,\varepsilon}| ^\beta
+1\big)dt + \gamma \int _0 ^T \big( {1 \over{|q_{\delta,\varepsilon}|
^{\gamma +1}}} + |q_{\delta,\varepsilon}|^{\gamma -1}\big )dt \\
 & \leq & C_8 \end{eqnarray*}
and
$$\int_0^T |\dot p_{\delta,\varepsilon}|^\gamma dt  \leq C_8.$$
So we can extract a subsequence -still indexed by $\varepsilon$- such that
\begin{equation}p_{\delta, \varepsilon} \to p_\delta \; {\hbox{strongly in}} \;L^\gamma (0,T;\mathbb{R}) \hbox{ and
uniformly in } [0,T].\label{eq:5.5} \end{equation}
By \ref{eq:5.3} and \ref{eq:5.5}, a passage to the limit on \ref{eq:4.1}-\ref{eq:4.2} similar as in Section 4 completes the proof.

\section{Proof of Theorem 1.1}

We take a limit as $\delta \to 0$ to obtain a $T$-periodic solution of (\ref{HS}). Let $(p_\delta,q_\delta)\in(
P \times\Lambda)\cap E_0$ be a critical point of $I_\delta (p,q)$ obtained by Proposition 5.1.
By Lemma 5.1, \ref{eq:5.3} and \ref{eq:5.5}, we have
\begin{lemma}
For any $\delta \in ]0,1]$,
$$ \displaylines{ \rlap{\rm (i)}\hfill
\int _0^T |q_\delta|^\alpha |p_{\delta}|^\beta dt+\int _0^T |q_{\delta}|
^\alpha dt \leq C_2, \hfill\cr
\rlap{\rm (ii)}\hfill
\delta \int_0^T {1\over {|q_{\delta}|^\gamma}}dt\leq C_2. \hfill
}$$
\end{lemma}
By (i) of Lemma 6.1, we can extract a subsequence -still indexed by $\delta$- such that
$$q_\delta \rightharpoonup q \quad {\hbox {weakly in }}\; L^\alpha (0,T;\mathbb{R}^N).$$
We also remark that $I'_\delta (p_\delta ,q_\delta)=0$ is equivalent to
\begin{equation}\dot q_\delta=H_p(t,p_\delta,q_\delta), \label{eq:6.1}\end{equation}
\begin{equation}\dot p_\delta =-[H_q(t,p_\delta,q_\delta)+\delta\gamma {q_\delta  \over  {|q_\delta|^{
\gamma+2}}}]. \label{eq:6.2}\end{equation}
\begin{lemma}$q_\delta \to q\in \Lambda$ uniformly in $[0,T].$ \end{lemma}
\paragraph{Proof.}
By (H4) and \ref{eq:6.1}, we have
$$\int _0^T |\dot q _\delta |dt \leq a_5 \int_0^T |q_\delta |^\alpha |p_\delta |^{\beta-1}    dt+ a_5 \int _0^T
|q_\delta|^\alpha dt+a_6 \int _0^T |q_\delta|dt.$$
Using (i) of Lemma 6.1, we can see that
$\|q_\delta\|_{W^{1,1}(0,T;\mathbb{R}^N)}$ is bounded.
Thus we can find a constant $C_9>0$ independent of $\delta$, such that
$$\int _0^T |\dot q_\delta |^{\beta \over {\beta -1}}dt \leq C_9.
$$
Consequently, we obtain $q_\delta \to q $ uniformly in $[0,T]$.

