Non-collision Orbits for a Class of Singular Hamiltonian Systems on the Plane with Weak Force Potentials

Abstract

We study the existence of non-collision orbits for a class of singular Hamiltonian systems

$$\begin{aligned} \ddot{q}+ V'(q)=0 \end{aligned}$$

where $q:{\mathbb {R}} \longrightarrow {\mathbb {R}}^2$ and $V\in C^2({\mathbb {R}}^2 {\setminus } \{e\},\, {\mathbb {R}})$ is a potential with a singularity at a point $e\not =0$. We consider V which behaves like $\displaystyle -1/|q-e|^\alpha$ as $q\rightarrow e$ with $\alpha \in ]0,2[.$ Under the assumption that 0 is a strict global maximum for V, we establish the existence of a homoclinic orbit emanating from 0. Moreover, in case $\displaystyle V(q) \longrightarrow 0$ as $|q|\rightarrow +\infty$, we prove the existence of a heteroclinic orbit “at infinity" i.e. a solution q such that

$$\begin{aligned} \lim _{t\rightarrow -\infty } q(t)=0,\,\, \lim _{t \rightarrow +\infty }|q(t)|=+\infty \,\, \hbox {and} \, \lim _{t \rightarrow \pm \infty }\dot{q}(t)=0. \end{aligned}$$

1 Introduction

In this paper we consider the second order Hamiltonian system

$$\begin{aligned} \ddot{q} + V'(q)=0 \end{aligned}$$

(HS)

where $q:{\mathbb {R}}\longrightarrow {\mathbb {R}}^2$ and $V\in C^2({\mathbb {R}}^2{\setminus }\{e\},\, {\mathbb {R}})$ has a singularity at a point $e\not =0$ such that

$$\begin{aligned} \displaystyle V(q) \sim -\frac{1}{|q-e|^\alpha } \quad \hbox {as} \, \, q\rightarrow e \, \hbox { with} \,\, \alpha \in ]0,2[. \end{aligned}$$

(1)

We will assume that V has a strict global maximum at $q=0$. So 0 is a equilibrium point for (HS).

Our goal in the first part is studying the existence of nontrivial homoclinic solutions to 0 of (HS), i.e. solutions q of (HS) such that

$$\begin{aligned} q\ne 0 \quad \hbox {and }\lim _{t\rightarrow \pm \infty } q(t)\equiv q(\pm \infty )=0=\dot{q} (\pm \infty ). \end{aligned}$$

We note that the order $\alpha$ in (1) plays an important role and we consider the existence of homoclinic solutions of (HS) under weak force case ($\alpha \in ]0,2[$). This case has been studied in several works which deal via variational methods with the periodic problem. See, e.g., [1, 2, 5, 9, 15, 17]. Let us now define the strong force condition:

(SF)    There exists a neighbourhood $\Omega$ of e in ${\mathbb {R}}^2$ and $U\in C^1(\Omega \setminus \{e\},{\mathbb {R}})$ such that

$$\begin{aligned}&|U(q)|\rightarrow \infty&\hbox {as }\, q\rightarrow e,\\&-V(q)\ge |U'(q)|^2&\hbox { for all } q\in \Omega \setminus \{e\}. \end{aligned}$$

Condition (SF) was introduced by Gordon [10]. For a potential $\displaystyle V(q) \sim -\frac{1}{|q-e|^\alpha }$ as $q\rightarrow e,$ (SF) is satisfied if and only if $\alpha \ge 2$. In fact, for $\alpha \ge 2$ we can take $U(q)=-2^{-1}\ln |q-e|$. The major role of (SF) is the following property.

Lemma 1.1

Assume (SF) and $V(q)\rightarrow -\infty$ as $q\rightarrow e$. Let $a<b \in R$ and $(q_m)\subset H^1([a,b],\Omega \setminus \{e\})$ which converges weakly in $H^1([a,b], {\mathbb {R}}^2)$ to q such that $q(t_0)=e$ for some $t_0\in [a,b]$. Then

$$\begin{aligned} -\int _a^b V(q_m)dt \longrightarrow +\infty \end{aligned}$$

(and therefore $\displaystyle \int _a^b\Big [ {1\over 2} |\dot{q}_m|^2 -V(q_m)\Big ]dt \longrightarrow +\infty$).

The proof of this lemma can be found in ([11], Lemma 2.1) or in [13]. As a consequence, if (SF) holds then functions with bounded energy are uniformly away from the singularity e. Therefore, in such case, a standard variational arguments in [13] provided the existence of a a pair of homoclinic orbits that wind respectively around the singularity e in a positive and negative sense. These solutions were obtained by minimizing the energy functional

$$\begin{aligned} I(q)=\displaystyle \int _{\mathbb {R}}\Big [\frac{1}{2}|\dot{q}|^2- V(q)\Big ]dt \end{aligned}$$

on classes of sets with a fixed winding number around e (see also [6, 7] for multiplicity results). If this condition is dropped (weak force case), Rabinowitz [13] proved the existence of a “generalized” homoclinic solution of (HS) which may pass through the singularity.

In ${\mathbb {R}}^N$ with $N\ge 3$, the existence of homoclinic solutions of (HS) was proved in [16] for strong force potentials (see also [8] in the case of time periodic potentials) and [3, 14] for weak force potentials like (1.1). In [3, 14], the authors introduced a strong force perturbed potential $V_\varepsilon$ for $\varepsilon \in ]0,1]$ such that $V_\varepsilon (q)= V(q)- \displaystyle {\varepsilon }/{|q-e|^2}$ near $q=e$ and proved through a min-max method from Bahri-Rabinowitz [4] the existence of non-collision solutions for approximated differential problems. Then they passed to the limit as $\varepsilon \rightarrow 0$ with the aid of appropriate estimates to obtain a generalized homoclinic solution. In [3] we studied the Morse index of approximated functionals at critical points to estimate the number of collisions. In particular we established the existence of non-collision homoclinic solution for $\alpha \in ]1,2[$ i.e. $q(t)\ne e$ for all $t\in {\mathbb {R}}$, while in [14] this result is obtained by assuming that V(q) is radially symmetric near $q=e$.

The main purpose of Sect. 2 is to prove the existence of non-collision homoclinic orbits of (HS) in ${\mathbb {R}}^2$ for weak force potentials. By exploiting the topology of the plane and using a minimization method, we first show the existence of a generalized homoclinic solution of (HS) as a limit of solutions of perturbed problems with boundary conditions. Then and for the regularity of this solution, we will use a Tanaka’s rescaling argument to prove some additional properties of approximated solutions near collisions, and we will prove how the generalized homoclinic solution obtained is actually a non-collision orbit in the case $\alpha \in ]1,2[$.

In Sect. 3, we assume that V has another global maximum at infinity i.e. $\displaystyle \lim _{|x|\rightarrow +\infty } V(x)=V(0)$ and we study the existence of a heteroclinic orbit “at infinity" i.e. a solution q of (HS) satisfying

$$\begin{aligned} q(-\infty )=0,\,\, |q(+\infty )|=+\infty \,\,\, \hbox { and } \, \,\dot{q}(\pm \infty )=0. \end{aligned}$$

The problem in ${\mathbb {R}}^N$ was treated by Serra in [14] for regular potentials where $V(q)\sim \displaystyle -{a}/{|q|^b } \; \hbox {as } \, |q|\rightarrow +\infty \, \hbox { with } \, a,\; b >0.$ He also treated the case of singular potentials which behaves like (1) when $N\ge 3$ and established the existence of non-collision orbits using some results from [15] on the analysis of collisions solutions of minimization problems. In the present paper we deal with the case $N=2$ and we will perturb V near e with a strong force term to get the existence of sequence $(q_n)$ of heteroclinic orbits at infinity for perturbed problems. We obtain uniform estimates to show that $(q_n)$ converges to a generalized solution. Some local properties of $q_n$ near collisions and the fact that $q_n$ is obtained via a minimization procedure permit us to obtain a non-collision heteroclinic solution at infinity.

