\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Non-collision solutions for Lagrangian systems \hfil EJDE--2000/75} {EJDE--2000/75\hfil Morched Boughariou \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~75, pp.~1--10. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Non-collision solutions for a class of planar singular Lagrangian systems % \thanks{ {\em Mathematics Subject Classifications:} 35D05, 35D10, 58E30. \hfil\break\indent {\em Key words:} Singular Lagrangian system, periodic solution, non-collision. \hfil\break\indent \copyright 2000 Southwest Texas State University. \hfil\break\indent Submitted September 18, 2000. Published December 15, 2000.} } \date{} % \author{ Morched Boughariou } \maketitle \begin{abstract} In this paper, we show the existence of non-collision periodic solutions of minimal period for a class of singular second order Hamiltonian systems in $\mathbb{R}^2$ with weak forcing terms. We consider the fixed period problem and the fixed energy problem in the autonomous case. \end{abstract} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Definition}{Definition}[section] \newtheorem{Proposition}{Proposition}[section] \newtheorem{Lemma}{Lemma}[section] \newtheorem{Corollary}{Corollary}[section] \newtheorem{Remark}{Remark}[section] \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \section {Introduction and statement of results} This paper deals with the existence of non-collision periodic solutions of minimal period for the problem $$\ddot q+ V_q(t,q)=0$$ where $q \in \mathbb{R}^N \setminus \{0\}$ with $N=2$, the potential $V$ is of the form $V(t,q)=-{1\over{|q|^\alpha}}+W(q)$ in a neighborhood of $q=0$ with $1<\alpha <2$ and $W$ is such that $ |q|^{\alpha} W(q), \; |q|^{\alpha+1} W'(q) \to 0 $ as $|q| \to 0$. \medskip We will consider to cases: {\it the fixed period problem} $$ \begin{array} {c} \ddot q+ V_q(t,q)=0\\[2pt] q(t+T)=q(t),\end{array}\eqno (P_T) $$ and {\it the fixed energy problem} (autonomous case) $$ \begin{array} {c} \ddot q+ V'(q)=0 \\[2pt] {1\over 2} |\dot q|^2 +V(q)=h\\[2pt] q {\hbox{ periodic.}} \end{array} \eqno(P_h)$$ The case $\alpha \geq 2$ ``Strong force" and $N\geq 2$ has been studied by many authors. The existence of classical (non-collision) solutions of $(P_T)$ and $(P_h)$ has been proved via variational methods( See \cite{A-CZ1, B-R, G1, G, P}). The case $0<\alpha <2$ ``weak force" is more complicated because the lose of control of the functional, whose critical points correspond to periodic solutions on the functions passing through the origin. Recently, there has been several works which deal with these two problems for $N\geq 3$( See also \cite{A-CZ1, A-CZ2, A-S, CZ-S,T1,T2}). In our situation ($N=2$), we refer for the study of $(P_T)$ to Degiovanni-Giannoni \cite{D-G}, Ambrosetti-Coti Zelati \cite{A-CZ3}, Serra-Terracini \cite{S-T} where they treated also case of $N \geq 3$, and to Coti Zelati \cite{CZ}. In \cite{D-G}, they obtained the existence of classical solutions under a global conditions like \begin{equation} {a\over{|q|^\alpha}}\leq -V(q)\leq {b\over{|q|^\alpha}},\;\;\forall \; q \not= 0.