Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems

Z. angew. Math. Phys. 50 (1999) 860–8910044-2275/99/060860-32 $ 1.50+0.20/0c© 1999 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

Multibump solutions to possibly degenerate equilibria foralmost periodic Lagrangian systems

Francesca Alessio1, Maria Letizia Bertotti and Piero Montecchiari

Abstract. We study via variational methods some chaotic features of a class of almost periodicLagrangian systems on a torus. In particular we show that slowly oscillating perturbations ofsuch systems admit a multibump dynamics relative to possibly degenerate equilibria.

Mathematics Subject Classification (1991). 34C37, 58F05, 70H35.

Keywords. Lagrangian systems, homoclinic, heteroclinic, multibump solutions, almost period-icity, variational methods.

1. Introduction

Starting with [11] and [19], in the last few years global variational methods havebeen successfully applied in studying homoclinic and heteroclinic motions for La-grangian and Hamiltonian systems (see more specifically [33], [34], [35] for theheteroclinic problem).

In particular, the study of the multiplicity problem led to the development ofnew techniques which have allowed to prove, via variational methods, shadowing-like lemmas (see [39]) and then to show the existence of a class of solutions, calledmultibump solutions, whose presence displays chaotic features of the dynamics.

The use of a global variational approach permitted to show the existence ofa multibump dynamics for a wide class of time dependencies of the Hamiltonianfunction under nondegeneracy assumptions on the set of homoclinic (heteroclinic)solutions weaker than the classical transversality one. We refer to [15], [14] for theautonomous case, to [39], [30], [9], [10], [16], [22], [18], [37], [32] for the periodicand asymptotically periodic cases, to [20], [36], [31], [2] for the almost periodicand recurrent cases, to [1], [13], [21] and [5] for the slowly oscillating case. Wemention also [4] and [6], which generalize via a variational approach the Melnikovmethod to systems of any dimension and with arbitrary time dependence of the

1Supported by Istituto Nazionale di Alta Matematica F. Severi (Borsa di Ricerca Senior1997/98) and by MURST Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”.

Vol. 50 (1999) Multibump solutions 861

perturbative term.In this paper we are concerned with Lagrangian systems described by a La-

grangian function L ∈ C1(RN ×RN ×R) of the form

L(q, v, t) = L2 + L1 + L0 =12〈A(q, t)v, v〉 + 〈b(q, t), v〉+ c(q, t), (1.1)

where A(q, t) is a symmetric N × N matrix, b(q, t) ∈ RN and c(q, t) ∈ R, for all(q, t) ∈ RN ×R. More precisely, we assume

(Hp 1) A, b, c ∈ C1(RN ×R) have bounded derivatives and(i) A(q + ξ, t) = A(q, t), b(q + ξ, t) = b(q, t) and c(q + ξ, t) = c(q, t) for allq ∈ RN , ξ ∈ ZN , t ∈ R,(ii) A(q, t), b(q, t), c(q, t) are almost periodic in t uniformly with respect toq ∈ RN ,

(Hp 2) there exist 0 < m < M such that m|v|2 ≤ 〈A(q, t)v, v〉 ≤ M |v|2 for all(q, v, t) ∈ RN ×RN ×R.

We note that setting γ(q, t) = 12〈A−1(q, t)b(q, t), b(q, t)〉 − c(q, t) we can write

the Lagrangian in the form

L(q, v, t) =12

∣∣∣A 12 (q, t) v +A−

12 (q, t) b(q, t)

∣∣∣2 − γ(q, t) (1.2)

and since the equations of motion

d

dt

∂L

∂v(q, q, t)− ∂L

∂q(q, q, t) = 0 (1.3)

are left invariant if we add to the Lagrangian a purely time dependent function,without loss of generality, we can assume that γ(0, t) = 0 for any t ∈ R. Inaddition to (Hp 1), (Hp 2) we ask that

(Hp 3) supt∈R γ(q, t) < 0 for any q ∈ RN \ ZN ,(Hp 4) there exists k > 0 such that |b(q, t)|2 ≤ −kγ(q, t) for all (q, t) ∈ RN ×R.

By (Hp 3), (Hp 4) and (1.2) we plainly recognize that the Lagrangian L(q, v, t)is not negative and that in fact the zeros set of the Lagrangian is exactly the set{(ξ, 0, t) : ξ ∈ ZN , t ∈ R}. Consequently any point in the phase space of the form(ξ, 0), ξ ∈ ZN , is a rest point for the system.

We consider the problem of the existence of heteroclinic solutions joining dif-ferent equilibria. More precisely, given ξ−, ξ+ ∈ ZN we look for solutions q(t) of(1.3) which satisfy the asymptotic conditions

q(t)→ ξ± and q(t)→ 0 as t→ ±∞.

862 F. Alessio et al. ZAMP

Obviously, the existence of heteroclinic motions will imply in particular that theequilibria (ξ, 0), ξ ∈ ZN , are unstable. We refer to [26] where it is proved thatat least in the autonomous case the assumption (Hp 3) is in fact sufficient for theinstability of these equilibria, i.e., no “gyrostatic stabilization”, due to the presenceof the linear in velocity term L1, occurs. To be more precise, also the requirementthat b(ξ) = 0 (ξ ∈ ZN ) is needed in [26]. Our assumption (Hp 4) strengthens infact the analogous (in the not autonomous setting) requirement that b(ξ, t) = 0(ξ ∈ ZN , t ∈ R). However we point out that, assuming b(ξ, t) = 0 (ξ ∈ ZN ,t ∈ R), (Hp 4) is just a consequence of (Hp 1) and (Hp 3) whenever the maximaξ of γ are nondegenerate.

The model problem we have in mind is a holonomic mechanical system, forwhich both the constraints and the forces (derivable from a potential) may betime dependent and whose configuration space at any time t is an N -dimensionaltorus TN . The form (1.1) is the most general for a Lagrangian arising in theframework of analytical mechanics and in fact, for problems with time-dependentconstraints the linear in velocity term L1 is typically different from zero. We referto [7] where the heteroclinic problem for this kind of almost periodic Lagrangianwas first considered.

As an example we can consider a planar N -pendulum whose suspension pointis moving vertically in a prescribed almost periodic fashion (see [12]). More simplywe can think to a particle P of mass m constrained to move along a frictionlessand massless closed and simple guide Γ which lies in a vertically oscillating plane.

An important class of examples is given by the cases in which the Lagrangianhas the simple form L(q, v, t) = 1

2 |v|2 − V (q, t), i.e., the cases A = 1, b = 0,γ = −c = V . Our assumptions then reduce to ask that the potential −V (q, t) isregular, periodic in q, almost periodic in t and that V (q, t) < V (0, t) = 0 uniformlywith respect to t ∈ R whenever q ∈ RN \ ZN . The heteroclinic problem for thiskind of systems has been already studied in [8] where the existence of infinitelymany solutions is proved.

In this simple case it is particularly evident the fact that, apart from the signcondition, no other assumption is made on the local behavior of the potential −V(and in the general case on γ) at the equilibria. In particular, no hyperbolicity ofthe equilibria is postulated. Even in the periodic, perturbative, one dimensionalcase, this generality gives rise to a series of difficulties in tackling the problem byusing the classical stable and unstable manifold theorem ([40]) employing in thisway Poincare Melnikov type methods.The problem of studying homoclinic motions relative to degenerate periodic orbits(or degenerate equilibria) arises for example in Celestial Mechanics, see e. g. [29],[23], [38], [41], [28]. In this context we quote [17] where a refined developmentof a Poincare Melnikov method is made to study a planar, periodic, perturbativedegenerate case.

In this paper, using variational methods, we firstly show that (1.3) always ad-mits infinitely many heteroclinic solutions and that in fact, given any two equilibria


there exists between them infinitely many chains of heteroclinics. Then, as a sec-ond step, we prove that if we add to the Lagrangian a slowly oscillating term, thenthe new system actually exhibits a multibump (chaotic) dynamics. This providesa large class of examples (a L∞-dense set of Lagrangians of the specified type) ofsystems presenting a multibump dynamics relative to a degenerate equilibrium.

To be more precise we first establish the following result.

Theorem 1.1. If (Hp 1) - (Hp 4) hold true, then the system (1.3) admits infinitelymany heteroclinic solutions. More precisely, there exist a m ∈ N, m ≥ 2N , andm points σ1, . . . , σm ∈ ZN \ {0} such that {

∑mi=1 niσi : ni ∈ N ∪ {0}} = ZN

and such that for any fixed j ∈ {1, . . . ,m} there exist infinitely many heteroclinicsolutions of (1.3) connecting 0 with σj .

As we said above this is the analogous of Theorem 1.1 in [8] for the generalclass of Lagrangians satisfying (Hp 1)–(Hp 4). Given any Lagrangian of this kind,by Theorem 1.1 we can select m points in ZN whose positive semigroup is allZN and such that for each of these points there are infinitely many heteroclinicsemanating from the origin and ending up exactly in this selected equilibrium. Byspace-periodicity of the Lagrangian this situation is repeated at any other pointξ ∈ ZN , i.e., for any σ ∈ {σ1, . . . , σm} and ξ ∈ ZN there are infinitely manyheteroclinics joining ξ with ξ + σ.Moreover, since the positive semigroup generated by {σ1, . . . , σm} is ZN we plainlyderive that given any two points ξ−, ξ+ we can connect them with infinitely manychains of heteroclinics. In fact ξ+ − ξ− ∈ ZN and there exist infinitely manyways to obtain this point as sums of the points σi. More precisely, there existsinfinitely many (n1, . . . , nm) ∈ (N ∪ {0})m such that ξ+ − ξ− =

∑mi=1 niσi. A

simple combinatorial computation sufficies to establish how many different chainsof heteroclinics correspond to each of such vectors (n1, . . . , nm).Finally, we can also consider infinite formal sums of heteroclinics obtaining curveswhich behave almost randomly in the phase space passing through a prefixedinfinite sequence of equilibria.

Clearly, the chains of heteroclinics are not solutions of (1.3). Our next resultestablishes that if we perturb the given Lagrangian in a suitable way then the newsystem admits a class of true solutions which behaves formally as the set of chains(finite or infinite) of heteroclinics described above.

