Prescribed energy connecting orbits for gradient systems

TL;DR

This paper proves the existence of energy-specific connecting orbits in gradient systems with certain potential functions, classifying their types and demonstrating applications to classical mechanical models.

Contribution

It introduces an energy constrained variational method to establish the existence of connecting orbits in gradient systems with general potentials, including applications to double-well, Duffing, and pendulum systems.

Findings

Existence of bounded connecting solutions with prescribed energy.

Classification of solutions into brake, heteroclinic, and homoclinic types.

Convergence results for families of solutions in specific potential cases.

Abstract

We are concerned with conservative systems $\ddot{q}=\nabla V(q), \; q\in\mathbb{R}^N$ for a general class of potentials $V\in C^1(\mathbb{R}^N)$. Assuming that a given sublevel set $\{V\leq c\}$ splits in the disjoint union of two closed subsets $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$, for some $c\in\mathbb{R}$, we establish the existence of bounded solutions $q_c$ to the above system with energy equal to $-c$ whose trajectories connect $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of $\nabla V$ on $\partial\mathcal{V}^c_{\pm}$. Next, we illustrate applications of the existence result to double-well potentials $V$, and for potentials associated to systems of Duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions $(q_c)$.

Equations

$$\ddot{q}=\nabla V(q),$$

$${\mathcal{V}}^c:=\{x\in {\mathbb{R}}^N:V(x)\le c\},$$

$$E_q(t):=\frac{1}{2}|\dot{q}(t){|}^2-V(q(t))=-c,for\; allt\in \mathbb{R},$$

$$\underset{t\in \mathbb{R}}{\inf }\mathrm{dist}(q(t),{\mathcal{V}}_{-}^c)=\underset{t\in \mathbb{R}}{\inf }\mathrm{dist}(q(t),{\mathcal{V}}_{+}^c)=0,$$

$$\underset{t\to -\infty }{\mathrm{liminf}}\mathrm{dist}(q(t),{\mathcal{V}}_{\pm }^c)=0,\text{ or }\underset{t\to +\infty }{\mathrm{liminf}}\mathrm{dist}(q(t),{\mathcal{V}}_{\pm }^c)=0.$$

$${\Gamma }_c:=\{q\in H_{loc}^1(\mathbb{R},{\mathbb{R}}^N):$$

$$J_c(q):={\int }_{-\infty }^{+\infty }\frac{1}{2}|\dot{q}(t){|}^2+(V(q(t))-c)dt.$$

$$\underset{t\to {\alpha }^{+}}{\lim }\mathrm{dist}(q(t),{\mathcal{V}}_{-}^c)=\underset{t\to {\omega }^{-}}{\lim }\mathrm{dist}(q(t),{\mathcal{V}}_{+}^c)=0.$$

$$\exists R>0\text{ s.t. }\underset{q\in {\Gamma }_c}{\inf }{J}_c(q)=\inf \{J_c(q):q\in {\Gamma }_c,\Vert q{\Vert }_{L^{\infty }(\mathbb{R},{\mathbb{R}}^N)}\le R\}.$$

$$E_{q_c}(t):=\frac{1}{2}|{\dot{q}}_c(t){|}^2-V(q_c(t))=-c,\text{ for all }t\in \mathbb{R}.$$

$$\underset{t\to -\infty }{\lim }\mathrm{dist}(q_c(t),\mathcal{A})=0and\underset{t\to +\infty }{\lim }\mathrm{dist}(q_c(t),\Omega )=0.$$

$$J_{c,I}(q):={\int }_I\frac{1}{2}|\dot{q}(t){|}^2+(V(q(t))-c)dt,$$

$${\mathcal{X}}_c:=\{q\in H_{loc}^1(\mathbb{R},{\mathbb{R}}^N):\underset{t\in \mathbb{R}}{\inf }V(q(t))\ge c\}.$$

$$J_{c,(\sigma ,\tau )}(q)$$

$$\ge \sqrt{\frac{2}{\tau -\sigma }{\int }_{\sigma }^{\tau }(V(q(t))-c)dt}|q(\tau )-q(\sigma )|.$$

$$J_{c,(\sigma ,\tau )}(q)\ge \sqrt{2\mu }|q(\tau )-q(\sigma )|.$$

$$4{\rho }_0:=\mathrm{dist}({\mathcal{V}}_{-}^c\cap B_R(0),{\mathcal{V}}_{+}^c\cap B_R(0))>0.$$

$$\inf \{V(x):|x|\le Cand\mathrm{dist}(x,{\mathcal{V}}^c)\ge r\}\ge c+h_{r,C}.$$

$$\Gamma_{c}:=\biggl{\{}u\in{\mathcal{X}}_{c}:\,\liminf_{t\to-\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{-})=\liminf_{t\to+\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{+})=0\biggr{\}},$$

$$q(t)=\{\begin{array}{ll}{\xi }_{-}& \text{if }t\le 0,\\ t{\xi }_{+}+(1-t){\xi }_{-}& \text{if }t\in (0,1),\\ {\xi }_{+}& \text{if }1\le t.\end{array}$$

$$m_c=\inf \{J_c(q):q\in {\Gamma }_{c}and\Vert q{\Vert }_{L^{\infty }(\mathbb{R},{\mathbb{R}}^N)}\le R\}.$$

$$\underset{t\to -\infty }{\mathrm{liminf}}\mathrm{dist}(q(t),{\mathcal{V}}_{-}^c)=\underset{t\to +\infty }{\mathrm{liminf}}\mathrm{dist}(q(t),{\mathcal{V}}_{+}^c)=0.$$

$$|q(\tau )-q(\sigma )|=2{\rho }_0,and\mathrm{dist}(q(t),{\mathcal{V}}^c)\ge {\rho }_0\text{ for any }t\in (\sigma ,\tau ).$$

$$J_{c,(\sigma ,\tau )}(q)\ge \sqrt{2h_{{\rho }_0}}|q(\tau )-q(\sigma )|\ge \sqrt{2h_{{\rho }_0}}2{\rho }_0.$$

$$\underset{t\to -\infty }{\lim }\mathrm{dist}(q(t),{\mathcal{V}}_{-}^c)=\underset{t\to +\infty }{\lim }\mathrm{dist}(q(t),{\mathcal{V}}_{+}^c)=0.$$

$$J_{c,({\sigma }_n,{\tau }_n)}(q)\ge \sqrt{2h_{\rho ,M}}|q({\tau }_n)-q({\sigma }_n)|=\sqrt{2h_{\rho ,M}}\rho ,\text{ for all }n\in \mathbb{N}.$$

$${\mu }_r$$

$$L_r$$

$${\nu }_r<\min \left\{\sqrt{h_{\frac{{\rho }_0}{2}}}\frac{{\rho }_0}{4},1\right\}for\; anyr\in (0,\bar{r}).$$

$$\sigma :=\sup \{t\in \mathbb{R}:\mathrm{dist}(q(t),{\mathcal{V}}_{-}^c)\le r\},\tau :=\inf \{t>\sigma :\mathrm{dist}(q(t),{\mathcal{V}}_{+}^c)\le r\}.$$

$$\mathrm{dist}(q(t),{\mathcal{V}}^c)>r\text{ for any }t\in (\sigma ,\tau ).$$

Full text

2010 Mathematics Classification. 34C25, 34C37, 49J40, 49J45.

Keywords. Conservative systems, energy constraints, variational methods, brake orbits,

homoclinic orbits, heteroclinic orbits, convergence of solutions.

1. Introduction

In the present paper we are concerned with second order conservative systems

[TABLE]

where potentials $V\in C^{1}({\mathbb{R}}^{N})$ are considered, for any dimension $N\geq 2$.

We study the existence of particular solutions to (1.1), for a class of potentials $V$ for which there exists some value $c\in{\mathbb{R}}$ so that the sublevel set

[TABLE]

is the union of two disjoint subsets. More precisely, we assume that for some $c\in{\mathbb{R}}$,

1. $(\mathbf{V}^{c})$

There exist $\mathcal{V}^{c}_{-},\,\mathcal{V}^{c}_{+}\subset{\mathbb{R}}^{N}$ closed sets, such that $\mathcal{V}^{c}=\mathcal{V}^{c}_{-}\cup\mathcal{V}^{c}_{+}$ and ${\rm dist}\,(\mathcal{V}^{c}_{-},\mathcal{V}^{c}_{+})>0$,

where ${\rm dist}\,(A,B):=\inf\{|x-y|:x\in A,\,y\in B\}$ refers to the Euclidean distance from a set $A\subset{\mathbb{R}}^{N}$ to a set $B\subset{\mathbb{R}}^{N}$.

Provided $(\mathbf{V}^{c})$ holds, we look for bounded solutions $q$ of (1.1) on ${\mathbb{R}}$ with prescribed mechanical energy at level $-c$

[TABLE]

which in addition connect the sets $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$:

[TABLE]

where ${\rm dist}\,(x,A):=\inf\{|x-y|:y\in A\}$ denotes the distance from a point $x\in{\mathbb{R}}^{N}$ to a set $A\subset{\mathbb{R}}^{N}$.

To better describe which kind of solutions of (1.1) satisfying (1.2) and (1.3) one can get, it is better to make some simple qualitative reasoning. Note that condition (1.3) imposes $\inf_{t\in{\mathbb{R}}}{\rm dist}\,(q(t),\mathcal{V}^{c}_{\pm})=0$; this is true if either the solution $q$ touches one (or both) of $\mathcal{V}^{c}_{\pm}$ in a point, or if it accumulates $\mathcal{V}^{c}_{\pm}$ at infinity, that is to say,

[TABLE]

In the first case, there exists a time $t_{0}$ such that $q(t_{0})\in\mathcal{V}^{c}_{\pm}$. In this situation we say that $q(t_{0})$ is a contact point between the trajectory $q$ and $\mathcal{V}^{c}_{\pm}$, and that $t_{0}$ is a contact time. Let us note right away that if $t_{0}$ is a contact time, since $V(q(t_{0}))\leq c$, then the energy condition (1.2) imposes that $V(q(t_{0}))=c$ and $\dot{q}(t_{0})=0$. Hence $t_{0}$ is a turning time, i.e., $q$ is symmetric with respect to $t_{0}$. From this we recover that $q$ has at most two contact points.

The connecting solutions between $\mathcal{V}^{c}_{\pm}$ can therefore be classified into three types, corresponding to the different number of contact points they exhibit. Precisely, we have

1. (I)

Two contact points: In this case the solution $q$ has one contact point $q(\sigma)$ with $\mathcal{V}^{c}_{-}$ and one contact point $q(\tau)$ with $\mathcal{V}^{c}_{+}$. We can assume that $\sigma<\tau$ (by reflecting the time if necessary), and that the interval $(\sigma,\tau)$ does not contain other contact times. Since the solution is symmetric with respect to both $\sigma$ and $\tau$, it then follows that it has to be periodic, with period $2(\tau-\sigma)$. The solution oscillates back and forth in the configuration space along the arc $q([\sigma,\tau])$, and verifies $V(q(t))>c$ for any $t\in(\sigma,\tau)$. This solution is said to be of brake orbit type (see [30], [32]). Let us remark that a brake orbit solution has only one contact point with each set $\mathcal{V}^{c}_{\pm}$. 2. (II)

One contact point: In this case the solution $q$ is symmetric with respect to the (unique) contact time $\sigma$, resulting that $V(q(t))>c$ for any $t\in{\mathbb{R}}\setminus\{\sigma\}$ and $q(\sigma)\in\mathcal{V}^{c}_{\pm}$. Moreover $\liminf_{t\to\pm\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{\mp})=0$. These solutions are said to be of homoclinic type. 3. (III)

No contact points: In this last case the absence of contact times implies that $V(q(t))>c$ for any $t\in{\mathbb{R}}$, being that $\liminf_{t\to-\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{\pm})=0$ and $\liminf_{t\to+\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{\mp})=0$. These solutions are said to be of heteroclinic type.

A great amount of work regards the existence and multiplicity of brake orbits when $c$ is regular for $V$, and the set $\{V\geq c\}$ is non-empty and bounded; see [9, 12, 13, 15, 17, 20, 21, 22, 24, 25].

A unified approach for the study of general connecting solutions was first made via variational arguments in [1] for systems of Allen-Cahn type equations, where the author already builds solutions in the PDE setting analogous to the ones of heteroclinic type, homoclinic type and brake type solutions (cf. [1, Theorem 1.2] for details, and also [3, 4, 5, 6, 7] for related results and techniques).

Concerning the ODE case, the problem of existence of connecting orbits of (1.1) and their classification into heteroclinic, homoclinic and periodic type has been recently studied in [11], and subsequently in [18] (see also [19]) for potentials $V\in C^{2}({\mathbb{R}}^{N})$ that in addition to $(\mathbf{V}^{c})$ satisfy $\partial\{x\in{\mathbb{R}}^{N}:V(x)>c\}$ is compact.

Our approach to the problem is variational and is an adaptation of the arguments developed in [1, 7] to the ODE setting. We work on the admissible class

[TABLE]

and we look for minimizers in $\Gamma_{c}$ of the Lagrangian functional

[TABLE]

Note that the set $\Gamma_{c}$ of admissible functions is defined via (i) and (ii). Condition (i) constitutes an energy constraint, in that, the function $V(q(t))-c$ is non-negative over ${\mathbb{R}}$, so the functional $J_{c}$ is well defined and bounded from below on $\Gamma_{c}$. If $q$ is a minimizer of $J_{c}$ on $\Gamma_{c}$, then $q$ is a solution of (1.1) on any interval $I\subset{\mathbb{R}}$ for which condition (i) is strictly satisfied, i.e., $V(q(t))>c$ for all $t\in I$ (see Lemma 2.10). Condition (ii) forces $q$ to connect $\mathcal{V}^{c}_{-}$ to $\mathcal{V}^{c}_{+}$. Indeed, if $q$ is a minimizer of $J_{c}$ on $\Gamma_{c}$ there exists an interval $I=(\alpha,\omega)\subset{\mathbb{R}}$ (possibly with $\alpha=-\infty$ or $\omega=+\infty$) for which $V(q(t))>c$ for any $t\in I$ (see Lemma 2.11), and

[TABLE]

Thus, $q$ is a solution of (1.1) on $I$ and $I$ is a connecting time interval, that is to say, an open interval $I\subset{\mathbb{R}}$ (not necessarily bounded) whose eventual extremes are contact times. The existence of a solution to our problem is then obtained by recognizing that the energy of such a minimizer $q$ restricted to $I$ equals $-c$ (see Lemma 2.14), from which we can proceed (by reflection and periodic continuation) to construct our entire connecting solution. Hence, we obtain brake orbit when the connecting interval $I$ is bounded ($\alpha,\omega\in{\mathbb{R}}$), a homoclinic when $I$ is an half-line (precisely one of $\alpha$ and $\omega$ is finite) and finally a heteroclinic if $I$ is the entire real line.

In the present paper, we first establish a general existence result for solutions satisfying the aforementioned properties (see section §2). In fact, the existence of a minimizer of $J_{c}$ on $\Gamma_{c}$ is obtained whenever $(\mathbf{V}^{c})$ holds and $J_{c}$ satisfies a mild coerciveness property on $\Gamma_{c}$, namely,

[TABLE]

From this minimizer we can reconstruct a solution $q_{c}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to the problem (1.1),(1.3) satisfying the energy constraint $E_{q_{c}}(t):=\frac{1}{2}|\dot{q}(t)|^{2}-V(q(t))=-c$ for all $t\in{\mathbb{R}}$, see Theorem 2.1.

In order to have a better understanding of the scope of Theorem 2.1, which is presented in a very general form, it might be useful to illustrate some specific situations in which we can verify condition $(\mathbf{V}^{c})$ and (1.7). This is done in §3 where more explicit assumptions on the potential $V$ are considered, including classical cases as double well, Duffing like and pendulum like potential systems. In all these situations the potential $V$ has isolated minima at the level $c=0$. The application of Theorem 2.1 to these cases allows us to obtain existence and multiplicity results of connecting orbits $q_{c}$ at energy level $-c$ whenever $c$ is sufficiently small (see propositions 3.1, 3.5 and 3.10). When $c=0$ the corresponding connecting orbits are homoclinic or heteroclinic solutions connecting the different minima of the potential, while we get brake orbit solutions when $c$ is a regular value for $V$. We then study convergence properties of the family of solutions $q_{c}$ to homoclinic type solutions or heteroclinic type solutions as the energy level $c$ goes to zero (see propositions 3.3, 3.7 and 3.12).

Our results extend recent studies made in [33] in the ODE framework, where for a certain class of two-well potentials, periodic orbits of (1.1) are shown to converge, in a suitable sense, to a heteroclinic solution joining the wells of such potential.

The issue of existence of heteroclinic solutions connecting the equilibria of multi-well potentials has been quite explored in the literature; the interested reader is referred to [8], [11], [19], [23], [28], and [26, 31] for different approaches on the subject.

Acknowledgement. This work has been partially supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP), and partially supported by the Projects EFI ANR-17-CE40-0030 (A.Z.) of the French National Research Agency. A.Z. also wishes to thank Peter Sternberg for fruitful discussions on the subject of this paper.

2. The general existence result

In this section we state and prove our general result concerning the existence of solutions to the conservative system (1.1) connecting the sublevels $\mathcal{V}^{c}_{\pm}$ and that satisfy a pointwise energy constraint, provided (1.7) and $(\mathbf{V}^{c})$ hold. The proof of Theorem 2.1 adapts, to the ODE case, arguments that were already developed in [1], [5] and [7] for (systems of) PDE.