2 Existence of Homoclinic Orbits

In this section, we consider the existence of a homoclinic solution of (HS) where the potential V satisfies the following assumptions:

• (V1) $V\in C^2 ({\mathbb {R}}^2\setminus \{e\},{\mathbb {R}})$ for some $e\ne 0$;

• (V2) $V(q)< V(0)=0$ for all $q\in {\mathbb {R}}^2\setminus \{0,e\}$;

• (V3) V is of the form

  $$\begin{aligned} \displaystyle V(q)=- \frac{1}{|q-e|^\alpha } + W(q), \end{aligned}$$

with $\alpha \in ]0,2[$ and W is such that

$$\begin{aligned} {|}q-e|^{\alpha -\nu } W(q), \;\; |q-e|^{\alpha -\nu +1} W'(q)\;\; \hbox {and }\; |q-e|^{\alpha -\nu +2} W''(q) \longrightarrow 0\, \text{ as } \, q\rightarrow e \end{aligned}$$

for some $\nu \in ]0,\alpha [$;

• (V4) There are $R>2|e|$ and a function $W_{\infty } \in C^1 ({\mathbb {R}}^2, {\mathbb {R}})$ such that

$|W_\infty (q)|\longrightarrow +\infty \quad \hbox {as}\quad |q|\rightarrow + \infty$ and $-V(q)\ge |W_\infty '(q)|$ for $|q|\ge R$.

Remark 2.1

• i) The condition (V3) remains valid when $\nu =0$. In particular it involves that $V\sim \displaystyle -1/|q-e|^\alpha$ near $q=e$ with $\alpha \in ]0,2[$.

• ii) The condition (V4) concerns the behavior of the potential at infinity. It will be satisfied if for example $V(q)\sim -a|q|^\beta$ as $|q|\rightarrow +\infty$ where $a>0$ and $\beta \ge -2$.

Our main result of this section is

Theorem 2.2

Assume (V1)-(V4).

• 1) If $\alpha \in ]1,2[$, then (HS) possesses at least one non-collision homoclinic solution.

• 2) If $\alpha \in ]0,1]$, then (HS) possesses a non trivial generalized homoclinic solution q having at most one collision. Moreover, if $q(t_0)=e$ then q(t) is a collision brake orbit, i.e. $q(t+t_0)=q(t_0-t)$ for all $t \in {\mathbb {R}}$.

Here, similarly as in [4, 17] for the periodic problem, we define a generalized homoclinic solution as a continuous function $q:{\mathbb {R}}\longrightarrow {\mathbb {R}}^2$ such that

• (i) $\dot{q} \in L^2({\mathbb {R}},\, {\mathbb {R}}^2)$ and $I(q)<\infty$;

• (ii) $D=\{t\in {\mathbb {R}}, \, q(t)=e\}$ is a set of measure 0;

• (iii) $q\in C^2 ({\mathbb {R}}\setminus D,\, {\mathbb {R}}^2) \quad \hbox {and satisfies (HS) on}\, {\mathbb {R}}\setminus D$;

• (iv) $\displaystyle \frac{1}{2}|\dot{q}(t)|^2 + V(q(t)) =0$ for $t \in {\mathbb {R}}{\setminus } D$;

• (v) $q(t)\longrightarrow 0\quad \hbox {and}\quad \dot{q}(t)\longrightarrow 0 \,\, \hbox {as}\;\, t\rightarrow \pm \infty$.

If $D=\emptyset$, q is a classical (non-collision) homoclinic solution.

Remark 2.3

Since V is independent of t, $q(-t)$ is a homoclinic solution of (HS) whenever q(t) is a homoclinic solution.

The proof of Theorem 2.2. is divided in various steps. We shall construct a homoclinic solution of (HS) as a limit of solutions of approximate value problems. We started by modifying the potential V near e. For $\varepsilon \in ]0,1]$, we define $V_\varepsilon \in C^2({\mathbb {R}}^2{\setminus }\{e\},\, {\mathbb {R}})$ such that $V_1\le V_\varepsilon \le V$ and

$$\begin{aligned} V_\varepsilon (q)= \left\{ \begin{array}{ll} V(q)-\displaystyle \frac{\varepsilon }{|q-e|^2} &{} \quad ~\text {if}~~\,\, 0<|q-e|\le |e|/4,\\ 0 &{} \quad ~\text {if}~~\,\, \displaystyle |q-e|\ge |e|/2.\\ \end{array} \right. \end{aligned}$$

Remark that $\displaystyle V_\varepsilon (q)\sim -\frac{\varepsilon }{|q-e|^2}\; \hbox {as}\; q\rightarrow e$. So $V_\varepsilon$ satisfies the strong force condition.

Let $(\varepsilon _n)_{n \in {\mathbb {N}}^*}\subset ]0,1]$ be a non-increasing sequence converging to 0. We consider for each $n\in {\mathbb {N}}^*$ the Dirichlet boundary value problem

The corresponding functional is

$$\begin{aligned} I_{0,n}(q)=\displaystyle \int _{0}^{n}\Big [\frac{1}{2}|\dot{q}|^2 -V_{\varepsilon _n}(q)\Big ]dt \in C^1 (\Lambda _n, {\mathbb {R}}) \end{aligned}$$

where

$$\begin{aligned} \Lambda _n= \{q\in H_0^1 ([0,n],\, {\mathbb {R}}^2); \quad q(t) \ne e,\, \forall \, t\in [0,n]\}. \end{aligned}$$

Let $\hbox {ind}_{z_0}(\gamma )$ denote the winding number of a closed curve in ${\mathbb {C}}$ around a point $z_0$. That is

$$\begin{aligned} \hbox {ind}_{z_0}(\gamma )={1\over {2 i \pi }}\int _\gamma {{dz}\over {z-z_0}} \end{aligned}$$

which is a integer representing the number of counterclockwise turns that $\gamma$ makes around $z_0$.

A critical point of $I_{0,n}$ will be found as a minimizer of $I_{0,n}$ over the set

$$\begin{aligned} \Gamma _n^{\pm 1} = \{ q\in \Lambda _n,\hbox { ind}_e(q)=\pm 1 \}. \end{aligned}$$

Clearly $\Gamma _n^{\pm 1} \ne \emptyset$, so we can define

$$\begin{aligned} c_n^{\pm 1} = \displaystyle \inf _{q\in \Gamma _n^{\pm 1}}I_{0,n}(q). \end{aligned}$$

(2)

We remark that, since $I_{0,n}(q)= I_{0,n}(q(n-.))\; \hbox {for all}\; q\in \Lambda _n$, then $c_n^1 = c_n^{-1}$.

Proposition 2.4

• 1) There exist $M_1, \, M_2 >0$ such that

  $$\begin{aligned} 0<M_1\le c_n^1\le M_2, \quad \forall n\in {\mathbb {N}}^*. \end{aligned}$$

  (3)

• 2) For every $n\in {\mathbb {N}}^*$, there is $q_n\in \Gamma _n^1$ such that $I_{0,n}(q_n)=c_n^1$. Moreover $q_n$ is a non trivial classical solution of $(D_n)$.