\label{C} \end{equation} In \cite{A-CZ3}, they found solutions of large period $T$. In \cite{S-T}-\cite{CZ}, they used a radially symmetric assumption on $V$ in a neighborhood of the singularity in order to get a non-collision solution of $(P_T)$. For the study of $(P_h)$, we know the result of Benci-Giannoni \cite{B-G} where the existence of classical solution strongly depend on the pertubation $W$. The other result has been obtained by Coti Zelati-Serra \cite{CZ-S2}. There arguments are based on the fact that the topology of $\{V \leq h \}$ is non trivial; We remark that the case $V(q)=-{1\over {|q|^\alpha }}$ is excluded in this work. In the present paper, we are able to find estimates in minima of suitable minimisation perturbed problems using a re-scaling argument. Such estimates give actually non-collision solutions with minimal period to our problems without assuming a radially symmetric condition on $V$. More precisely, in section 2, we study the fixed period problem; We deal with non-autonomous potentials $V$ satisfying the hypotheses: \begin{enumerate} \item[(V0)] $V\in C^1(\mathbb{R}\times \mathbb{R}^N \setminus \{0\};\mathbb{R})$ and $T$-periodic in $t$; \item[(V1)] $V(t,q)<0 ,\; \forall \;(t,q)\in [0,T]\times \mathbb{R}^N \setminus \{0\}$; \item[(V2)] $|{\partial V \over \partial t }(t,q)|\leq -V(t,q) ,\; \forall \; (t,q) \in [0,T] \times \mathbb{R}^N \setminus \{0\};$ \item[(V3)] There exist $r>0,1<\alpha<2$ and $W \in C^1( \mathbb{R}^N \setminus \{0\},\mathbb{R})$ satisfying $|q|^{\alpha}W(q), \; |q|^{\alpha+1}W'(q) \to 0$ as $|q|\to 0$ such that: $$V(t,q)= -{1\over{|q|^\alpha}}+W(q),\; \forall\; 0<|q|0$, $(P_T)$ possesses at least one non-collision solution having $T$ as minimal period. \end{Theorem} \begin{Remark} For $N\geq 3$, Theorem 1.1 was proved in \cite {T1} under condition (V3) by Morse theoretical arguments. \end{Remark} In section 3, we study the fixed energy problem. Here, we assume: \begin{enumerate} \item[(V'0)] $V\in C^2( \mathbb{R}^N \setminus \{0\},\mathbb{R})$; \item[(V'1)] $3V'(q)q+ V''(q)qq>0,\; \forall \;q\not=0; $ \item[(V'2)] There exists an constant $\alpha_1\in ]0,2[$ such that: $$V'(q)q\geq -\alpha_1 V(q)>0,\; \forall \;q\not=0;$$ \item[(V'3)] $\lim \inf[V(q) +{1\over 2}V'(q)q] \geq 0 $ as $|q| \to \infty$; \item[(V'4)] The same as (V3) with $V(t,q)=V(q)$. \end{enumerate} \begin{Theorem} Assume (V'0)-(V'4) with $N=2$. Then for any $h<0$, $(P_h)$ possesses at least one classical solution with a minimal period. \end{Theorem} \begin{Remark} i) For $N\geq 3$, (V'1) is used in \cite{A-CZ2} to prove existence of a generalized solution (that may enter the singularity) and in \cite{T2} to avoid collision solutions in the case $N=3$ and $1<\alpha <{4\over 3}$. \\ ii) Assumptions (V'1)-(V'2) can be made only in $\{ V\leq h\}$ (See \cite{A-CZ2}).\end{Remark} \medskip \noindent {\bf Notation.