Theorem 2.1**.**

Assume $V\in C^{1}({\mathbb{R}}^{N})$, and that there exists $c\in{\mathbb{R}}$ such that $(\mathbf{V}^{c})$ and the coercivity condition (1.7) of the energy functional $J_{c}$ over $\Gamma_{c}$ hold true. Then there exists a solution $q_{c}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to (1.1)-(1.3) which in addition satisfies

[TABLE]

Furthermore, any such solution is classified in one of the following types

1. (a)

$q_{c}$* is of brake orbit type: There exist $-\infty<{\sigma<\tau}<+\infty$ so that*

**

1. (a.i)

$V(q_{c}(\sigma))=V(q_{c}(\tau))=c$, $V(q_{c}(t))>c$ for every $t\in(\sigma,\tau)$ and $\dot{q_{c}}(\sigma)=\dot{q_{c}}(\tau)=0$,

** 2. (a.ii)

$q_{c}(\sigma)\in\mathcal{V}^{c}_{-}$, $q_{c}(\tau)\in\mathcal{V}^{c}_{+}$, $\nabla V(q_{c}(\sigma))\neq 0$ and $\nabla V(q_{c}(\tau))\neq 0$,

** 3. (a.iii)

$q_{c}(\sigma+t)=q_{c}(\sigma-t)$* and $q_{c}(\tau+t)=q_{c}(\tau-t)$ for all $t\in{\mathbb{R}}$.* 2. (b)

$q_{c}$ is of homoclinic type:* There exist $\sigma\in{\mathbb{R}}$ and a component $\mathcal{V}^{c}_{\pm}$ of $\mathcal{V}^{c}$ so that*

**

1. (b.i)

$V(q_{c}(\sigma))=c$, $V(q_{c}(t))>c$ for every $t\in{\mathbb{R}}\setminus\{\sigma\}$, $\dot{q_{c}}(\sigma)=0$ and $\lim_{t\to\pm\infty}\dot{q}_{c}(t)=0$,

** 2. (b.ii)

$q_{c}(\sigma)\in\mathcal{V}^{c}_{\pm}$, $\nabla V(q_{c}(\sigma))\neq 0$ and there exists a closed connected set $\Omega\subset\mathcal{V}^{c}_{\mp}\cap\{x\in{\mathbb{R}}^{N}:V(x)=c,\,\nabla V(x)=0\}$ so that $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c}(t),\Omega)=0$,

** 3. (b.iii)

$q_{c}(\sigma+t)=q_{c}(\sigma-t)$* for all $t\in{\mathbb{R}}$.* 3. (c)

$q_{c}$ is of heteroclinic type: There holds* *

**

1. (c.i)

$V(q_{c}(t))>c$* for all $t\in{\mathbb{R}}$ and $\lim_{t\to\pm\infty}\dot{q}_{c}(t)=0$,*

** 2. (c.ii)

There exist closed connected sets $\mathcal{A}\subset\mathcal{V}^{c}_{-}\cap\{x\in{\mathbb{R}}^{N}:V(x)=c,\,\nabla V(x)=0\}$ and $\Omega\subset\mathcal{V}^{c}_{+}\cap\{x\in{\mathbb{R}}^{N}:V(x)=c,\,\nabla V(x)=0\}$ such that

[TABLE]

Remark 2.2**.**

Note that if $c$ is a regular value for $V$ then the corresponding solution $q_{c}$ given by Theorem 2.1 is of brake type, while it may be of the heteroclinic or homoclinic type if $c$ is a critical value of $V$.

To prove Theorem 2.1, given $c\in{\mathbb{R}}$ and an interval $I\in{\mathbb{R}}$, we consider the action functional

[TABLE]

defined on the space

[TABLE]

We will write henceforth $J_{c}(q):=J_{c,{\mathbb{R}}}(q)$.

Remark 2.3**.**

Note that since the lower bound $V(q(t))\geq c$ for all $t\in{\mathbb{R}}$ holds for any $q\in{\mathcal{X}}_{c}$, then we readily see that $J_{c,I}$ is non-negative on ${\mathcal{X}}_{c}$ for any given real interval $I$. Moreover, ${\mathcal{X}}_{c}$ is sequentially closed, and for any interval $I\subset{\mathbb{R}}$, $J_{c,I}$ is lower semicontinuous with respect to the weak topology of $H^{1}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$.

Remark 2.4**.**

If $q\in{\mathcal{X}}_{c}$ and $(\sigma,\tau)\subset{\mathbb{R}}$, then

[TABLE]

In particular, if there exists some $\mu>0$ for which $V(q(t))-c\geq\mu\geq 0$ for all $t\in(\sigma,\tau)$, then we have

[TABLE]

Remark 2.5**.**

In view of $(\mathbf{V}^{c})$, the sets $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$ are disjoint and closed, and so they are locally well separated. Hence, if $R$ denotes the constant introduced in the coerciveness assumption (1.7), we have

[TABLE]

The continuity of $V$ ensures that for any $r>0$ and $C>0$, there exists $h_{r,C}>0$ in such a way that

[TABLE]

In what follows, we will simply denote $h_{r}:=h_{r,R}$, where $R$ is the constant given in (1.7).

The variational problem we are interested in studying involves the following admissible set

[TABLE]

and we will denote $m_{c}:=\inf_{q\in\Gamma_{c}}J_{c}(q)$.

The first observation in place is

Lemma 2.6**.**

There results $m_{c}\in(0,+\infty)$.

Proof of Lemma 2.6.

It is plain to observe that $m_{c}<+\infty$. Indeed, by $(\mathbf{V}^{c})$ we can choose two points $\xi_{-}\in\mathcal{V}^{c}_{-}$ and $\xi_{+}\in\mathcal{V}^{c}_{+}$ such that $V(t\xi_{+}+(1-t)\xi_{-})>c$ for any $t\in(0,1)$. Then defining

[TABLE]

one plainly recognizes that $q\in{\Gamma}_{c}$ and $m_{c}\leq J_{c}(q)<+\infty$.

To show that $m_{c}>0$, let us observe that in light of (1.7)

[TABLE]

Also, by Sobolev embedding theorems, any $q\in\Gamma_{c}$ is continuous over ${\mathbb{R}}$ and it verifies

[TABLE]

Recalling that $\mathcal{V}^{c}=\mathcal{V}^{c}_{-}\cup\mathcal{V}^{c}_{+}$, we deduce from (2.9) and the fact that $\|q\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R$, that there exists a nonempty open interval $(\sigma,\tau)\subset{\mathbb{R}}$ depending on $q$, in such a way that

[TABLE]

But then Remark 2.5 yields a uniform lower bound $V(q(t))-c\geq h_{\rho_{0}}$, for any $t\in(\sigma,\tau)$. This fact, combined with Remark 2.4 yields

[TABLE]

Therefore, $m_{c}=\inf_{\Gamma_{c}}J_{c}(q)\geq\inf_{\Gamma_{c}}J_{c,(\sigma,\tau)}(q)\geq\sqrt{2h_{\rho_{0}}}\,2\rho_{0}>0$. ∎

We now argue that finite energy elements in the admissible class $\Gamma_{c}$ asymptotically approach the sublevel sets $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$ in the following sense

Lemma 2.7**.**

Suppose $q\in\Gamma_{c}$ satisfies $\|q\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}<+\infty$ and $J_{c}(q)<+\infty$. Then

[TABLE]

Proof of Lemma 2.7.

Let us argue the case of the limit as $t\to-\infty$, the other limit can be argued similarly. By definition of $\Gamma_{c}$, $q$ satisfies $\liminf_{t\to-\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{-})=0$. Let us assume by contradiction that $\limsup_{t\to-\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{-})>0$, so there must exist $\rho\in(0,\rho_{0})$ and two sequences $\sigma_{n}\to-\infty$, ${\tau_{n}\to-\infty}$ such that $\tau_{n+1}<\sigma_{n}<\tau_{n}$ for which there results $|q(\tau_{n})-q(\sigma_{n})|=\rho$ and $\rho\leq{\rm dist}\,(q(t),\mathcal{V}^{c}_{-})\leq 2\rho$ for any $t\in(\sigma_{n},\tau_{n})$. In particular, since $\rho<\rho_{0}$ it follows that ${\rm dist}\,(q(t),\mathcal{V}^{c})>\rho$ for any $t\in(\sigma_{n},\tau_{n})$ and $n\in{{\mathbb{N}}}$. Since $M:=\|q\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}<+\infty$, Remark 2.5 yields $V(q(t))>c+h_{\rho,M}$ for any $t\in(\sigma_{n},\tau_{n})$ and so, by Remark 2.4, we conclude

[TABLE]

But then $J_{c}(q)\geq\sum_{n=1}^{\infty}J_{c,(\sigma_{n},\tau_{n})}(q)=+\infty$, thus contradicting the assumption $J_{c}(q)<+\infty$. ∎

Moreover, by (2.9) we obtain the following concentration result

Lemma 2.8**.**

There exists $\bar{r}\in(0,\frac{\rho_{0}}{2})$ so that for any $r\in(0,\bar{r})$, there exist $L_{r}>0$, $\nu_{r}>0$, in such a way that for any $q\in\Gamma_{c}$ satisfying: ${\rm dist}\,(q(0),\mathcal{V}^{c})\geq\rho_{0}$, $\|q\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R$ and $J_{c}(q)\leq m_{c}+\nu_{r}$, one has

1. (i)

There is $\tau\in(0,L_{r})$ so that $\,{\rm dist}\,(q(\tau),\mathcal{V}^{c}_{+})\leq r$, and ${\rm dist}\,(q(t),\mathcal{V}^{c}_{+})<\rho_{0},\,$ for all $t\geq\tau$. 2. (ii)

There is $\sigma\in(-L_{r},0)$ so that $\,{\rm dist}\,(q(\sigma),\mathcal{V}^{c}_{-})\leq r$, and ${\rm dist}\,(q(t),\mathcal{V}^{c}_{-})<\rho_{0},\,$ for all $t\leq\sigma$.

Proof of Lemma 2.8.

Given any $r\in(0,\frac{\rho_{0}}{2})$ we define the following quantities

[TABLE]

In view of the continuity of $V$ we have $\lim_{r\to 0^{+}}\nu_{r}=0$. Hence, we can choose $\bar{r}\in(0,\frac{\rho_{0}}{2})$ so that

[TABLE]

For $q\in\Gamma_{c}$ satisfying the assumptions of this lemma, let us define

[TABLE]

We observe that $-\infty<\sigma<\tau<+\infty$, since $\lim_{t\to-\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{-})=\lim_{t\to+\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{+})=0$, in light of the fact that the hypotheses of Lemma 2.7 are fulfilled for any $q$ as above. Furthermore, the definition of $\sigma$ and $\tau$ yield

[TABLE]

Now we claim that

[TABLE]

Indeed, we can fix $\xi_{\sigma}\in\mathcal{V}^{c}_{-}$ so that $|q(\sigma)-\xi_{\sigma}|\leq r$ and

[TABLE]

Let us define

[TABLE]

and

[TABLE]

First we note that by (2.13) and since $q\in\Gamma_{c}$, we have $\overline{q}\in\Gamma_{c}$ and so $J_{c}(\overline{q})\geq m_{c}$. The latter, combined with the following inequality

[TABLE]

shows that $J_{c,(\sigma,+\infty)}(\overline{q})\geq m_{c}-\nu_{r}$, from which it follows

[TABLE]

thus proving

[TABLE]

To finish the proof of claim (2.12), let us assume by contradiction that there is $t_{*}<\sigma$ such that ${\rm dist}\,(q(t_{*}),\mathcal{V}^{c}_{-})\geq\rho_{0}$. Since ${\rm dist}\,(q(\sigma),\mathcal{V}^{c}_{-})\leq r<\tfrac{\rho_{0}}{2}$, we deduce that there exists an interval $(\gamma,\delta)\subset(-\infty,\sigma)$ such that $|\underline{q}(\delta)-\underline{q}(\gamma)|=\frac{\rho_{0}}{2}$ and $\tfrac{\rho_{0}}{2}\leq{\rm dist}\,(\underline{q}(t),\mathcal{V}^{c}_{-})\leq\rho_{0}$ for all $t\in(\gamma,\delta)$. Then, estimate (2.14) combined with Remark 2.4 and Remark 2.5 (since $\|q\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R$), yields

[TABLE]

which contradicts (2.10) in view of the definition of $\bar{r}$. An analogous argument proves that

[TABLE]

In this way, we have argued that the conditions (i)-(ii) are satisfied for the choice of $\tau$ and $\sigma$ as above. We are left to prove the chain of inequalities$-L_{r}<\sigma<0<\tau<L_{r}$, for the choice of $L_{r}$ as in the beginning of the proof. To see this, let us first note that $0\in(\sigma,\tau)$. This follows from the way the time $t=0$ was chosen: ${\rm dist}\,(q(0),\mathcal{V}^{c})\geq\rho_{0}$ and from (2.12)-(2.15) combined. Also, from Remark 2.5 and (2.11) we have $V(q(t))-c\geq h_{r}$ for $t\in(\sigma,\tau)$. This, and (2.10) yield the lower bound

[TABLE]

In other words, we have proved that $0<\tau<\frac{m_{c}+1}{h_{r}}=:L_{r}$. Analogously, we derive that $m_{c}+1>m_{c}+\nu_{r}\geq J_{c,(\sigma,0)}(q)\geq-\sigma h_{r}$ from which $0<-\sigma<L_{r}$. The proof of Lemma 2.8 is now complete. ∎

We can now conclude that the minimal level $m_{c}$ is achieved in $\Gamma_{c}$. Indeed, we have

Lemma 2.9**.**

There exists $q_{0}\in\Gamma_{c}$ such that $J_{c}(q_{0})=m_{c}$.

Proof of Lemma 2.9.

Let $(q_{n})\subset\Gamma_{c}$ be a minimizing sequence, so $J_{c}(q_{n})\to m_{c}$. The coerciveness assumption (1.7) allows us to assume that

[TABLE]

and hence Lemma 2.7 yields

[TABLE]

Thus, (2.9) combined with continuity arguments shows that there is $(t_{n})\subset{\mathbb{R}}$ so that ${\rm dist}\,(q_{n}(t_{n}),\mathcal{V}^{c})=\rho_{0}$. Since the variational problem is invariant under time translations, we can assume that

[TABLE]

But then conditions (2.17),(2.16) together with $J_{c}(q_{n})\to m_{c}$ allow us to use Lemma 2.8 to deduce the existence of $L>0$, in a such way that

[TABLE]

for all but finitely many terms in the sequence $(q_{n})$. Observe now that $\inf_{t\in{\mathbb{R}}}V(q_{n}(t))\geq c$ for all $n\in{{\mathbb{N}}}$, since $q_{n}\in{\mathcal{X}}_{c}$, whence

[TABLE]

By (2.16) and (2.19) we obtain the existence of $q_{0}\in H^{1}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$, such that along a subsequence (which we continue to denote $q_{n}$) $q_{n}\rightharpoonup q_{0}$ weakly in $H^{1}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$. As $q_{0}\in{\mathcal{X}}_{c}$, in view of Remark 2.3, we deduce $J_{c}(q_{0})\leq m_{c}=\lim_{n\to\infty}J_{c}(q_{n})$. On the other hand, the pointwise convergence, (2.16) and (2.18) yield that $V(q_{0}(t))\geq c$ for any $t\in{\mathbb{R}}$, that $\|q_{0}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R$ and

[TABLE]

Since $\int_{L}^{+\infty}(V(q_{0}(t))-c)\,dt\leq J_{c,(L,+\infty)}(q_{0})\leq m_{c}$, we obtain that

[TABLE]

in view of (2.20). Analogously, we deduce that $\liminf_{t\to-\infty}{\rm dist}\,(q_{0}(t),\mathcal{V}^{c}_{-})=0$. Thus, we have argued that $q_{0}\in\Gamma_{c}$, which in turn shows the reverse inequality $J_{c}(q_{0})\geq m_{c}$. The proof of Lemma 2.9 is now complete. ∎

It will be convenient to introduce the following set of minimizers to the variational problem studied in Lemma 2.9,

[TABLE]

The proof of the above lemma reveals that ${\mathcal{M}}_{c}\neq\emptyset$.

For any $q\in{\mathcal{M}}_{c}$, we introduce the contact times of the trajectory of $q$ with the sublevel sets $\mathcal{V}^{c}_{+}$ and $\mathcal{V}^{c}_{-}$ of the potential by letting

[TABLE]

and

[TABLE]

Note that for all $q\in{\mathcal{M}}_{c}$, by Lemma 2.8 and by the definition of $\rho_{0}$ (2.9), it is simple to verify that

[TABLE]

Moreover, by definition of $\alpha_{q}$ and $\omega_{q}$, since ${\rm dist}\,(q(0),\mathcal{V}^{c})=\rho_{0}$ for every $q\in{\mathcal{M}}_{c}$, we have $q(t)\in{\mathbb{R}}^{N}\setminus(\mathcal{V}^{c}_{-}\cup\mathcal{V}^{c}_{+})$ for any $\alpha_{q}<t<\omega_{q}$, that is

[TABLE]

Therefore we obtain

Lemma 2.10**.**

If $q\in{\mathcal{M}}_{c}$ then $q\in{C}^{2}((\alpha_{q},\omega_{q}),{\mathbb{R}}^{N})$. Furthermore, any such $q$ is a solution to the system

[TABLE]

Proof of Lemma 2.10.