Given any Lagrangian L which verifies (Hp 1) - (Hp 4) we consider the per-turbed Lagrangian

Lω(q, v, t) = L(q, v, t) + α(ωt)V (q)

where ω > 0, α ∈ C(R,R) is any not constant and almost periodic function suchthat infR α > 0, V ∈ C1(RN ,R) is such that V (q + ξ) = V (q) for all q ∈ RN ,ξ ∈ ZN , and V (q) > V (0) = 0 for any q ∈ RN \ ZN . Then, regarding the new


system of equations

d

dt

∂Lω∂v

(q, q, t)− ∂Lω∂q

(q, q, t) = 0, (1.4)

we have

Theorem 1.2. There exists ω > 0 such that for all ω ∈ (0, ω) the system (1.4)admits multibump dynamics.More precisely, there exist σ1, . . . , σm ∈ ZN \ {0} with {

∑mi=1 niσi : ni ∈ N ∪

{0}} = ZN such that for all ρ > 0 there exists lρ > 0 for which, given l ≥ lρ, forany k± ∈ N∪ {0,+∞}, {pj}−k−<j<k+ ⊂ Z with pj < pj+1 for all −k− < j < k+

(we agree that p−k− = −∞ and pk+ = +∞) and {zj}−k−<j<k+ ⊂ {σ1, . . . , σm},there exists q ∈ C2(R,RN) solution of (1.4) which, for any k ∈ {−k−+1, . . . , k+},satisfies

|q(t) +∑−kj=1 z−j| ≤ ρ if k ≤ −1,

|q(t)| ≤ ρ if k = 0,

|q(t)−∑k−1j=0 zj| ≤ ρ if k ≥ 1

for any t ∈ [pk−1l+ 13 l, pkl−

13 l] and |q(t)| ≤ ρ for any t ∈ [pk−1l+ 1

2 l− 1, pk−1l+12 l]. Moreover, if k+ < +∞ then q(t) →

∑k+−1j=0 zj, q(t) → 0 as t → +∞ and,

analogously, if k− < +∞ then q(t)→ −∑k−−1j=1 z−j, q(t)→ 0 as t→ −∞.

As we easily see, among these solutions we find heteroclinics connecting 0 withany ξ ∈ ZN , and in particular even homoclinics joining a point with itself afterpassing near an arbitrary but finite number of other equilibria in the phase space.Moreover, Theorem 1.2 states that for any ρ > 0 there exists an uncountableset of neither heteroclinic nor homoclinic motions of (1.4) each of which entersthe ρ-neighborhood of a prefixed infinite sequence of equilibria in the phase spacerespecting, roughly speaking, a certain time-table. This shows that the systemexhibits sensitive dependence on the initial conditions and in fact in the periodiccase it is possible to prove that the system actually has positive topological entropy(see [39]).

As conclusive remarks we point out that even on the perturbative term V (q)nothing is asked about the local behavior around the origin apart again from thesign condition. As we said above, this provides a wide class of systems exhibitinga chaotic dynamics relative to a non hyperbolic equilibrium. Moreover, nothing isasked about the L∞ norm of the functions α and V and we see that as close as wewant to any Lagrangian satisfying (Hp 1)–(Hp 4) we find a family of Lagrangianssatisfying the same assumptions and whose associated systems exhibit a chaoticdynamics (see also [2], [3] where the use of slowly oscillating perturbations was firstintroduced). Finally we point out that even if we state the theorem for factorizedperturbations the proof can be easily extended to the non factorized case.


The work is organized as follows. In Section 2 we build up the variationalsetting and we study a series of preliminaries. In Section 3 we prove Theorem1.1. In Section 4 we introduce and study a discreteness condition (assumption(∗)) on the sublevel of the action functional associated to (1.3), proving that it isactually sufficient for having a multibump dynamics. In Section 5 we show thatthe perturbed system verifies the condition (∗) if ω is sufficiently small, provingin this way Theorem 1.2. We point out that it is possible to obtain a multibumpdynamics under discreteness assumptions weaker than our (∗). We choose to treatthis simpler situation since it turns out to be sufficient to prove Theorem 1.2.

We finally recall that a function f(q, t) is almost periodic in t uniformly in q ∈RN if for any ε > 0 there is a number l > 0 such that each interval of length l con-tains at least one ε-period of f , i.e., a number τ satisfying supt∈R supq∈RN |f(q, t)−f(q, t− τ)| ≤ ε (see e.g. [24]). For any ε > 0 we will denote with Pε(L) the set ofcommon ε-periods of the functions A(q, t), b(q, t), c(q, t) and γ(q, t).

2. Variational setting and some preliminary results

We consider the action functional

ϕ(q) =∫

RL(q, q, t) dt

on the Hilbert space

E = {q ∈W 1,2loc (R,RN ) :

∫R|q(t)|2 dt < +∞}

equipped with the norm

‖q‖2 = |q(0)|2 +∫

R|q(t)|2 dt.

The assumptions (Hp 2) and (Hp 3) imply that L(q, v, t) ≥ 0 and L(q, 0, t) = 0for any t ∈ R whenever q ∈ ZN . This implies in particular that the functional ϕis bounded below.

The regularity of the function L(q, v, t) and its convexity with respect to thevariable v are more than sufficient conditions to prove that the functional ϕ isweakly lower semicontinuous on E (see e.g. [25]). We give for completeness aproof.

Lemma 2.1. If qn ⇀ q0 in E (i.e., qn → q0 weakly in E), then ϕ(q0) ≤lim infn→∞ ϕ(qn).

Proof. Let T > 0. Since qn ⇀ q0 in E we have qn → q0 in L∞((−T, T ),RN) andqn ⇀ q0 in L2((−T, T ),RN).


By Lusin Theorem, given any ε > 0 there exists a compact set K ⊂ (−T, T ) suchthat q0 is continuous on K and∫

K

L(q0, q0, t)dt ≥∫ T

−TL(q0, q0, t)dt− ε.

Finally, since v → L(q, v, t) is convex, we haveL(q, v1, t)−L(q, v2, t) ≥ ∂L∂v (q, v2, t)(v1−

v2) whenever t ∈ R, q, v1, v2 ∈ RN . Then, as n→∞, we obtain

ϕ(qn) ≥∫K

L(qn, qn, t)dt

≥∫K

L(qn, q0, t)dt+∫K

∂L

∂v(qn, q0, t)(qn − q0)dt

=∫K

L(qn, q0, t)dt+∫K

∂L

∂v(q0, q0, t)(qn − q0)dt

+∫K

(∂L

∂v(qn, q0, t)−

∂L

∂v(q0, q0, t))(qn − q0)dt

=∫K

L(q0, q0, t)dt+ o(1) ≥∫ T

−TL(q0, q0, t)dt+ o(1)− ε.

Since ε is arbitrary, we conclude

lim infn→∞

ϕ(qn) ≥∫ T

−TL(q0, q0, t)dt, ∀T > 0

and the lemma follows. �

Next lemma shows that ϕ is in some sense coercive on E, i.e., if |qn(0)| isbounded and ‖qn‖ → ∞, then ϕ(qn)→∞.

Lemma 2.2. There exists % > 0 such that for all (q, v, t) ∈ RN ×RN ×R,

L(q, v, t) ≥ %|v|2.

Proof. We have that L(q, v, t) ≥ %|v|2 if and only if

〈(A− 2%I)v, v〉+ 2〈b, q〉+ 2c ≥ 0

(here and below we will omit to explicitly write the dependencies of q and t) whereI denotes the identity matrix. If 4% ≤ m we have A − 2%I ≥ m

2 I (recall that by(Hp 2) A ≥ mI). Then A−2%I is invertible and we can write the above inequalityin the form

|(A− 2%I)12 v + (A− 2%I)−

12 b|2 + 2(c− 1

2〈(A− 2%I)−1b, b〉) ≥ 0. (2.1)


Then, to conclude, it suffices to show that there exists % ≤ m4 such that c− 1

2〈(A−2%I)−1b, b〉 ≥ 0. Since (Hp 4) gives |b|2 ≤ −kγ, we have

c− 12〈(A− 2%I)−1b, b〉 = −γ − 〈%A−1(A− 2%I)−1b, b〉

≥ −γ − ‖%A−1(A− 2%I)−1‖ |b|2

≥ −γ(1− % 2km2 )

and the lemma follows with % = min{m4 ,m2

2k }. �

For any r ∈ (0,√N2 ), setting

µr = infq/∈Br(ZN )

t∈R

(−γ(q, t)),

by (Hp 3) we have µr > 0. Then, denoting ϕU (q) =∫U L(q, q, t) dt for any mea-

surable subset U of R, we have that if q ∈ E and (p, s) ⊂ R are such thatq(t) /∈ Br(ZN ) for all t ∈ (p, s), then

ϕ(p,s)(q) =∫ s

p

L(q, q, t) dt ≥ µr(s− p). (2.2)

In particular, we obtain that a function q ∈ E can be outside of an r-neighborhoodof Zn for an amount of time not greater than ϕ(q)/µr.

Now, observe that we have also

L(q, v, t) ≥ −m12 γ(q, t)

(2c(q, t))12|v| ∀ (q, v, t) ∈ (RN \ ZN )×RN ×R. (2.3)

Indeed, recalling that c = −γ + 12 |A−

12 b|2, we obtain

L(q, v, t) =12|A 1

2 v|2 + 〈b, v〉+ c ≥ (2c)12 |A 1

2 v|+ 〈A− 12 b, A

12 v〉

≥ ((2c)12 − |A− 1

2 b|)|A 12 v| = −2γ

(2c)12 + |A− 1

2 b||A 1

2 v|

and (2.3) plainly follows.Then, given r ∈ (0,

√N2 ), we set

νr = infq/∈Br(ZN )

t∈R

−m 12 γ(q, t)

(2c(q, t))12,


noting that νr > 0. Then, if q ∈ E and (p, s) ⊂ R are such that q(t) /∈ Br(ZN )for all t ∈ (p, s), by (2.3), we obtain

ϕ(p,s)(q) =∫ s

p

L(q, q, t) dt ≥ νr∫ s

p

|q| dt ≥ νr|q(s)− q(p)|. (2.4)

Using these estimates we obtain

Lemma 2.3. If q ∈ E and ϕ(q) < +∞, there exist ξ− and ξ+ ∈ ZN such thatq(t)→ ξ± as t→ ±∞.

Proof. By (2.4) we have that if ϕ(q) < +∞, then q ∈ L∞(R,RN) and the setof accumulation points of q(t) as t → +∞ is compact and nonempty. Therefore,it contains at most a finite number of points of ZN . Moreover, by (2.2), weobtain lim inft→+∞ dist(q(t),ZN ) = 0. Therefore there exists ξ+ ∈ ZN such thatlim inft→+∞ |q(t)− ξ+| = 0. Let ε ∈ (0, 1

4). Then, if lim supt→∞ |q(t) − ξ+| > 2ε,there exist infinitely many disjoint intervals (pi, si), i ∈N, such that |q(pi)−ξ+| =ε, |q(si)−ξ+| = 2ε and if t ∈ (pi, si) then ε < dist(q(t),ZN ) < 2ε for any i ∈ N. Onthe other hand, the inequality (2.4) implies that this is possible only if ϕ(q) = +∞.So, lim supt→+∞ |q(t)− ξ+| ≤ 2ε for any ε > 0, i.e., q(t)→ ξ+ as t→ +∞.In a similar way one argues to show that there exists ξ− ∈ ZN such that q(t)→ ξ−

as t→ −∞. �

In what follows, if the limits exist, we will denote q(±∞) = limt→±∞ q(t).

Lemma 2.4. For any β > 0 there exists R(β) > 0 such that if ϕ(q) ≤ β, then‖q − q(−∞)‖L∞ ≤ R(β).