Proof

1) Let $q\in \Gamma _n^1$. The fact that ind$_e(q) =1$ implies that $||q||_{L^{\infty } ( [0,n],\, {\mathbb {R}}^2)}\ge |e|$. Since $q(0)=q(n)=0$, there exist $s_q<t_q$ such that

$$\begin{aligned} |q(s_q)|=\frac{|e|}{3},\,\, |q(t_q)|= \frac{2|e|}{3}\hbox { and } \frac{|e|}{3}\le |q(t)|\le \frac{2|e|}{3}\,\, \hbox {for all}\,\, t\in [s_q,t_q]. \end{aligned}$$

Using the Cauchy-Schwartz inequality, we have the general formula

$$\begin{aligned} \int _{t_1}^{t_2} \Big [\frac{1}{2}|\dot{u}|^2 -V(u)\Big ]dt&\ge \displaystyle {{|u(t_2)-u(t_1)|^2} \over {2(t_2-t_1)}} +(t_2-t_1)\min _{t\in [t_1,t_2]}-V(u(t)) \nonumber \\&\ge |u(t_2)-u(t_1)| \sqrt{2\min _{t\in [t_1,t_2]}-V(u(t))} \end{aligned}$$

(4)

where $u\in H^1([t_1,t_2],{\mathbb {R}}^2)$.

We denote $c=\displaystyle \min _{\frac{|e|}{3}\le |x|\le \frac{2|e|}{3}} -V(x)>0$. Then from (4), we get

$$\begin{aligned} I_{0,n}(q)&\ge \int _{s_q}^{t_q} \Big [\frac{1}{2}|\dot{q}|^2- V(q)\Big ]dt\\&\ge {{|e|}\over 3}\sqrt{2c}=M_1. \end{aligned}$$

Thus by the arbitrariness of q, we obtain $c_n ^1 \ge M_1 >0$ for any $n\in {\mathbb {N}}^*$.

In order to prove that $c_n ^1$ is bounded from above, let $q\in \Gamma _1^1$ and define

$$\begin{aligned} v_n(t)= \left\{ \begin{array}{ll} q(t) &{} \quad ~\text {if}~~ t\in [0,1],\\ 0 &{} \quad ~\text {if}~~ t\in ]1,n].\\ \end{array} \right. \end{aligned}$$

Clearly $v_n \in \Gamma _n ^1$ and then

$$\begin{aligned} c_n ^1&\le I_{0,n}(v_n)=\int _{0}^{1}\Big [\frac{1}{2}|\dot{q}|^2-V_{\varepsilon _n}(q)\Big ]dt\\&\le I_{0,1}(q). \end{aligned}$$

Therefore

$$\begin{aligned} c_n^1 \le \displaystyle \inf _{ q\in \Gamma _1 ^1} I_{0,1}(q)=M_2. \end{aligned}$$

2) Let $(u_m )$ be a minimizing sequence for $c_n^1$. We have from (3), $(u_m)$ is bounded in $H^1_0([0,n],\, {\mathbb {R}}^2)$. It follows that along a subsequence $(u_m )$ converge weakly in $H_0^1 ([0,n], \; {\mathbb {R}}^2)$ and uniformly in [0, n] to a function $q_n$. Since $\int _{0}^{n} -V_{\varepsilon _n}(u_m) dt$ is bounded independently of m and $V_{\varepsilon _n}$ is a strong force, Lemma 1.1 shows that $q_n\in \Lambda _n$. Moreover we know that the winding number is continuous with respect to uniform convergence of curves. Therefore $\hbox {ind}_e(q_n)=\lim _{m\rightarrow +\infty } \hbox {ind}_e (u_m) = 1$ and so $q_n\in \Gamma _n ^1$. Using the lower semi continuity of $I_{0,n}$, we get $I_{0,n}(q_n) \le \liminf _{m\rightarrow +\infty }I_{0,n}(u_m)= c_n^1$. That is $I_{0,n}(q_n)=c_n^1$. Now in a standard way, we can see that $q_n$ is a critical point of $I_{0,n}$ and then a nontrivial classical solution of $(D_n)$. $\square$

As a consequence of Proposition 2.4, we get the following estimates:

Lemma 2.5

• (i) There is a constant $C>0$ which is independent of n such that for any $n\in {\mathbb {N}}^*$,

  $$\begin{aligned} ||\dot{q}_n||_{ L^2 ([0,n], \, {\mathbb {R}}^2)} \le C;\quad \displaystyle \int _{0}^{n}-V(q_n) dt \le C; \quad ||q_n||_{L^{\infty } ([0,n],\, {\mathbb {R}}^2)} \le C. \end{aligned}$$

• (ii) For every $n\in {\mathbb {N}}^*$, there is a constant $h_n>0$ such that

  $$\begin{aligned} \displaystyle \frac{1}{2}|\dot{q}_n(t)|^2 + V_{\varepsilon _n}(q_n(t))= h_n, \quad \forall \, t\in [0,n]. \end{aligned}$$

  Moreover, $\displaystyle h_n={1\over 2} |\dot{q}_n(0)|^2= {1\over 2} |\dot{q}_n(n)|^2 \longrightarrow 0$.

Since $q_n \in \Gamma _n^1$, we have $\displaystyle \max _{t\in [0,n]} |q_n(t)| >{{|e|}/ 4}$. Otherwise we would have ind$_e( q_n) =0$.

Then we can find numbers $\tau _n^1,\, \tau _n^2 \in ]0,n[$ such that

$$\begin{aligned} \displaystyle |q_n(\tau _n^1)|=|q_n(\tau _n^2)|=|e|/4 \displaystyle \quad \hbox {and } |q_n(t)|< |e|/4\;\, \hbox { if }\;\, t\in [0,\tau _n^1[\cup ]\tau _n^2,n]. \end{aligned}$$

Note that in [3], it was also proved the existence of approximated solution $q_n$ of $(D_n)$ in ${\mathbb {R}}^N$ ($N\ge 3)$ such that

*:

$\displaystyle \max _{t\in [0,n]} |q_n(t)| >{\rho }$ where $\rho >0$ is a constant;

*:

$|\dot{q}_n (0)| \rightarrow 0$ and $|\dot{q}_n(n)|\rightarrow 0$.

Using the continuity theorem of solutions with respect to initials conditions, we can see in a similar way to Lemma 2.7 in [3],

$$\begin{aligned} \tau _n^1\rightarrow \infty \quad \hbox {and} \, \, n-\tau _n^2 \rightarrow \infty . \end{aligned}$$

Next we define

$$\begin{aligned} \tilde{q}_n(t)= \left\{ \begin{array}{ll} q_n(t+\tau _n ^1) &{} \quad ~\text {if}~~ t\in [-\tau _n ^1,n-\tau _n ^1] ,\\ 0 &{} \quad ~\text {if}~~ t\in {\mathbb {R}}\setminus [-\tau _n ^1,n-\tau _n ^1] .\\ \end{array} \right. \end{aligned}$$

(5)

Clearly $|{\tilde{q}}_n (0)|=|e|/4$ and ${\tilde{q}}_n$ verifies

$$\begin{aligned}{} & {} \ddot{\tilde{q}}_n+V_{\varepsilon _n}'(\tilde{q}_n)=0\quad \hbox { in }] -\tau _n^1,\, n-\tau _n^1[,\\{} & {} \displaystyle \frac{1}{2}|\dot{\tilde{q}}_n|^2 +V_{\varepsilon _n}(\tilde{q}_n)= h_n \quad \hbox { in }] -\tau _n^1,\, n-\tau _n^1[. \end{aligned}$$

By (i) of Lemma 2.5, we can extract a subsequence -still denoted by ${\tilde{q}}_n$- which converges in $C_{loc}({\mathbb {R}},\, {\mathbb {R}}^2)$ to some function $\tilde{q}\in C({\mathbb {R}},\, {\mathbb {R}}^2)\cap L^\infty ({\mathbb {R}},\,{\mathbb {R}}^2)$ with $\dot{\tilde{q}}\in L^2({\mathbb {R}},\,{\mathbb {R}}^2)$. Since $-\tau _n^1 \rightarrow -\infty$ and $n-\tau _n^1 \rightarrow +\infty$, we can see $\tilde{q}$ is a non trivial generalized homoclinic solution of (HS). The complete proofs to Lemma 2.5 and the last statements are ommited as they are similar to its analogues in [3].