Let $\psi\in C_{0}^{\infty}({\mathbb{R}})$ be so that $\hbox{supp}\,\psi\subset[a,b]\subset(\alpha_{q},\omega_{q})$. Since $V(q(t))>c$ for any $t\in(\alpha_{q},\omega_{q})$ and $t\mapsto V(q(t))$ is continuous on ${\mathbb{R}}$, we derive that there exists $\lambda_{0}>0$ such that $\min_{t\in[a,b]}V(q(t))=c+\lambda_{0}$. The continuity of $V$ ensures that there exists $h_{\psi}>0$ such that

[TABLE]

In other words, for any $\psi\in C_{0}^{\infty}({\mathbb{R}})$ with $\hbox{supp}\,\psi\subset[a,b]\subset(\alpha_{q},\omega_{q})$ there exists $h_{\psi}>0$ in such a way that $q+h\,\psi\in\Gamma_{c}$, provided $h\in(0,h_{\psi})$. Since $q$ is a minimizer of $J_{c}$ over $\Gamma_{c}$, then

[TABLE]

Writing the inequality explicitly, and using the Dominated Convergence Theorem as $h\to 0^{+}$, we readily see

[TABLE]

The same argument with $-\psi$ as test function shows $\int_{{\mathbb{R}}}\dot{q}\cdot\dot{\psi}+\nabla V(q)\cdot\psi\,dt=0$, so $q$ is a weak solution of $\ddot{q}=\nabla V(q)$ on $(\alpha_{q},\omega_{q})$. Standard regularity arguments show that $q\in C^{2}((\alpha_{q},\omega_{q}),{\mathbb{R}}^{N})$, whence $q$ is a strong solution to the above system. ∎

Moreover, we have

Lemma 2.11**.**

If $q\in{\mathcal{M}}_{c}$, then

1. (i)

$\lim\limits_{t\to\alpha_{q}^{+}}{\rm dist}\,(q(t),\mathcal{V}^{c}_{-})=0$, and if $\alpha_{q}>-\infty$ then $V(q(\alpha_{q}))=c$ with $q(\alpha_{q})\in\mathcal{V}^{c}_{-}$, 2. (ii)

$\lim\limits_{t\to\omega_{q}^{-}}{\rm dist}\,(q(t),\mathcal{V}^{c}_{+})=0$, and if $\omega_{q}<+\infty$ then $V(q(\omega_{q}))=c$ with $q(\omega_{q})\in\mathcal{V}^{c}_{+}$.

Proof of Lemma 2.11.

If $\alpha_{q}=-\infty$, by Lemma 2.7 we obtain $\lim_{t\to-\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{-})=0$. If $\alpha_{q}>-\infty$, then the continuity of $q$ and the definition of $\alpha_{q}$ imply that $q(\alpha_{q})\in\mathcal{V}^{c}_{-}$, and $V(q(\alpha_{q}))=c$. In particular, $\lim_{t\to\alpha_{q}^{+}}V(q(t))=c$, and (i) follows. One argues (ii) in a similar fashion. ∎

By the previuos result we obtain

Lemma 2.12**.**

Any $q\in{\mathcal{M}}_{c}$ satisfies $J_{c}(q)=J_{c,(\alpha_{q},\omega_{q})}(q)=m_{c}$. Moreover, $q(t)\equiv q(\alpha_{q})$ on $(-\infty,\alpha_{q})$ if $\alpha_{q}\in{\mathbb{R}}$, and $q(t)\equiv q(\omega_{q})$ on $(\omega_{q},+\infty)$ if $\omega_{q}\in{\mathbb{R}}$.

Proof of Lemma 2.12.

Let us define $\tilde{q}$ to be equal to $q$ on the interval $(\alpha_{q},\omega_{q})$, and such that $\tilde{q}(t)=q(\alpha_{q})$ on $(-\infty,\alpha_{q})$ if $\alpha_{q}\in{\mathbb{R}}$, while $\tilde{q}(t)=q(\omega_{q})$ on $(\omega_{q},+\infty)$ if $\omega_{q}\in{\mathbb{R}}$ (so if neither of $\alpha_{q}$ or $\omega_{q}$ is finite, then $\tilde{q}=q$). In view of Lemma 2.11 we see that $\tilde{q}\in\Gamma_{c}$, whence $J_{c}(\tilde{q})\geq m_{c}$. The latter implies that $\tilde{q}$ is also a minimizer of $J_{c}$, as $J_{c}(\tilde{q})=J_{c,(\alpha_{q},\omega_{q})}(\tilde{q})=J_{c,(\alpha_{q},\omega_{q})}(q)\leq m_{c}$, from which we deduce $m_{c}=J_{c}(\tilde{q})=J_{c,(\alpha_{q},\omega_{q})}(q)$. In particular, since $\inf_{t\in{\mathbb{R}}}V(q(t))\geq c$ we obtain $J_{c,(-\infty,\alpha_{q})}(q)=J_{c,(\omega_{q},+\infty)}(q)=0$, which shows $\|\dot{q}\|_{L^{2}((-\infty,\alpha_{q}),{\mathbb{R}}^{N})}=\|\dot{q}\|_{L^{2}((\omega_{q},+\infty),{\mathbb{R}}^{N})}=0$. Therefore, $q$ must be constant on $(-\infty,\alpha_{q})$, and on $(\omega_{q},+\infty)$, so the lemma is established. ∎

By the previous result, we obtain

Lemma 2.13**.**

Consider $q\in{\mathcal{M}}_{c}$, and let $(\tau,\sigma)\subseteq(\alpha_{q},\omega_{q})$ be arbitrary. Then,

[TABLE]

Proof of Lemma 2.13.

Given any $\tau\in(\alpha_{q},\omega_{q})$, we will prove that

[TABLE]

For any $s>0$ and $\tau$ as above, we define

[TABLE]

It is easy to check that $q_{s}\in\Gamma_{c}$, by Lemma 2.11 using that $q\in{\mathcal{M}}_{c}$. Furthermore, by Lemma 2.12, we deduce

[TABLE]

In particular, for all $s>0$ we obtain the following

[TABLE]

Hence, setting $T:=\int_{\tau}^{\omega_{q}}\tfrac{1}{2}|\dot{q}(t)|^{2}\,dt$ and $U:=\int_{\tau}^{\omega_{q}}(V(q(t))-c)\,dt$, we get that the real function $s\mapsto f(s)=(\tfrac{1}{s}-1)T+(s-1)U$ is non-negative over $(0,+\infty)$. Since it achieves a non-negative minimum at $s=\sqrt{T/U}$, where

[TABLE]

we conclude $T=U$, i.e. (2.23). A similar argument also shows that for any $\sigma\in(\alpha_{q},\omega_{q})$ one has the identity

[TABLE]

Then, by (2.23) and (2.24), using the additivity property of the integral, we conclude the proof. ∎

We are now able to prove that every $q\in{\mathcal{M}}_{c}$ satisfies the following pointwise energy constraint:

Lemma 2.14**.**

Every $q\in{\mathcal{M}}_{c}$ verifies

[TABLE]

Proof of Lemma 2.14.

By Lemma 2.10, $q$ solves the system of differential equations (1.1) on $(\alpha_{q},\omega_{q})$ and so the energy $E_{q}(t)=\frac{1}{2}|\dot{q}(t)|^{2}-V(q(t))$ must be constant on $(\alpha_{q},\omega_{q})$. We are left to show that the value of this constant is precisely $-c$. Let us first treat the case $\alpha_{q}=-\infty$. We observe that $J_{c}(q)=\int_{\mathbb{R}}\frac{1}{2}|\dot{q}(t)|^{2}+(V(q(t))-c)=m_{c}<+\infty$ direcly yields

[TABLE]

Since $V(q(t))\geq c$ for any $t\in{\mathbb{R}}$, we deduce that there exists a sequence $(t_{n})$ such that $t_{n}\to-\infty$ for which $\lim_{n\to+\infty}\tfrac{1}{2}|\dot{q}(t_{n})|^{2}=0$ and $\lim_{n\to\infty}V(q(t_{n}))=c$. So necessarily

[TABLE]

Hence $E_{q}(t)=-c$ for all $t\in(-\infty,\omega_{q})$, proving the lemma in the case $\alpha_{q}=-\infty$. Clearly, the argument above can be easily applied when $\omega_{q}=+\infty$, to show $E_{q}(t)=-c$ for all $t\in(\alpha_{q},+\infty)$.

Let us consider now the case $-\infty<\alpha_{q}<\omega_{q}<+\infty$. As $q\in{\mathcal{M}}_{c}$, Lemma 2.11 tells us that $V(q(\omega_{q}))=c$, and so by continuity of the potential it follows ${\lim_{t\to\omega_{q}^{-}}V(q(t))=c}$. A similar continuity argument shows

[TABLE]

which, in light of the identity of Lemma 2.13, directly proves that

[TABLE]

For then, $\liminf_{y\to\omega_{q}^{-}}\tfrac{1}{2}|\dot{q}(t)|^{2}=0$, from which

[TABLE]

proving that $E_{q}(t)=-c$ for every $t\in(\alpha_{q},\omega_{q})$. ∎

We are now able to construct the connecting solutions, concluding the Proof of Theorem 2.1

For $q\in{\mathcal{M}}_{c}$ and provided $\omega_{q}<+\infty$, we denote $q_{+}$ the extension of $q$ by reflection with respect to $\omega_{q}$

[TABLE]

Similarly, for $q\in{\mathcal{M}}_{c}$ with $\alpha_{q}>-\infty$, let $q_{-}$ be the extension of $q$ by reflection with respect to $\alpha_{q}$

[TABLE]

We have

Lemma 2.15**.**

For $q\in{\mathcal{M}}_{c}$, the following properties hold:

1. $\bullet$

If $\omega_{q}<+\infty$, then $\lim_{t\to\omega_{q}^{-}}\dot{q}_{+}(t)=0$, and $q_{+}$ is a solution of (1.1) on $(\alpha_{q},2\omega_{q}-\alpha_{q})$. Furthermore, there results $\nabla V(q_{+}(\omega_{q}))\neq 0$. 2. $\bullet$

If $\alpha_{q}>-\infty$, then $\lim_{t\to\alpha_{q}^{+}}\dot{q}_{-}(t)=0$, and $q_{-}$ is a solution of (1.1) on $(2\alpha_{q}-\omega_{q},\omega_{q})$. Furthermore, there results $\nabla V(q_{-}(\alpha_{q}))\neq 0$.

Proof of Lemma 2.15.

Given $q\in{\mathcal{M}}_{c}$, let us assume $\omega_{q}<+\infty$; the other case where $\alpha_{q}>-\infty$ can be treated analogously. By Lemma 2.11 we already know that $V(q(\omega_{q}))=c$, so by continuity, $\lim_{t\to\omega_{q}^{-}}V(q(t))-c=0$, which in turn shows $\lim_{t\to\omega_{q}^{-}}|\dot{q}(t)|^{2}=\lim_{t\to\omega_{q}^{-}}2(V(q(t)-c)=0$, due to Lemma 2.14.

The system (1.1) is of second order and autonomous, so starting from the solution $q$ of (1.1) over $(\alpha_{q},\omega_{q})$ (in view of Lemma 2.10) we immediately get that $q_{+}$ is a solution of (1.1) on $(\alpha_{q},\omega_{q})\cup$ ${(\omega_{q},2\omega_{q}-\alpha_{q})}$. Since $q_{+}$ is continuous on the entire interval $(\alpha_{q},2\omega_{q}-\alpha_{q})$ and $\lim_{t\to\omega_{q}}\dot{q}_{+}(t)=0$ as argued in the preceding paragraph, we deduce that $q_{+}\in C^{1}(\alpha_{q},2\omega_{q}-\alpha_{q})$. Using now the fact that $q_{+}$ solves (1.1) on $(\alpha_{q},2\omega_{q}-\alpha_{q})\setminus\{\omega_{q}\}$, we readily see that the second derivative exists $\ddot{q}_{+}(\omega_{q}):=\lim_{t\to\omega_{q}}\ddot{q}_{+}(t)=\nabla V(q_{+}(\omega_{q}))$. For then $q_{+}\in C^{2}(\alpha_{q},2\omega_{q}-\alpha_{q})$, and furthermore, it solves (1.1) on the entire interval $(\alpha_{q},2\omega_{q}-\alpha_{q})$.

In order to conclude the proof, we need to argue that $\nabla V(q_{+}(\omega_{q}))\neq 0$. Suppose on the contrary that $\nabla V(q_{+}(\omega_{q}))=0$, then $q(\omega_{q})$ is an equilibrium of (1.1). Since $\dot{q}_{+}(\omega_{q})=0$ and $q_{+}(\omega_{q})=q(\omega_{q})$, the uniqueness of solutions to the Cauchy problem shows $q_{+}(t)=q(\omega_{q})$ for any $t\in(\alpha_{q},2\omega_{q}-\alpha_{q})$.

However, this contradicts Lemma 2.11, for which $q_{+}(\omega_{q})=q(\omega_{q})\in\mathcal{V}^{c}_{+}$ and

[TABLE]

∎

Thanks to Lemma 2.15, we know that in the event $q\in{\mathcal{M}}_{c}$ satisfies $-\infty<\alpha_{q}<\omega_{q}<+\infty$, then $q_{+}$ is a solution on the bounded interval $(\alpha_{q},2\omega_{q}-\alpha_{q})$, and it verifies

[TABLE]

This property implies, in particular, that the $2(\omega_{q}-\alpha_{q})$-periodic extension of $q_{+}$ is well defined. In fact, by Lemma 2.10 this extension is a classical $2(\omega_{q}-\alpha_{q})$-periodic solution of (1.1). Clearly, one also makes analogous statements for $q_{-}$.

Hence, in the case $q\in{\mathcal{M}}_{c}$ is so that $-\infty<\alpha_{q}<\omega_{q}<+\infty$, we denote $T:=2(\omega_{q}-\alpha_{q})$, and we let $q_{c}$ be the T-periodic extension of $q_{+}$ (or $q_{-}$) over ${\mathbb{R}}$, obtaining that $q_{c}$ is a $T$-periodic classical solution of (1.1) over ${\mathbb{R}}$, that satisfies the pointwise energy constraint $E_{{q_{c}}}(t)=-c$ for all $t\in{\mathbb{R}}$. Furthermore, by Lemma 2.11 and Lemma 2.15, it connects $\mathcal{V}^{c}_{-}$ to $\mathcal{V}^{c}_{+}$, in the following sense

1. (i)

$V(q_{c}(\alpha_{q}))=V(q_{c}(\omega_{q}))=c$, $V(q_{c}(t))>c$ for any $\,t\in(\alpha_{q},\omega_{q})$ and $\dot{q_{c}}(\alpha_{q})=\dot{q_{c}}(\omega_{q})=0$, 2. (ii)

$q_{c}(\alpha_{q})\in\mathcal{V}^{c}_{-}$, $q_{c}(\omega_{q})\in\mathcal{V}^{c}_{+}$, $\nabla V(q_{c}(\alpha_{q}))\neq 0$ and $\nabla V(q_{c}(\omega_{q}))\neq 0$, 3. (iii)

$q_{c}(\alpha_{q}-t)=q_{c}(\alpha_{q}+t)\,$ and $\;q_{c}(\omega_{q}-t)=q_{c}(\omega_{q}+t)$, for any $t\in{\mathbb{R}}$.

Therefore, when a minimizer $q\in{\mathcal{M}}_{c}$ satisfies $-\infty<\alpha_{q}<\omega_{q}<+\infty$, then it generates a solution which periodically oscillates back and forth between the boundary of the two sets $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$, that is a brake orbit type solution of (1.1) connecting $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$. Moreover, it is bounded, and in fact, verifies (1.3). Hence, denoting $\sigma=\alpha_{q}$ and $\tau=\omega_{q}$, the assertion (a) in Theorem 2.1 is proved.

The remaining cases, where the minimizer $q\in{\mathcal{M}}_{c}$ satisfies either $\omega_{q}=+\infty$, or $\alpha_{q}=-\infty$, are dealt in Lemma 2.16 below. Before stating this lemma, let us introduce some notation. For $q\in{\mathcal{M}}_{c}$ having $\omega_{q}=+\infty$, we will denote the $\omega$-limit set of $q$ by

[TABLE]

The boundedness of any minimizer $q\in{\mathcal{M}}_{c}$, $\|q\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R$, shows that $\Omega_{q}\subset\overline{B_{R}(0)}$. Also $\Omega_{q}$ is a closed, connected subset of ${\mathbb{R}}^{n}$, being the intersection of closed connected sets. Analogously, for $q\in{\mathcal{M}}_{c}$ having $\alpha_{q}=-\infty$, we will write

[TABLE]

for the $\alpha$-limit set of $q$, which is a closed, connected subset of $\overline{B_{R}(0)}$. Hence, we have

Lemma 2.16**.**

Suppose $q\in{\mathcal{M}}_{c}$ has either $\alpha_{q}=-\infty$, or $\,\omega_{q}=+\infty$. Then, $\dot{q}(t)\to 0$ as $t\to-\infty$, or as $t\to+\infty$, respectively. Moreover the $\alpha$-limit set of $q$, or the $\omega$-limit set of $q$, respectively, is constituted by critical points of $V$ at level $c$, namely

1. $\bullet$

$\mathcal{A}_{q}\subset\mathcal{V}^{c}_{-}\cap\{\xi\in{\mathbb{R}}^{N}:V(\xi)=c,\,\nabla V(\xi)=0\}\,$, or respectively, 2. $\bullet$

$\Omega_{q}\subset\mathcal{V}^{c}_{+}\cap\{\xi\in{\mathbb{R}}^{N}:V(\xi)=c,\,\nabla V(\xi)=0\}$.