\ } For any $u \in H ^1([0,T]; \mathbb{R}^2) $, we note $u(t)=(|u(t)|,\theta (u)(t))$ in polar coordinates. We consider the following function space: $$E_0^T=\{ u \in H^1 ([0,T];\mathbb{R}^2 );\;u(0)=u(T);\; \int _0^T \dot \theta (u)(t)dt =2 \pi \}. $$ i.e., $E_0^T$ is the set of $T$-periodic functions $u \in H^1 ([0,T];\mathbb{R}^2 )$ such that $\theta:[0,T]/\{0,T\} \sim S^1 \to S^1 $ has degre $1$. We shall work in the function set: $$\Lambda_0^T=\{ u \in E_0^T ;\; u(t) \not=0 \; \forall \; t\}.$$ \section{The fixed period problem} In this section we proof Theorem 1.1. Let us define $$f(q)={1\over 2}\int_0^T |\dot q|^2 dt -\int_0^T V(t,q)dt.$$ It is well known that $f \in C^1 (\Lambda _0^T;\mathbb{R})$ and any critical point $u \in \Lambda _0^T$ is a solution of $(P_T)$. Since we deal with ``weak force" potentials, we know the existence of situation where the minimum of $f$ is assumed on functions going through the origin( See \cite{G2}). For any $\varepsilon \in ]0,1]$, we introduce the perturbed potential: $$V_\varepsilon (t,q)=V(t,q)-{\varepsilon \over {|q|^2}}.$$ The corresponding Lagrangian systems are $$ \begin{array}{c} \ddot q+ (V_\varepsilon)_q(t,q)=0\\ q(t+T)=q(t)\end{array}\eqno (P_T)_\varepsilon $$ and the associated functionals are $$f_\varepsilon(q)={1\over 2}\int_0^T |\dot q|^2 dt -\int_0^T V_\varepsilon(t,q)dt.$$ One has that $f_\varepsilon (q_n) \to +\infty$ as $q_n \to \partial \lambda_0^T$ weakly in $H^1([0,T];\mathbb{R}^2)$. We recall that in $\Lambda_0^T$, $$|| \dot u||_2=(\int_0^T |\dot u|^2 dt )^{1\over 2}$$ is a norm. Set $$m_\varepsilon ={\inf_{q \in \Lambda_0^T}} f_\varepsilon (q).$$ The following result is closely related to this of \cite{G1} (See \cite{A-CZ1}). \begin{Lemma} For any $\varepsilon \in ]0,1],\; m_\varepsilon$ is a critical value for $f_\varepsilon$; i.e. there exists $q_\varepsilon \in \Lambda_0^T$ such that $f_\varepsilon (q_\varepsilon )=m_\varepsilon$ and $f'_\varepsilon (q_\varepsilon )=0$. \end{Lemma} The fact that $f_\varepsilon (q_\varepsilon )=m_\varepsilon \leq m_1$ implies \begin{equation}{1\over 2}\int_0^T |\dot q_\varepsilon|^2 dt \leq m_1 \label{eq 2.1} \end{equation} and \begin{equation} \int_0^T V(t,q_\varepsilon )dt \leq m_1.\label{eq 2.2}\end{equation} It follows from \ref{eq 2.1} the existence of $\varepsilon _n \to 0$ such that $$q_n=q_{\varepsilon_n} \to q \hbox{ weakly in }H^1([0,T];\mathbb{R}^2) \hbox{ and uniformly in }[0,T].$$ We say that $q$ is a weak solution of $(P_T)$ in the sense of \cite{A-CZ1}. Setting $C(q)=\{t \in [0,T],\; q(t)=0 \}$, one can see from \ref{eq 2.2} and (V3), that mes$C(q)=0$ (Lebesgue measure). Moreover, we have \begin{equation} q_n \to q \hbox{ in } C^2 (K; \mathbb{R}^2),\; \forall \; K \hbox{ compact } \subset [0,T] \setminus C(q). \label{eq 2.3}\end{equation} Hence, we have that $$\ddot q+ V_q(t,q)=0, \; \forall \; t \in [0,T] \setminus C(q).$$ Therefore $q$ is a generalized solution of $(P_T)$ in the sense of \cite{B-R}. Now, we state these properties of approximated solutions $q_n$: \begin{Lemma}(i) There exists an constant $C_1 >0$ independent of $n$, such that $$|{1\over 2} |\dot q_n|^2 + V(t,q_n) -{\varepsilon_n \over { |q_n |^2}}|\leq C_1;$$ (ii)There exist constants $0<\mu 0$ independent of $n$, such that: $${1\over 2 }{d^2 \over {dt^2}}|q_n(t)|^2 \geq C_2 ,\; \forall \;t:\; |q_n(t)|<\mu.