Proof. Assume by contradiction that there exist a sequence (qn) ⊂ E and a se-quence (tn) ⊂ R such that ϕ(qn) ≤ β and |qn(tn)− qn(−∞)| ≥ n

√N . Then, fixed

r ∈ (0, 14 ), there exist at least n disjoint intervals (pi, si) ⊂ (−∞, tn) such that

qn(t) /∈ Br(ZN ) for t ∈ ∪i(pi, si) and |qn(si) − qn(pi)| ≥ 1 − 2r. Then, by (2.4),ϕ(qn) ≥ nνr (1− 2r) for any n ∈N, a contradiction. �

For any ξ ∈ ZN , we define the class

Γξ = {q ∈ E : q(−∞) = 0, q(+∞) = ξ}.

Note that since ϕ(q) = ϕ(q+ ζ) for ζ ∈ ZN , the behavior of ϕ over Γξ is the sameas the behavior of ϕ over ζ + Γξ for any ζ ∈ ZN .

Remark 2.1. By Lemmas 2.2 and 2.4, we have that for every β > 0 and ξ ∈ ZN ,there exists M = M(β, ξ) > 0 such that if q ∈ Γξ ∩ {ϕ ≤ β} then ‖q‖ ≤ M andtherefore every sequence (qn) ⊂ Γξ ∩ {ϕ ≤ β} is weakly precompact in E.


Setting cξ = infΓξ ϕ(q), as an immediate consequence of (2.4) and Lemma 2.4,we obtain

Lemma 2.5. cξ > 0 for any ξ 6= 0 and cξ → +∞ as |ξ| → +∞.

Next lemma shows that there exists a finite set in ZN whose elements satisfya suitable minimality property.

Lemma 2.6. There exist σ1, . . . σm ∈ ZN \ {0} and c > 0 such that ZN =[σ1, . . . ,σm] = {ξ ∈ ZN : ξ =

∑mj=1 njσj , nj ∈N ∪ {0}} and

if ξ1, ξ2 ∈ ZN \ {0} satisfy ξ1 + ξ2 = σ ∈ {σ1, . . . ,σm},then cξ1 + cξ2 ≥ cσ + c.

(2.5)

Proof. We construct the elements σ1, . . . σm of ZN as in [8] (and, originally, in [12])according to the following rule.By Lemma 2.5, there exists σ1 ∈ ZN \ {0} such that

cσ1 = min{cξ : ξ ∈ ZN \ {0}}.

Suppose that the elements σ1, . . . ,σi−1 have been defined. If [σ1, . . . ,σi−1] 6= ZN ,then by Lemma 2.5, there exists σi ∈ ZN \ [σ1, . . . ,σi−1] such that

cσi = min{cξ : ξ ∈ ZN \ [σ1, . . . ,σi−1]}.

In this way we generate, by Lemma 2.5, a finite set {σ1, . . . σm}, with m ≥ 2N ,such that [σ1, . . . ,σm] = ZN .The elements σ1, . . . ,σm satisfy the minimality property (2.5). Indeed, if ξ1, ξ2 ∈ZN satisfy ξ1 + ξ2 = σi ∈ {σ1, . . . ,σm}, then max{cξ1 , cξ2} ≥ cσi since it is notpossible that ξ1, ξ2 ∈ [σ1, . . . ,σi−1]. Therefore, setting c = cσ1 = min{cξ : ξ ∈ZN \ {0}}, we obtain cξ1 + cξ2 ≥ cσi + c. �

In the sequel we will denote c = max{cσi : i = 1, . . . ,m}.In the following, to estimate the value of the functional ϕ on different functions

q, we will often make use of a suitable cutting procedure. For sake of brevity wedefine the following cut off operators. Given T ∈ R, ξ ∈ ZN and q ∈ E, we let

χ+T,ξ(q)(t) =

q(t) if t < T,

(T + 1− t)q(T ) + (t− T )ξ if T ≤ t ≤ T + 1ξ if t > T + 1,

χ−T,ξ(q)(t) =

ξ if t < T − 1,(T − t)ξ + (t− T + 1)q(T ) if T − 1 ≤ t ≤ Tq(t) if t > T.


To control the error made in the cutting procedure it is useful to introduce, givenδ > 0, the quantity

λδ =12

supq∈Bδ(ZN )

t∈R

|A(q, t)|δ2 + supq∈Bδ(ZN )

t∈R

|b(q, t)|δ + supq∈Bδ(ZN )

t∈R

|c(q, t)|.

Note that λδ → 0 as δ → 0. With a direct computation one can easily show thefollowing result.

Lemma 2.7. Let (p, s) ⊂ R, p < t− < t+ < s, ξ−, ξ+ ∈ ZN , δ > 0. If q ∈ E issuch that |q(t±)− ξ±| ≤ δ then

(i) ϕ(p,s)(χ+t+,ξ+

(q)) ≤ ϕ(p,t+)(q) + λδ, ϕ(p,s)(χ−t−,ξ−(q)) ≤ ϕ(t−,s)(q) + λδ,

(ii) ϕ(p,s)(χ−t−,ξ− ◦ χ

+t+,ξ+

(q)) ≤ ϕ(t−,t+)(q) + 2λδ.

Proof. We have

ϕ(p,s)(χ−t−,ξ−(q)) ≤ ϕ(t−,s)(q) + ϕ(t−−1,t−)(χ

−t−,ξ−(q)).

Then (i) follows since L(q, v, t) ≤ λδ for any t ∈ R, |v| ≤ δ, q ∈ Bδ(ZN ), and|χ−t−,ξ−(q)(t) − ξ−| = |(t− − t)ξ− + (t − t− + 1)q(t−) − ξ−| ≤ δ, | ddt [(t− − t)ξ− +(t− t− + 1)q(t−)]| = |q(t−)− ξ−| ≤ δ for any t ∈ (t− − 1, t−).We argue in the same way to prove the other inequalities. �

In the next lemma we will prove that functions in Γσ ∩ {ϕ ≤ cσ + λδ}, with δsmall enough, satisfy a non dichotomy property which will be used as a key argu-ment in the sequel. This property is essentially a consequence of the minimalityproperty (2.5) satisfied by σ1, . . . , σm.

Lemma 2.8. There exists δ0 ∈ (0, 16) such that for any δ ∈ (0, δ0) there exists

ρδ ∈ (δ, 13 ) for which if (p, s) ⊂ R, σ ∈ {σ1, . . . ,σm}, q ∈ E, ϕ(p,s)(q) ≤ cσ + λδ

and if (t−, t+) ⊂ (p, s) is such that |q(t−)| = |q(t+)− σ| = δ then

|q(t)| ≤ ρδ ∀ t ∈ (p, t−) and |q(t)− σ| ≤ ρδ ∀ t ∈ (t+, s).

Moreover, dist(q(t),ZN \ {0, σ}) > δ for any t ∈ (p, s) and ρδ → 0 as δ → 0.

Proof. Given δ > 0, let rδ = inf{r > 0 : νr ≥√λδ}. Since λδ → 0 as δ → 0,

there exists δ1 ∈ (0, 16) such that {r > 0 : νr ≥

√λδ} 6= ∅ and so rδ is well defined

for any δ ∈ (0, δ1). In fact, rδ is not decreasing in (0, δ1) and rδ → 0 as δ → 0.Set rδ = max{δ, rδ} and ρδ = rδ + 4 λδ

νrδ. Since by definition, νrδ ≥

√λδ, we have

that ρδ → 0 as δ → 0. Let δ0 ∈ (0, δ1) be such that ρδ < 13 and λδ ≤ 1

8c for allδ ∈ (0, δ0).


Let q ∈ E, δ ∈ (0, δ0) and p < t− < t+ < s be such that |q(t−)| = |q(t+)− σ| = δ.Then q = χ−t−,0 ◦ χ

+t+,σ

(q) ∈ Γσ and ϕ(q) ≥ cσ. By Lemma 2.7 we obtain

cσ ≤ ϕ(q) ≤ ϕ(p,s)(q)− ϕ(p,t−)(q)− ϕ(t+,s)(q) + 2λδ,

from which we derive that if ϕ(p,s)(q) ≤ cσ + λδ then

max{ϕ(p,t−)(q), ϕ(t+,s)(q)} ≤ ϕ(q)− cσ + 2λδ ≤ 3λδ. (2.6)

Now, assume by contradiction that there exists t ∈ (p, t−) such that |q(t)| > ρδ.Then there exist (p0, p1) ⊂ (t, t−) such that |q(p0)| = ρδ, |q(p1)| = rδ and ρδ ≥|q(t)| ≥ rδ for all t ∈ (p0, p1). Then, by (2.4), we obtain

ϕ(p,t−)(q) ≥ ϕ(p0,p1)(q) ≥ νrδ(ρδ − rδ) = 4λδ,

in contradiction with (2.6). This proves that |q(t)| ≤ ρδ for any t ∈ (p, t−).Analogously we obtain that |q(t)− σ| ≤ ρδ for any t ∈ (t+, s).To prove the lemma it remains to show that dist(q(t),ZN \ {0, σ}) > δ for anyt ∈ (t−, t+). Indeed, if not, we have that there exists t ∈ (t−, t+) and ζ ∈ZN \{0, σ} such that |q(t)−ζ| = δ. We consider the functions q− = χ−t−,0 ◦χ

+t,ζ

(q),q+ = χ−

t,ζ◦χ+

t+,σ(q), observing that q− ∈ Γζ , q+ ∈ ζ+Γσ−ζ . Since ζ+(σ−ζ) = σ,

by (2.5), we obtain ϕ(q−) + ϕ(q+) ≥ cζ + cσ−ζ ≥ cσ + c. On the other hand,by Lemma 2.7, we have ϕ(q−) + ϕ(q+) ≤ ϕ(a,b)(q) + 4λδ which leads to thecontradiction cσ + c ≤ cσ + 4λδ ≤ cσ + 1

2c, by the choice of δ. �

Fixed any δ ∈ (0, δ02 ), let λ = λδ > 0 and ρ = ρδ ∈ (δ, 13 ) be fixed according to

Lemma 2.8. By (2.2) we also fix l > 0 such that, if q ∈ E and q(t) ∈ RN \Bδ(ZN )for all t ∈ I, where I is any real interval with length |I| ≥ l, then ϕI(q) ≥ c+ λ.

As first consequence of Lemma 2.8 we can describe the weak (sequential) closureof the sets Γσ ∩ {ϕ ≤ cσ + λ}.

Lemma 2.9. Let σ ∈ {σ1, . . . ,σm} and (qn) ⊂ Γσ be such that ϕ(qn) ≤ cσ+ λ andqn ⇀ q0. If there exists t0 ∈ R such that dist(q0(t0), {0, σ}) > ρ, then q0 ∈ Γσ.

Proof. By Lemma 2.1 we have that ϕ(q0) ≤ cσ+λ. Then, by Lemma 2.3, q0(±∞) ∈ZN . Since by Lemma 2.8, dist(qn(t),ZN \{0, σ}) > δ and qn → q0 in L∞loc(R,RN ),we have dist(q0(t),ZN\{0, σ}) ≥ δ for all t ∈ R and so q0(±∞) ∈ {0, σ}. If q0 /∈ Γσwe have either q0(−∞) 6= 0 or q0(+∞) 6= σ. Assume q0(−∞) = σ. This impliesthat if tj → −∞, then for any j ∈N there exists nj ∈N such that qn(tj) ∈ Bδ(σ)for any n ≥ nj. Then, by Lemma 2.8, we obtain qn(t) ∈ Bρ(σ) for any t ∈ (tj ,+∞)and n ≥ nj . This is impossible since, by assumption, there exists t0 ∈ R such thatdist(q0(t0), σ) > ρ. In a similar way one argues to prove that q0(+∞) = σ. �


Lemma 2.9 gives a criterium to select the sequences in Γσ which weakly con-verge to functions in Γσ. In this direction we define the function T : {ϕ < +∞} →R as

T (q) = sup{t ∈ R : dist(q(t),ZN ) =12}.