Proof of Lemma 2.16.

Given $q\in{\mathcal{M}}_{c}$, let us assume $\omega_{q}=+\infty$; the case $\alpha_{q}=-\infty$ is treated similarly. First, note that $\lim_{t\to+\infty}\dot{q}(t)=0$. Indeed, the fact that $\lim_{t\to+\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{+})=0$ (cf. Lemma 2.7) combined with the uniform continuity of $V$ on $\overline{B_{R}(0)}$, where $\|q\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R$, proves that $\lim_{t\to+\infty}V(q(t))=c$. But this together with Lemma 2.14 show that $\lim_{t\to+\infty}|\dot{q}(t)|^{2}=\lim_{t\to+\infty}2(V(q(t))-c)=0$.

To prove the second statement, let $\xi\in\Omega_{q}$ so there is a sequence $t_{n}\to+\infty$ such that $q(t_{n})\to\xi$ as $n\to+\infty$. Since $\lim_{t\to+\infty}{\rm dist}\,(q(t),\mathcal{V}^{c}_{+})=0$, it follows ${\rm dist}\,(\xi,\mathcal{V}^{c}_{+})=$ $\lim_{n\to\infty}{\rm dist}\,(q(t_{n}),\mathcal{V}^{c}_{+})=0$, whereby $\xi\in\mathcal{V}^{c}_{+}$ in light of the closeness of $\mathcal{V}^{c}_{+}$. On the other hand, we have already seen that $\lim_{t\to+\infty}V(q(t))=c$, so the continuity of the potential yields $V(\xi)=\lim_{n\to\infty}V(q(t_{n}))=c$. All of this shows that $\Omega_{q}\subset\mathcal{V}^{c}_{+}\cap\{\xi\in{\mathbb{R}}^{N}:V(\xi)=c\}$. There just remains to be shown that $\nabla V(\xi)=0$.

For $(t_{n})\subset{\mathbb{R}}$ given as above, consider the following sequence of translates of $q$,

[TABLE]

and note that for any bounded interval $I\subset{\mathbb{R}}$ we have that $\sup_{t\in I}|\dot{q}_{n}(t)|\to 0$ as $n\to+\infty$, because $\dot{q}(t)\to 0$ as $t\to+\infty$. In particular, for any $t\in{\mathbb{R}}$ we deduce that

[TABLE]

Put another way, $q(\cdot+t_{n})\to\xi$ as $n\to+\infty$, with respect to the $C^{1}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$-topology. Since $\ddot{q}(t+t_{n})=\nabla V(q(t+t_{n}))$ for $t\in(\alpha_{q}-t_{n},+\infty)$, we conclude that $\ddot{q}_{n}(t)\to\nabla V(\xi)$ as $n\to+\infty$, uniformly on bounded subsets of ${\mathbb{R}}$. On the other hand, given any $t\neq 0$, we can take the limit as $n\to+\infty$ in the identity

[TABLE]

As argued before, the left side of the equation converges to [math], while the right side converges to $t\nabla V(\xi)$. Therefore, $\nabla V(\xi)=0$, which concludes the proof. ∎

We remark that the previous result proves, in particular, that if $c$ is a regular value for $V$ then for every $q\in{\mathcal{M}}_{c}$ both $\alpha_{q}$ and $\omega_{q}$ must be finite.

For $q\in{\mathcal{M}}_{c}$ with $\alpha_{q}=-\infty$ or $\omega_{q}=+\infty$, Lemma 2.15 and Lemma 2.16 allow us to construct from it an entire solution of (1.1) with energy at level $-c$, connecting $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$ in the sense of (1.3). This entire solution is either a homoclinic type solution or a heteroclinic type solution, depending on the finiteness of $\alpha_{q}$ and $\omega_{q}$. Indeed, let us define

[TABLE]

and observe that in light of Lemma 2.14 and Lemma 2.15, every subcase in the definition of $q_{c}$ solves (1.1) and has energy $E_{q_{c}}(t)=-c$ for all $t\in{\mathbb{R}}$. The way the remaining condition (1.3) is fulfilled, depends on whether the trajectory of $q$ has finite contact times with $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$, or if it accumulates at infinity, as in (1.4).

In the case where $q\in{\mathcal{M}}_{c}$ has precisely one of $\alpha_{q}$ and $\omega_{q}$ finite, we say that $q_{c}$ is a homoclinic type solution connecting $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$. This solution satisfies the following properties

1. $\bullet$

If $\alpha_{q}=-\infty$ and $\omega_{q}<+\infty$, then we have $q_{c}:=q_{+}$. In particular, adopting the notation of Theorem 2.1 we denote $\Omega:=\mathcal{A}_{q}$ (=$\mathcal{A}_{q_{c}}=\Omega_{q_{c}}$) and $\sigma:=\omega_{q}$, thus $\Omega$ is a closed connected set and $\sigma<+\infty$. From Lemma 2.11, Lemma 2.15 and Lemma 2.16, the definition of $q_{+}$ in (2.27) together with $\sigma=\omega_{q}$, it follows that

• (i)

$V(q_{c}(\sigma))=V(q(\omega_{q}))=c$, $V(q_{c}(t))>c$ for any $t\neq\sigma$ and $\lim_{t\to\pm\infty}\dot{q_{c}}(t)=0$,

• (ii)

$q_{c}(\sigma)\in\mathcal{V}^{c}_{+}$ and $\nabla V(q_{c}(\sigma))=0$. Moreover, $\Omega\subset{\mathcal{V}^{c}_{-}\cap\{\xi\in{\mathbb{R}}^{N}:{V(\xi)=c},\,\nabla V(\xi)=0\}}$, and $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c}(t),\Omega)=0$,

• (iii)

$q_{c}(\sigma+t)=q_{c}(\sigma-t)$ for any $t\in{\mathbb{R}}$.

Finally we remark that (1.3) is satisfied. Indeed, by Lemma 2.11, we get that the infimum $\inf_{t\in{\mathbb{R}}}{\rm dist}\,(q_{c}(t),\mathcal{V}^{c}_{+})=0$ is achieved at $t=\sigma$ and ${\lim_{t\to\pm\infty}{\rm dist}\,(q_{c}(t),\mathcal{V}^{c}_{-})=0}$, so in particular, $\inf_{t\in{\mathbb{R}}}{\rm dist}\,(q_{c}(t),\mathcal{V}^{c}_{-})=0$. 2. $\bullet$

A similar reasoning allows us to conclude that, when $\alpha_{q}>-\infty$ and $\omega_{q}=+\infty$, the function $q_{c}:=q_{-}$ is a homoclinic type solution connecting $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$, for which we have that $\Omega:=\Omega_{q}$ ($=\Omega_{q_{c}}=\mathcal{A}_{q_{c}}$) is a connected closed set and setting $\sigma:=\alpha_{q}$, we get

• (i)

$V(q_{c}(\sigma))=V(q(\alpha_{q}))=c$, $V(q_{c}(t))>c$ for any $t\neq\sigma$ and $\lim_{t\to\pm\infty}\dot{q_{c}}(t)=0$,

• (ii)

$q_{c}(\sigma)\in\mathcal{V}^{c}_{-}$ and $\nabla V(q_{c}(\sigma))=0$. Moreover, $\Omega\subset{\mathcal{V}^{c}_{+}\cap\{\xi\in{\mathbb{R}}^{N}:{V(\xi)=c},\,\nabla V(\xi)=0\}}$, and $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c}(t),\Omega)=0$,

• (iii)

$q_{c}(\sigma+t)=q_{c}(\sigma-t)$ for any $t\in{\mathbb{R}}$.

Moreover, also in this case (1.3) is satisfied.

In the remaining case, where $q\in{\mathcal{M}}_{c}$ has $\alpha_{q}=-\infty$ and $\omega_{q}=+\infty$, we say that $q_{c}$ is a heteroclinic type solution connecting $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$.

• •

Clearly $q_{c}=q$. Adopting the notation of Theorem 2.1, we will write $\mathcal{A}:=\mathcal{A}_{q}\,(=\mathcal{A}_{q_{c}})$ and $\Omega:=\Omega_{q}\,(=\Omega_{q_{c}})$, whence $\mathcal{A}$ and $\Omega$ are closed connected sets. Then, since $\alpha_{q}=-\infty$ and $\omega_{q}=+\infty$, by definition and Lemma 2.16, we obtain

• (i)

$V(q_{c}(t))>c$ holds for any $t\in{\mathbb{R}}$ and $\lim_{t\to\pm\infty}\dot{q}_{c}(t)=0$,

• (ii)

$\mathcal{A}\subset\mathcal{V}^{c}_{-}\cap\{\xi\in{\mathbb{R}}^{N}:V(\xi)=c,\,\nabla V(\xi)=0\}$, $\Omega\subset\mathcal{V}^{c}_{+}\cap\{\xi\in{\mathbb{R}}^{N}:V(\xi)=c,\,\nabla V(\xi)=0\}$ and $\lim_{t\to-\infty}{\rm dist}\,(q_{c}(t),\mathcal{A})=\lim_{t\to+\infty}{\rm dist}\,(q_{c}(t),\Omega)=0$.

Finally, by Lemma 2.11 we get $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c}(t),\mathcal{V}^{c}_{\pm})=0$, from which, condition (1.3) is satisfied.

The proof of Theorem 2.1 is now completed.

Remark 2.17**.**

Note that, by construction, the solution $q_{c}$ given by Theorem 2.1 has a connecting time interval $(\alpha_{q_{c}},\omega_{q_{c}})\subseteq{\mathbb{R}}$, with $-\infty\leq\alpha_{q_{c}}<\omega_{q_{c}}\leq+\infty$ (coinciding in the statement with $(\sigma,\tau)$ in the case (a), with $(-\infty,\sigma)$ in the case (b) and with ${\mathbb{R}}$ in the case (c), respectively), such that

1. (1)

$V(q_{c}(t))>c$* for every $t\in(\alpha_{q_{c}},\omega_{q_{c}})$,* 2. (2)

$\lim\limits_{t\to\alpha_{q_{c}}^{+}}{\rm dist}\,(q_{c}(t),\mathcal{V}^{c}_{-})=0$, and if $\alpha_{q_{c}}>-\infty$ then $\dot{q}_{c}(\alpha_{q_{c}})=0$, $V(q_{c}(\alpha_{q_{c}}))=c$ with $q_{c}(\alpha_{q_{c}})\in\mathcal{V}^{c}_{-}$, 3. (3)

$\lim\limits_{t\to\omega_{q_{c}}^{-}}{\rm dist}\,(q_{c}(t),\mathcal{V}^{c}_{+})=0$, and if $\omega_{q_{c}}<+\infty$ then $\dot{q}_{c}(\omega_{q_{c}})=0$, $V(q_{c}(\omega_{q_{c}}))=c$ with $q_{c}(\omega_{q_{c}})\in\mathcal{V}^{c}_{+}$, 4. (4)

$J_{c,(\alpha_{q_{c}},\omega_{q_{c}})}(q_{c})=m_{c}$.

Finally, the behavior of $q_{c}$ on ${\mathbb{R}}$ is obtained by (eventual) reflection and periodic continuation of its restriction to the interval $(\alpha_{q_{c}},\omega_{q_{c}})\subseteq{\mathbb{R}}$. In particular, we have

1. (5)

$\|q_{c}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}=\|q_{c}\|_{L^{\infty}((\alpha_{q_{c}},\omega_{q_{c}}),{\mathbb{R}}^{N})}$.

3. Some applications

In this last section we illustrate some applications of Theorem 2.1 to certain classes of potentials $V$, extensively studied in the literature. More precisely, we establish existence of connecting orbits to (1.1), in the case of double-well potentials, as well as potentials associated to duffing-like systems and multiple-pendulum-like systems. Additionally, we show in each one of these cases that solutions of heteroclinic type and homoclinic type, at energy level [math], can be obtained as limits of sequences $(q_{c})$ of solutions to (1.1), as $c\to 0^{+}$.

3.1. Double-well potential systems.

As a first example we consider double-well potential systems like the ones considered e.g. in [16] in the PDE (non-autonomous) setting and in [8, 11, FGN, 31] in the ODE setting, among others. Precisely, we assume that $V\in C^{1}({\mathbb{R}}^{N})$ satisfies

1. (V1)

There exist $a_{-}\not=a_{+}\in{\mathbb{R}}^{N}$ such that $V(a_{+})=V(a_{-})=0$, and $V(x)>0$ for $x\in{\mathbb{R}}^{N}\setminus\{a_{-},a_{+}\}$, 2. (V2)

$\liminf_{|x|\to+\infty}V(x)=:\nu_{0}>0$.

As a consequence of Theorem 2.1 we have the following result.

Proposition 3.1**.**

Assume that $V\in C^{1}({\mathbb{R}}^{N})$ satisfies (V1) and (V2). If $c\in[0,\nu_{0})$ is such that $(\mathbf{V}^{c})$ holds, then the coercivity condition (1.7) of the energy functional $J_{c}$ over $\Gamma_{c}$ holds true. In particular, for such value of $c\in[0,\nu_{0})$, Theorem 2.1 gives a solution $q_{c}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to the problem (1.1)-(1.3) that satisfies the pointwise energy constraint $E_{q_{c}}(t)=-c$ for all $t\in{\mathbb{R}}$.

Proof of Proposition 3.1.

In order to prove that (1.7) holds true for $c\in[0,\nu_{0})$ for which  $(\mathbf{V}^{c})$ holds, we show that there exists $R>0$ such that any minimizing sequence $(q_{n})\subset\Gamma_{c}$, $J_{c}(q_{n})\to m_{c}=\inf_{\Gamma_{c}}J_{c}(q)$, verifies the uniform bound $\|q_{n}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R$. Arguing by contradiction, let us assume that there is a value $c_{*}\in[0,\nu_{0})$ for which $({\bf V}^{c_{*}})$ holds true and that there exists a sequence $(q_{n})\subset\Gamma_{c_{*}}$ for which $J_{c_{*}}(q_{n})\to m_{c_{*}}$ but $\|q_{n}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\to+\infty$.

Since $c_{*}<\nu_{0}$ and $\liminf_{|x|\to+\infty}V(x)=\nu_{0}$, we have that, denoting $\mu_{0}:=\frac{1}{2}(\nu_{0}-c_{*})$, there exists $R_{0}>0$ such that

[TABLE]

Consequently, $\mathcal{V}^{c_{*}}=\mathcal{V}^{c_{*}}_{-}\cup\mathcal{V}^{c_{*}}_{+}\subset B_{R_{0}}(0)$. Since $(q_{n})\subset\Gamma_{c_{*}}$ with $\|q_{n}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\to+\infty$, we deduce that for any $R>R_{0}$ there exists $\bar{n}\in{{\mathbb{N}}}$ such that if $n\geq\bar{n}$ then $q_{n}$ crosses (at least two times) the annulus $B_{R}(0)\setminus B_{R_{0}}(0)$. In particular, by (3.31), we obtain that for any $n\geq\bar{n}$ there is an interval $(s_{n},t_{n})\subset{\mathbb{R}}$ such that

[TABLE]

By Remark 2.4 we then recover that for $n\geq\bar{n}$

[TABLE]

Since $R$ is arbitrary, the latter contradicts the finiteness of $m_{c_{*}}$. Indeed, by Lemma 2.6 we know that $m_{c_{*}}<+\infty$ (in particular, the proof of Lemma 2.6 shows that this does not depend on (1.7)). ∎

Remark 3.2**.**

The continuity of $V$ and the assumptions (V1) and (V2) imply that there always exists $c_{\it dw}\in(0,\nu_{0})$ for which the condition $(\mathbf{V}^{c})$ is satisfied for every $c$ in the interval $[0,{c_{\it dw}})$. Indeed, as seen in (3.31), from (V2) we obtain $\exists R_{0}>0$ large so that $\mathcal{V}^{c}\subset B_{R_{0}}(0)$ for any $c\in(0,\nu_{0})$. Hence, $\mathcal{V}^{c}$ is compact. Also, in view of (V1) and $V\in C({\mathbb{R}}^{N})$, we can choose $c_{\it dw}$ sufficiently small so that $\mathcal{V}^{c}$ splits in the disjoint union of two compact sets $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$, for any $c\in[0,c_{\it dw})$. In particular, we can assume that for $\varrho_{0}:=\frac{1}{4}|a_{-}-a_{+}|$ the condition below holds

[TABLE]

by reducing the value of $c_{\it dw}\in(0,\nu_{0})$, if necessary. It follows that ${\rm dist}\,(\mathcal{V}^{c}_{-},\mathcal{V}^{c}_{+})\geq\frac{1}{2}|a_{-}-a_{+}|>0$.