\noindent
We now argue indirectly and suppose that
$$q(t_0)=0 \quad {\hbox{for some }}\; t_0 \in [0,T].$$
We may assume $t_0=0.$ By \ref{eq:6.1}, for any $t \in ]0,T]$ we have
\begin{equation}
|\log |q_\delta(t)|-\log |q_\delta (0)\|
\leq  \int_0^t{{|\dot q_\delta (s)|} \over{|q_\delta (s)|}}ds
= \int_0^t{{|H_p(s,p_\delta ,q_\delta )|}\over{|q_\delta |}}ds.\label{eq:6.3}
\end{equation}
By (H4),
$$\int_0^t{{|H_p(s,p_\delta ,q_\delta )|}\over{|q_\delta |}}ds\leq a_5\int_0^t |q_\delta|^{\alpha-1}|p_\delta |^{
\beta -1}ds + a_5 \int _0^t |q_\delta |^{\alpha-1}ds+a_6 T. $$
 Since $\alpha > \beta >1$ and $\int_0^T |q_\delta |^\alpha|p_\delta |^\beta dt \leq C_2$, there exists a
constant $C_{10}>0$ independent of $\delta$, such that
\begin{equation}\int_0^t{{|H_p(s,p_\delta ,q_\delta )|}\over{|q_\delta |}}ds \leq C_{10}.  \label{eq:6.4}
\end{equation}
Passing to the limit in \ref{eq:6.3}, we see that
$q_\delta  \to 0$ uniformly in  $[0,T]$.
By \ref{eq:6.1}-\ref{eq:6.2}, we have
\begin{eqnarray*}
I_\delta (p_\delta,q_\delta) &=& \int _0^TH_p(t,p_\delta ,q_\delta)p_\delta dt-\int _0^TH(t,p_\delta,q_\delta)
dt+\delta \int _0^T {1\over {|q_\delta|^\gamma }}dt \\
\nonumber&=& \int_0^T H_q(t,p_\delta,q_\delta)q_\delta dt-\int_0^T H(t,p_\delta,q_\delta)dt+\delta (
\gamma +1) \int_0^T{1\over  {|q_\delta|^\gamma }}dt .
\end{eqnarray*}
Hence
$$\int_0^T[H_q(t,p_\delta,q_\delta)q_\delta - H_p(t,p_\delta,q_\delta)p_\delta]dt+\delta \gamma \int_0^T {
1 \over {|q_\delta|}}dt=0.$$
From (H6)(i) and (H2)(ii), it follows that
$$a_7 a_2 \int_0^T |q_\delta |^\alpha |p_\delta |^\beta dt -a_7 \int_0^T K_1(q_\delta )dt
+\int_0^T K_2(q_\delta )dt+\delta\gamma   \int_0^T {1 \over{|q_\delta|^\gamma}}dt \leq 0$$
 for small $\delta$.
Since $q_\delta \to 0$ uniformly in $[0,T]$, we find
\begin{equation}\int _0^T |q_\delta |^\alpha |p_\delta |^\beta dt \to 0 \; {\hbox{as}} \;  \delta \to 0.
 \label{eq:6.5}
\end{equation}
Thus we can see from \ref{eq:6.1}, \ref{eq:6.5} and (H6)(ii),
\begin{eqnarray}
\nonumber
\int_0^T {|\dot q_\delta |^{\beta \over {\beta -1}}\over |q_\delta |^{\alpha \over {\beta -1}}}
dt&=&\int_0^T{{|H_p(t,p_\delta ,q_\delta )|^{\beta \over {\beta -1}}}\over{|q_\delta |^{\alpha
\over {\beta -1}}}}dt\\
&\leq&a_8\int_0^T [|q_\delta |^\alpha |p_\delta |^\beta +K_3(q_\delta)]dt
\nonumber\\
&& \to 0 \hbox{ as } \delta \to 0.\label{eq:6.6}
\end{eqnarray}
In other hand, we have from Lemma 2.1
\begin {eqnarray*}
\int_0^T {|\dot q_\delta |^{\beta \over {\beta -1}}\over |q_\delta |^{\alpha \over {\beta -1}}}dt &\geq&{{ \big(
\int_0^T | \dot q_\delta|dt\big)^{\beta \over {\beta-1}}}\over {T^{1\over {\beta-1}}\|q_\delta \|_\infty ^{\alpha
 \over {\beta -1}}}}\\&\geq&{1\over {T^{1\over{ \beta-1}}\|q_\delta \|_\infty ^{{\alpha-\beta} \over {\beta -1}}}}\\
&&\to +\infty \hbox{  as  } \delta \to 0.
\end{eqnarray*}
This is a contradiction to  \ref{eq:6.6} which proves the Lemma 6.2.