Then, we have

Lemma 2.10. Let σ ∈ {σ1, . . . ,σm} and (qn) ⊂ Γσ be such that ϕ(qn) ≤ cσ + λ,T (qn)→ T0 ∈ R and qn ⇀ q0. Then q0 ∈ Γσ and 0 ≤ T (q0)− T0 ≤ l.

Proof. Since qn → q0 in L∞loc(R,RN ), we have dist(q0(T0),ZN ) = 12 > ρ. Therefore

T (q0)−T0 ≥ 0 and, by Lemma 2.9, q0 ∈ Γσ. If T (q0) = T0, nothing remains to beproved. If T (q0) 6= T0, by Lemma 2.9, we obtain that dist(q0(t),ZN ) > δ for anyt ∈ (T0, T (q0)). Therefore, since ϕ(q0) ≤ cσ + λ, by the choice of l we concludeT (q0)− T0 < l. �

For any δ ∈ (0, δ), σ ∈ ZN and q ∈ Γσ we define

T−δ (q) = sup{t ≤ T (q) : |q(t)| = δ}, T+δ (q) = inf{t ≥ T (q) : |q(t)− σ| = δ}.

and the corresponding truncated function qδ = χ−T−δ

(q),0 ◦ χ+T+δ

(q),σ(q).

Remark 2.2. Using Lemma 2.7, one can easily see that we have(i) qδ ∈ Γσ and T (qδ) = T (q),(ii) T−δ (q) < T (q) < T+

δ (q),(iii) ϕ(qδ(·+ τ)) ≤ ϕ(q(· + τ)) + 2λδ for any τ ∈ R.

We use the described cut off procedure to study the recurrence properties ofthe functional ϕ along the translates of a function q.

Lemma 2.11. For every δ ∈ (0, δ), β > 0 and l > 0 there exists ε(δ, β, l) > 0such that if σ ∈ ZN , q ∈ Γσ ∩ {ϕ ≤ β} and T+

δ (q)− T−δ (q) ≤ l then

|ϕ(qδ(·+ τ))− ϕ(qδ)| ≤ λδ ∀ τ ∈ Pε(L).

Proof. Let σ ∈ ZN and q ∈ Γσ ∩ {ϕ ≤ β} with T+δ (q) − T−δ (q) ≤ l. Setting

T± = T±δ (q), since qδ(t) ∈ ZN for t /∈ (T− − 1, T+ + 1), if ε > 0 and τ ∈ Pε(L),


we obtain

|ϕ(qδ(·+ τ)) − ϕ(qδ)| ≤∫

R|L(qδ, qδ, t− τ) − L(qδ, qδ, t)| dt

=∫ T++1

T−−1|L(qδ, qδ, t− τ) − L(qδ, qδ, t)| dt

≤ 12

supx∈RNt∈R

|A(x, t − τ)−A(x, t)|∫

R|qδ|2 dt

+ supx∈RNt∈R

|b(x, t− τ) − b(x, τ)|∫ T++1

T−−1|qδ| dt

+ supx∈RNt∈R

|c(x, t− τ)− c(x, τ)|∫ T++1

T−−1dt

≤ 12ε‖qδ‖2L2 + ε(l + 2)

12 ‖qδ‖L2 + ε(l + 2).

Then, the lemma follows choosing ε small enough since, by Lemma 2.2 and Remark2.2, we have %‖qδ‖2L2 ≤ ϕ(qδ) ≤ β + 2λδ. �

3. Existence of infinitely many solutions

As we have seen in Lemma 2.10, the function T provides a useful tool for selectingamong the weakly convergent sequences in Γσ the ones which converge to elementsof Γσ. In this section we use the function T to locate the “masses” of the elementsof {ϕ ≤ cσ + λ} ∩ Γσ, σ ∈ {σ1, . . . , σm}, and then to build up a constrainedminimization procedure which allows us to show the existence of infinitely manysolution of (1.3) in any one of the classes Γσ.

Given σ ∈ {σ1, . . . ,σm}, two alternatives are possible:

(i) for any t0 ∈ R there exists q ∈ Γσ with ϕ(q) = cσ and T (q) ∈ [t0, t0 + 2l];(ii) there exists t0 ∈ R such that if q ∈ Γσ and T (q) ∈ [t0, t0 + 2l], then ϕ(q) > cσ.

If (i) holds true, we conclude that the functional ϕ admits infinitely manyminima in Γσ and so, by classical arguments, that the system (1.3) admits infinitelymany solutions in Γσ.

Let us consider the case (ii). Setting I0 = [t0, t0 + l], we obtain

Lemma 3.1. If (ii) holds true, then there exists λ ∈ (0, λ8 ) such that

inf{ϕ(q) : q ∈ Γσ, T (q) ∈ I0} ≥ cσ + 4λ.


Proof. Arguing by contradiction, assume that there exists a sequence (qn) ∈ Γσsuch that T (qn)→ T0 ∈ I0 and ϕ(qn)→ cσ. Then, by Remark 2.1, up to a subse-quence, qn ⇀ q0. Then, by Lemma 2.10, we obtain q0 ∈ Γσ and T (q0) ∈ [t0, t0+2l].This is in contradiction with (ii) since, by Lemma 2.1 ϕ(q0) ≤ limn→∞ ϕ(qn) = cσ.�

Let us fix δ ∈ (0, δ2) such that 16λδ ≤ 16λ2δ ≤ λ. By (2.2), there exists l ≥ l forwhich, if q ∈ E is such that q(t) /∈ Bδ(ZN ) for all t ∈ I, where I is a real intervalwith |I| ≥ l, then ϕ(q) > c + λ. Let also ε = ε(δ, c + λ, l) be fixed according toLemma 2.11.

The use of Lemma 2.11 allows us to show that an analogous of Lemma 3.1 holdsfor infinitely many translated of the interval I0. Precisely, the intervals Iτ = τ+I0with τ ∈ Pε(L).

Lemma 3.2. If (ii) holds true and τ ∈ Pε(L), then

inf{ϕ(q) : q ∈ Γσ, T (q) ∈ Iτ} ≥ cσ + 3λ.

Proof. Assume by contradiction that there exist τ ∈ Pε(L) and q ∈ Γσ withT (q) ∈ Iτ and ϕ(q) < cσ + 3λ. Then, considering the truncated function qδ, byRemark 2.2, we obtain ϕ(qδ) < cσ + 3λ+ 2λδ. Moreover, by Lemma 2.8 and thechoice of l, we obtain T+

δ(q) − T−

δ(q) ≤ l. Then, by Lemma 3.1, ϕ(qδ(· + τ)) <

cσ + 3λ + 3λδ < cσ + 4λ. On the other hand, since T (qδ) = T (q) ∈ Iτ , wehave T (qδ(· + τ)) = T (qδ) − τ ∈ Iτ − τ = I0 and, by Lemma 3.1, we concludeϕ(qδ(·+ τ)) ≥ cσ + 4λ, a contradiction. �

Let q ∈ Γσ be such thatϕ(q) ≤ cσ + λ.

By Lemma 2.11 and Remark 2.2, for all τ ∈ Pε(L) we have ϕ(qδ(· − τ)) ≤ ϕ(qδ) +λδ ≤ ϕ(q) + 3λδ ≤ cσ + 2λ. Then, by Lemma 3.2, we obtain

T (qδ(· − τ)) = T (q) + τ ∈ R \ (∪τ∈Pε(L)Iτ ) ∀ τ ∈ Pε(L).

Denote by Jτ the connected component of T (q) + τ in R \ ∪τ∈Pε(L)Iτ . Then Jτis a real interval which has on the right and on the left two adjacent intervalsI± ∈ {Iτ : τ ∈ Pε(L)}.

Now, we are able to prove the existence of infinitely many solutions of (1.3) ineach Γσ obtained as constrained minima of ϕ.

Proposition 3.1. If (ii) holds true, then for every τ ∈ Pε(L) there exists qτ ∈C2(R,RN ) ∩ Γσ, with T (qτ ) ∈ Jτ and ϕ(qτ ) ≤ cσ + λ, which verifies (1.3).


Proof. Let cσ,τ = inf{ϕ(q) : q ∈ Γσ, T (q) ∈ Jτ}. Then, since T (qδ(· − τ)) ∈ Jτ ,we obtain cσ ≤ cσ,τ ≤ cσ + 2λ.Let qn ∈ Γσ with ϕ(qn) → cσ,τ and T (qn) ∈ Jτ . Then, by Remark 2.1, up toa subsequence, qn ⇀ qτ and since T (qn) ∈ Jτ , we have T (qn) → T0 ∈ Jτ . ByLemma 2.10, we have qτ ∈ Γσ and 0 ≤ T (qτ) − T0 ≤ l. If T (qτ ) /∈ Jτ , sinceT (qτ )−T0 ≤ l, we have that T (qτ ) ∈ I± but this contradicts Lemma 3.2 since, bythe weak lower semicontinuity of ϕ, ϕ(qτ ) ≤ cσ,τ ≤ cσ + 2λ. Then T (qτ ) ∈ Jτ andϕ(qτ ) = cσ,τ .To prove that qτ ∈ C2(R,RN) and it verifies (1.3) by classical arguments, itis sufficient to prove that ϕ(qτ + ψ) ≥ ϕ(qτ ) for any ψ ∈ C∞c (RN ,R) withmaxR |ψ(t)| ≤ δ.If [t0, t1] denotes the connected component of {t ∈ R : dist(qτ (t),ZN ) ≥ δ}containing T (qτ), by definition of l, we have t1 − t0 ≤ l. Moreover, by Lem-ma 2.8, since δ < 1

6 , we have that dist(qτ (t) + ψ(t),ZN ) ≤ ρ + δ < 12 for any

t ∈ (−∞, t0) ∪ (t1,+∞). This implies that T (qτ + ψ) ∈ (t0, t1) and therefore|T (qτ + ψ)− T (qτ )| ≤ l. Since T (qτ ) ∈ Jτ , we have T (qτ + ψ) ∈ I− ∪ Jτ ∪ I+. Inany one of the three cases T (qτ + ψ) ∈ I−, T (qτ + ψ) ∈ Jτ or T (qτ + ψ) ∈ I+, wehave that ϕ(qτ + ψ) ≥ ϕ(qτ ). �

Next lemma shows that the solutions we found in Γσ are actually heteroclinicsolutions of (1.3) joining 0 with σ.

Lemma 3.3. If q ∈ C2(R,RN)∩ {ϕ < +∞} is a solution of (1.3), then q(t)→ 0as t→ ±∞.