In what follows, we focus our attention to study the regularity of $\tilde{q}$. First we state some further properties of $\tilde{q}_n$ and $\tilde{q}$ near the singularity.

Let $t \in {\mathbb {R}}$ such that $|\tilde{q}_n(t)-e|< |e|/4$. From the definition of $V_{\varepsilon _n},\, \, \tilde{q}_n(t)$ verifies

$$\begin{aligned}&\displaystyle \ddot{\tilde{q}}_n+ \alpha \frac{{\tilde{q}}_n-e}{|{\tilde{q}}_n -e|^{\alpha +2}}+ W'(\tilde{q}_n) +2\varepsilon _n \frac{\tilde{q}_n-e}{|\tilde{q}_n-e|^4}=0, \end{aligned}$$

(6)

$$\begin{aligned}&\displaystyle \frac{1}{2}|\dot{\tilde{q}}_n|^2- \frac{1}{|\tilde{q}_n-e|^\alpha }+ W(\tilde{q}_n)-\frac{\varepsilon _n}{|\tilde{q}_n-e|^2}=h_n. \end{aligned}$$

(7)

Then

$$\begin{aligned} {1\over 2} \displaystyle \frac{d^2}{dt^2}|{\tilde{q}}_{n}(t)-e|^2&= < \ddot{\tilde{q}}_n,\tilde{q}_n-e> +|\dot{\tilde{q}}_n|^2 \\&= \displaystyle \frac{2-\alpha }{|\tilde{q}_n-e|^\alpha }-W'(\tilde{q}_n)(\tilde{q_n}-e) - 2W(\tilde{q}_n)+2h_n \\&= \displaystyle \frac{1}{|\tilde{q}_n-e|^\alpha } [ 2-\alpha -|\tilde{q}_n-e|^\alpha W'(\tilde{q}_n)(\tilde{q_n}-e) -2|\tilde{q}_n-e|^\alpha W(\tilde{q}_n)\\&\quad +2h_n |\tilde{q}_n-e|^\alpha ]. \end{aligned}$$

By (V3) (see Remark 2.1 i)) and the fact that $h_n \rightarrow 0$, we can find $0<\delta < |e|/4$ such that for sufficiently large n,

$$\begin{aligned} \displaystyle \frac{1}{2}\frac{d^2}{dt^2}|\tilde{q}_n(t)-e|^2>0 \quad \hbox {if}\quad |\tilde{q}_n(t)-e|<\delta . \end{aligned}$$

(8)

Similarly, if $\tilde{q}(t) \not = e$ then $\tilde{q}(t)$ satisfies (HS) and of energy 0. From this, we obtain

$$\begin{aligned} {1\over 2} \frac{d^2}{dt^2}| {\tilde{q}}(t)-e|^2 = \displaystyle \frac{1}{|\tilde{q} -e|^\alpha } [ 2-\alpha -|\tilde{q}-e|^\alpha W'(\tilde{q})(\tilde{q}-e)-2|\tilde{q}-e|^\alpha W(\tilde{q})]. \end{aligned}$$

Thus the property (8) holds also for $\tilde{q}$, i.e.

$$\begin{aligned} \displaystyle \frac{1}{2}\frac{d^2}{dt^2}|\tilde{q}(t)-e|^2 >0 \quad \hbox {if}\quad 0< |\tilde{q}(t)-e|<\delta . \end{aligned}$$

(9)

Taking into account the property (ii) of a generalized solution, (9) implies that the collisions times of $\tilde{q}$ (if they exist) are isolated.

Now we suppose that $\tilde{q}$ has a collision at $t=\tilde{t}$ i.e. $\tilde{q}(\tilde{t})=e$ for some ${ \tilde{t}} \in {\mathbb {R}}$. We will study the angle which describes $\tilde{q}_n(t)$ around e when t is near $\tilde{t}$. In particular we will show that $\tilde{q}_n$ have one self intersection if $\alpha \in ]1,2[$.

Since $\tilde{q}(t)\longrightarrow 0$ as $t\rightarrow \pm \infty$, there exist $\tau _1< \tilde{t}<\tau _2$ such that

$$\begin{aligned} |\tilde{q}(\tau _1)-e|= |\tilde{q}(\tau _2)-e|= \displaystyle \frac{\delta }{2} \quad \hbox {and }\,\, 0< |\tilde{q}(t)-e|<\displaystyle \frac{\delta }{2}\quad \forall \, t\in ]\tau _1,\tau _2[\setminus \{\tilde{t}\}. \end{aligned}$$

Thus for sufficiently large n, we have

$$\begin{aligned} |\tilde{q}_n(\tau _i)-e|\ge \displaystyle \frac{\delta }{4}\quad \hbox {for}\quad i=1,2 \end{aligned}$$

(10)

and

$$\begin{aligned} |\tilde{q}_n(t)-e|<\delta \quad \forall \; t\in [\tau _1,\tau _2]. \end{aligned}$$

(11)

Let $t_n \in [\tau _1,\tau _2]$ and $\delta _n>0$ such that $\delta _n = |\tilde{q}_n(t_n)-e|= \displaystyle \min _{t\in [\tau _1,\tau _2]} |\tilde{q}_n(t)-e|.$

Clearly $\delta _n \le |\tilde{q}_n({\tilde{t}})-e|\longrightarrow |\tilde{q}(\tilde{t})-e|=0$. So $\delta _n\longrightarrow 0$. Moreover, up a subsequence, we have $t_n\longrightarrow \tilde{t}$.

By (8), we have

$$\begin{aligned}&\displaystyle \frac{d}{dt}|\tilde{q}_n(t)-e| < 0 \quad \forall \, t\in [\tau _1,t_n [, \end{aligned}$$

(12)

$$\begin{aligned}&\displaystyle \frac{d}{dt}|\tilde{q}_n(t)-e| > 0 \quad \forall \, t\in ]t_n,\tau _2]. \end{aligned}$$

(13)

In the sequel we use a rescaling argument as in [17] and we introduce the function

$$\begin{aligned} x_n(s)= \delta _n ^{-1} \Big [ \tilde{q}_n \left( \delta _n^{\frac{\alpha +2}{2}} s +t_n \right) -e \Big ], \quad s\in {\mathbb {R}}. \end{aligned}$$

Remark that

$$\begin{aligned} |x_n(0)|=1 \quad \hbox {and} \quad (x_n(0),\, \dot{x} _n (0))=0. \end{aligned}$$

(14)

Let $l>0$. For sufficiently large n, since $\delta _n^{\frac{\alpha +2}{2}} s +t_n \in [\tau _1,\tau _2]$ for $s\in [-l,l]$, we have from (11) and (6)-(7),

$$\begin{aligned}&\ddot{x}_n(s)+ \alpha \frac{x_n}{|x_n|^{\alpha +2}}+\delta _n^{\alpha +1} W'(\delta _nx_n+e)+ \frac{2\varepsilon _n}{\delta _n^{2-\alpha }}\frac{x_n}{|x_n|^4}=0 \quad \hbox {in}\; [-l,l], \end{aligned}$$

(15)

$$\begin{aligned}&\frac{1}{2}|\dot{x}_n|^2- \frac{1}{|x_n|^\alpha }+ \delta _n^\alpha W(\delta _nx_n+e)- \frac{\varepsilon _n}{\delta _n^{2-\alpha }} \frac{1}{|x_n|^2}= \delta _n^\alpha h_n \quad \hbox { in} \; [-l,l]. \end{aligned}$$

(16)

We extract a subsequence still indexed by n such that

$$\begin{aligned} d= \displaystyle \lim _{n\rightarrow +\infty }\frac{\varepsilon _n}{\delta _n^{2-\alpha }}\in [0, +\infty ] \end{aligned}$$

(17)

exists. For d we need to show that

Lemma 2.6

The quantity d defined in (17) is a finite one.