$$ \end{Lemma} \paragraph{Proof.} (i) follows from (V2) and \ref{eq 2.2}, while for (ii), it is a consequence of (i) and (V3). For more details, we refer to \cite {CZ-S,A-CZ1}. \begin{Remark} (ii) of Lemma 2.2 does not hold in general when $q$ is merely a generalized solution of $(P_T)$ as in \cite{T1}.\end{Remark} \paragraph{Proof of theorem 1.1.} We will prove how the function $q$ is actually a non-collision solution of $(P_T)$. We suppose that $q$ has a collision in $\bar t$. The contradiction will be showed in two steps. \paragraph{Step 1.} The solution $q_n$ have a self-intersection. We study the angle that the approximated solution $q_n$ describes close to the singularity. By (ii) of Lemma 2.2 and \ref{eq 2.3}, we get $${1\over 2 }{d^2 \over {dt^2}}|q(t)|^2 \geq C_2 >0,\; \forall \;t:\; 0<|q(t)|<\mu.$$ Take $\mu _0<\hbox{min}(\mu ,r)$ and $t_1 <\bar t 0,\; \forall t\in]t_n,t_2] .$$ Now, we will use a re-scaling argument as in (\cite{T1}-\cite{T2}). We set for any $L>0$, $$x_n (s)=\delta_n ^{-1} q_n(\delta_n ^{{\alpha +2}\over 2}s + t_n),\; s \in [-L,L]$$ when $\delta_n=|q_n(t_n)|\to 0$. Let us remark that for sufficiently large $n$, $\delta_n ^{{\alpha +2}\over 2}s + t_n \in [t_1,t_2]$ for $s \in [-L,L]$ and then $\delta_n |x_n(s)|<\mu$. Hence, $x_n (s) $ satisfies \begin{enumerate} \item[(i)] $|x_n(0)|=1;\; x_n (0).\dot x_n (0)=0$; ${d \over {ds}}|x_n(s)|<0,\; \forall \; s\in [-L,0[$; $ {d \over {ds}}|x_n(s)|>0,\;\forall \;s \in ]0,L]$; \item[(ii)] $\ddot x_n +{\alpha x_n \over {|x_n|^{\alpha +2}}} +\delta_n ^{\alpha +1} W'(\delta_n x_n )+{{2 \varepsilon}_n \over{\delta _n^{2-\alpha}}}{x_n\over {|x_n |^4}}=0$; \item[(iii)] $|{1\over 2}|\dot x_n|^2 -{1 \over {|x_n|^{\alpha }}} +\delta_n ^\alpha W(\delta_n x_n )-{\varepsilon _n \over{\delta _n^{ 2-\alpha}}|x_n|^2}|\leq C_1 \delta_n ^\alpha$. \end{enumerate} We may assume the existence -up a subsequence- of $$d=\lim_{n\to \infty}{\varepsilon _n \over{\delta _n^{2-\alpha}}} \in [0,\infty].$$ We consider the following two cases: \paragraph{Case1: $d<\infty$} From (i) and (iii), we may assume \begin{eqnarray*}x_n(0)&\to &e_1\\ \dot x_n (0) &\to& \sqrt{2(1+d)}e_2 \end{eqnarray*} where $(e_1,e_2)$ is an orthogonal basis of $\mathbb{R}^2$. By the continuous dependence of solutions in initial data and equations, one can see from (V3) that, $x_n (s) $ converge to a function $y_{\alpha,d}$ in $C^2(-L,L;\mathbb{R}^2)$ where $y_{\alpha,d} $ is the solution of $$\displaylines{ \ddot y +{{\alpha y} \over {|y|^{\alpha+2}}}+{{dy} \over {|y|^4}}=0 \cr y(0)=e_1, \quad \dot y (0)= \sqrt{2(1+d)}e_2 \, . }$$ Here we state some properties of $y_{\alpha,d}$ (c.f. \cite{T1}-\cite{T2}). \begin{eqnarray} &|y_{\alpha,d}(s)|=|y_{\alpha,0}(s)| \geq 1,\;\; \forall s \in \mathbb{R};& \label{p1}\\ &|y_{\alpha,d}(s)|^2 \dot \theta (y_{\alpha,d})(s)=\sqrt{2(1+d)},\;\; \forall s\in \mathbb{R};&\label{p2} \\ &\lim_{s \to -\infty}\theta (y_{\alpha,0} )(s) =-{\pi \over {2-\alpha}};&\label{p3}\\ &\lim_{s \to +\infty}\theta (y_{\alpha,0} )(s)=+{\pi \over {2-\alpha}}.