Proof. First note that since ϕ(q) < +∞, by Lemma 2.2, we know thatlim inf |t|→+∞ |q(t)| = 0. Then, if |q(t)| does not tend to 0, there exist two sequences(rn) and (sn) with rn < sn < rn+1 and ε > 0, such that |q(rn)| = ε

2 , |q(sn)| = εand ε

2 ≤ |q(t)| ≤ ε for rn < t < sn. So, by Lemma 2.2, we obtain

%ε2

4

+∞∑n=1

(sn − rn) ≤ ϕ(q) < +∞.

But, the equation (1.3) and (Hp 1) in conjunction with the boundedness of |q(t)|on the intervals (rn, sn) imply that |q(t)| ≤ K for some K > 0 and any t ∈ (rn, sn).Then, since

ε

2(sn − rn)=|q(sn)| − |q(rn)|

sn − rn≤ |q(θn)|,

for some θn ∈ (rn, sn), this yields sn − rn ≥ ε2K > 0. This is in contrast with the

convergence of∑+∞n=1(sn − rn). �


4. Existence of infinitely many multibump solutions

In the previous section we showed that system (1.3) admits infinitely many solu-tions in each class Γσ in both the alternative cases (i) or (ii). In this section wewill show that if (ii) holds then the system has a whole class of solutions whosepresence displays chaotic features of the dynamics.

Let us assume that

(*) ∃m0 ∈ {1, . . . ,m}, ∃ξ1, . . . , ξm0 ∈ {σ1, . . . , σm}, ξi 6= ξj if i 6= j, and∃t1, . . . , tm0 ∈ R such that if i ∈ {1, . . . ,m0}, q ∈ Γξi and T (q) ∈ [ti, ti + 2l]then ϕ(q) > cξi .

We set Γi = Γξi and ci = cξi , i = 1, . . . ,m0. The arguments developed in theprevious section apply now for any one of the classes Γi. To fix some constant andnotation we recall below some results.

By Lemma 3.1, if i ∈ {1, . . . ,m0}, then

inf{ϕ(q) : q ∈ Γi, T (q) ∈ Ii,0 = [ti, ti + l]} ≥ ci + 4λ.

Moreover, setting Ii,τ = τ + Ii,0 (τ ∈ R), Lemma 3.2 tells us that, if i ∈{1, . . . ,m0}, then

inf{ϕ(q) : q ∈ Γi, T (q) ∈ Ii,τ} ≥ ci + 3λ ∀ τ ∈ Pε(L).

By definition of ci, for any i ∈ {1, . . . ,m0} there exists qi ∈ Γi with ϕ(qi) ≤ ci+ λ.Setting qi,τ = (qi)δ(· − τ), by Lemma 2.11 and Remark 2.2, we obtain

ϕ(qi,τ ) ≤ ci + 2λ ≤ ci +λ

4∀ τ ∈ Pε(L), i ∈ {1, . . . ,m0}. (4.1)

By (Hp 1), there exists ` > 0 such that the set Pε(L) is `-dense in R, i.e., any realinterval of length ` intersects Pε(L).Let ˜= 12 max{`, l+ 1} and note that, since ˜≥ 6` and Pε(L) is `-dense in R, wehave that for any p ∈ Z, the interval [p˜− 1

12˜, p˜+ 1

12˜] contains m0 points of the

form T (qi) + τi,p with τi,p ∈ Pε(L), i = 1, . . . ,m0.We set qi,p = qi,τi,p and we denote with Ji,p the connected component of T (qi,p) =T (qi) + τi,p in R \ ∪τ∈Pε(L)Ii,τ .We remark that since Pε(L) is `-dense in R, the interval Ji,p cannot have lengthgreater than `. Then we have Ji,p ⊂ [p˜− 1

6˜, p˜+ 1

6˜] for any p ∈ Z and i ∈

{1, . . . ,m0}. If we denote with I−i,p and I+i,p the intervals of length l adjacent to

the interval Ji,p (respectively on the left and on the right), we have that I±i,p ⊂∪τ∈Pε(L)Ii,τ and I±i,p ⊂ [p˜− 1

3˜, p˜+ 1

3˜] for any p ∈ Z and i ∈ {1, . . . ,m0}.

Given p ∈ Z and q ∈ E, we set


Up = [p˜− 12

˜, p˜+ 12

˜],ϕp(q) =

∫UpL(q, q, t) dt,

Tp(q) = sup{t ∈ Up : dist(q(t),ZN ) = 12}.

We plainly recognize that ∪p∈ZUp = R and that dist(I−i,p∪Ji,p∪I+i,p,R\Up) ≥

16

˜≥ l + 1 for any i ∈ {1, . . . ,m0} and p ∈ Z. Then, since Tp(qi,p) ∈ Ji,p, by thechoice of l and the definition of qi,p, we obtain qi,p(t) ∈ ZN for any t /∈ Up. Moreprecisely, qi,p(t) = 0 for any t ≤ p˜− 1

3˜ and qi,p(t) = ξi for any t ≥ p˜+ 1

3˜.

For what concerns the functionals ϕp, we observe that the proof of Lemma 2.1shows in fact that each functional ϕp is weakly lower semicontinuous on E, i.e., ifqn ⇀ q0 as n→∞, then lim infn→∞ ϕp(qn) ≥ ϕp(q0) for any p ∈ Z. Moreover, by(4.1), we have that

ϕp(qi,p) = ϕ(qi,p) ≤ ci +λ

4∀ i ∈ {1, . . . ,m0}, ∀ p ∈ Z.

Remark 4.1. By Lemma 2.8 it is possible to have information on the behaviorof a function q ∈ E on the interval Up by looking at the value ϕp(q). Precisely, letp ∈ Z, σ ∈ {σ1, . . . , σm}, ξ ∈ ZN and q ∈ E. If there exists t− < t+ ∈ Up suchthat |q(t−)− ξ| = |q(t+)− (ξ + σ)| = δ and if ϕp(q) ≤ cσ + λ, then

|q(t)− ξ| ≤ ρ ∀ t ∈ Up, t ≤ t− and |q(t)− (ξ + σ)| ≤ ρ ∀ t ∈ Up, t ≥ t+.

Moreover dist(q(t),ZN \ {ξ, ξ + σ}) > δ, for any t ∈ Up.

Given σ ∈ {σ1, . . . σm} and p ∈ Z, we define

Γp,σ = {q ∈ E : ∃t− < t+ ∈ Up s.t. |q(t−)| = |q(t+)− σ| = δ}.

Lemma 4.1. Let q ∈ ξ + Γp,σ be such that ϕp(q) ≤ cσ + λ and Tp(q) ∈ [p˜−13

˜, p˜+ 13

˜] for some ξ ∈ ZN , σ ∈ {σ1, . . . , σm} and p ∈ Z. Then, there exist

s− ∈ [p˜− 12

˜+ 1, p˜− 13

˜] and s+ ∈ [p˜+13

˜, p˜+12

˜− 1]

such that |q(s−)− ξ|, |q(s+)− (ξ + σ)| ≤ δ.

Proof. Since q ∈ ξ + Γp,σ, there exist t− < t+ ∈ Up such that |q(t−) − ξ| =|q(t+)−(ξ+σ)| = δ. Moreover, since ˜≥ 6(l+1) and ϕp(q) ≤ cσ+ λ, by the choiceof l we obtain that there exist s− ∈ [p˜− 1

2˜+1, p˜− 1

3˜], s+ ∈ [p˜+ 1

3˜, p˜+ 1

2˜−1]

and z−, z+ ∈ ZN such that |q(s−) − z−|, |q(s+) − z+| ≤ δ < δ. By Remark 4.1,we know that z−, z+ ∈ {ξ, ξ + σ}.If t− > s−, by Remark 4.1, we obtain that |q(s−) − ξ| ≤ ρ and therefore that


z− = ξ. Instead, if t− < s−, assume by contradiction that z− = ξ + σ. Then thepath q(Up) is in the δ-neighborhood of ξ at time t−, it is in the δ-neighborhoodof ξ + σ at time s− and it is outside the ρ-neighborhood of ξ + σ at time Tp(q).By Remark 4.1, this is impossible since Tp(q) > t− and so we conclude also in thiscase that z− = ξ.The same argument shows that z+ = ξ + σ. �

Using Remark 4.1 and arguing as in the first section one can easily establishthe following result regarding the functions Tp.

Lemma 4.2. Let p ∈ Z, σ ∈ {σ1, . . . ,σm} and ξ ∈ ZN . Let qn ∈ ξ + Γp,σ besuch that ϕp(qn) ≤ cσ + λ, Tp(qn) → T0 ∈ [p˜− 1

3˜, p˜+ 1

3˜] and qn ⇀ q0. Then

q0 ∈ ξ + Γp,σ and 0 ≤ Tp(q0)− T0 ≤ l.

Now, let us fix k ∈ N, p1 < p2 < . . . < pk ∈ Z and z1, . . . , zk ∈ {ξ1, . . . , ξm0}.For any ι ∈ {1, . . . , k}, let iι ∈ {1, . . . ,m0} be such that zι = ξiι . With theagreement p0 = −∞ and pk+1 = +∞, we set

Γk,p,z = {q ∈ E : q ∈∑

1≤i≤ι−1

zi + Γpι,zι , Tpι(q) ∈ Jiι,pι , ∀ ι = 1, . . . , k}

and ck,p,z = infΓk,p,z ϕ.We remark that if q ∈ Γk,p,z and ϕpι(q) ≤ ciι + λ for any ι ∈ {1, . . . , k}, then

for any ι ∈ {1, . . . , k} the path q(Upι) starts from a ρ-neighborhood of the point∑1≤i≤ι−1 zi and ends up in a ρ-neighborhood of the point

∑1≤i≤ι zi. Therefore,

we look for “k-bumps” solutions of (1.3) as minima of ϕ on Γk,p,z .Before proving that in fact ϕ attains the value ck,p,z on Γk,p,z we need to

establish a technical lemma which will be useful in the following.

Lemma 4.3. Let q ∈ E be such that(i) q ∈ ∩kι=1(

∑1≤i≤ι−1 zi + Γpι,zι),

(ii) Tpι(q) ∈ I−iι,pι ∪ Jiι,pι ∪ I+iι,pι

for any ι ∈ {1, . . . , k},(iii) for any ι ∈ {1, . . . , k} there exist s−ι ∈ [pι ˜− 1

2˜+ 1, pι ˜− 1

3˜] and s+

ι ∈ [pι ˜+13

˜, pι ˜+ 12

˜−1] such that |q(s−ι )−∑

1≤i≤ι−1 zi| ≤ 2δ and |q(s+ι )−

∑1≤i≤ι zi| ≤

2δ.Then, if q /∈ Γk,p,z, there exists q ∈ Γk,p,z such that ϕ(q) ≤ ϕ(q)− λ

4 .