For any $c\in[0,\nu_{0})$ for which $(\mathbf{V}^{c})$ is satisfied, Proposition 3.1 provides a solution $q_{c}$ with energy $-c$ which connects $\mathcal{V}^{c}_{-}$ with $\mathcal{V}^{c}_{+}$. As noted in Remark 2.2, such a solution is of brake orbit type when $c$ is a regular value for $V$, while it may be of the homoclinic or heteroclinic type if $c$ is a critical value of $V$. Of particular interest is the case when $c=0$, where we see that $\mathcal{V}^{c}_{-}=\{a_{-}\}$ with $\mathcal{V}^{c}_{+}=\{a_{+}\}$ (or viceversa) and since $a_{\pm}$ are critical points of $V$, we are in the case (c) of Theorem 2.1. Therefore the solution given by Proposition 3.1 for $c=0$ is of heteroclinic type connecting the equilibria $a_{-}$ and $a_{+}$.

We continue our analysis by studying the behavior of the solution $q_{c}$ given by Proposition 3.1 as $c\to 0^{+}$ and we will prove that they converge, in a suitable sense, to a heteroclinic solution connecting the equilibria $a_{\pm}$. Precisely, we have

Proposition 3.3**.**

Assume that $V\in C^{1}({\mathbb{R}}^{N})$ satisfies (V1)-(V2) and that $\nabla V$ is locally Lipschitz continuous in ${\mathbb{R}}^{N}$. Let $c_{n}\to 0^{+}$ be any sequence and $(q_{c_{n}})$ be the sequence of solutions to the system (1.1) given by Proposition 3.1. Then, up to translations and a subsequence, $q_{c_{n}}\to q_{0}$ in $C^{2}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$, where $q_{0}$ is a solution to (1.1) of heteroclinic type between $a_{-}$ and $a_{+}$, i.e. it satisfies

[TABLE]

To prove Proposition 3.3, we begin by establishing in the next lemma a uniform estimate of the $L^{\infty}$-norm of the solutions $(q_{c})$ for $0\leq c<c_{\it dw}$, where $c_{\it dw}$ is given in Remark 3.2.

Lemma 3.4**.**

Assume that $V\in C^{1}({\mathbb{R}}^{N})$ satisfies (V1) and (V2) and let $c_{\it dw}\in(0,\nu_{0})$. Then there exists $R_{\it dw}>0$ such that $\|q_{c}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{\it dw}$ for any $c\in[0,c_{\it dw})$.

Proof of Lemma 3.4.

First we claim that there exists $M_{\it dw}>0$ such that

[TABLE]

Indeed, consider the function $\xi(t)=(1-t)a_{-}+ta_{+}$ for $t\in[0,1]$. From (3.32) and the compactness of $\mathcal{V}^{c}_{\pm}$ we see that for each $c\in[0,c_{\it dw})$ there exist $0\leq\sigma_{c}<\tau_{c}\leq 1$ that satisfy

[TABLE]

Then, the function

[TABLE]

belongs to $\Gamma_{c}$, and so our claim (3.33) follows by a plain estimate:

[TABLE]

If $q_{c}$ denotes the solution in Proposition 3.1, corresponding to a $c\in[0,c_{\it dw})$, by Remark 2.17 there is a connecting time interval $(\alpha_{q_{c}},\omega_{q_{c}})\subset{\mathbb{R}}$ for which properties (1)-(5) hold true. In particular, by (3.33)

[TABLE]

holds true. We now claim that there exists $R_{\it dw}>0$ in such a way that

[TABLE]

The proof of Lemma 3.4 will be concluded upon establishing (3.35), since condition (5) in Remark 2.17 gives $\|q_{c}\|_{L^{\infty}((\alpha_{q_{c}},\omega_{q_{c}}),{\mathbb{R}}^{N})}=\|q_{c}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}$. To prove (3.35), arguing by contradiction, we assume that for every $R>0$ there exists $c_{R}\in[0,c_{\it dw})$ such that

[TABLE]

Let us observe that assumption (V2) ensures that for $h_{0}:=\frac{1}{2}(\nu_{0}-c_{\it dw})$ there exists $R_{0}>0$ for which

[TABLE]

In particular, this shows that $\mathcal{V}^{c}=\mathcal{V}^{c}_{-}\cup\mathcal{V}^{c}_{+}\subset B_{R_{0}}(0)$ for any $c\in[0,c_{\it dw})$. In the remaining of the proof we choose $R\in(R_{0},+\infty)$ to satisfy $\sqrt{2h_{0}}(R-R_{0})>M_{\it dw}$. For this choice of $R$, the contradiction assumption (3.36) implies that the trajectory of $q_{c_{R}}\in\mathcal{M}_{c_{R}}$ crosses the annulus $B_{R}(0)\setminus B_{R_{0}}(0)$; thus, there is an interval $[\sigma,\tau]\subset(\alpha_{q_{c_{R}}},\omega_{q_{c_{R}}})$ in such a way that

[TABLE]

Hence, Remark 2.4 along with the properties above, show the strict lower bound on the energy of $q_{c_{R}}$

[TABLE]

However, this contradicts the upper bound (3.34). In this way, we have argued that (3.35) follows, which in turn completes the proof of this lemma. ∎

We can now prove Proposition 3.3. Without loss of generality, let the sequence $c_{n}\to 0^{+}$ be so that $(c_{n})\subset(0,c_{\it dw})$. Since $c_{n}<c_{\it dw}$, we know from Remark 3.2 that $(\mathbf{V}^{c_{n}})$ holds true for all $n\in{{\mathbb{N}}}$. By Remark 2.17, for each $n\in{{\mathbb{N}}}$, the solution $q_{c_{n}}$ given by Proposition 3.1 has a connecting time interval $(\alpha_{n},\omega_{n})\subset{\mathbb{R}}$, with $-\infty\leq\alpha_{n}<\omega_{n}\leq+\infty$, in such a way that

1. $(1_{n})$

$V(q_{c_{n}}(t))>c_{n}$ for every $t\in(\alpha_{n},\omega_{n})$, 2. $(2_{n})$

$\lim\limits_{t\to\alpha_{n}^{+}}{\rm dist}\,(q_{c_{n}}(t),\mathcal{V}^{c_{n}}_{-})=0$, and if $\alpha_{n}>-\infty$ then $\dot{q}_{c_{n}}(\alpha_{n})=0$, $V(q_{c_{n}}(\alpha_{n}))=c_{n}$ with $q_{c_{n}}(\alpha_{n})\in\mathcal{V}^{c_{n}}_{-}$, 3. $(3_{n})$

$\lim\limits_{t\to\omega_{n}^{-}}{\rm dist}\,(q_{c_{n}}(t),\mathcal{V}^{c_{n}}_{+})=0$, and if $\omega_{n}<+\infty$ then $\dot{q}_{c_{n}}(\omega_{n})=0$, $V(q_{c_{n}}(\omega_{n}))=c_{n}$ with $q_{c_{n}}(\omega_{n})\in\mathcal{V}^{c_{n}}_{+}$. 4. $(4_{n})$

$J_{c_{n},(\alpha_{n},\omega_{n})}(q_{c_{n}})=m_{c_{n}}=\inf\limits_{q\in\Gamma_{c_{n}}}J_{c_{n}}(q)$.

We first start by renormalizing the sequence $(q_{c_{n}})$ using the following phase shift procedure. In light of the properties $(2_{n})$ and $(3_{n})$, for any $n\in{{\mathbb{N}}}$ there exists $\zeta_{n}\in(\alpha_{n},\omega_{n})$ such that

[TABLE]

for $\varrho_{0}:=\frac{1}{4}|a_{-}-a_{+}|$ as in (3.32). Hence, up to translations, eventually renaming $q_{c_{n}}$ to be $q_{c_{n}}(\cdot-\zeta_{n})$, we can assume

[TABLE]

We now argue that $(q_{c_{n}})$ converges, in the $C^{2}$-topology on compact sets, to an entire solution $q_{0}$ of the system $\ddot{q}=\nabla V(q)$ over ${\mathbb{R}}$. To see this, let us observe that $(q_{c_{n}})$, $(\dot{q}_{c_{n}})$ and $(\ddot{q}_{c_{n}})$ are uniformly bounded in $L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})$, given the fact that $\|q_{c_{n}}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{\it dw}$ for any $n\in{{\mathbb{N}}}$ (by Lemma 3.4). More precisely, for every $n\in{{\mathbb{N}}}$ we have the bounds

[TABLE]

where the former is a consequence of the pointwise energy constraint $E_{q_{c_{n}}}(t)=-c_{n}$ for all $t\in{\mathbb{R}}$, and the latter follows since $\ddot{q}_{c_{n}}=\nabla V(q_{c_{n}})$ on ${\mathbb{R}}$.

An application of the Ascoli-Arzelà Theorem shows that there exists $q_{0}\in C^{1}({\mathbb{R}},{\mathbb{R}}^{N})$ and a subsequence of $(q_{c_{n}})$, still denoted $(q_{c_{n}})$, such that

[TABLE]

In particular, (3.37) implies that

[TABLE]

Moreover, the above convergence can be improved to $q_{c_{n}}\to q_{0}$ in $C_{loc}^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ as $n\to+\infty$, by using once more that $\ddot{q}_{c_{n}}=\nabla V(q_{c_{n}})$ on ${\mathbb{R}}$, for any $n\in{{\mathbb{N}}}$. In fact, the latter convergence shows, in turn, that

[TABLE]

To conclude the proof of Proposition 3.3, it will be enough to establish

[TABLE]

Indeed, once the validity of (3.40) has been proved, we then get $\ddot{q}(t)\to 0$ as $t\to\pm\infty$ from (3.39), which in turn shows that $\dot{q}(t)\to 0$ as $t\to\pm\infty$ by interpolation inequalities.

To prove (3.40), let us first note that conditions (3.37), (3.38) and (3.39) imply altogether

[TABLE]

Indeed, arguing by contradiction, assume that along a subsequence $\alpha_{n}$ is bounded. As $\alpha_{n}<0$ then, up to a subsequence, we deduce that $\alpha_{n}\to\alpha_{0}$ for some $\alpha_{0}\leq 0$. By ($2_{n}$) and (3.32) we have $q_{c_{n}}(\alpha_{n})\in\mathcal{V}^{c_{n}}_{-}\subset B_{\varrho_{0}}(a_{-})$, which combined with $c_{n}\to 0$, (V1) and (V2) then yields $q_{c_{n}}(\alpha_{n})\to a_{-}$. This, $\dot{q}_{c_{n}}(\alpha_{n})=0$ for any $n\in{{\mathbb{N}}}$, and the fact that $q_{c_{n}}\to q_{0}$ in $C^{1}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$ allow us to conclude $q_{0}(\alpha_{0})=a_{-}$ and $\dot{q}_{0}(\alpha_{0})=0$. Nonetheless, the uniqueness of solutions to the Cauchy problem would then imply that $q_{0}(t)=a_{-}$ for any $t\in{\mathbb{R}}$. This is contrary to (3.38), thus showing $\alpha_{n}\to-\infty$, as claimed. Analogously, one can prove that $\omega_{n}\to+\infty$. Therefore, (3.41) follows.

In order to establish (3.40), let us observe that it is sufficient to prove: for any $r\in(0,\rho_{0})$ there exists $L_{r}^{\pm}>0$ and $n_{r}\geq\bar{n}$ such that for any $n\geq n_{r}$, there hold

[TABLE]

Indeed, by taking the limit as $n\to+\infty$ in (3.42), we then conclude in view of the pointwise convergence $q_{c_{n}}\to q_{0}$ and (3.41), that for any $r\in(0,\varrho_{0})$ there exists $L_{r}:=\max\{L^{-}_{r},L^{+}_{r}\}$ such that

[TABLE]

therefore, (3.40) follows.

To obtain the first estimate in (3.42) we argue by contradiction assuming that there exists $\bar{r}\in(0,\rho_{0})$, a subsequence of $(q_{c_{n}})$, still denoted $(q_{c_{n}})$, and a sequence $s_{n}\to-\infty$ such that for any $n\in{{\mathbb{N}}}$

[TABLE]

Then, by observing that $(4_{n})$ and (3.34) imply the inequality

[TABLE]

we obtain that the contradiction hypothesis $s_{n}\to-\infty$ yields

[TABLE]

In particular, we deduce that for every $n\in{{\mathbb{N}}}$ there exists $t_{n}\in(s_{n},0)$ so that $V(q_{c_{n}}(t_{n}))\to 0\hbox{ as }n\to+\infty$. This, in light of (V1) and (V2), shows

[TABLE]

Let us show, if fact, that

[TABLE]

Indeed, if (3.45) fails, along a subsequence (still denoted $q_{c_{n}}$) we have $q_{c_{n}}(t_{n})\to a_{+}$. We connect the point $q_{c_{n}}(t_{n})$ with $a_{+}$ with the segment $\{(1-\sigma)q_{c_{n}}(t_{n})+\sigma a_{+}:\,\sigma\in[0,1]\}$. By continuity, since $V(q_{c_{n}}(t_{n}))>c_{n}>0=V(a_{+})$, there is $\sigma_{n}\in(0,1)$ such that

[TABLE]

In particular $(1-\sigma_{n})q_{c_{n}}(t_{n})+\sigma_{n}a_{+}\in\mathcal{V}^{c_{n}}_{+}$ and the function

[TABLE]

(where we agree to omit the first item in the definition if $\alpha_{n}=-\infty$) is by construction an element of $\Gamma_{c_{n}}$ for any $n\in{{\mathbb{N}}}$, whence $J_{c_{n}}(q_{-,n})\geq m_{c_{n}}$. Since $q_{c_{n}}(t_{n})\to a_{+}$ we have $V((1-\sigma)q_{c_{n}}(t_{n})+\sigma a_{+})\to 0$ as $n\to+\infty$ uniformly for $\sigma\in[0,1]$. Thus, we derive that,

[TABLE]

as $n\to+\infty$. Since $q_{-,n}=q_{c_{n}}$ on $(\alpha_{n},t_{n})$ and since $q_{-,n}$ is constant outside $(\alpha_{n},t_{n}+\sigma_{n})$, the latter shows

[TABLE]

This bound with $(4_{n})$ implies that

[TABLE]

and so

[TABLE]

On the other hand, since $q_{c_{n}}(t_{n})\to a_{+}$ and, by (3.37), ${\rm dist}\,(q_{c_{n}}(0),\{a_{-},a_{+}\})=\varrho_{0}$, when $n$ is large the trajectory of $q_{c_{n}}$ crosses the annulus $B_{\varrho_{0}/2}(a_{+})\setminus B_{\varrho_{0}/4}(a_{+})$ at least one in the interval $(t_{n},0)$, i.e., there exists $(t_{1,n},t_{2,n})\subset(t_{n},0)$ such that $|q_{c_{n}}(t_{2,n})-q_{c_{n}}(t_{1,n})|=\tfrac{\varrho_{0}}{4}$ and ${\rm dist}\,(q_{c_{n}}(t),\{a_{-},a_{+}\})\geq\tfrac{\varrho_{0}}{4}$ for any $t\in(t_{1,n},t_{2,n})$. If we set $\mu(\tfrac{\varrho_{0}}{4}):=\inf_{{\mathbb{R}}^{N}\setminus(B_{\varrho_{0}/4}(a_{-})\cup B_{\varrho_{0}/4}(a_{+}))}V$ it follows that for every $t\in(t_{1,n},t_{2,n})$, $V(q_{c_{n}}(t))\geq\mu(\tfrac{\varrho_{0}}{4})$, and hence Remark 2.4 yields the bound

[TABLE]

for $n$ large enough. This last inequality contradicts (3.47) proving (3.45).

By (3.44) and (3.45) we obtain $q_{c_{n}}(t_{n})\to a_{-}$. We now show that this case is not possible either, thus obtaining a contradiction with (3.44). This will establish the first estimate of (3.42).

To prove that $q_{c_{n}}(t_{n})\to a_{-}$ cannot occur, we use an argument similar to the one used above. If $q_{c_{n}}(t_{n})\to a_{-}$, we fix $\sigma_{n}\in(0,1)$ such that

[TABLE]

Then the function

[TABLE]

(where we omit the last item if $\omega_{n}=+\infty$) belongs to $\Gamma_{c_{n}}$ and $J_{c_{n}}(q_{+,n})\geq m_{c_{n}}$. Since $q_{c_{n}}(t_{n})\to a_{-}$, with an argument analogous to the one used for $q_{-,n}$, we obtain

[TABLE]

A reasoning similar to the one that lead to (3.47), then shows

[TABLE]

By (3.43) and (3.45), we are now in the situation where

[TABLE]

Then, when $n$ is large, the trajectory of $q_{c_{n}}$ crosses the annulus $B_{\bar{r}/2}(a_{-})\setminus B_{\bar{r}/4}(a_{-})$ in the interval $(s_{n},t_{n})$ and so there exists $(t_{3,n},t_{4,n})\subset(s_{n},t_{n})$ such that $|q_{c_{n}}(t_{3,n})-q_{c_{n}}(t_{4,n})|=\tfrac{\bar{r}}{4}$ and ${\rm dist}\,(q_{c_{n}}(t),\{a_{-},a_{+}\})\geq\tfrac{\bar{r}}{4}$ for $t\in(t_{3,n},t_{4,n})$. As above, since $V(q(t))\geq\mu(\tfrac{\bar{r}}{4}):=\inf_{{\mathbb{R}}^{N}\setminus(B_{\bar{r}/4}(a_{-})\cup B_{\bar{r}/4}(a_{+}))}V$ for any $t\in(t_{3,n},t_{4,n})$, we can use Remark 2.4 to obtain

[TABLE]

This contradicts (3.48) and so the case $q(t_{n})\to a_{-}$ cannot occur either. Then the first estimate of (3.42) follows.