Proof

On the contrary, we assume that $d= +\infty$. We will prove that $\tilde{q}_n$ has a self intersection around e to find a contradiction. Let consider another rescaling of $\tilde{q}_n$:

$$\begin{aligned} y_n(s)=\displaystyle \delta _n ^{-1} \Big [\tilde{q}_n \left( \varepsilon _n^{-\frac{1}{2}}\delta _n^2 s +t_n\right) -e \Big ], \quad s\in {\mathbb {R}}. \end{aligned}$$

(18)

Since $\displaystyle \varepsilon _n ^{-\frac{1}{2}}\delta _n^2 = \Big ( \varepsilon _n ^{-1}\delta _n^{2-\alpha } \Big ) ^{\frac{1}{2}}\delta _n ^{1+ \frac{\alpha }{2}}\longrightarrow 0$, then for sufficiently large n, we have $\varepsilon _n^{-\frac{1}{2}}\delta _n^2\,s +t_n \in [\tau _1,\tau _2], \,\forall \, s \in [-l,l]$. From (12)-(13), we get

$$\begin{aligned}{} & {} \displaystyle \frac{d}{ds}|y_n(s)|<0 \quad \forall \, \, s\in [-l,0[,\\{} & {} \displaystyle \frac{d}{ds}|y_n(s)|>0 \quad \forall \, \, s\in ]0,l]. \end{aligned}$$

As in [3], we can see that -up a subsequence-

$$\begin{aligned} y_n\longrightarrow \cos (\sqrt{2}s)e_1 + \sin (\sqrt{2}s)e_2 \quad \hbox { in } \, C^2_{\hbox {\tiny {loc}}} ({\mathbb {R}}, \, {\mathbb {R}}^2) \end{aligned}$$

where $(e_1, e_2)$ is an orthonormal basis of ${\mathbb {R}}^2$. Using polar coordinates, there exists a function $\alpha _n \in C^2({\mathbb {R}},\, {\mathbb {R}})$ such that

$$\begin{aligned} y_n(s)= |y_n(s)|\Big ( \cos (\alpha _n(s))e_1 + \sin (\alpha _n(s))e_2\Big ). \end{aligned}$$

We take $l> \sqrt{2}\pi$. Since $\dot{\alpha }_n\longrightarrow \sqrt{2}$ uniformly on $[-l,l]$, then for sufficiently large n, there exist $s_1<0<s_2$ such that

$$\begin{aligned} \alpha _n(0)-\alpha _n(s_1)=\alpha _n(s_2)-\alpha _n(0)=2\pi . \end{aligned}$$

(19)

We may assume that $1< |y_n(s_1)|\le |y_n(s_2)|$. By continuity, there exists $s_3\in ]0,s_2]$ such that $|y_n(s_1)|=|y_n(s_3)|$. Since ${\dot{\alpha }}_n >0$, it follows from (19) that

$$\begin{aligned} \alpha _n(s_3)-\alpha _n(s_1)&=\alpha _n(s_3)-\alpha _n(0)+\alpha _n(0)-\alpha _n(s_1)\\&>2\pi . \end{aligned}$$

This implies the existence of $s_1', s_2' \in [s_1, s_3]$ such that $y_n(s_1')=y_n(s_2')$ and

ind$_0 {y_n}|_{[s'_1,s'_2]}=1$. From (5) and (18), it follows the existence of $t',\, t''\in ]0,n[$ such that $q_n(t')=q_n(t'')$ and ind$_e q_n|{_{[t',t'']}}=1$. But this contradicts the fact that $q_n$ is a minimum of $I_{0,n}$ over $\Gamma ^1_n$. Indeed, let consider the function

$$\begin{aligned} \underline{q_n}(t)= \left\{ \begin{array}{ll} q_n(t) &{} \quad ~\text {if}~~ t\in [0,n]\setminus [t',t''],\\ q_n(t'+t''-t) &{} \quad ~\text {if}~~ t\in [t',t''].\\ \end{array} \right. \end{aligned}$$

Then $\underline{q_n} \in \Gamma _n^{-1}$ and $I_{0,n}(\underline{q_n})=I_{0,n}(q_n)=c_n^1=c_n^{-1}$. Therefore $\underline{q_n}$ is a classical solution of $(D_n)$. By the uniqueness of the solution of ordinary differential equation, we deduce that $\underline{q_n}=q_n$: clearly this is a contradiction since $\hbox {ind}_e(\underline{q_n})=-1$ and ind$_e(q_n)=1$. $\square$

Since $d< +\infty$, by the continuity dependence of solutions on initial data and equation, we can see from (14)-(16) and (V3) the existence of an orthonormal basis $(e_1,e_2)$ of ${\mathbb {R}}^2$ such that

$$\begin{aligned} x_n(s)\longrightarrow x_{\alpha ,d}(s)\quad \hbox {in } \, \, C^2_{\hbox {\tiny {loc}}} ({\mathbb {R}},\, {\mathbb {R}}^2) \end{aligned}$$

where $x_{\alpha ,d}(s)$ is the solution of the initial value problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \ddot{x}+\frac{\alpha x}{|x|^{\alpha +2}} +2d \frac{x}{|x|^4}=0, \\ x(0)=e_1,\quad \dot{x}(0)=\sqrt{2(1+d)}e_2. \\ \end{array}\right. \end{aligned}$$

We use polar coordinates and write

$$\begin{aligned}{} & {} \tilde{q}_n(t)-e =|\tilde{q}_n(t)-e| \Big ( \cos (\tilde{\theta }_n(t))e_1 + \sin (\tilde{\theta }_n(t))e_2\Big ),\\{} & {} x_{\alpha ,d}(s)= |x_{\alpha ,d}(s)|\Big (\cos (\theta _{\alpha ,d}(s))e_1 + \sin (\theta _{\alpha ,d}(s))e_2\Big ), \end{aligned}$$

where $\tilde{\theta }_n(s),\, \theta _{\alpha ,d}(s)\in {\mathbb {R}}$ with $\theta _{\alpha ,d}(0)=0$. In [18] we observed the following properties for $\theta _{\alpha ,d}$

$$\begin{aligned}&\dot{\theta }_{\alpha ,d}(s)>0 \quad \forall \; s\in {\mathbb {R}}, \end{aligned}$$

(20)

$$\begin{aligned}&\Delta \theta _{\alpha ,d}=\displaystyle \lim _{s\rightarrow +\infty } (\theta _{\alpha ,d}(s) -\theta _{\alpha ,d}(-s))= \frac{2\pi \sqrt{1+d}}{2-\alpha }. \end{aligned}$$

(21)

We remark that $\Delta \theta _{\alpha ,d}>\pi \, \, \forall \, \alpha \in ]0,1]$ and if $\alpha \in ]1,2[$ then $\Delta \theta _{\alpha ,d}>2\pi$.

Let $B_r(e)$ denote the open ball of radius r about e. We will give a estimate of $\tilde{\theta }_n(t)$ when $\tilde{q}_n(t) \in B_\mu (e)\setminus B_{L\delta _n}(e)$ for sufficiently small $\mu >0$ and large $L>1$ and n.