Proof. Let us denoteI = {ι ∈ {1, . . . , k} : Tpι(q) ∈ I−iι,pι ∪ I

+iι,pι},

I+ = {ι ∈ {1, . . . , k} \ I : ι+ 1 ∈ I},I− = {ι ∈ {1, . . . , k} \ I : ι− 1 ∈ I}.Then, if ι ∈ I, consider the function

qι(t) = χ−s−ι ,∑

1≤i≤ι−1zi◦ χ+

s+ι ,∑

1≤i≤ιzi

(q)(t)


and note that qι ∈∑

1≤i≤ι−1 zi + Γzι and T (qι) ∈ I−iι,pι ∪ I+iι,pι

. Then, by Lemmas

2.7 and 3.2, we obtain ϕpι(q) ≥ ϕpι(qι)−2λ2δ ≥ ciι+5λ2 . Since ϕpι(qiι,pι) ≤ ciι+2λ

we conclude that

if ι ∈ I, then ϕpι(q)− ϕpι(qiι,pι) ≥λ

2. (4.2)

For any ι ∈ {1, . . . k} and t ∈ [pι−1 ˜+ 12

˜, pι+1 ˜− 12

˜] we set

q(t) =

∑1≤i≤ι−1 zi + qiι,pι(t) if ι ∈ I,

χ−s−ι ,∑

1≤i≤ι−1zi◦ χ+

s+ι ,∑

1≤i≤ιzi

(q)(t) if ι ∈ I+ ∩ I−,

χ+s+ι ,∑

1≤i≤ιzi

(q)(t) if ι ∈ I+ \ I−,

χ−s−ι ,∑

1≤i≤ι−1zi

(q)(t) if ι ∈ I− \ I+,

q(t) otherwise.

We plainly recognize that Tpι(q) ∈ Jiι,pι for any ι ∈ {1, . . . , k} and then q ∈ Γk,p,z .Moreover we observe that either q(t) = q(t) or q(t) ∈ ZN for any t ∈ [pi−1 ˜+12

˜, pi ˜− 12

˜] and i ∈ {1, . . . , k + 1}. Then

ϕR\∪kι=1Uι

(q) ≤ ϕR\∪kι=1Uι

(q).

Now, note that by (4.2), if ι ∈ I, then ϕpι(q) ≤ ϕpι(q)− λ2 . Moreover, by Lemma

2.7 and (iii) we have also that if ι ∈ I+∩I−, then ϕpι(q) ≤ ϕpι(q)+ 2λ2δ, while ifι ∈ (I+∪I−)\ (I+∩I−), then ϕpι(q) ≤ ϕpι(q)+λ2δ and finally if ι /∈ I ∪I−∪I+,then ϕpι(q) = ϕpι(q). Therefore, we obtain

ϕ(q) ≤ ϕR\∪kι=1Uι

(q) +k∑ι=1

ϕpι(q)

= ϕR\∪kι=1Uι

(q) +∑ι∈I

ϕpι(q) +∑

ι∈I+∪I−

ϕpι(q) +∑

ι/∈I∪I+∪I−

ϕpι(q)

≤ ϕR\∪kι=1Uι

(q) +∑ι∈I

(ϕpι(q)−λ

2)

+∑

ι∈I+∪I−

(ϕpι(q) + 2λ2δ) +∑

ι/∈I∪I+∪I−

ϕpι(q)

≤ ϕ(q)− 12

∑ι∈I

λ+∑

ι∈I+∪I−

2λ2δ.

Since 2card(I) ≥ card(I+ ∪ I−) and 16λ2δ ≤ λ, if I 6= ∅ we conclude ϕ(q) ≤ϕ(q) − λ

4 and the lemma is proved. �


We are now able to prove that the infimum on Γk,p,z is achieved.

Lemma 4.4. There exists qk,p,z ∈ Γk,p,z with ϕ(qk,p,z) = ck,p,z and ϕpι(qk,p,z) ≤ciι + λ for any ι ∈ {1, . . . , k}.

Proof. First of all we claim that for any q ∈ Γk,p,z there exists q ∈ Γk,p,z such thatϕ(q) ≤ ϕ(q) and ϕpι(q) ≤ ciι + λ for any ι ∈ {1, . . . , k}.We construct a function q with the claimed properties by using a cut off proceduresimilar to the one used in the proof of Lemma 4.3. LetI = {ι ∈ {1, . . . , k} : ϕpι(q) > ciι + λ

2},I+ = {ι ∈ {1, . . . , k} \ I : ι+ 1 ∈ I},I− = {ι ∈ {1, . . . , k} \ I : ι− 1 ∈ I}.

If ι ∈ I we observe that ϕpι(qiι,pι) ≤ ciι + λ4 ≤ ϕpι(q)−

λ4 . On the other hand,

if ι /∈ I, we have that ϕpι(q) ≤ ciι + λ2 and since q ∈ Γk,p,z, by Lemma 4.1, there

exist s−ι ∈ [pι ˜− 12

˜+ 1, pι ˜− 13

˜] and s+ι ∈ [pι ˜+ 1

3˜, pι ˜+ 1

2˜− 1] such that

|q(s−ι )−∑

1≤i≤ι−1 zi|, |q(s+ι )−

∑1≤i≤ι zi| ≤ δ.

For any ι ∈ {1, . . . k} and t ∈ [pι−1 ˜+ 12

˜, pι+1 ˜− 12

˜] we set

q(t) =

∑1≤i≤ι−1 zi + qiι,pι(t) if ι ∈ I,

χ−s−ι ,∑

1≤i≤ι−1zi◦ χ+

s+ι ,∑

1≤i≤ιzi

(q)(t) if ι ∈ I+ ∩ I−,

χ+s+ι ,∑

1≤i≤ιzi

(q)(t) if ι ∈ I+ \ I−,

χ−s−ι ,∑

1≤i≤ι−1zi

(q)(t) if ι ∈ I− \ I+,

q(t) otherwise.

Arguing as in the proof of Lemma 4.3 we recognize that ϕpι(q) ≤ ciι + λ for anyι ∈ {1, . . . , k}, q ∈ Γk,p,z and ϕR\∪k

ι=1Uι(q) ≤ ϕR\∪k

ι=1Uι(q). Moreover, if ι ∈ I,

then ϕpι(q) = ϕpι(qiι,pι) ≤ ϕpι(q) − λ4 . By Lemma 2.7, we obtain also that if

ι ∈ I+ ∪ I−, then ϕpι(q) ≤ ϕpι(q) + 2λδ and finally, if ι /∈ I ∪ I− ∪ I+ thenϕpι(q) = ϕpι(q). As in Lemma 4.3 we obtain

ϕ(q) ≤ ϕ(q)− 14

∑ι∈I

λ+∑

ι∈I+∪I−

2λδ.

Since 2card(I) ≥ card(I− ∪ I+) and 4λδ ≤λ16 , we conclude, as we claimed, that

ϕ(q) ≤ ϕ(q).We can now proceed in the proof considering a minimizing sequence (qn) in Γk,p,zwith the additional property ϕpι(qn) ≤ ciι + λ for any ι ∈ {1, . . . , k}.Since the sequence (qn) is bounded in E, up to a subsequence, it weakly convergesto some qk,p,z ∈ E. Since the functionals ϕ and ϕpι are weakly lower semicontin-uous we have ϕ(qk,p,z) ≤ ck,p,z and ϕpι(qk,p,z) ≤ ciι + λ for any ι ∈ {1, . . . , k}.


Since ϕpι(qn) ≤ ciι+λ, Tpι(qn) ∈ Jiι,pι ⊂ [pι ˜−13

˜, pι ˜+13

˜] and qn ∈∑

1≤i≤ι−1 zi+Γpι,zι for any ι ∈ {1, . . . , k}, by Lemma 4.2 we know that

qk,p,z ∈ ∩kι=1(∑

1≤i≤ι−1

zi + Γpι,zι). (4.3)

By Lemma 4.2 we have dist(Tpι(qk,p,z), Jiι,pι) ≤ l and then

Tpι(qk,p,z) ∈ I−iι,pι ∪ Jiι,pι ∪ I+iι,pι

for any ι ∈ {1, . . . , k}. (4.4)

Therefore, by Lemma 4.1, for any ι ∈ {1, . . . , k} there exist s−ι ∈ [pι ˜− 12

˜+1, pι ˜− 1

3˜] and s+

ι ∈ [pι ˜+ 13

˜, pι ˜+ 12

˜− 1] such that

|qk,p,z(s−ι )−∑

1≤i≤ι−1

zi|, |qk,p,z(s+ι )−

∑1≤i≤ι

zi| ≤ δ. (4.5)

By (4.3), (4.4), and (4.5), since ϕ(qk,p,z) ≤ ck,p,z , we can conclude that qk,p,z ∈Γk,p,z . Indeed, if not, by Lemma 4.3 there exists q ∈ Γk,p,z with ϕ(q) < ck,p,z, acontradiction. �

Finally, we have that the minimum on Γk,p,z is a classical “k-bumps” solutionof (1.3).

Lemma 4.5. Let qk,p,z be given by Lemma 4.4. Then qk,p,z ∈ C2(R,RN ) ∩Γ∑k

i=1zi

verifies (1.3) and ‖qk,p,z‖∞ < C and ‖qk,p,z‖∞ < C where C is a

positive constant depending only on the data of the problem. Moreover, for anyι ∈ {1, . . . , k + 1} and t ∈ [pι−1 ˜+ 1

3˜, pι ˜− 1

3˜] we have

|qk,p,z(t)−∑

1≤i≤ι−1

zi| ≤ ρ.

Proof. The fact that qk,p,z ∈ C2(R,RN ) and it verifies (1.3) will follow by classicalarguments whenever we show that ϕ(qk,p,z+ψ) ≥ ϕ(qk,p,z) for any ψ ∈ C∞c (R,RN )with ‖ψ‖∞ ≤ δ.Let q = qk,p,z + ψ with ψ ∈ C∞c (R,RN) and ‖ψ‖∞ ≤ δ. Denoting with [t0,ι, t1,ι]the connected component of Tpι(qk,p,z) in {t ∈ R : dist(qk,p,z(t),ZN ) > δ}, sinceϕpι(qk,p,z) ≤ ciι + λ, by definition of l we deduce that t1,ι− t0,ι ≤ l and then that[t0,ι, t1,ι] ⊂ I−iι,pι ∪ Jiι,pι ∪ I

+iι,pι

.By Remark 4.1, we have dist(qk,p,z(t),ZN ) ≤ ρ for any t ∈ Upι \ [t0,ι, t1,ι]. Sinceby the choice of ρ, ρ+ δ < 1

2 , it is simple to recognize that also

Tpι(q) ∈ [t0,ι, t1,ι] ⊂ I−iι,pι ∪ Jiι,pι ∪ I+iι,pι

for any ι ∈ {1, . . . , k}. (4.6)


We recall now that in the proof of Lemma 4.4 we showed that for any ι ∈ {1, . . . , k}there exists s−ι ∈ [pι ˜− 1

2˜+ 1, pι ˜− 1

3˜] and s+

ι ∈ [pι ˜+ 13

˜, pι ˜+ 12

˜− 1] suchthat |qk,p,z(s−ι ) −

∑1≤i≤ι−1 zi| = |qk,p,z(s+

ι ) −∑

1≤i≤ι zi| = δ. Since ‖ψ‖∞ ≤ δ

and since 2δ ≤ δ, we have

q ∈ ∩kι=1(∑

1≤i≤ι−1

zi + Γpι,zι), (4.7)

|q(s−ι )−∑

1≤i≤ι−1

zi|, |q(s+ι )−

∑1≤i≤ι

zi| ≤ 2δ (4.8)

for any ι ∈ {1, . . . , k}. If q ∈ Γk,p,z then ϕ(q) ≥ ck,p,z = ϕ(qk,p,z). If q /∈ Γk,p,z, by(4.6), (4.7), (4.8) and Lemma 4.3, we have that there exists q ∈ Γk,p,z such thatϕ(q) ≥ ϕ(q) + λ

4 > ck,p,z = ϕ(qk,p,z).This shows that qk,p,z is actually a classical heteroclinic solution of (1.3) joining 0with

∑ki=1 zi. In fact, we observe that by Lemma 2.2, for any j ∈ Z we have

‖qk,p,z‖2L2(j,j+1) ≤1%ϕ(j,j+1)(qk,p,z) ≤

2%

maxι∈{1,... ,k}

ϕpι(qk,p,z) ≤4c%.