& \label{p4} \end{eqnarray} Since $1<\alpha<2$, we get from \ref{p1}-\ref{p4}, the existence of $\bar L>0$ such that \begin{eqnarray*} \lim_{n \to \infty}[\theta(x_n)(\bar L)-\theta(x_n)(- \bar L)] &=&\theta(y_{\alpha,d})(\bar L)-\theta (y_{\alpha,d})(- \bar L)\\ &\geq &\theta(y_{\alpha,0})(\bar L)-\theta(y_{\alpha,0})(- \bar L)\\ &>& 2 \pi. \end{eqnarray*} Thus, for sufficiently large $n$, there exist $-\bar L 0\hbox{ for }s=s_0,s_1.$$ \paragraph{Case 2: $d=+\infty$} In this case, we set for $L>0$ $$z_n (s)=\delta_n ^{-1} q_n(\varepsilon_n ^{-{1\over 2}}\delta_n ^2 s + t_n),\; s \in [-L,L].$$ Since $\varepsilon_n ^{-{1\over 2}}\delta_n ^2 \to 0$, we see that $\delta_n |z_n(s)|<\mu$ for sufficiently large $n$ for any $L>0$. As in case 1, we find: \begin{eqnarray*} &|z_n(0)|=1,\; z_n (0).\dot z_n (0)=0;&\\ &{d \over {ds}}|z_n(s)|<0, \; \forall \; s\in [-L,0[;\;\; {d \over {ds}}|z_n(s)|>0, \; \forall \; s\in ]0,L];&\\ &z_n(s) \to y_\infty (s) \hbox{ in } C^2([-L,L];\mathbb{R}^2 )& \end{eqnarray*} where $y_\infty$ is the solution of the system $$\displaylines{ \ddot y+{{2y} \over {|y|^4}}=0 \cr y(0)=e_1\,\quad \dot y (0)= \sqrt 2e_2 }$$ for a suitable orthogonal basis $(e_1,e_2)$ of $\mathbb{R}^2$. Then, $$y_\infty (s)= e_1 \hbox{ cos} \sqrt 2 s +e_2 \hbox{ sin} \sqrt 2 s.$$ We remark that $\dot \theta (z_n) \to \sqrt 2$ uniformly in $[-L,L]$. So $z_n$ has at least a self intersection for $L> {{\sqrt 2 \pi } \over 2}$. From the two cases, it follows the existence of $t_{1,n},t_{2,n} \in ]t_1,t_2[$ such that \begin{eqnarray*} &q_n (t_{1,n})=q_n(t_{2,n});&\\ &{d\over dt}|q_n (t)| \not=0 \hbox{ and } \dot \theta (q_n)(t) >0 \hbox{ for }t=t_{1,n},t_{2,n}.& \end{eqnarray*} \paragraph{Step 2.} The solution $q_n$ cannot have a self intersection. Let $${q_n }^*(t) = \left \{ \begin{array}{ll} q_n(t) &\hbox{if } t \not \in [t_{1,n},t_{2,n}]\\ q_n(t_{1,n}+t_{2,n}-t) & \hbox{if } t \in [t_{1,n},t_{2,n}]. \end{array} \right. $$ We have $$\int_0^T \dot \theta ({q_n }^* )(t)dt=\int_0^T \dot \theta ({q_n})(t)dt =2\pi.$$ Hence ${q_n}^* \in \Lambda_0^T$. Since $f_{\varepsilon _n} ({q_n}^* )=f_{\varepsilon _n} (q_n)= m_{\varepsilon_n}$, ${q_n}^* $ must be a solution of $(P_T)_{\varepsilon_n}$ and then of class $C^1$. This is a contradiction with the fact $$\lim_{t \to {t_{1,n}}^{-}} \dot {q_n} ^* (t)=\dot q_n (t_{1,n})\not = -\dot q_n (t_{2,n}) = \lim_{t \to {t_{1,n}}^{+}} \dot {q_n}^* (t).$$ Therefore, we proved that $q$ is a non-collision solution of $(P_T)$. The minimality of the period $T$ follows from the fact that $q_n \to q \in \Lambda _0^T$. \section{The fixed energy problem} We give an outline of the proof of Theorem 1.2. According to the variational principle given by \cite{A-CZ2}, we define $$I(u)= {1\over 2} \int_0^1 |\dot u |^2 dt \int_0^1 [h-V(u) ]dt$$ on the set $M_h =\{ u \in \Lambda _0^1;\; g(u)=h \}$ where $$g(u)=\int_0^1 [V(u)+{1\over 2}V'(u)u] dt.$$ We know, if $u \in \Lambda_0^1$ is any possible solution of $(P_h)$, then $g(u)=h$. Moreover, under assumptions (V'0)-(V'4), $M_h \not = \emptyset $ is a $C^1$ manifold of codimension $1$ and if $u \in M_h$ is a critical point of $I$ constrained on $M_h$ such that $I(u)>0$, set $$w^2={\int_0^1V'(u)u dt \over {\int_0^1 |\dot u|^2 dt}}\,,$$ then $q(t)=u(wt)$ is a non-constant classical solution of $(P_h)$. We modify $V$, as in section 2, setting $$V_\varepsilon (u)=V(u) -{\varepsilon \over{|u|^2}},\;\; \varepsilon \in]0,1].