One can readily verify that a strategy similar as the one above can be used to establish the second estimate of (3.42). In conclusion, (3.42) follows, and as a consequence (3.40) has been established. The proof of Proposition 3.3 is now complete.

3.2. Duffing like systems.

A second application of Theorem 2.1 include Duffing like systems. More precisely, we follow the assumptions made in [10] and [27]: let $V$ be a $C^{1}({\mathbb{R}}^{N})$ potential satisfying

1. (V3)

$V$ has a strict local minimum at $x_{0}:=0$, with value $V(0)=0$:

[TABLE] 2. (V4)

The set $\mathcal{C}_{0}:=\{x\in{\mathbb{R}}^{N}:V(x)>0\}\cup\{0\}$ is bounded, and such that $\nabla V(x)\neq 0$ for any $x\in\partial\mathcal{C}_{0}$.

We observe that for any $c\geq 0$, $J_{c}$ satisfies the coercivity property (1.7) on $\Gamma_{c}$. To see this, in view of (V3) and (V4), note that $\{x\in{\mathbb{R}}^{N}:V(x)\geq c\}\subset\overline{\mathcal{C}_{0}}$ thus, by definition of $\Gamma_{c}$, $q({\mathbb{R}})\subset\overline{\mathcal{C}_{0}}$ for any $q\in\Gamma_{c}$. Hence, since $\mathcal{C}_{0}$ is a bounded set, we conclude that (1.7) holds.

This discussion shows that Theorem 2.1 applies, and so we have

Proposition 3.5**.**

Assume that $V\in C^{1}({\mathbb{R}}^{N})$ satisfies (V3) and (V4). If $c\geq 0$ is such that $(\mathbf{V}^{c})$ holds true then Theorem 2.1 gives a solution $q_{c}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to the problem (1.1)-(1.3) with energy $E_{q_{c}}(t)=-c$ for all $t\in{\mathbb{R}}$.

We now remark that condition $(\mathbf{V}^{c})$ is verified if $c>0$ is chosen sufficiently small. Indeed, by (V3) there exists $\nu_{1}>0$ such that $V(x)\geq\nu_{1}$ for $|x|=r_{0}$. Then $\mathcal{V}^{c}\cap\partial B_{r_{0}}(0)=\emptyset$ for any $c\in[0,\nu_{1})$ and $(\mathbf{V}^{c})$ is satisfied by the sets

[TABLE]

In particular, the fact that ${\rm dist}\,(\mathcal{V}^{c}_{-},\mathcal{V}^{c}_{+})>0$ is guaranteed by the continuity of $V$.

It is worth distinguishing among the different type of solutions $q_{c}$ that we can obtain from Proposition 3.5, for suitable choices of $c\in[0,+\infty)$. In the case $c=0$, we have from (3.49) that $\mathcal{V}^{c}_{-}=\{0\}$ and $\mathcal{V}^{c}_{+}={\mathbb{R}}^{N}\setminus\mathcal{C}_{0}$. Then Proposition 3.5 above states the existence of a solution $q_{0}$ to (1.1) connecting ${0}$ with $\partial\mathcal{C}_{0}$. This solution cannot be of brake orbit type since $\nabla V({0})=0$, and so by Theorem 2.1-(a) it cannot have a contact point with $\mathcal{V}^{c}_{-}$. Analogously, $q_{0}$ cannot be of heteroclinic type, since in this case, by Theorem 2.1-(c), there must exist a set $\Omega\subset\partial\mathcal{C}_{0}$ consisting of critical points of $V$; thus contradicting the hypothesis made on $\partial\mathcal{C}_{0}$ in (V4). We conclude that $q_{0}$ must be a homoclinic type solution with energy $E_{q_{0}}(t)=0$ for all $t\in{\mathbb{R}}$ satisfying $q_{0}(t)\to 0$, $\dot{q}_{0}(t)\to 0$ as $t\to\pm\infty$, and that reaches $\partial\mathcal{C}_{0}$ at a contact time $\sigma\in{\mathbb{R}}$, with respect to which it is symmetric (see Figure 1(a)). This gives back the result already proved in [10], [11],[18] and [27]. In contrast, let us now consider the cases $0<c<\nu_{1}$, where $\nu_{1}:=\min\{V(x):|x|=r_{0}\}$ as above. Relative to the separation property (3.49), Proposition 3.5 provides for any such value $c$ the existence of a connecting orbit $q_{c}$ between $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$. Arguing as above, we recognize that if $c$ is a regular value of $V$ then $q_{c}$ is a brake orbit type solution of (1.1) with energy $E_{q_{c}}(t)=-c$ for all $t\in{\mathbb{R}}$, connecting $\mathcal{V}^{c}_{-}$ and $\mathcal{V}^{c}_{+}$ (see Figure 1(b)).

To analyze the case where $c$ is not a regular value of $V$, we use an approximation argument. Let us first note that ${\rm dist}\,(\mathcal{V}^{c}_{+},\partial\mathcal{C}_{0})\to 0$ as $c\to 0^{+}$ in view of compactness of these sets, and the continuity of $V$. Since $\nabla V\not=0$ on $\partial\mathcal{C}_{0}$, the continuity of $\nabla V$ and compactness of $\mathcal{V}^{c}_{+}$ show $\nabla V\not=0$ on $\partial\mathcal{V}^{c}_{+}$ if $c$ is small, say $c<c_{\it D}$ for some number $c_{\it D}\in(0,\nu_{1})$. So by arguing as above, the solution $q_{c}$ can be either of brake orbit type or of homoclinic type (depending on whether $\nabla V\not=0$ on $\partial\mathcal{V}^{c}_{-}$), reaching $\partial\mathcal{V}^{c}_{+}$ at a contact time with respect to which it is symmetric. Using Remark 2.17 we conclude that for any $c\in(0,c_{\it D})$ the solution $q_{c}$ given by Proposition 3.5 relative to the decomposition (3.49) has a connecting time interval $(\alpha_{q_{c}},\omega_{q_{c}})\subset{\mathbb{R}}$, with $-\infty\leq\alpha_{q_{c}}<\omega_{q_{c}}<+\infty$, such that

1. $(1_{\it D})$

$V(q_{c}(t))>c$ for every $t\in(\alpha_{q_{c}},\omega_{q_{c}})$, 2. $(2_{\it D})$

$\lim\limits_{t\to\alpha_{q_{c}}^{+}}{\rm dist}\,(q_{c}(t),\mathcal{V}^{c}_{-})=0$, and if $\alpha_{q_{c}}>-\infty$ then $\dot{q}_{c}(\alpha_{q_{c}})=0$, $V(q_{c}(\alpha_{q_{c}}))=c$ with $q_{c}(\alpha_{q_{c}})\in\mathcal{V}^{c}_{-}$, 3. $(3_{\it D})$

$\dot{q}_{c}(\omega_{q_{c}})=0$, $V(q_{c}(\omega_{q_{c}}))=c$ with $q_{c}(\omega_{q_{c}})\in\mathcal{V}^{c}_{+}$, 4. $(4_{\it D})$

$J_{c,(\alpha_{q_{c}},\omega_{q_{c}})}(q_{c})=m_{c}$.

We continue by observing that

Remark 3.6**.**

There exists $M_{\it D}>0$ such that

[TABLE]

This is obtained along the lines of the proof of (3.33). Let us fix $\zeta\in{\mathbb{R}}^{N}$ with $|\zeta|=1$, then we may consider the ray $\{t\zeta:t\geq 0\}\subset{\mathbb{R}}^{N}$. From (V3), there must exist $\bar{t}>0$ such that $\{t\zeta:0\leq t<\bar{t}\}\subset\mathcal{C}_{0}$, while $\bar{t}\zeta\in\partial\mathcal{C}_{0}$. Moreover for any $c\in[0,c_{\it D})$ there exist $0\leq\sigma_{c}<\tau_{c}\leq\bar{t}$ such that

[TABLE]

As in the proof of (3.33), we readily see the function

[TABLE]

belongs to $\Gamma_{c}$. Hence, setting ${\rm diam}\,(\mathcal{C}_{0}):=\sup\{|x-y|\,:x,y\in\mathcal{C}_{0}\}$, we obtain that

[TABLE]

Arguments similar to the ones used in the case of double well potentials show that the solutions $(q_{c})$ accumulate a near solution of homoclinic type, as $c$ approaches zero.

Proposition 3.7**.**

Assume that $V\in C^{1}({\mathbb{R}}^{N})$ satisfies (V3)-(V4), and that $\nabla V$ is locally Lipschitz continuous in ${\mathbb{R}}^{N}$. For any sequence $c_{n}\to 0^{+}$, consider the sequence of solutions $(q_{c_{n}})$ to the system (1.1) given by Proposition 3.5. Then, up to translations and a subsequence, $q_{c_{n}}\to q_{0}$ in $C^{2}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$ where $q_{0}$ is a solution to (1.1) of homoclinic type at $x_{0}=0$.

Proof of Proposition 3.7.

Without loss of generality assume the sequence $c_{n}\to 0^{+}$ is so $(c_{n})\subset(0,c_{\it D})$. Since $c_{n}<c_{\it D}$ we have $(\mathbf{V}^{c})$ is satisfied by (3.49), and let us denote $q_{n}:=q_{c_{n}}$ the solution given by Proposition 3.5. Recalling $(1_{\it D})$ through $(4_{\it D})$, we denote $(\alpha_{n},\omega_{n}):=(\alpha_{c_{n}},\omega_{c_{n}})$ the connecting time interval of $q_{n}$. Since $\omega_{n}\in{\mathbb{R}}$ for $n\in{{\mathbb{N}}}$, we can assume, up to translations, that $\omega_{n}=0$. That is to say,

[TABLE]

Since $\mathcal{C}_{0}$ is bounded and by construction $q_{n}({\mathbb{R}})\subset\mathcal{C}_{0}$ for every $n\in{{\mathbb{N}}}$, there exists $R_{\it D}>0$ such that $\sup_{n\in{{\mathbb{N}}}}\|q_{n}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{\it D}$. Since $\ddot{q}_{n}=\nabla V(q_{n})$ on ${\mathbb{R}}$, the same arguments in the proof of Proposition 3.3 yield that there exists $q_{0}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ such that $\ddot{q}_{0}=\nabla V(q_{0})$ on ${\mathbb{R}}$, and $q_{n}\to q_{0}$ in $C_{loc}^{2}({\mathbb{R}},{\mathbb{R}}^{N})$, along a subsequence, as $n\to+\infty$. The pointwise convergence and (3.50), together with the fact that ${\rm dist}\,(\partial\mathcal{V}^{c_{n}}_{+},\partial\mathcal{C}_{0})\to 0$ as $n\to+\infty$, imply furthermore that

[TABLE]

Our next goal is to show, similarly to (3.41), that

[TABLE]

If (3.52) were false then we could assume, up to a subsequence, that $\alpha_{n}\to\alpha_{0}\in{\mathbb{R}}$. By $(2_{\it D})$ and (3.49), since $c_{\it D}\in(0,\nu_{1})$, then yields $q_{n}(\alpha_{n})\in\mathcal{V}^{c_{n}}_{-}\subset B_{r_{0}}(0)$. Since $c_{n}\to 0$, it would follow that $q_{n}(\alpha_{n})\to 0$ and, recalling $\dot{q}_{n}(\alpha_{n})=0$ for any $n\in{{\mathbb{N}}}$, we would obtain $q_{0}(\alpha_{0})=0$ and $\dot{q}_{0}(\alpha_{0})=0$. By uniqueness of solutions to the Cauchy problem, necessarily $q_{0}(t)\equiv 0$ for any $t\in{\mathbb{R}}$, a contradiction with (3.51). This shows (3.52).

To prove the proposition, we are left to show that

[TABLE]

Note that by (3.51), $q_{0}$ is symmetric with respect to the contact time $\omega_{0}=0$. Hence, (3.53) follows once we show that

[TABLE]

The latter reduces to prove that for any $r\in(0,r_{0})$, with $r_{0}$ as in (V3), there exist $L_{r}>0$ and $n_{r}\in{{\mathbb{N}}}$ s.t.

[TABLE]

In order to establish (3.55) we assume by contradiction that there exist $\bar{r}\in(0,r_{0})$, a subsequence of $(q_{n})$, still denoted $(q_{n})$ and a sequence $(s_{n})\subset{\mathbb{R}}$, in such a way that

[TABLE]

The rest of the proof is devoted to obtain a contradiction with (3.56), following very similar steps as the proof of (3.42) in Proposition 3.3. We will only sketch the main ideas and spare some details. The contradiction will be reached by arguing that if (3.56) holds true, then there exists $(t_{n})$ so that $t_{n}\in(s_{n},0)$ for $n\in{{\mathbb{N}}}$ with

[TABLE]

But for such sequence there hold furthermore

[TABLE]

which is in contradiction with (3.57). This will prove (3.54), completing the proof of Proposition 3.7.

To establish (3.57), we remark that the uniform bound on the energies: $J_{c_{n},(\alpha_{n},0)}(q_{n})=m_{c_{n}}\leq M_{\it D}$ for all $n\in{{\mathbb{N}}}$ (see Remark 3.6 since $c_{n}<c_{\it D}$) together with $s_{n}\to-\infty$ yield that for any $n\in{{\mathbb{N}}}$ there exists $t_{n}\in(s_{n},0)$ so that $V(q_{n}(t_{n}))\to 0$ as $n\to+\infty$. This, in light of (V3)-(V4), shows (3.57).

We next argue (3.58). If $\liminf_{n\to+\infty}{\rm dist}\,(q_{n}(t_{n}),\partial\mathcal{C}_{0})>0$ fails to hold, then there exists $\xi_{0}\in\partial\mathcal{C}_{0}$ so that $q_{n}(t_{n})\to\xi_{0}$ as $n\to+\infty$, along a subsequence that we continue to denote $(q_{n})$. We first note that this behavior of $q_{n}$ is energetically inexpensive, in that

[TABLE]

This is a consequence of an energy analysis with a suitable competitor $(q_{-,n})$ as follows. By continuity, since $V(q_{n}(t_{n}))>c_{n}>0=V(\xi_{0})$, there is $\sigma_{n}\in(0,1)$ such that

[TABLE]

Hence, the curve

[TABLE]

is an element of $\Gamma_{c_{n}}$, thus $J_{c_{n}}(q_{-,n})\geq m_{c_{n}}$. Since $V((1-\sigma)q_{n}(t_{n})+\sigma\xi_{0})\to 0$ as $n\to+\infty$ uniformly for $\sigma\in[0,1]$, we derive that

[TABLE]

as $n\to+\infty$. Hence, $J_{c_{n},(\alpha_{n},t_{n})}(q_{n})=J_{c_{n},(\alpha_{n},t_{n})}(q_{-,n})=J_{c_{n}}(q_{-,n})-J_{c_{n},(t_{n},+\infty)}(q_{-,n})\geq m_{c_{n}}-o(1)$, and since $m_{c_{n}}=J_{c_{n},(\alpha_{n},t_{n})}(q_{n})+J_{c_{n},(t_{n},0)}(q_{n})$ we conclude (3.59).

As an intermediate step, we now claim that

[TABLE]

Indeed, if not, there is a $\rho_{*}\in(0,r_{0}/3)$ and a sequence $(\tau_{n})$ with $\tau_{n}\in(t_{n},0)$ for $n\in{{\mathbb{N}}}$, such that ${\rm dist}\,(q_{n}(\tau_{n}),\partial\mathcal{C}_{0})=3\rho_{*}$, up to subsequence. In light of (3.50), this implies that there exists $(t_{1,n},t_{2,n})\subset(\tau_{n},0)$ such that $|q_{n}(t_{2,n})-q_{n}(t_{1,n})|=\rho_{*}$ and $2\rho_{*}\geq{\rm dist}\,(q_{n}(t),\partial\mathcal{C}_{0})\geq\rho_{*}$ for every $t\in(t_{1,n},t_{2,n})$. Hence

[TABLE]

and by Remark 2.4 we obtain

[TABLE]

for $n$ large enough. This last inequality contradicts (3.59), proving (3.60).