\begin{lemma}
There exists a constant $C_{11}$ independent of $\delta  \in ]0,1]$
such that
$$\|p_\delta\|_{W^{1,\gamma}(0,T;\mathbb{R}^N)} \leq C_{11}.$$
 \end{lemma}

\paragraph{Proof.} Since $q_\delta \to q \in \Lambda $ uniformly in $[0,T]$ and $\int _0^T |q_\delta|^\alpha |
p_\delta|^\beta dt \leq C_2$, there exists a constant $C_{12}>0$ independent of
$\delta \in]0,1]$ such that
$$ \int _0^T |p_\delta|^\beta dt \leq C_{12}.$$
By (H5) and \ref{eq:6.2}, one deduce that $\int _0^T |\dot p_\delta|dt $
is bounded.
Thus we can see for some constant $C_{11}>0$ independent of $\delta \in ]0,1]$
$$\|p_\delta\|_{W^{1,\gamma}(0,T;\mathbb{R}^N)} \leq C_{11}.$$
We complete the proof of Theorem 1.1 as follows:
By Lemmas 6.2 and 6.3, we can extract a subsequence -still
indexed by $\delta$- such that $p_\delta \to p$ strongly in
$L^\gamma (0,T;\mathbb{R}^N)$ and
$(p_\delta,q_\delta) \to (p,q) \in (P\times \Lambda )\cap E_0$ uniformly
in $[0,T]$. Since $I'_\delta (p_\delta,q_\delta)=0$, we get
$$I'(p,q)(h,k)=0 \quad {\hbox {for all }}\; (h,k) \in E.$$
That is $(p,q) \in( P \times \Lambda)\cap E_0 $ is a non-constant
$T$-periodic solution of (\ref{HS}).

\section{Remarks on the prescribed energy problem}

If $H(t,p,q)$ does not depend on $t$, then the energy surface
$$S_h=H^{-1}(h)=\{(p,q)\in
\mathbb{R}^N\times \mathbb{R}^N ; \; H(p,q)=h\}\; (h>0)$$
is not compact for such Hamiltonian functions. Moreover, $S_h$ is equal to $$\tilde H ^{-1}(1)=\{(p,q)
\in\mathbb{R}^N\times \mathbb{R}^N\setminus \{0\}; \;  \tilde H(p,q)=1\}$$
where
\begin{equation}
\tilde H(p,q)={ {H(p,q)-h} \over {|q|^\alpha}}+1.\label{eq:7.1}
\end{equation}
It is clear that, if
$H(p,q) \sim |q|^\alpha(|p|^\beta -1)$, then
\begin{equation}
\tilde H (p,q) \sim |p|^\beta - {h\over {|q|^\alpha}}.\label{eq:7.2}
\end{equation}
In the last few years, the existence of periodic solutions of singular
Hamiltonian systems has been studied via variational methods under the
situation related to two-body problem in celestial mechanics.
That is, situation  $\tilde H(p,q)$ is of the form
$$\tilde H(p,q)={1\over 2}|p|^2+V(q)
$$
where $V(q)\in C^1( \mathbb{R}^N\setminus \{0\},\mathbb{R})$
and $V(q)\to -\infty $ as $q\to 0$. See \cite{G,Ba-R,A-Z1} and references
therein. Results dealing with more general singular Hamiltonians of the
form (\ref{eq:7.2}) can be found in  \cite{T1,Bou1} for fixed period
problems,  and in  \cite {T-S-C,Bou2} for fixed energy problems.