Then also ‖qk,p,z‖L1(j,j+1) ≤ (4c% )

12 for any j ∈ Z. Since qk,p,z is a solution of (1.3)

we plainly obtain by (Hp1) that there exists C = C(A, b, c) ≥ 1 such that

‖qk,p,z‖L1(j,j+1) ≤ C(1 + ‖qk,p,z‖L1(j,j+1) + ‖qk,p,z‖2L2(j,j+1))

‖qk,p,z‖W1,1(j,j+1) ≤ 2C(1 +4c%

+4c%

12)

for any j ∈ Z. The Sobolev Immersion Theorem together with the fact thatqk,p,z satisfies (1.3), permits us to conclude that there exists a positive constant Cdepending only on the data for which ‖qk,p,z‖∞ ≤ C and ‖qk,p,z‖∞ ≤ C.

To prove the lemma it remains to show that |qk,p,z(t) −∑

1≤i≤ι−1 zi| ≤ ρ forany ι ∈ {1, . . . , k + 1} and t ∈ [pι−1 ˜ + 1

2˜, pι ˜− 1

2˜]. This is done again by

contradiction using a simple cut off procedure.Let us assume that there exist ι ∈ {1, . . . , k + 1} and t ∈ [pι−1 ˜ + 1

3˜, pι ˜−

13

˜] such that |qk,p,z(t) −∑

1≤i≤ι−1 zi| > ρ. Then, by definition of (t0,ι, t1,ι),qk,p,z([t1,ι−1, pι ˜− 1

2˜]) crosses the annulus Bρ(

∑1≤i≤ι−1 zi) \Bδ(

∑1≤i≤ι−1 zi) at

least one time in [t1,ι−1, t] and as in the proof of Lemma 2.8 we obtain that

ϕ[t1,ι−1,pι ˜− 12

˜](qk,p,z) ≥ 4λ. (4.9)

Then we define

q(t) =

χ+t1,ι−1,

∑1≤i≤ι−1

zi(qk,p,z)(t) if t ≤ t,

χ−t0,ι,∑

1≤i≤ι−1zi

(qk,p,z)(t) if t ≥ t,


observing that q ∈ Γk,p,z and, by Lemma 2.7,

ϕ(−∞,pι−1 ˜+ 12

˜)(q) ≤ ϕ(−∞,t1,ι−1)(qk,p,z) + λδ, (4.10)

ϕ(pι ˜− 12

˜,+∞)(q) ≤ ϕ(pι ˜− 12

˜,+∞)(qk,p,z) + λδ. (4.11)

By (4.9), (4.10) and (4.11), we obtain ϕ(q) ≤ ϕ(qk,p,z) + 2λδ − 4λ < ck,p,z , acontradiction. �

Lemma 4.5 states the existence of infinitely many heteroclinic solutions of (1.3)in each class Γζ whenever ζ belongs to the positive semigroup generated by theelements ξ1, . . . , ξm0 . Moreover, each of these solutions is in the set C2(R,RN ) ∩{q ∈ E : ‖q‖∞ ≤ C, ‖q‖∞ ≤ C} and considering its C1

loc-closure, the Ascoli ArzelaTheorem permits us to establish the existence of a multibump dynamics for (1.3).

Proposition 4.1. If (∗) holds, then for all ρ > 0 there exists `ρ > 0 such thatgiven ` ≥ `ρ, for any k± ∈N∪{0,+∞}, {pj}−k−<j<k+ ⊂ Z with pj < pj+1 for all−k− < j < k+ (we agree that p−k− = −∞ and pk+ = +∞) and {zj}−k−<j<k+ ⊂{ξ1, . . . , ξm0}, there exists q ∈ C2(R,RN ) solution of (1.3) which satisfies

|q(t) +∑−kj=1 z−j| ≤ ρ if k ≤ −1,

|q(t)| ≤ ρ if k = 0,

|q(t)−∑k−1j=0 zj| ≤ ρ if k ≥ 1

(4.12)

for any k ∈ {−k− + 1, . . . , k+} and t ∈ [pk−1` + 13`, pk` −

13 `]. Moreover, if

k+ < +∞ then q(t) →∑k+−1j=0 zj, q(t) → 0 as t → +∞ and, analogously, if

k− < +∞ then q(t)→ −∑k−−1j=1 z−j, q(t)→ 0 as t→ −∞.

Proof. Given any ρ > 0 let us fix δ ∈ (0, δ0) as in Section 2 such that ρδ < ρ.Then, considering ˜ fixed in Section 2 corresponding to this value, we set `ρ = ˜.Let us fix ` ≥ `ρ and let {pj}−k−<j<k+ ⊂ Z and {zj}−k−<j<k+ ⊂ {ξ1, . . . , ξm0}be given.

For all n ∈ N, let k±n = min{n, k± − 1} and kn = k+n + k−n + 1. We set

pn = (p−k−n , . . . , p−1, p0, p1, . . . , pk+n

),

zn = (z−k−n , . . . , z−1, z0, z1, . . . , zk+n

)

and we denote pnj and znj the j-th component of pn and zn, respectively, with theusual convention pn0 = −∞ and pnkn+1 = +∞.By Lemma 4.5 for any n ∈ N there exists qkn,pn,zn ∈ C2(R,RN ) ∩ {q ∈ E :‖q‖∞ ≤ C, ‖q‖∞ ≤ C} solution of (1.3) verifying for any ι ∈ {1, . . . , kn + 1},

|qkn,pn,zn(t)−∑

1≤j≤ι−1

znj | ≤ ρ ∀ t ∈ [pnι−1`+13`, pnι `−

13`]. (4.13)


Then, for any n ∈N we define

qn = qkn,pn,zn −k−n∑j=1

z−j .

Note that for all n ∈ N and j ∈ {1, . . . , kn}, we have znj = zj−k−n−1 and pnj =pj−k−n−1. Then, by (4.13), one can easily see that for any k ∈ {−k−n , . . . , k+

n + 1}and t ∈ [pk−1`+ 1

3`, pk`−13 `]

|qn(t) +∑−kj=1 z−j | ≤ ρ if k ≤ −1,

|qn(t)| ≤ ρ if k = 0,

|qn(t)−∑k−1j=0 zj | ≤ ρ if k ≥ 1.

(4.14)

Since max{‖qn‖∞, ‖qn‖∞} ≤ C, by the Ascoli Arzela Theorem we have thatqn → q in C1

loc(R,RN) and q ∈ C2(R,RN) is a solution of (1.3). To show that qverifies (4.12), it is sufficient to check that fixed any k ∈ {−k−+ 1, . . . , k+} theseproperties are actually true for any function qn whenever n is sufficiently large.Indeed, given any k ∈ {−k−+1, . . . , k+}, let n ≥ |k|. Then k ∈ {−k−n , . . . , k+

n +1}and therefore, passing to the limit in (4.14), we obtain the required properties.

Finally assume that k+ < +∞. Then we know that ϕ(pk+−1,+∞)(qn) ≤ β

for a certain constant β > 0 independent of n. By semicontinuity we have alsoϕ(p

k+−1,+∞)(q) ≤ β and then dist(q(t),ZN ) → 0 as t → +∞. By (4.12) we

conclude that q(t) →∑k+−1j=0 zj and q(t) → 0 as t → +∞ follows. Analogous

arguments apply to prove that if k− < +∞ then q(t)→ −∑−k−+1j=−1 zj and q(t)→ 0

as t→ −∞. �

Remark 4.2. There exists C(`) > 0, C(`)→ 0 as `→∞, such that the solution qgiven by Proposition 4.1 corresponding to ` verifies the following further property

|q(t)| ≤ C(`) ∀ t ∈ [pk−1`+12`− 1, pk−1`+

12`].

We only give a sketch of the proof which can be based on a cutting proceduresimilar to the ones frequently used above.

We first observe that it suffices to prove this for the solutions qk,p,z . These arelocal minima for the functional ϕ. Then, setting Mk = (pk−1`+ 1

2`−1, pk−1`+ 12`),

we have ϕMk(qk,p,z) ≤ 2λδ (otherwise, roughly speaking, we can cut away this part

without perturbing the constraints and obtaining a value of the functional less thanthe minimum). This implies that ‖qk,p,z‖2L2(Mk) ≤

2λδ

% which is small as we want

if δ is sufficiently small (and so ` sufficiently large). The estimate now follows bya bootstrap argument.


5. Proof of the main theorem

In the previous section we have proved that if condition (∗) holds then the system(1.3) admits infinitely many multibump solutions. In this section, we will provethat as close as we want to a given LagrangianL which satisfies (Hp1)−(Hp4) thereexists a whole class of Lagrangians for which the corresponding functionals satisfycondition (∗). As a consequence the corresponding systems admit multibumpdynamics.

Let α ∈ C(R,R) be any nonconstant almost periodic function such that infR α =α > 0 and let V ∈ C1(RN ,R) be such that V (q+ξ) = V (q) for all q ∈ RN , ξ ∈ ZN

with V (q) > V (ξ) = 0 for ξ ∈ ZN and q ∈ RN \ ZN .Let L be a Lagrangian which satisfies (Hp1)−(Hp4) and for all ω > 0, consider

the Lagrangian

Lω(q, v, t) = L(q, v, t) + α(ωt)V (q), (q, v, t) ∈ RN ×RN ×R,

and the corresponding functional

ϕω(q) =∫

RLω(q, q, t) dt = ϕ(q) +

∫Rα(ωt)V (q) dt, q ∈ E.

Remark 5.1. Clearly Lω satisfies (Hp1)− (Hp3) for all ω > 0. Moreover, settingcω(q, t) = c(q, t) + α(ωt)V (q), we have

γω(q, t) =12〈A(q, t)b(q, t), b(q, t)〉 − cω(q, t) = γ(q, t)− α(ωt)V (q) ≤ γ(q, t)

and we deduce that Lω satisfies also the assumption (Hp4).

By the previous remark we obtain in particular that for every ω > 0 thefunctional ϕω satisfies all the properties stated in the previous sections for thefunctional ϕ. Moreover we have ϕω(q) ≥ ϕ(q) for all q ∈ E and ω > 0.

We will show that for ω small enough the functional ϕω satisfies condition(∗) and therefore that the system corresponding to the Lagrangian Lω admitsinfinitely many multibump solutions.