We have for $t <t'$,

$$\begin{aligned} |\tilde{\theta }_n(t')-\tilde{\theta }_n(t)|&\le \int _t^{t'}|\dot{\tilde{\theta }}_n(\tau )|d\tau \nonumber \\&=\int _t^{t'}\Big |{d\over {dt}} {{{\tilde{q}}_n(\tau ) -e}\over {|{{\tilde{q}}_n(\tau ) -e}|}}\Big |d\tau \end{aligned}$$

(22)

On the other hand, Tanaka [18] studied under the condition (V3) with $e=0$ the behavior of generalized periodic solutions of singular Hamiltonian systems in ${\mathbb {R}}^N$. In a neighborhood of the singularity, the generalized solution is a limit of classical solutions of perturbed problems with potentials $V_\varepsilon$ as in our case, so we can apply some locally property of approximated solutions near a collision. More precisely, modifying the argument in Proposition 1.5 slightly, we can see that for any $\eta >0$ there exist constants $\mu ,\, S>0$ and $n_0\in {\mathbb {N}}^*$ such that for $n\ge n_0$,

$$\begin{aligned}{} & {} \displaystyle \int _t^{t'}\Big |{d\over {dt}} {{{\tilde{q}}_n(\tau ) -e}\over {|{{\tilde{q}}_n(\tau ) -e}|}}\Big |d\tau \le \frac{\eta }{2} \quad \hbox {if } \,\, {\tilde{q}}_n(t),\,\, {\tilde{q}}_n(t')\in B_\mu (e) \ \ \hbox {and} \nonumber \\{} & {} \tau _1<t<t'< t_n-S\delta _n^{\frac{\alpha +2}{2}}\, \, \hbox {or }\,\, t_n+S \delta _n^{\frac{\alpha +2}{2}}<t<t'<\tau _2. \end{aligned}$$

(23)

Combining (22) and (23), we get

Lemma 2.7

For any $\eta >0$, there are constants $\mu \in ]0,\, \delta /4[,\, S>0$ such that for sufficiently large n, if $\, {\tilde{q}}_n(t),\,\, {\tilde{q}}_n(t')\in B_\mu (e) \,\, \hbox {and}$

$$\begin{aligned} \tau _1<t<t'< t_n-S\delta _n^{\frac{\alpha +2}{2}}\, \, \hbox {or } \,\, t_n+S \delta _n^{\frac{\alpha +2}{2}}<t<t'<\tau _2, \end{aligned}$$

then

$$\begin{aligned} |\tilde{\theta }_n(t')-\tilde{\theta }_n(t)|\le \displaystyle \frac{\eta }{2}. \end{aligned}$$

End of the proof of Theorem 2.2. 1) If $\alpha \in ]1,2[$, there exists from (21) $\eta >0$ such that $\Delta \theta _{\alpha ,d}> 2\pi +\eta$. For this $\eta$, we choose $\mu \in ]0,\; \delta /4[,\; S>0$ and n sufficiently large as in Lemma 2.7.

From (21) again we can take $S_1>S$ such that

$$\begin{aligned} \theta _{\alpha ,d}(S_1)- \theta _{\alpha ,d}(-S_1)> 2\pi +\eta . \end{aligned}$$

Then we obtain for sufficiently large n,

$$\begin{aligned} \displaystyle \tilde{\theta }_n\left( t_n+\delta _n^{\frac{\alpha +2}{2}}S_1\right) -\tilde{\theta }_n\left( t_n-\delta _n^{\frac{\alpha +2}{2}}S_1\right) >2\pi + \eta . \end{aligned}$$

(24)

On the other hand, since $\displaystyle | \tilde{q}_n(t_n \pm S_1\delta _n^{\frac{\alpha +2}{2}})-e|\longrightarrow |\tilde{q}({\tilde{t}})-e|=0,$ we can assume that

$$\begin{aligned} \displaystyle |\tilde{q}_n\left( t_n \pm S_1\delta _n ^{\frac{\alpha +2}{2}}\right) -e|<\mu . \end{aligned}$$

We set $t'_{1,n}= t_n - S_1\delta _n^{\frac{\alpha +2}{2}}$. Then we have from (10)

$$\begin{aligned} \displaystyle |\tilde{q}_n(t'_{1,n})-e|<\mu <{\delta \over 4} \le |\tilde{q}_n(\tau _{1})-e|. \end{aligned}$$

Therefore there exists $t_{1,n}\in ]\tau _1,\, t'_{1,n}[$ such that

$$\begin{aligned} |\tilde{q}_n(t_{1,n})-e|= \mu . \end{aligned}$$

Similarly we set $t_{2,n}= t_n +S_1\delta _n^{\frac{\alpha +2}{2}}$. Since $\displaystyle |\tilde{q}_n(t_{2,n})-e|<\mu <{\delta \over 4} \le |\tilde{q}_n(\tau _{2})-e|$, there exists $t'_{2,n}\in ] t_{2,n},\, \tau _2[$ such that

$$\begin{aligned} |\tilde{q}_n(t'_{2,n})-e|= \mu . \end{aligned}$$

Applying lemma 2.7 for $t=t_{i,n}$ and $t'=t'_{i,n}\,\, (i=1,2)$, we obtain

$$\begin{aligned} |\tilde{\theta }_n (t'_{i,n}) -\tilde{\theta }_n(t_{i,n})|\le \displaystyle \frac{\eta }{2} \quad \hbox { for}\quad i=1,\, 2. \end{aligned}$$

(25)

It follows from (24)-(25),

$$\begin{aligned} \tilde{\theta }_n(t'_{2,n})-\tilde{\theta }_n(t_{1,n})&= \tilde{\theta }_n(t'_{2,n}) -\tilde{\theta }_n(t_{2,n}) +\tilde{\theta }_n(t_{2,n}) -\tilde{\theta }_n(t'_{1,n})+\tilde{\theta }_n(t'_{1,n}) -\tilde{\theta }_n(t_{1,n}) \\&> \displaystyle -\frac{\eta }{2} +2\pi + \eta - \frac{\eta }{2}=2\pi . \end{aligned}$$

That is $\tilde{q}_n$ describes an angle greater than $2\pi$ in going from $\partial B_\mu (e)\; \hbox {back to}\, \partial B_\mu (e)$ which implies the existence of $t''_{1,n},\; t''_{2,n}$ with $\tau _1<t''_{1,n}<t''_{2,n}<\tau _2$ such that

$$\begin{aligned} {\tilde{q}}_{n}(t''_{1,n})={\tilde{q}}_{n}(t''_{2,n}) \quad \hbox {and ind}_e {{\tilde{q}}_n|}_{{[t''_{1,n},t''_{2,n}]}}=1. \end{aligned}$$

Thus we deduce that ${q_n}$ has a self intersection around e for sufficiently large n. As in the proof of Lemma 2.6, we get a contradiction and then we conclude that $\tilde{q}$ is a non collision homoclinic solution of (HS).

2) In the case $\alpha \in ]0,1]$, the angle which describes $\tilde{q}_n$ near e is greater than $\pi$ and $\tilde{q}_n$ cannot have a self intersection. The fact that the collisions times of $\tilde{q}$ are isolated and since $\tilde{q}(t)\longrightarrow 0$ as $t\rightarrow \pm \infty$, we get that the number of collisions of $\tilde{q}$ is finite. Assume $\tilde{q}(t)$ enters the singularity e and let

$$\begin{aligned} {t_0}=\displaystyle \min \{ t\in {\mathbb {R}}, \,\, \tilde{q}(t)=e\}. \end{aligned}$$

Since (HS) is time reversible, the function

$$\begin{aligned} {q}(t)= \left\{ \begin{array}{ll} \tilde{q}(t) &{} \quad ~\text {if}~~ t\le {t_0},\\ \tilde{q}(2 {t_0}-t) &{} \quad ~\text {if}~~ t\ge {t_0}, \end{array} \right. \end{aligned}$$

is a generalized homoclinic solution of (HS) and satisfies ${q}(t+ {t_0})= {q}({t_0}-t)$ for all t. Moreover q has one collision in ${\mathbb {R}}$. The proof of Theorem 2.2 is finally complete.