According to the fundamental lemma of Rabinowitz ( see \cite{R}
and \cite[lemma 3.1]{MW}),
it follows that the Hamiltonian system (\ref{HS}) has, for $H$ and
$\tilde H$ which are related by
\ref{eq:7.1}, the same orbits on $S_h$. Therefore, under suitable conditions
on $H$ including $|q|^\alpha (|p|^\beta -1)$ with $\alpha >\beta>1$, the  theorem of \cite{T-S-C} carries a non-collision orbit of
the singular Hamiltonian system
$$\displaylines{\dot q =\tilde H_p (p(t),q(t))\cr
\dot p = -\tilde H_q(p(t), q(t))\cr
\tilde H(p,q) =1,}$$
which corresponds to a non-constant periodic solution of (\ref{HS}) with
energy $h$.

\section*{Appendix: Proof of Lemma 3.2}

The proof of Lemma 3.2 is a special case of \cite[lemma 3.1]{T1}.
We fix $A \in \Gamma_{m} $ and take $R>0$ such that
$$R> \lambda \max _{\xi \in S^{N-2}}\|{\mathop{\rm proj}}_m|
{d \over {dt}}(\sigma_0(\xi))(t)|^{{1\over {\gamma -1}}-1}{d \over {dt}}
(\sigma_0(\xi))(t)\|_\beta,$$
$$A (p,\xi)=(p,\sigma_0 (\xi))\quad{\hbox{if}} \; \; \|p\|_\beta \geq R.
$$
We note that
$$A(p,\xi)=\big(x(p,\xi),y(p,\xi)\big),\eqno{(A.1)}$$
$$B(\rho)=\{ p \in P_m;\; \|p\|_\beta \leq \rho \},\quad \rho >0.$$
Then we define the function $\phi(\rho)\in C(\mathbb{R},[0,1])$ such that
$$ \phi(\rho)=\left \{  \begin{tabular}{l}$1,\quad \rho\leq R,$ \\$0, \quad \rho\geq 2R. $\end{tabular}\right.$$
Using the notation (A.1), we define a  mapping
$$F:P_m\times S^{N-2}\times [0,T]/{\{0,T}\} \sim  P_m \times S^{N-2}\times S^1 \to P_m
 \times S^{N-1}$$
by
$$F(p,\xi,t)=\big(x(p,\xi)-\lambda\phi(\|p\|_\beta){{\mathop{\rm proj}}}_m(|\dot y(p,\xi)|^{{1\over
{\gamma-1}}-1}\dot y(p,\xi)),\tilde{\sigma}(\xi)(t) \big)$$
where $\tilde{\sigma}(\xi)(t) ={{\sigma (\xi)(t)}\over |{\sigma (\xi)(t)}|}$ and
$$ \sigma(\xi)(t) =(3+\hbox{cos} {2\pi t\over T})({\xi}_1,\dots ,{\xi}_{N-1},0)-(3,0,\dots ,0)+(0,\dots ,0,\hbox{sin} {2\pi
t\over T }).$$
We remark that $F(p, \xi,t)=(p, \tilde \sigma (\xi)(t))$ for $\|p\|_\beta \geq 2R$ and the
degree of the map $\tilde \sigma:S^{N-2} \times  S^1 \to S^{N-1}$ is not equal to zero.

