\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 $$I'_{\delta,\varepsilon,m}(p,q)(h,k)=0 \quad \hbox{for all } (h,k)\in E_m. \label{eq:2.1}$$ 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 $$I'_{\delta,\varepsilon,m}(p_{\delta,\varepsilon,m},q_{\delta, \varepsilon,m})=0,\label{eq:2.2}$$ $$I_{\delta,\varepsilon,m}(p_{\delta,\varepsilon,m},q_{\delta,\varepsilon,m})\leq \bar c \label{eq:2.3}$$ 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 $$I_\delta (p_\delta,q_\delta)\leq \bar c. \label{eq:2.4}$$ 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 $(p,q) \in P_m\times \Lambda _m$ $$I_{\delta,\varepsilon,m}(p,q) \geq \int_0^T [ p\dot q-a_1 |q|^\alpha |p|^\beta +\varepsilon (|q|^\alpha-|p|^\gamma)] dt+\delta \int _0^T{1\over{|q|^\gamma}}dt. \label{eq3.3'}$$ 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 $$I_{\delta,\varepsilon,m}(p_j,q_j) \to c, \label{eq:3.4}$$ $$\|I'_{\delta,\varepsilon,m}(p_j,q_j)\|_{E_m^\star}\to 0. \label{eq:3.5}$$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 $(p,q) \in {\cal D}_{m,\lambda}$, $$\int _0^T p \dot q dt = \lambda \int _0^T |\dot q|^{{\gamma\over {\gamma -1}}}dt. \label{eq:3.6}$$ Moreover, by Lemma 2.1 and Lemma 2.3 $$T^{1\over \gamma} \|\dot q\|_{\gamma \over {\gamma -1}} \geq \int_0^T |\dot q|dt \geq\|q\|_\infty, \label{eq:3.7}$$ $$\int_0^T |p|^\gamma dt = \lambda ^\gamma \|{{\mathop{\rm proj}}}_m(|\dot q|^{{1\over {\gamma -1}}-1} \dot q )\|_\gamma^\gamma \leq \lambda ^\gamma K_{\gamma} ^\gamma\|\dot q\|_{\gamma \over {\gamma -1}}^{\gamma \over {\gamma-1}}. \label{eq:3.8}$$ 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 $$\dot q_{\delta,\varepsilon, m} = {\mathop{\rm proj}}_m[H_p (t,p_{\delta,\varepsilon, m},q_{\delta,\varepsilon , m})+ \varepsilon \gamma |p_{\delta, \varepsilon, m}|^{\gamma-2}p_{\delta,\varepsilon, m}] , \label{eq:4.3}$$ $$\dot p_{\delta,\varepsilon, m} = \hbox{-proj}_m [H_q (t,p_{\delta,\varepsilon, m}, q_{\delta,\varepsilon , m}) + \delta \gamma {q_{\delta,\varepsilon, m} \over|q_{\delta,\varepsilon, m}|^{\gamma+ 2} }-\varepsilon \gamma |q_{\delta,\varepsilon, m}|^{\gamma -2}q_{\delta,\varepsilon, m}].\label{eq:4.4}$$ 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 $$q_{\delta,\varepsilon,m}\to q_{\delta,\varepsilon}\in \Lambda \hbox{ uniformly in } [0,T]. \label{eq:4.5}$$ 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 $$p_{\delta,\varepsilon,m}\to p_{\delta,\varepsilon} \hbox{ strongly in } L^\gamma (0,T ;\mathbb{R}^N) \hbox{ and uniformly in } [0,T].\label{eq:4.6}$$ 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 $$\dot q_{\delta,\varepsilon} = H_p(t,p_{\delta,\varepsilon},q_{\delta,\varepsilon })+\varepsilon \gamma | p_{\delta, \varepsilon}|^{\gamma-2}p_{\delta,\varepsilon} , \label{eq:5.1}$$ $$\dot p_{\delta,\varepsilon} = -[H_q(t,p_{\delta,\varepsilon}, q_{\delta,\varepsilon }) +\delta \gamma { q_{\delta,\varepsilon} \over|q_{\delta,\varepsilon}|^{\gamma+ 2} }-\varepsilon \gamma |q_{\delta,\varepsilon} |^{\gamma -2}q_{\delta,\varepsilon}].\label{eq:5.2}$$ 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 $$q_{\delta,\varepsilon}\to q_\delta \in \Lambda \quad {\hbox {uniformly in}}\; [0,T]. \label{eq:5.3}$$ Since $\int _0^T|q_{\delta,\varepsilon}|^\alpha |p_{\delta,\varepsilon}|^\beta dt\leq C_2$, we get $$\int _0^T |p_{\delta,\varepsilon}|^\beta dt\leq C_7 \label{eq:5.4}$$ 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 $$p_{\delta, \varepsilon} \to p_\delta \; {\hbox{strongly in}} \;L^\gamma (0,T;\mathbb{R}) \hbox{ and uniformly in } [0,T].\label{eq:5.5}$$ 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 $$\dot q_\delta=H_p(t,p_\delta,q_\delta), \label{eq:6.1}$$ $$\dot p_\delta =-[H_q(t,p_\delta,q_\delta)+\delta\gamma {q_\delta \over {|q_\delta|^{ \gamma+2}}}]. \label{eq:6.2}$$ \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 $$|\log |q_\delta(t)|-\log |q_\delta (0)\| \leq \int_0^t{{|\dot q_\delta (s)|} \over{|q_\delta (s)|}}ds = \int_0^t{{|H_p(s,p_\delta ,q_\delta )|}\over{|q_\delta |}}ds.\label{eq:6.3}$$ 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 $$\int_0^t{{|H_p(s,p_\delta ,q_\delta )|}\over{|q_\delta |}}ds \leq C_{10}. \label{eq:6.4}$$ 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 $$\int _0^T |q_\delta |^\alpha |p_\delta |^\beta dt \to 0 \; {\hbox{as}} \; \delta \to 0. \label{eq:6.5}$$ 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 $$\tilde H(p,q)={ {H(p,q)-h} \over {|q|^\alpha}}+1.\label{eq:7.1}$$ It is clear that, if $H(p,q) \sim |q|^\alpha(|p|^\beta -1)$, then $$\tilde H (p,q) \sim |p|^\beta - {h\over {|q|^\alpha}}.\label{eq:7.2}$$ 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. 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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}