Remark 2.8

The assumption (V3) is far too restrictive in the case $\alpha \in ]0,1]$ and the existence of a generalized homoclinic solution with finite number of collisions and then a solution as in Theorem 2.2 2) still holds if we replace (V3) by

• (V’3)

  • (i) $V(q)\rightarrow -\infty$ as $q \rightarrow e$;

  • (ii) There exists a constant $\delta \in ]0,|e|/4[$ such that

    $$\begin{aligned} V(q)+ \displaystyle \frac{1}{2}V'(q)(q-e)<0\quad \hbox { for} \,\, 0<|q-e|\le \delta . \end{aligned}$$

We have kept (V3) in the case $\alpha \in ]0,1]$, on the one hand to obtain a certain symmetry in the statements of Theorem 2.2, on the other hand the study of approximated solutions near collisions under (V3) will be useful in Sect. 3 to prove the existence of a non-collision heteroclinic orbit at infinity for every $\alpha \in ]0,2[$ (see Theorem 3.1 below).

3 Existence of Heteroclinic Orbits

In this section, the existence of non-collision heteroclinic orbits at infinity for (HS) will be established. Consider the problem

$$\begin{aligned} \left\{ \begin{array}{ll} \ddot{q}+V'(q)=0, \\ q(-\infty )=0, \quad |q(+\infty )|= +\infty ,\\ \dot{q}(\pm \infty )=0,\\ \end{array}\right. \end{aligned}$$

(P)

where V behaves like (1) near e and satisfies the assumptions (V1)-(V3) of Theorem 2.2.

The natural condition for V at infinity for (P) is $\displaystyle \lim _{|q|\rightarrow +\infty }V(q)=0$. More precisely, we assume

• (V’4) $V(q)\sim -\displaystyle \frac{a}{|q|^b} \quad \hbox {as} \; |q|\rightarrow +\infty \,\hbox { for some }\, a>0,\, b>2$.

When $\alpha \in ]0,1]$, we need a further property of V near e

• (V5) there exists $\phi \in C^2 (]0,\, r[, \, {\mathbb {R}})$ for some $r\in ]0,|e|/4[$ such that

  $$\begin{aligned} V(q)= \phi (|q-e|)\quad \forall \, \, q \in B_r(e). \end{aligned}$$

Theorem 3.1

Suppose (V1)-(V3), (V’4) and (V5)(only when $\alpha \in ]0,1]$).

Then there exists at least one non-collision orbit of (P).

We now pass to the proof of Theorem 3.1. Solutions of (P) can be found as critical points of the functional

$$\begin{aligned} \displaystyle I(q)= \int _{{\mathbb {R}}}\Big [ \frac{1}{2}|\dot{q}|^2 - V(q) \Big ]dt \end{aligned}$$

defined on the set

$$\begin{aligned} \displaystyle \Lambda _0^{\infty } =\{ q\in H;\; \, q(-\infty )=0, \, |q(+\infty )|=+\infty , \, q(t)\ne e\, \forall \,\, t\in {\mathbb {R}}\} \end{aligned}$$

where

$$\begin{aligned} H=\Big \{ q\in H_{loc} ^1 ({\mathbb {R}}, {\mathbb {R}}^N), \displaystyle \int _{{\mathbb {R}}} |\dot{q}|^2 dt < +\infty \Big \}. \end{aligned}$$

In [14] the case $\alpha \ge 2$ (strong force case) has been studied and the existence of one classical solution of (P) has been found as a minimizer of I on $\Lambda _0^{\infty }$. In our situation where $0<\alpha <2$, we make a perturbation to the potential as in Theorem 2.2 and we consider for every n the problem

Since $V_{\varepsilon _n}$ is a strong force, we can use Lemma 1.1, and a standard compactness argument provides the existence of a classical (non-collision) solution $q_n$ of $(P_n)$ as a minimizer of the functional

$$\begin{aligned} \displaystyle I_n (q)=\int _{{\mathbb {R}} }\Big [ \frac{1}{2}|\dot{q}|^2 -V_{\varepsilon _n}(q) \Big ]dt \end{aligned}$$

on $\Lambda _0^{\infty }$, i.e. $q_n \in \Lambda _0 ^{\infty }$ such that

$$\begin{aligned} \displaystyle I_n(q_n)=\inf _{q\in \Lambda _0 ^{\infty }}I_n(q). \end{aligned}$$

(26)

By normalization, we can assume that

$$\begin{aligned} |q_n(0)|=\displaystyle \frac{|e|}{4}\quad \hbox {and}\quad |q_n(t)|< \frac{|e|}{4}\quad \forall \, t<0. \end{aligned}$$

Remark also that $q_n$ has energy zero.

Now we observe that $I_n(q_n ) \le \inf _{q\in \Lambda _0 ^{\infty }} I_1(q)=c_1< +\infty$. We deduce then the existence of a constant $C>0$ independent of n such that $||q_n||_H \le C$ and $\int _{{\mathbb {R}}}- V(q_n)dt\le C$. Thus there is a subsequence still denoted by $(q_n)$ and a function $q\in H$ such that $q_n$ converges weakly in H and uniformly in $C_{loc}({\mathbb {R}},\, {\mathbb {R}}^2)$ to q. By Fatou’s lemma $\int _{{\mathbb {R}}}-V(q) dt\le C$, so the set of collisions $D=\{t\in {\mathbb {R}},\, q(t)=e \}$ is of measure 0. In a standard way, we can see that $q\in C^2 ({\mathbb {R}}\setminus D, {\mathbb {R}}^2)$, satisfies (HS) and has energy zero in ${\mathbb {R}}\setminus D$, that is q is a generalized solution of (HS).

Lemma 3.2

$q(t)\ne e$ for all $\, t\in {\mathbb {R}}$.

Proof

We prove by contradiction assuming $q({\tilde{t}})=e$ for some $\tilde{t}\in {\mathbb {R}}$. From (V3) and the conservation of the energy, q satisfies the property (9) and then we can see that the collisions times of q are isolated. Moreover there is a sequence $(t_n)$ such that $t_n\longrightarrow \tilde{t}$ and $|q_n(t)-e|$ takes its local minimum at $t=t_n$.

As in Theorem 2.2 we define $\delta _n = |q_n(t_n)-e|$ and $d=\displaystyle \lim _{n\rightarrow +\infty } \frac{\varepsilon _n}{\delta _n ^{2-\alpha }}\in [0, +\infty ]$ (we extract a subsequence if necessary).

If we suppose that $d=+\infty$, we can see as in Lemma 2.6 that $q_n$ has a self intersection i.e. there exist $\sigma _1<\sigma _2$ such that $q_n(\sigma _1)=q_n(\sigma _2)$ and ind$_e q_n|_{[\sigma _1,\sigma _2]}=1$. Here we consider the function

$$\begin{aligned} {u_n}(t)= \left\{ \begin{array}{ll} q_n(t+\sigma _1-\sigma _2) &{} \quad ~\text {if}~~ t\le \sigma _2,\\ q_n(t) &{} \quad ~\text {if}~~ t \ge \sigma _2.\\ \end{array} \right. \end{aligned}$$

Then $u_n \in \Lambda _0 ^\infty$ and it is easy to see that $I_n(u_n)<I_n(q_n)$, which contradicts (26).

Therefore we get $d<+\infty$. In that case, there is a function $x_{\alpha ,d}$ such that after taking a subsequence still denoted by n,

$$\begin{aligned} \delta _n ^{-1} \Big [ q_n\left( \delta _n^{\frac{\alpha +2}{2}} s+ t_n\right) -e \Big ] \longrightarrow x_{\alpha ,d}(s)=|x_{\alpha ,d}(s)|\Big ( \cos (\theta _{\alpha ,d}(s)) e_1+\sin (\theta _{\alpha ,d}(s))e_2\Big ) \end{aligned}$$

in $C_{loc}^2 ({\mathbb {R}}, {\mathbb {R}}^2)$ where $(e_1,e_2)$ is an orthonormal basis of ${\mathbb {R}}^2$ and $\theta _{\alpha ,d}:{\mathbb {R}}\longrightarrow {\mathbb {R}}$ satisfies $\theta _{\alpha ,d} (0)=0$ and the properties (20)-(21).