$$ Let $$I_\varepsilon(u)= {1\over 2} \int_0^1 |\dot u |^2 dt \int_0^1 [h-V_\varepsilon (u) ]dt.$$ We remark that $$g(u)=\int_0^1 [V_\varepsilon(u)+{1\over 2}V'_\varepsilon(u)u ]dt.$$ It follows from (V'2) that $$I_\varepsilon(u)\geq {h\over {{1\over 2}-{1\over {\alpha_1}}}} \int_0^1 |\dot u |^2 dt,\;\; \forall u \in M_h.$$ Therefore, $I_\varepsilon$ is bounded below and coercive on $M_h$. Since $V_\varepsilon$ is a ``strong force" potential, one can see that $I_\varepsilon$ is lower semi continuous on $M_h$ and has a minimum $u_\varepsilon$ on $M_h$. Set $${w_\varepsilon}^2={\int_0^1V_\varepsilon'(u_\varepsilon) u_\varepsilon dt \over {\int_0^1 |\dot u_\varepsilon|^2 dt}},$$ the function $q_\varepsilon(t)=u_\varepsilon(w_\varepsilon t)$ is a solution of the modified system $(P_h)_\varepsilon$. Uniform estimates with respect to $\varepsilon$ allow to show that $u_\varepsilon$ converges uniformly on $[0,1]$ to $u$, ${w_\varepsilon} ^2 \to w^2 >0$ and that $q(t)=u(wt)$ satisfies the equations of the system $(P_h)$ for any $t \in \{ t \in [0,{1\over w}],\; u(t) \not =0 \}.$ Repeating the argument of section 2, one prove that $q$ is in fact a non-collision solution of $(P_h)$ with minimal period. If not, a new minimizer ${u_n}^* \in M_h$ for large $n$ can be constructed; But ${u_n}^*$ being a minimum of $I_{\varepsilon _n}$ on $M_h$ correspond to a solution of $(P_h)_{\varepsilon_n}$, on the other hand it does not have the required regularity. \begin{Remark} \rm (i)The existence of solutions $q_\varepsilon$ of $(P_h)_\varepsilon$ can be found without assuming condition (V'1). The proof relies on an application of the mountain-pass theorem to $ I_\varepsilon$. However, $q(t)=\lim q_\varepsilon (t)$ is a generalized solution of $(P_h)$ and collisions are possible. \\ (ii) Theorem 1.2 can be related to the work of Rabinowitz \cite{R} (see also \cite{CZ-S2}). He prove under a less restrictive setting than (V'0)-(V'4) that there exists a collision orbit of $(P_h)$. Combining this result with Theorem 1.2 shows the existence of a collision and a non-collision periodic solution of $(P_h)$ for a suitable class of planar singular potentials. \end{Remark} \paragraph{Acknowledgments.} I would like to express my gratitude to professor P. H. Rabinowitz for his interest in this work. \begin{thebibliography}{99} \bibitem{A-CZ1} A. Ambrosetti and V. Coti Zelati, ``Periodic solutions of singular Lagrangian systems", Birkhauser, Boston, Basel, Berlin, 1993. \bibitem{A-CZ2} A. Ambrosetti and V. Coti Zelati, {\it Closed orbits of fixed energy for singular Hamiltonian systems}, Arch. Rat. Mech. Anal. 112(1990), 339-362. \bibitem{A-CZ3} A. Ambrosetti and V. Coti Zelati, {\it Non-collision orbits for a class of Keplerian- like potentials,} Ann. Inst. H. Poincarre. Anal. Non lineaire, 5 (1988), 287-295. \bibitem{A-S} A.Ambrosetti and M. 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