As in Section 2, for ξ ∈ ZN consider the class Γξ and define

cξ(ω) = infq∈Γξ

ϕω(q), ω > 0.

As in Lemma 2.6, for every ω > 0, we can construct a finite set of elements inZN \{0} whose positive semigroup generates ZN and which satisfy the minimalityproperty (2.5). In the following lemma we prove that this property holds in same


sense uniformly with respect to ω > 0. Precisely, considering the constant c > 0given in Lemma 2.6, we have

Lemma 5.1. For every ω > 0 there exist σω1 , . . . , σωmω ∈ ZN \ {0} such that

[σω1 , . . . , σωmω ] = ZN and

if ξ1, ξ2 ∈ ZN \ {0} satisfy ξ1 + ξ2 = σ ∈ {σω1 , . . . ,σωmω},then cξ1(ω) + cξ2(ω) ≥ cσ(ω) + c.

(5.1)

Moreover, there exists C > 0 such that cσ(ω) ≤ C for all ω > 0 and σ ∈{σω1 , . . . , σωmω}.

Proof. As in the proof of Lemma 2.6, for all ω > 0 we can construct a finite set{σω1 , . . . , σωmω} in ZN \ {0} having the following properties:1. cσω1 (ω) = min{cξ(ω) : ξ ∈ ZN \ {0}},2. σωi /∈ [σω1 , . . . ,σ

ωi−1] and cσω

i(ω) = min{cξ(ω) : ξ ∈ ZN \ [σω1 , . . . ,σ

ωi−1]} for all

i = 2, . . . ,mω.This implies that if ξ1, ξ2 ∈ ZN \ {0} satisfy ξ1 + ξ2 = σ ∈ {σω1 , . . . ,σωmω} thencξ1(ω) + cξ2(ω) ≥ cσ(ω) + cσω1 (ω). Hence we obtain (5.1) simply noting thatcσω1 (ω) ≥ cσω1 ≥ c for all ω > 0, since ϕω(q) ≥ ϕ(q) for every q ∈ E.Let us consider the functional ϕ(q) = ϕ(q) + α

∫R V (q)dt, where α = supR α. For

ξ ∈ ZN , we set cξ = infΓξ ϕ and note that, since ϕω(q) ≤ ϕ(q) for all q ∈ E,ω > 0, we have cξ(ω) ≤ cξ. Let σ1, . . . , σm ∈ ZN be the elements constructed asabove and corresponding to the functional ϕ.For any ω > 0, note that since [σ1, . . . , σm] = ZN and [σω1 , . . . , σ

ωmω−1] 6= ZN ,

there exists σ ∈ {σ1, . . . , σm} such that σ 6∈ [σω1 , . . . , σωmω−1]. Then, by definition

of σωmω and σm, we conclude

cσωmω (ω) ≤ cσ(ω) ≤ cσ ≤ cσm

and the lemma follows setting C = cm. �

Remark 5.2. By the previous lemma we can prove that Lemma 2.8 holds truefor the functionals ϕω uniformly with respect to ω > 0. More precisely, for allδ > 0 let

Λδ = α supq∈Bδ(ZN)

V (q)

where α = supR α. Then, as in Lemma 2.8, we can prove that there existsδ0 ∈ (0, δ0) such that for any δ ∈ (0, δ0) there exists ρδ ∈ (δ, 1

3 ) for which if(p, s) ⊂ R, σ ∈ {σω1 , . . . ,σωmω}, q ∈ E, ϕω,(p,s)(q) ≤ cσ(ω)+λδ+Λδ for some ω > 0and if (t−, t+) ⊂ (p, s) is such that |q(t−)| = |q(t+)− σ| = δ then

|q(t)| ≤ ρδ ∀ t ∈ (p, t−) and |q(t)− σ| ≤ ρδ ∀ t ∈ (t+, s).


Moreover, dist(q(t),ZN \ {0, σ}) > δ for any t ∈ (p, s) and ρδ → 0 as δ → 0.

We need also the following estimate.

Lemma 5.2. There exists V0 > 0 such that

infq∈Γσ∩{ϕω≤2C}

∫RV (q)dt ≥ V0 > 0

for all σ ∈ ZN \ {0} and ω > 0.

Proof. By Lemma 2.2 we have that there exists % > 0 such that

Lω(q, v, t) ≥ L(q, v, t) ≥ %|v|2, ∀ (q, v, t) ∈ RN , ω > 0.

In particular this implies ϕω(q) ≥ %‖q‖2L2 for all q ∈ E and ω > 0. Now, let

q ∈ Γσ ∩ {ϕω ≤ 2C} for some ω > 0, σ ∈ ZN \ {0}. Then, there exist (p, s) ⊂ Rsuch that |q(p)| = |q(s) − σ| = 1

4 and q(t) 6∈ B 14(ZN ) for all t ∈ (p, s). By the

previous estimate, we obtain

12≤ |q(p)− q(s)| ≤

∫ s

p

|q(t)|dt ≤ (s− p) 12 ‖q‖L2

≤ (ϕω(q)%

)12 (s− p) 1

2 ≤ (2C%

)12 (s− p) 1

2

which implies s− p ≥ %

8C. Therefore we obtain∫

RV (q)dt ≥

∫ s

p

V (q)dt ≥ %

8Cinf

x 6∈B 14

(ZN)V (x).

Setting V0 = %

8Cinfx 6∈B 1

4(ZN) V (x), the lemma follows. �

Now, note that since α is almost periodic and not constant, we can assumethat α(0) is a local maximum for α and that there exist 0 < t0 < t1 < t2 suchthat

α0 = min|t|≤t0

α(t) > α1 = maxt1≤|t|≤t2

α(t).

Let us fix δ ∈ (0, δ0) such that Λ = λδ

+ Λδ≤ min{C, α0−α1

8 V0}, where C and V0are given in Lemmas 5.1 and 5.2, respectively.Corresponding to this value, let l > 0 be such that ϕ(q) ≥ C + Λ for all q ∈ Ewhich verify q(t) 6∈ B

δ(ZN ) for all t ∈ I with |I| ≥ l.

Remark 5.3. As in Lemma 2.10, by the choice of l, one can easily see that forall ω > 0, if σ ∈ {σω1 , . . . ,σωmω}, qn ∈ Γσ, ϕω(qn) ≤ cσ(ω) + Λ, T (qn) → T0 ∈ Rand qn ⇀ q0, then q0 ∈ Γσ and 0 ≤ T (q0)− T0 ≤ l.


For any σ ∈ {σω1 , . . . , σωmω} and q ∈ Γσ ∩ {ϕω ≤ cσ(ω) + Λ}, for some ω > 0,we set qδ = χ−

T−δ

(q),0 ◦ χ+T+δ

(q),σ(q). Then, by Remark 2.2, we obtain qδ ∈ Γσ,

T (qδ) = T (q) and

ϕω(qδ) ≤ ϕω(q) + 2Λ, ∀ω > 0. (5.2)

Note that by Remark 5.2, T+δ

(q) − T−δ

(q) ≤ l and moreover ϕ(q) ≤ ϕω(q) ≤cσ(ω) + Λ ≤ 2C. Then, according to Lemma 2.11, setting ε = ε(δ, 2C, l), weobtain

|ϕ(qδ(·+ τ)) − ϕ(qδ)| ≤ λδ, ∀ τ ∈ Pε(L). (5.3)

Finally, let ˆ≥ l + 2 be such that Pε(L) is ˆ-dense.Using these fixed values we can prove that condition (∗) is satisfied by the

functional ϕω with respect to any σ ∈ {σω1 , . . . , σωmω}, for all ω small enough.

Proposition 5.1. There exists ω > 0 such that for all ω ∈ (0, ω) and σ ∈{σω1 , . . . , σωmω}

(∗)ω if q ∈ Γσ and T (q) ∈ [0, 2ˆ], then ϕω(q) > cσ(ω) + Λ.

Proof. Let ω = 14ˆ min{t0, t2−t12 }. Arguing by contradiction, suppose that there

exist ω ∈ (0, ω), σ ∈ {σω1 , . . . , σωmω} and q ∈ Γσ with T (q) ∈ [0, 2ˆ] such thatϕω(q) ≤ cσ(ω) + Λ. Let q = q

δand note that q ∈ Γσ, T (q) ∈ [0, 2ˆ] and q(t) ∈ ZN

whenever t 6∈ I = [T−δ

(q) − 1, T+δ

(q) + 1]. Moreover, by the choice of ˆ, we have

|I| ≤ ˆ and since T (q) ∈ (T−δ

(q), T+δ

(q)) we obtain I ⊂ [−ˆ, 3ˆ].

Choose τ ∈ Pε(L) ∩ [− t2ω + 3ˆ,− t1ω − ˆ]. This τ exists since, by the choice of ω,the interval has length greater than ˆ and Pε(L) is ˆ-dense.Then, by (5.2) and (5.3), we obtain

ϕω(q(·+ τ)) = ϕ(q(·+ τ)) +∫

Rα(ω(t− τ))V (q) dt

≤ ϕ(q) + λδ +∫I

α(ω(t− τ))V (q) dt

≤ ϕω(q) + λδ +∫I

[α(ω(t− τ)) − α(ωt)]V (q) dt

≤ ϕω(q) + λδ

+ 2Λ + [ maxt∈ω(I−τ)

α(t) − mint∈ωI

α(t)]∫

RV (q) dt

≤ cσ(ω) + 4Λ + [ maxt∈ω(I−τ)

α(t)− mint∈ωI

α(t)]∫

RV (q) dt.


By the choice of ω and τ , since I ⊂ [−ˆ, 3ˆ], we have ωI ⊂ [−t0, t0] and ω(I− τ) ⊂[t1, t2]. Moreover, by Lemma 5.2, we have

∫R V (q)dt ≥ V0. Therefore, by the

choice of δ, we conclude

ϕω(q(·+ τ)) ≤ cσ(ω) + 4Λ− (α0 − α1)V0 ≤ cσ(ω)− α0 − α12

V0 < cσ(ω)

which is a contradiction since q(·+ τ) ∈ Γσ. �

By the previous results we obtain that for each ω ∈ (0, ω) we can exactly repeatfor the functional ϕω the arguments developed in Section 4 obtaining the existenceof infinitely many multibump solutions for the corresponding system.Moreover, we remark that since no condition on the L∞-norm of the perturba-tion term α(ωt)V (q) is needed, we plainly obtain that as close as we want tothe Lagrangian L there exists a family of Lagrangians (of the form Lω) whosecorresponding systems admit a multibump dynamics. Hence, Theorem 1.2 holds.

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Francesca AlessioDipartimento di Matematica“R. Caccioppoli”Universita di Napoli “Federico II”Via CintiaI–80126 Napoli(e-mail: [email protected])

Maria Letizia BertottiDipartimento di IngegneriaMeccanica e StrutturaleUniversita di TrentoVia Mesiano 77I–38050 Trento(e-mail: [email protected])

Piero MontecchiariDipartimento di MatematicaUniversita di TriestePiazzale Europa 1I–34100 Trieste(e-mail: [email protected])

(Received: April 29, 1998)

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Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems

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