In polar coordinates, there exists $\theta _n:{\mathbb {R}}\rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} q_n (t) -e=|q_n(t)-e|\Big ( \cos (\theta _{n}(t)) e_1 +\sin (\theta _{n}(t))e_2\Big ). \end{aligned}$$

For $\alpha \in ]1,2[$, we have from (21) $\Delta \theta _{\alpha ,d}>2\pi$. Repeating the argument of Theorem 2.2, we get that $q_n$ has a self intersection around e which is a contradiction as above.

For $\alpha \in ]0,1]$, we will use (V5) to get a contradiction. Here $\Delta \theta _{\alpha ,d}>\pi$ and $q_n$ cannot have a self intersection. However there exists $L>0$ such that $\theta _{\alpha ,d}(L)-\theta _{\alpha ,d}(-L)> \pi$. Setting $\sigma _{1,n}= t_n-\delta _n^{\frac{\alpha +2}{2}}L$ and $\sigma _{2,n}=t_n+ \delta _n ^{\frac{\alpha +2}{2}}L$, for sufficiently large n we have

$$\begin{aligned}{} & {} |q_n(t)-e|\le r,\quad \forall \, \, t \in [\sigma _{1,n},\sigma _{2,n}], \nonumber \\{} & {} \theta _n(\sigma _{2,n})-\theta _n(\sigma _{1,n})>\pi ,\nonumber \\{} & {} \dot{ \theta }_n(t)>0 \quad \forall \, t\in [\sigma _{1,n},\sigma _{2,n}]. \end{aligned}$$

(27)

Let $\sigma '_{1,n}, \sigma '_{2,n} \in [\sigma _{1,n}, \sigma _{2,n}]$ such that

$$\begin{aligned} \theta _n(\sigma '_{2,n})- \theta _n(\sigma '_{1,n})=\pi . \end{aligned}$$

We consider the function $\hat{q}_n$ defined by

$$\begin{aligned}{} & {} \hat{q}_n(t)=q_n(t)\quad \hbox { if} \quad t\in {\mathbb {R}}\setminus [\sigma '_{1,n},\sigma '_{2,n}], \\{} & {} \hat{q}_n(t)-e = |q_n(t)-e|\Big ( \cos \big (-\theta _n(t)+2 \theta _n(\sigma '_{1,n})\big ) e_1 + \sin \big (-\theta _n(t)+2\theta _n(\sigma '_{1,n})\big )e_2\Big )\\{} & {} \quad \hbox {if}\quad t\in [\sigma '_{1,n},\sigma '_{2,n}]. \end{aligned}$$

That is ${\hat{q}_n|}_{[\sigma '_{1,n},\sigma '_{2,n}]}$ and ${{q}_n|}_{[\sigma '_{1,n},\sigma '_{2,n}]}$ are axially symmetric with respect to the axis joining the two points $q_n(\sigma '_{1,n})$ and $q_n(\sigma '_{2,n})$.

Clearly $\hat{q}_n\in \Lambda _0 ^{\infty }$ and from (V5), since V is radially symmetric about e in $B_r(e)$, we get that $I_n(q_n)=I_n(\hat{q}_n)=\displaystyle \inf _{q\in \Lambda _0 ^{\infty }}I_n(q)$. It follows that $\hat{q}_n$ is of class $C^2$ and satisfies the equation $\ddot{q}+V_{\varepsilon _n}'(q)=0$. By the uniqueness of solution of ordinary differential equation, we deduce that $q_n=\hat{q}_n$, which enters in contradiction with (27). Therefore we conclude that $q(t)\ne e$ for all $t\in {\mathbb {R}}$. $\square$

End of the proof of Theorem 3.1. To prove that q is a solution of (P), it remains to show that $q(-\infty )=0,\, |q(+\infty )|= +\infty$ and $\dot{q}( \pm \infty )=0$. Using the formula (4) and the fact that $I(q)<+\infty$ one can see that $|q(-\infty )|$ and $|q(+\infty )|$ exist and they are 0 or $+\infty$. Since $|q(t)|=\lim |q_n(t)| \le |e|/4\, \, \forall \, t\le 0$, then $q(-\infty )=0$.

To show that $|q(+\infty )|=+\infty$, we suppose that $|q(+\infty )|=0$. We will construct as in [12] a function $Q_n\in \Lambda _0 ^{\infty }$ such that $I_n(Q_n) < I_n(q_n)$. Indeed, let $\varepsilon \in ]0,|e|/16[$ and $T_\varepsilon >0$ such that $q(T_\varepsilon ) \in B_\varepsilon$ the open ball of radius $\varepsilon$ about 0. For sufficiently large n we have $q_n(T_\varepsilon )\in B_{2\varepsilon }$. We consider the function $Q_n\in \Lambda _0^{\infty }$ different from $q_n$ for $t<T_\varepsilon$ such that

$$\begin{aligned} Q_n(t)=\left\{ \begin{array}{lll} 0 &{}\hbox {if }\,\, t<T_\varepsilon -1,\\ (t-T_\varepsilon +1)q_n(T_\varepsilon ) &{}\hbox {if } \,\,t\in [T_\varepsilon -1,T_\varepsilon ],\\ q_n(t)&{}\hbox {if }\,\, t\ge T_\varepsilon . \end{array}\right. \end{aligned}$$

Since $V_{\varepsilon _n}=V$ in $B_{2\varepsilon }$ and $V_{\varepsilon _n}\le V$, we have

$$\begin{aligned} I_n(Q_n)-I_n(q_n)\le 2 \varepsilon ^2 +\max _{x\in B_{2\varepsilon }}-V(x) -\int _{-\infty }^{T_\varepsilon } \Big [ \frac{1}{2}|\dot{q}_n|^2 - V(q_n) \Big ]dt. \end{aligned}$$

(28)

On the other hand, since $|q_n(0)|=|e|/4$ and $|q_n(T_\varepsilon )|\le 2\varepsilon <|e|/8$, there are $t_1<t_2$ in $[0,T_\varepsilon ]$ such that

$$\begin{aligned} |q_n(t_1)|=\frac{|e|}{4},\,\, |q_n(t_2)|=\frac{|e|}{8}\hbox { and } \frac{|e|}{8}\le |q_n(t)| \le \frac{|e|}{4}\,\, \hbox {for all}\,\, t\in [t_1,t_2]. \end{aligned}$$

By the formula (4), it holds that

$$\begin{aligned} \int _{-\infty }^{T_\varepsilon } \Big [ \frac{1}{2}|\dot{q}_n|^2 - V(q_n) \Big ]dt&\ge \int _{t_1}^{t_2} \Big [ \frac{1}{2}|\dot{q}_n|^2 - V(q_n) \Big ]dt\nonumber \\&\ge {{|e|}\over 8} \sqrt{2m_0} \end{aligned}$$

(29)

where $m_0=\displaystyle \min _{\frac{|e|}{8}\le |x|\le \frac{|e|}{4}} -V(x)>0$.

Then combining (28) and (29), we get $I_n(Q_n)-I_n(q_n)<0$ for sufficiently small $\varepsilon$, which contradicts (26). We conclude that $|q(+\infty )|=+\infty$.

From the conservation of energy and the fact that $V(q(t))\longrightarrow 0$ as $t \rightarrow \pm \infty$, it follows that $\displaystyle \frac{1}{2}|\dot{q}(t)|^2 = -V(q(t))\longrightarrow 0$ as $t\rightarrow \pm \infty$, that is $\dot{q}( \pm \infty )=0$. The proof is complete.