Since $t_{n}\to-\infty$ and $q_{n}\to q_{0}$ in $C^{2}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$, by (3.60) we conclude

[TABLE]

Also, the energy constraint $E_{q_{n}}(t)=-c_{n}$ for every $t\in{\mathbb{R}}$, together with the pointwise convergence show that $E_{q_{0}}(t)=0$ for any $t\in{\mathbb{R}}$. That is to say, $\frac{1}{2}|\dot{q}_{0}(t)|^{2}=V(q_{0}(t))$ for $t\in{\mathbb{R}}$ and since $V(x)=0$ for $x\in\partial\mathcal{C}_{0}$, by (3.61) we obtain $\dot{q}_{0}(t)=0$ for any $t<0$. Thus, $q_{0}$ is constant with $\ddot{q}(t)=0$ for $t<0$. Nonetheless, using that $\nabla V\not=0$ on $\partial\mathcal{C}_{0}$, see (V4), we would simultaneously have that $\ddot{q}_{0}(t)=\nabla V(q_{0}(t))\not=0$ for $t<0$ (by (3.61)). This contradiction proves that

[TABLE]

To conclude the proof, let us finally show that $\liminf_{n\to+\infty}{\rm dist}\,(q_{n}(t_{n}),0)>0$. Assume by contradiction that $q_{n}(t_{n})\to 0$ and, as in the two-well case, for any $n\in{{\mathbb{N}}}$ we fix $\sigma_{n}\in(0,1)$ such that

[TABLE]

Then

[TABLE]

is in $\Gamma_{c_{n}}$, whence $J_{c_{n}}(q_{+,n})\geq m_{c_{n}}$, and just as in the two-well case, we obtain as $n\to+\infty$,

[TABLE]

In summary, we are in a situation where

[TABLE]

Then, when $n$ is large, there exists $(t_{3,n},t_{4,n})\subset(s_{n},t_{n})$ such that $|q_{n}(t_{4,n})-q_{n}(t_{3,n})|=\tfrac{\bar{r}}{4}$ and $|q_{n}(t)|\geq\tfrac{\bar{r}}{4}$ for any $t\in(t_{3,n},t_{4,n})$. Letting $\bar{\mu}(\bar{r}):=\min\{V(x):\tfrac{\bar{r}}{4}\leq|x|\leq\bar{r}\}$ we can apply Remark 2.4 similarly as above, to obtain for $n$ sufficiently large:

[TABLE]

This contradicts (3.62), which proves that $q_{n}(t_{n})\to 0$ is not possible either; $\liminf_{n\to+\infty}{\rm dist}\,(q_{n}(t_{n}),0)>0$ is now established. ∎

3.3. The multiple pendulum type systems.

As a last classical example, we consider the case of multiple pendulum type systems. That is, we assume (see e.g. [2], [14],[29], for analogous assumptions)

1. (V5)

$V\in C^{1}({\mathbb{R}}^{N})$ is ${{\mathbb{Z}}}^{N}$-periodic, 2. (V6)

$V(x)\geq 0$, and $V(x)=0$ if and only if $x\in{{\mathbb{Z}}}^{N}$.

Upon assuming (V5) and (V6) we observe that $\mathcal{V}^{c}={\mathbb{Z}}^{N}$ for $c=0$. Thus, by continuity and periodicity there exists $c_{p}>0$ so that

[TABLE]

Hence, by denoting for any such $c\in[0,c_{p})$

[TABLE]

we observe the following properties hold:

1. ($\mathcal{V}$i)

$\mathcal{V}^{c}={\textstyle\bigcup_{\xi\in{\mathbb{Z}}^{N}}}\mathcal{V}^{c}_{\xi}$, 2. ($\mathcal{V}$ii)

$\mathcal{V}^{c}_{\xi}$ is compact for any $\xi\in{\mathbb{Z}}^{N}$, 3. ($\mathcal{V}$iii)

$\exists r_{c}>0$ such that ${\rm dist}\,(\mathcal{V}^{c}_{\xi},\mathcal{V}^{c}_{\xi^{\prime}})\geq 2r_{c}$ for any pair $\xi\not=\xi^{\prime}\in{\mathbb{Z}}^{N}$, 4. ($\mathcal{V}$iv)

$\forall r\in(0,r_{c})$, $\exists\mu_{r}>0$ such that $V(x)>c+\mu_{r}$ for any $x\in{\mathbb{R}}^{N}\setminus{\textstyle\bigcup_{\xi\in{\mathbb{Z}}^{N}}}B_{r}(\mathcal{V}^{c}_{\xi})\,$.

Remark 3.8**.**

Let us denote the elements of the canonical basis of ${\mathbb{R}}^{N}$ by ${\bm{e}}_{\ell}$, for $\ell=1,\ldots,N$, and define associated functions ${\zeta}_{\ell}:{\mathbb{R}}\to{\mathbb{R}}^{N}$ by $\zeta_{\ell}(t)=t{\bm{e}}_{\ell}$ for $t\in[0,1]$, $\zeta_{\ell}(t)=0$ for $t\leq 0$ and $\zeta_{\ell}(t)={\bm{e}}_{\ell}$ for $t\geq 1$. Since by definition $\mathcal{V}^{c}_{\xi}\subset B_{1/3}(\xi)$ for any $\xi\in{\mathbb{Z}}^{N}$ and $c\in[0,c_{p})$, elementary geometric considerations give that for any $\ell\in\{1,\ldots,N\}$ and $c\in[0,c_{p})$ there exists $(s_{c,\ell},t_{c,\ell})\subset[0,1]$ in such a way that

[TABLE]

Next, it will be convenient to introduce the following test function $\eta_{c,\ell}:{\mathbb{R}}\to{\mathbb{R}}^{N}$ given by $\eta_{c,\ell}(t)=\zeta_{\ell}(t)$ for $t\in(s_{c,\ell},t_{c,\ell})$, ${\eta_{c,\ell}}(t)=\zeta_{\ell}(s_{c,\ell})$ for $t\leq s_{c,\ell}$ and ${\eta_{c,\ell}}(t)=\zeta_{\ell}(t_{c,\ell})$ for $t\geq t_{c,\ell}$. We readily get the bound

[TABLE]

for any $\ell\in\{1,\ldots,N\}$ and $c\in[0,c_{p})$ as above.

We first observe

Lemma 3.9**.**

There exists $R_{p}>0$ so that for any $c\in[0,c_{p})$, if $q\in W^{1,2}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$ and $(\sigma,\tau)\subset{\mathbb{R}}$ are such that $V(q(t))\geq c$ for $t\in(\sigma,\tau)$ and $J_{c,(\sigma,\tau)}(q)\leq M_{p}+1$, then

[TABLE]

Proof of Lemma 3.9.

Let us write $r_{p}:=\frac{1}{2}r_{c_{p}}$, and $\mu_{p}:=\mu_{r_{p}}$ as in ($\mathcal{V}$iii)-($\mathcal{V}$iv). Then it follows that $V(x)>c_{p}+{\mu_{p}}$ for any $x\in{\mathbb{R}}^{N}\setminus{{\textstyle\bigcup_{\xi\in{\mathbb{Z}}^{N}}}B_{r_{p}}(\mathcal{V}^{c_{p}}_{\xi})}$. Now, to establish Lemma 3.9 let us assume by contradiction that there are sequences $(c_{n})\subset[0,c_{p})$ and $(q_{n})\subset W^{1,2}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$ with corresponding intervals $(\sigma_{n},\tau_{n})\subset{\mathbb{R}}$ such that $V(q_{n}(t))\geq c_{n}$ for any $t\in(\sigma_{n},\tau_{n})$, $J_{{c_{n}},(\sigma_{n},\tau_{n})}(q_{n})\leq M_{p}+1$, and

[TABLE]

In particular, since $c_{n}\leq c_{p}$ we see from ($\mathcal{V}$iii) that ${\rm dist}\,(\mathcal{V}^{c_{n}}_{\xi},\mathcal{V}^{c_{n}}_{\xi^{\prime}}){\geq 2r_{c_{p}}}$ for all $n\in{{\mathbb{N}}}$, if $\xi\neq\xi^{\prime}$. These inequalities, along with basic geometric considerations, cf. (V5), imply the existence of $n$ disjoint intervals $(s_{i},t_{i})\subset(\sigma_{n},\tau_{n})$ for $1\leq i\leq n$, such that $q_{n}(t)\notin{{\textstyle\bigcup_{\xi\in{\mathbb{Z}}^{N}}}B_{r_{p}}(\mathcal{V}^{c_{p}}_{\xi})}$ if $t\in{\textstyle\bigcup_{i}}(s_{i},t_{i})$, while $|q_{n}(t_{i})-q_{n}(s_{i})|\geq{2(r_{c_{p}}-r_{p})=r_{c_{p}}}$. Then, by Remark 2.4, $J_{c_{n},(\sigma_{n},\tau_{n})}(q_{n})\geq{n\sqrt{2\mu_{p}}r_{c_{p}}}$ for any $n\in{{\mathbb{N}}}$. But this goes in contradiction with $J_{{c_{n}},(\sigma_{n},\tau_{n})}(q_{n})\leq M_{p}+1$, and the Lemma follows. ∎

The above properties allow us to apply Theorem 2.1, giving the next

Proposition 3.10**.**

Assume that $V$ satisfies (V5) and (V6). Then for every $c\in[0,{c_{p}})$ there exists $k_{c}\in{{\mathbb{N}}}$ and a finite set $\{\xi^{1},\ldots,{\xi^{k_{c}}}\}\subset{\mathbb{Z}}^{N}\setminus\{0\}$, satisfying

[TABLE]

for which, given any $j\in\{1,\ldots,k_{c}\}$, there is a solution $q_{c,j}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to (1.1) with energy $E_{q_{c,j}}(t)=-c$ for any $t\in{\mathbb{R}}$, verifying

[TABLE]

where $R_{p}$ is given by Lemma 3.9.

Proof of Proposition 3.10.

For any $c\in[0,{c_{p}})$, let us set

[TABLE]

In particular, we observe in light of ($\mathcal{V}$iii) that

[TABLE]

so $(\mathbf{V}^{c})$ holds. Then defining $\Gamma_{c,1}$ as in (1), relative to the partition (3.66), we let

[TABLE]

Recall the definition of $\eta_{c,\ell}$ in Remark 3.8 and observe that $\eta_{c,1}\in\Gamma_{c,1}$. This, together with (3.63), yields

[TABLE]

Consider now any minimizing sequence $(q_{n})\subset\Gamma_{c,1}$, so that $J_{c}(q_{n})\to m_{c,1}$. Eventually passing to a subsequence, we can assume that $J_{c}(q_{n})\leq M_{p}+1$ for any $n\in{{\mathbb{N}}}$, and so by Lemma 3.9

[TABLE]

Therefore the coercivity condition (1.7) of $J_{c}$ is satisfied for the division (3.66), hence Theorem 2.1 gives the existence of a solution $q_{c,1}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to (1.1) with energy $E_{q_{c,1}}(t)=-c$ for all $t\in{\mathbb{R}}$, satisfying

[TABLE]

This solution is either of brake orbit type, case (a) of the theorem, of homoclinic type, case (b), or of heteroclinic type in the case (c). We now continue with the proof of Proposition 3.10 by checking that (3.65) is satisfied, regardless of the case in consideration.

If we are in the case (a) there exists $-\infty<\sigma<\tau<+\infty$ such that

1. $(a_{0})$

$q_{c,1}(\sigma)\in\mathcal{V}^{c}_{-,1}$, $q_{c,1}(\tau)\in\mathcal{V}^{c}_{+,1}$, $V(q_{c,1}(t))>c$ for $t\in(\sigma,\tau)$, $q_{c,1}(\sigma+t)=q_{c,1}(\sigma-t)$ and $q_{c,1}(\tau+t)=q_{c,1}(\tau-t)$ for all $t\in{\mathbb{R}}$.

Let us now point out that $(a_{0})$, (3.66) and ($\mathcal{V}$i) yield that $q_{c,1}(\sigma)\in\mathcal{V}^{c}_{0}$ and that there exists $\xi^{1}\in{\mathbb{Z}}^{N}\setminus\{0\}$ with $q_{c,1}(\tau)\in\mathcal{V}^{c}_{\xi^{1}}$. Moreover, from Remark 2.17 we have that for any $(s,t)\subset(\sigma,\tau)$, $J_{c,(s,t)}(q_{c,1})\leq J_{c,(\sigma,\tau)}(q_{c,1})=m_{c,1}\leq M_{p}$ and so Lemma 3.9 gives in particular $|q_{c,1}(\sigma)-q_{c,1}(t)|\leq R_{p}$ for any $t\in(\sigma,\tau)$. By periodicity we then obtain ${\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{0})\leq R_{p}$ for any $t\in{\mathbb{R}}$. Since $\mathcal{V}^{c}_{0}\subset B_{1/3}(0)$ we get $\|q_{c,1}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$, therefore property (3.65) is satisfied by $q_{c,1}$.

If we are in the case (b), then there exists $\sigma\in{\mathbb{R}}$ such that

1. $(b_{0})$

$q_{c,1}(\sigma)\in\mathcal{V}^{c}_{\pm,1}$, $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{\mp,1})=0$, $V(q_{c,1}(t))>c$ for $t\in{\mathbb{R}}\setminus\{\sigma\}$, and $q_{c,1}(\sigma+t)=q_{c,1}(\sigma-t)$ for all $t\in{\mathbb{R}}$.

Once again, we point out that $(b_{0})$, (3.66) and ($\mathcal{V}$i) imply that there is $\xi^{1}\in{\mathbb{Z}}^{N}\setminus\{0\}$ such that either $q_{c,1}(\sigma)\in\mathcal{V}^{c}_{0}$ and $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{\xi^{1}})=0$, or $q_{c,1}(\sigma)\in\mathcal{V}^{c}_{\xi^{1}}$ and $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{0})=0$. By Remark 2.17 we have $J_{c,(s,t)}(q_{c,1})\leq J_{c,(-\infty,\sigma)}(q_{c,1})=m_{c,1}\leq M_{p}$ for any $(s,t)\subset(-\infty,\sigma)$. This, combined with Lemma 3.9 and the reflection $q_{c,1}(\sigma+t)=q_{c,1}(\sigma-t)$ for all $t\in{\mathbb{R}}$, shows that $|q_{c,1}(t)-q_{c,1}(s)|\leq R_{p}$ for any $s<t\in{\mathbb{R}}$. But since $q_{c,1}(\sigma)\in\mathcal{V}^{c}_{\pm,1}$ and $\lim_{t\to\pm\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{\mp,1})=0$ we obtain ${\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{0})\leq R_{p}$ for any $t\in{\mathbb{R}}$ and so $\|q_{c,1}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$. This shows again (3.65) for $q_{c,1}$.

Finally if we are in the case (c) we have

1. $(c_{0})$

$V(q_{c,1}(t))>c$ for all $t\in{\mathbb{R}}$, $\lim\limits_{t\to-\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{-,1})=0\,$ and $\lim\limits_{t\to+\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{+,1})=0$.

As before $(c_{0})$, (3.66) and ($\mathcal{V}$i) show that there is $\xi^{1}\in{\mathbb{Z}}^{N}\setminus\{0\}$ such that $\lim\limits_{t\to+\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{\xi^{1}})=0$. From Remark 2.17 we have $J_{c,(s,t)}(q_{c,1})\leq J_{c}(q_{c,1})=m_{c,1}\leq M_{p}$ for any $(s,t)\subset{\mathbb{R}}$, which combined with Lemma 3.9 gives $|q_{c,1}(t)-q_{c,1}(s)|\leq R_{p}$ for any $s<t\in{\mathbb{R}}$. Since $\lim_{t\to-\infty}{\rm dist}\,(q_{c,1}(t),\mathcal{V}^{c}_{0})=0$, we deduce that ${\rm dist}\,(q_{c,1}(s),\mathcal{V}^{c}_{0})\leq R_{p}$ for any $s\in{\mathbb{R}}$ and hence $\|q_{c,1}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$. Whence, even in case (c) the condition (3.65) holds for $q_{c,1}$.

The above argument shows the existence of a solution $q_{c,1}$ with energy $E_{q_{c,1}}=-c$ with

[TABLE]

In order to establish the multiplicity of solutions in Proposition 3.10, we will proceed by induction. Let us assume that for $j\geq 1$ we have

1. (I)

There are $\xi^{1},\xi^{2},...,\xi^{j}\in{\mathbb{Z}}^{N}\setminus\{0\}$ such that $\xi^{h}\neq\xi^{k}$ for $h\neq k$ and if we set $\mathcal{L}_{j}:=\{\sum_{i=1}^{j}n_{i}\xi^{i}:\,n_{1},\ldots,n_{j}\in{\mathbb{Z}}\}$, then $\mathcal{L}_{j}\not={\mathbb{Z}}^{N}.$ 2. (II)

For any $i\in\{1,\ldots,j\}$ there exists $q_{c,i}$ with energy $E_{q_{c,i}}=-c$ such that

[TABLE]

Proposition 3.10 will follow once we show that (I) and (II) together imply the existence of $\xi^{j+1}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$ and a solution $q_{c,j+1}$ with energy $E_{q_{c,j+1}}=-c$, in such a way that

[TABLE]

To see this, we consider in view of (I), the following decomposition of $\mathcal{V}^{c}$:

[TABLE]

In light of (I) both sets $\mathcal{V}^{c}_{-,j+1}$, $\mathcal{V}^{c}_{+,j+1}$ are non-empty, they clearly verify ${\rm dist}\,(\mathcal{V}^{c}_{-,j+1},\mathcal{V}^{c}_{+,j+1})\geq r_{c},$ and

[TABLE]

see Figure 2 below.