\noindent
Thus, there exists $R' \geq 2R$ such that the degree of the mapping
$$F : \big( B(R')\times S^{N-2}\times S^1;\partial B(R') \times S^{N-2}  \times
S^1\big) \to \big(B(R') \times S^{N-1}; \partial B(R') \times S^{N-1}\big)$$
is not equal to zero.
Then it follows the existence of $(p,\xi)$ such that
$$x(p,\xi)-\lambda \phi (\|p\|_\beta) {{\mathop{\rm proj}}}_m
\big(|\dot y(p,\xi)|^{{1\over{\gamma-1}}-1}\dot y(p,\xi)\big)=0.
$$
By the definition of $R$, we have necessarily $\|p\|_\beta \leq R$. That is
$$x(p,\xi)=\lambda {\mathop{\rm proj}}_m\big(|\dot y(p,\xi)|^{{1\over{\gamma-1}}-1}\dot y(p,\xi)\big)$$
and then
$$A (P_m\times S^{N-2})\bigcap {\cal D}_{m,\lambda} \not =\emptyset.$$

\paragraph{Acknowledgments.}
The author wishes to thank Professors Abbas Bahri,
Leila Lasssoued, and Eric S\'er\'e for their helpful discussions .

\begin{thebibliography}{99} {\frenchspacing

\bibitem{R} P. H. Rabinowitz, {\it Periodic solutions of Hamiltonian systems},
Comm.  Pure Appl.  Math., 31,1978, pp.157-184.

\bibitem{BR} V.  Benci and P.  H.  Rabinowitz, {\it Critical point theorems for
indefinite functionals}, Inventiones, Vol.52, fasc.2, 1979, pp.241-274.

\bibitem{E} I.  Ekeland, {\it Periodic solutions of Hamiltonian equations and a
theorem of Rabinowitz}, J.  Diff.Eq.  , Vol.34, 1979, pp.523-534.

\bibitem{CE} F.  H.  Clarke and I.  Ekeland, {\it Hamiltonian trajectories
having prescribed minimal period}, Comm.  Pure Appl.  Math., 33, 1980,
pp.103-116.

\bibitem{B} H.  Berestycki, {\it Solutions p\'eriodiques de syst\`emes
 hamiltoniens}, S\'eminaire Bourbaki (603), 1983.

\bibitem{EG} R.  E.  Edwards and G.  I.  Gandry, ``Littlewood-Paleyand
multiplier theory", Springer-Verlag, Berlin, Heidelberg, NewYork,1977.

\bibitem{T1} K. Tanaka, {\it Periodic solutions of first order singular
Hamiltonian systems},  Nonlinear Analysis: T.M.A. 26, (1996) 691-704.

\bibitem{G} C.  Greco, {\it Periodic solutions of a class of singular
Hamiltonian systems}, Nonlinear Analysis; T.M.A.  12 (1988), 259-269.

\bibitem{Ba-R} A.  Bahri and P.  H.  Rabinowitz, {\it A minimax method for a
class of Hamiltonian systems with singular potentials}, Journal of functional
analysis, 82, 412-428 (1989).

\bibitem{A-Z1} A. Ambrosetti and V. Coti Zelati, ``Periodic solutions of
singular Lagrangian systems", Birkhauser, Boston, Basel, Berlin, 1993.

\bibitem{Bou1} M.  Boughariou, {\it Generalized solutions of first order
singular Hamiltonian systems}, Nonlinear Analysis:  T.M.A.  31, (1998) 431-444.

\bibitem{T-S-C} K. Tanaka, E. Sere and C. Carminati, {\it A prescribed energy
problem for first order singular Hamiltonian systems}, preprint.

\bibitem{Bou2} M.  Boughariou, {\it Solutions p\'eriodiques g\'en\'eralis\'ees
\`a \'energie fix\'ee pour un syst\`eme hamiltonien singulier du premier ordre,}
C.R.Acad.Sci.Paris t.328, S\' erieI, (1999) p.  17-22.

\bibitem{MW} J.  Mawhin and M.  Willem, ``Critical point Theory and Hamiltonian
systems", App.  Math.  Sci.  74, Springer-Verlag.

}\end{thebibliography}

\noindent\textsc{Morched Boughariou}\\
Facult\'e des Sciences de Tunis \\
D\'epartement de Math\'ematiques\\
Campus Universitaire, 1060 Tunis, Tunisie.\\
e-mail: Morched.Boughariou@fst.rnu.tn
\end{document}