Then, according to the partition (3.68) of $\mathcal{V}^{c}$, define $\Gamma_{c,j+1}$ as in (1) and let $m_{c,j+1}:=\inf_{q\in\Gamma_{c,j+1}}{J_{c}}(q)$. Since $\mathcal{L}_{j}\not={\mathbb{Z}}^{N}$ there must exist $\ell_{j+1}\in\{1,\ldots,N\}$ so that ${\bm{e}}_{\ell_{j+1}}\notin\mathcal{L}_{j}$. Then, by Remark 3.8, $\eta_{c,\ell_{j+1}}\in\Gamma_{c,j+1}$, and so by (3.63)

[TABLE]

Let $(q_{n})\subset\Gamma_{c,j+1}$ be such that $J_{c}(q_{n})\to m_{c,j+1}$. With no loss of generality we can assume that $J_{c}(q_{n})<M_{p}+1$ for any $n\in{{\mathbb{N}}}$, and so by Lemma 3.9 we obtain

[TABLE]

Since $q_{n}\in\Gamma_{c,j+1}$ we have $\liminf_{t\to-\infty}{\rm dist}\,(q_{n}(t),\mathcal{V}^{c}_{-,j+1})=0$ for any $n\in{{\mathbb{N}}}$. Hence, there exist sequences $(s_{n})\subset{\mathbb{R}}$ and $(\zeta_{n})\subset\mathcal{L}_{j}$ for which

[TABLE]

By periodicity of $V$, (3.69) and by (3.70)-(3.71), we have

[TABLE]

from which we conclude that

[TABLE]

namely, the coercivity condition (1.7) of $J_{c}$ over $\Gamma_{c,j+1}$ follows. Thus, Theorem 2.1 yields the existence of a solution $\bar{q}_{c,j+1}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to (1.1) with energy $E_{\bar{q}_{c,j+1}}=-c$, satisfying

[TABLE]

This solution is either of brake orbit type, case (a), of homoclinic type, case (b), or of heteroclinic type in the case (c). If we are in the case (a), then there exists $-\infty<\sigma<\tau<+\infty$ such that

1. $(a_{j})$

$\bar{q}_{c,j+1}(\sigma)\in\mathcal{V}^{c}_{-,j+1}$, $\bar{q}_{c,j+1}(\tau)\in\mathcal{V}^{c}_{+,j+1}$, $V(\bar{q}_{c,j+1}(t))>c$ for $t\in(\sigma,\tau)$, $\bar{q}_{c,j+1}(\sigma+t)=\bar{q}_{c,j+1}(\sigma-t)$ and $\bar{q}_{c,j+1}(\tau+t)=\bar{q}_{c,j+1}(\tau-t)$ for all $t\in{\mathbb{R}}$.

By $(a_{j})$ there exists $\xi_{-}\in\mathcal{L}_{j}$, $\xi_{+}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$ such that $\bar{q}_{c,j+1}(\sigma)\in\mathcal{V}^{c}_{\xi_{-}}$ and $\bar{q}_{c,j+1}(\tau)\in\mathcal{V}^{c}_{\xi_{+}}$. Then $\xi^{j+1}=\xi_{+}-\xi_{-}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$, and by periodicity of $V$, the function $q_{c,j+1}:=\bar{q}_{c,j+1}-\xi_{-}$ is a solution to (1.1) with energy $E_{q_{c,j+1}}=-c$. By Remark 2.17 we have moreover $J_{c,(s,t)}(q_{c,j+1})\leq J_{c,(\sigma,\tau)}(q_{c,j+1})=m_{c,j+1}\leq M_{p}$ for any $(s,t)\subset(\sigma,\tau)$, and so, arguing as in the case $(a_{0})$ above we deduce $\|q_{c,j+1}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$. Then property (3.67) with respect to $\xi^{j+1}$ is satisfied by $q_{c,j+1}$.

In case (b) there exists $\sigma\in{\mathbb{R}}$ such that

1. $(b_{j})$

$\bar{q}_{c,j+1}(\sigma)\in\mathcal{V}^{c}_{\pm,j+1}$, $\lim_{t\to\pm\infty}{\rm dist}\,(\bar{q}_{c,j+1}(t),\mathcal{V}^{c}_{\mp,j+1})=0$, $V(\bar{q}_{c,j+1}(t))>c$ for $t\in{\mathbb{R}}\setminus\{\sigma\}$, and $\bar{q}_{c,j+1}(\sigma+t)=\bar{q}_{c,j+1}(\sigma-t)$ for all $t\in{\mathbb{R}}$.

In particular, from $(b_{j})$ we deduce the existence of $\bar{\xi}\in{\mathbb{Z}}^{N}$ so that $\bar{q}_{c,j+1}(\sigma)\in\mathcal{V}^{c}_{\bar{\xi}}$. Let us say that $\bar{\xi}\in\mathcal{L}_{j}$, then $\lim_{t\to\pm\infty}{\rm dist}\,(\bar{q}_{c,j+1}(t),\mathcal{V}^{c}_{+,j+1})=0$. The symmetry of $\bar{q}_{c,j+1}$ with respect to $\sigma$ together with the discreteness of $\mathcal{V}^{c}_{+,j+1}$ implies that there is $\xi_{\infty}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$ so that $\lim_{t\to\pm\infty}{\rm dist}\,(\bar{q}_{c,j+1}(t),\mathcal{V}^{c}_{\xi_{\infty}})=0$. In this case we set $\xi^{j+1}:=\xi_{\infty}-\bar{\xi}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$ and the function $q_{c,j+1}:=\bar{q}_{c,j+1}-\bar{\xi}$ is a solution to (1.1) with energy $E_{\bar{q}_{c,j+1}}=-c$, which verifies the first part of (3.67) with respect to $\xi^{j+1}$. Otherwise, if $\bar{\xi}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$, then $\lim_{t\to\pm\infty}{\rm dist}\,(\bar{q}_{c,j+1}(t),\mathcal{V}^{c}_{-,j+1})=0$. The symmetry of $\bar{q}_{c,j+1}$ with respect to $\sigma$ and the discreteness of $\mathcal{V}^{c}_{-,j+1}$ imply the existence of $\xi_{\infty}\in\mathcal{L}_{j}$ so that $\lim_{t\to\pm\infty}{\rm dist}\,(\bar{q}_{c,j+1}(t),\mathcal{V}^{c}_{\xi_{\infty}})=0$. In this case we let $\xi^{j+1}:=\bar{\xi}-\xi_{\infty}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$, and again the function $q_{c,j+1}:=\bar{q}_{c,j+1}-\xi_{\infty}$ is a solution to (1.1) with energy $E_{q_{c,j+1}}=-c$ which verifies the first part of (3.67) with respect to $\xi^{j+1}$.

To get the second part of (3.67) observe that by Remark 2.17 we have $J_{c,(s,t)}(q_{c,j+1})\leq J_{c,(-\infty,\sigma)}(q_{c,j+1})=m_{c,j+1}\leq M_{p}$ for any $(s,t)\subset(-\infty,\sigma)$. Then Lemma 3.9 and the symmetry property of $q_{c,j+1}$ with respect to $\sigma$ allow us to argue as in the case $(b_{0})$ above to deduce $\|q_{c,j+1}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$, thus showing (3.67).

Finally, let us assume that we are in case (c). Then

1. $(c_{j})$

$V(\bar{q}_{c,j+1}(t))>c$ for all $t\in{\mathbb{R}}$, and $\lim\limits_{t\to\pm\infty}{\rm dist}\,(\bar{q}_{c,j+1}(t),\mathcal{V}^{c}_{\pm,j+1})=0$.

By invoking once again the discreteness of the sets $\mathcal{V}^{c}_{\pm}$, $(c_{j})$ shows the existence of $\xi_{-}\in\mathcal{L}_{j}$ and $\xi_{+}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$ in such a way that $\lim\limits_{t\to\pm\infty}{\rm dist}\,(\bar{q}_{c,j+1}(t),\mathcal{V}^{c}_{\xi_{\pm}})=0$. Setting $\xi^{j+1}:=\xi_{+}-\xi_{-}\in{\mathbb{Z}}^{N}\setminus\mathcal{L}_{j}$ we see from the periodicity of $V$ that the function $q_{c,j+1}:=\bar{q}_{c,j+1}-\xi_{-}$ is a solution to (1.1) with energy $E_{q_{c,j+1}}=-c$, verifying the first part of (3.67) with respect to $\xi^{j+1}$. By Remark 2.17 we have $J_{c,(s,t)}(q_{c,j+1})\leq J_{c}(q_{c,j+1})=m_{c,j+1}\leq M_{p}$ for any $(s,t)\subset{\mathbb{R}}$. Using Lemma 3.9 and arguing as in the case $(c_{0})$ above we obtain again $\|q_{c,j+1}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$. Then (3.67) follows for $q_{c,j+1}$.

This concludes the proof of the inductive step, and hence the proof of Proposition 3.10. ∎

Remark 3.11**.**

Proposition 3.10 constitutes a multiplicity result. It asserts the existence of $k_{c}$ elements $\xi^{1},\ldots,\xi^{k_{c}}$ in the lattice ${\mathbb{Z}}^{N}\setminus\{0\}$, for each of which there exists a connecting orbit between $\mathcal{V}^{c}_{0}$ and $\mathcal{V}^{c}_{\xi^{j}}$. In particular, we necessarily have $k_{c}\geq N$, due to (3.64).

As for the preceding cases, we finalize by analyzing the convergence properties of the family of solutions $q_{c,j}$ given by Proposition 3.10 as $c\to 0^{+}$.

Proposition 3.12**.**

Assume that $V\in C^{1}({\mathbb{R}}^{N})$ satisfies (V5)-(V6) and that $\nabla V$ is locally Lipschitz continuous in ${\mathbb{R}}^{N}$. For any sequence $c_{n}\to 0^{+}$, let $k_{c_{n}}\in{{\mathbb{N}}}$, $\{\xi_{n}^{1},\ldots,\xi_{n}^{k_{c_{n}}}\}\subset{\mathbb{Z}}^{N}\setminus\{0\}$ and $q_{c_{n},j}$ be the solution given by Proposition 3.10 associated to $\xi_{n}^{j}$ for $j=1,\ldots,k_{c_{n}}$. Then, along a subsequence, we have that

1. (i)

There exists $\kappa\in{{\mathbb{N}}}$ such that $k_{c_{n}}=\kappa\,$ for all $n\in{{\mathbb{N}}};$ 2. (ii)

There exist distinct elements $\hat{\xi}^{1},\ldots,\hat{\xi}^{\kappa}$ in ${\mathbb{Z}}^{N}\setminus\{0\}$ so that $\{n_{1}\hat{\xi}^{1}+\ldots+n_{\kappa}\hat{\xi}^{\kappa}:n_{1},\ldots,n_{\kappa}\in{\mathbb{Z}}\}={\mathbb{Z}}^{N}$, and

[TABLE] 3. (iii)

For any $j\in\{1,\ldots,\kappa\}$ there is a solution $q^{j}\in C^{2}({\mathbb{R}},{\mathbb{R}}^{N})$ to (1.1) of heteroclinic type between [math] and $\hat{\xi}^{j}$, and there exists $(\tau_{n,j})\subset{\mathbb{R}}$ such that

[TABLE]

Proof of Proposition 3.12.

With no loss of generality we can assume $c_{n}<c_{p}$ for all $n\in{{\mathbb{N}}}$. Let $k_{c_{n}}\in{{\mathbb{N}}}$, $\{\xi_{n}^{1},\ldots,{\xi_{n}^{k_{c_{n}}}}\}\subset{\mathbb{Z}}^{N}\setminus\{0\}$ and $q_{c_{n},j}$, for $j=1,\ldots,k_{c_{n}}$ be given by Proposition 3.10 associated to $\xi_{n}^{j}$. We know $\|q_{c_{n},j}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$ for any $n\in{{\mathbb{N}}}$. Moreover $\inf_{t\in{\mathbb{R}}}{\rm dist}\,(q_{c_{n},j}(t),{\mathcal{V}^{c_{n}}_{0}})=\inf_{t\in{\mathbb{R}}}{\rm dist}\,(q_{c_{n},j}(t),\mathcal{V}^{c_{n}}_{\xi_{n}^{j}})=0$ for any $j\in\{1,\ldots,k_{c_{n}}\}$ and all $n\in{{\mathbb{N}}}$.

In particular, the above shows that $\{\xi_{n}^{1},\ldots,\xi_{n}^{k_{c_{n}}}\}\subset B_{R_{p}+1}(0)\cap{\mathbb{Z}}^{N}$ for any $n\in{{\mathbb{N}}}$. Since $B_{R_{p}+1}(0)\cap{\mathbb{Z}}^{N}$ is a finite set, all the sequences $(k_{c_{n}})$, $(\xi_{n}^{j})$ for $1\leq j\leq k_{c_{n}}$ take their values in a finite set, hence they are all constant along a common subsequence: there exists $\kappa\in{{\mathbb{N}}}$, $\{\hat{\xi}^{1},\ldots,\hat{\xi}^{\kappa}\}\subset{\mathbb{Z}}^{N}\setminus\{0\}$ and an increasing sequence $(n_{i})\subset{{\mathbb{N}}}$, such that $k_{c_{n_{i}}}=\kappa$, $\xi_{n_{i}}^{j}=\hat{\xi}^{j}$ for any $i\in{{\mathbb{N}}}$ and $1\leq j\leq\kappa$. Thus (i) and (ii) follow.

For $j\in\{1,\ldots,\kappa\}$ fixed, and let us simplify the notation by allowing

[TABLE]

Since any $q_{i}$ is given by Theorem 2.1, we can invoke Remark 2.17 to see that, for each $i\in{{\mathbb{N}}}$, $q_{i}$ has a connecting time interval $(\alpha_{i},\omega_{i})\subset{\mathbb{R}}$ with $-\infty\leq\alpha_{i}\leq\omega_{i}\leq+\infty$ in such a way that

1. $(1_{i})$

$V(q_{i}(t))>c_{i}$ for every $t\in(\alpha_{i},\omega_{i})$, 2. $(2_{i})$

$\lim\limits_{t\to\alpha_{i}^{+}}{\rm dist}\,(q_{i}(t),\mathcal{V}^{c_{i}}_{0})=0$, and if $\alpha_{i}>-\infty$ then $\dot{q}_{i}(\alpha_{i})=0$, $V(q_{i}(\alpha_{i}))=c_{i}$ with $q_{i}(\alpha_{i})\in\mathcal{V}^{c_{i}}_{0}$, 3. $(3_{i})$

$\lim\limits_{t\to\omega_{i}^{-}}{\rm dist}\,(q_{i}(t),\mathcal{V}^{c_{i}}_{\xi})=0$, and if $\omega_{i}<+\infty$ then ${\dot{q}_{i}}(\omega_{i})=0$, $V(q_{i}(\omega_{i}))=c_{l}$ with $q_{i}(\omega_{i})\in\mathcal{V}^{c_{i}}_{\xi}$, 4. $(4_{i})$

$J_{c_{i},(\alpha_{i},\omega_{i})}(q_{i})=m_{c_{i}}=\inf\limits_{q\in\Gamma_{c_{i}}}J_{c_{i}}(q)$.

The rest of our argument goes along the same lines as the proof convergence to a heteroclinic type solution in Proposition 3.3, so we will briefly review it. We first renormalize the sequence $(q_{i})$ by a phase shift procedure. From $(1_{i})$-$(2_{i})$-$(3_{i})$ it follows that for all $i\in{{\mathbb{N}}}$ there is $t_{i}\in(\alpha_{i},\omega_{i})$ so that $|q_{i}(t_{i})|=\tfrac{1}{2}$. Renaming, if necessary, $q_{i}$ to be $q_{i}(\cdot-t_{i})$, we can assume

[TABLE]

Just like before, the bound $\sup_{i\in{{\mathbb{N}}}}\|q_{i}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$ combined with energy constraint $E_{q_{i}}=-c_{i}$ and the fact that $q_{i}$ solves the system $\ddot{q}=\nabla V(q)$ on ${\mathbb{R}}$, allows us to conclude that $(\dot{q}_{i})$ and $(\ddot{q}_{i})$ are uniformly bounded in $L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})$. Whence, by the Ascoli-Arselà Theorem there is $q_{0}\in C^{1}({\mathbb{R}},{\mathbb{R}}^{N})$ so that $(q_{i})$ has a subsequence, yet denoted by $(q_{i})$, for which $q_{i}\to q_{0}$ in $C^{1}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$ as $i\to+\infty$. This convergence is then bootstrapped into the equation $\ddot{q}_{i}=\nabla V(q_{i})$ in order to enhance it to $C^{2}_{loc}({\mathbb{R}},{\mathbb{R}}^{N})$. This shows, in turn, that

[TABLE]

In addition, by taking the limit $i\to+\infty$ in $|q_{i}(0)|=\tfrac{1}{2}$ and $\|q_{i}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{N})}\leq R_{p}+1$, we learn that this solution satisfies

[TABLE]

Furthermore, the same argument used to establish (3.41) can be applied to our case, to yield

[TABLE]

for potentials satisfying (V5)-(V6): arguing by contradiction, we show that $q_{0}$ solves the Cauchy problem $\ddot{q}=\nabla q$ and $q(0)=0,\dot{q}(0)=0$, whence $q_{0}\equiv 0$, which is contrary to (3.73).

To conclude the proof of Proposition 3.7 we are left to show that $q_{0}$ is of heteroclinic type. More precisely, our goal is to show that

[TABLE]

As before, interpolation inequalities would then prove that $q_{0}$ is a solution to (1.1) of heteroclinic type between [math] and $\xi$. By analogy with our previous analysis, (3.74) reduces to proving that for any $r\in(0,\frac{1}{3})$ there exist $L^{-}_{r},L^{+}_{r}>0$ and $i_{r}\in{{\mathbb{N}}}$, in such a way that

[TABLE]

for all $i\geq i_{r}$. The proof of this assertion can be obtained by rephrasing the argument used to prove (3.42) in Proposition 3.3, and we omit it. ∎