R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI P ADOVA
A NTONIO A MBROSETTI V ITTORIO C OTI Z ELATI
Multiple homoclinic orbits for a class of conservative systems
Rendiconti del Seminario Matematico della Università di Padova, tome 89 (1993), p. 177-194
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http://www.numdam.org/
for
aClass of Conservative Systems.
ANTONIO AMBROSETTI - VITTORIO COTI
ZELATI(*)
1. Introduction.
This paper deals with the existence of
multiple
homoclinic orbits for second order autonomoussystems
where
q E R N
and is smooth and such thatV’(0)=0.
Ahomoclinic orbit is a
solution q
eC2 (R, RN)
of(1)
which satisfies theasymptotic
conditionsThe
typical
situation where homoclinics arise is the case in which theorigin (p, q) = (0, 0)
E is ahyperbolic point
for the hamiltoniannamely
when q = 0 is a strict local maximum for thepotential
V.The existence of homoclinic motions has been
deeply investigated.
For
example,
we refer to[2], [7], [15], [16]
for resultsconcerning
sec-ond order
systems
and[11], [18]
and[19]
for first ordersystems.
When the
potential
Vdepends
on time in aperiodic fashion,
multi-ple
homoclinic orbits arise.Actually,
the existence of many homoclinics is a very classicalproblem
and the firstmultiplicity
results go back to(*) Indirizzo
degli
AA.: A. AMBROSETTI: Scuola NormaleSuperiore,
56100 Pisa; V. COTI ZELATI: Istituto di Matematica, FacoltA di Architettura, 81034Napoli.
Poincaré
[14]
and Melnikov[12]. By
means ofperturbation techniques they proved
that in the one dimensional case(N
=1)
and when V de-pends
in aperiodic
fashionon t,
forcedsystems
likepossess
finitely
many homoclinics. The extension of this kind of results to any finite number ofdegrees
of freedom(namely,
when N ~1)
is aquite
non trivial matter.Recently,
there has been agreat
progress in such aproblem by using
criticalpoint theory. Actually equations (1)
or(2)
are variational in nature and homoclinics can be found as criticalpoints
ofon the Sobolev space E =
HI, 2(R, R N).
Thegroundwork
foremploying
variational tools has been
given
in[8]
andexpecially
in the remarkablepaper
[17], dealing
with first order convex forced hamiltonian sys- temsSubsequently,
theconvexity assumption
has beendropped in [9],
inthe case of second order
systems
like(2).
Both in[17]
and in[9]
the existence ofinfinitely
many homoclinics is established. It is worth re-marking
that such amultiplicity
result isnot
a consequence of any sym-metry
or invarianceproperty
of thefunctional f. Rather,
it is obtainedby using
in astriking
way theperiodic
timedependence
of thehamiltonian.
On the
contrary,
in the autonomous case, nomultiplicity
result isknown,
and in fact one does notexpect,
ingeneral,
to findinfinitely
many
homoclinics,
normultiple
solutions in such a case.The purpose of the
present
paper is to showthat,
for a class of sec-ond order autonomous
systems, multiple
homoclinic orbitsactually
occur.
Roughly,
we deal with hamiltonians likewhere
W(q)
satisfiesOur main result states that
(1)
possesses at least two homoclinics pro- vided b2°‘-2/2 ,
C~. Othermultiplicity
results are also discussed.Our
approach
is stillvariational,
the main toolbeing
a Lusternik-Schnirelman
type
lemma(Lemma
5below)
which allows us to take ad-vantage
of the«pinching»
condition(4).
Such a lemma is in a certainsense a
comparison result,
somewhat related with[4,
Theorem1.2],
and we think it can be of
possible
interest in itself. Itpermits
to esti=mate the number of critical
points
of a functionalf
on a manifoldM,
inrelationship
with those of a related functional g, whenever the sub- levels off
and g are boxed in a suitable manner.The main results of the
present
paper have been outlined in the pre-liminary
note[3].
2. Main results.
Let us consider a
potential
V of the formwhere
q ERN, I q I
denotes the Euclidean norm inRN
and W satis- fies :(W3) 3!Pb!P2
cC2 (RN, R), homogeneus
ofdegree
>2,
such that 0
REMARK. From the
preceding hypotheses
it follows that q = 0 is astrict local maximum for V. Moreover the set
V(q) ~ 01
iscompact
and q = 0belongs
to the interior.We set
Our main result is the
following
oneTHEOREM 1. Let V
of
theform (5)
with Wsatisfying (Wo)-(W3),
and let
Then
(1)
possesses at least two homoclinic orbits.REMARK. A condition like
(7)
recalls a similar one used in a cele- bratedmultiplicity
resultby Ekeland-Lasry [10],
see inparticular
theproof given
in[5]. Actually,
even if theapproach
in thepresent
paper is somewhat similar to that oneof [5],
the discussion in thesequel
willmake it clear that the two results are different in nature.
When W is even, the
preceding
theorem can beimproved.
Fork
N,
letIIk
denote the class of k-dhnensional linearsubspaces
ofR N
and let us set
THEOREM 2. Let V be
of
theform (5)
with Wsatisfying (WO)-(W3)
and suppose, in
addition,
that W is even.If, for
some k ~N,
thereresults
then
( 1 )
possesses at least k hornoctinic orbits. Inparticular, if (7) holds,
then(1)
has at least N homoclinic orbits.As a consequence of the
preceding theorems,
one can deduce multi-plicity results, perturbative
in nature. Forexample,
one hasTHEOREM 3. Let V be
of
theform
with a > 2 and R
satisfying (Wo)
andThen there exists eo > 0 such that
for
all 0 ~ ~ ~ eo one has:(a) (1 )
possesses at least two homoclinicorbits;
(b) if,
inaddition,
R is even, then( 1 )
has at least Npairs of homo-
clinic orbits.
3. The variational
setting.
Let E denote the Sobolev space with scalar
prod-
uct
and norm
Let V be of the form
(5).
With thepreceding notation,
the functional F defined in(3)
becomesIn order to prove the
preceding
theorems it is convenient to use a dif- ferent variationalprinciple, substituting
the search of criticalpoint
ofF on E with a constrained
problem.
This device isclosely
related tothat one
used,
forexample,
in[5].
See also[6]
and[ 1 ], Proposition
1.4.
Let us consider the unit
sphere
,S =ju
e E: =1};
for u E S andÀ E
R + ,
one findsLet
~1o
be such that = 0. Then(13) yields
and from
(14)
it followsUsing ( W2 )
we infer from(15)
thatMoreover
(Wi) (or
else(W3)) readily implies
thatThen
(16)
and(17) imply
that for all u E S there exists aunique ~ (u)
such that
The
preceding
discussion allows us to define a new functionf
S ~ Rby setting
Let
Vfls
denote the contrainedgradient of f on
S.LEMMA 4.
Suppose
that Wsatisfies (W,)-(W2).
ThenPROOF. From the
preceding
discussion it follows thatÀ(u)
is theunique
solution ofThen the
regularity
ofF,
and(16) imply that À - À(u)
isC1
and hence(a)
follows.From
(W1)-(W2)
we infer thatW(q)
=o( 1 q 2)
as q - 0 and thus there is ~ > 0 such thatSince for all v e E one has
c111vll I
then for any v e E such that a1 _alcl (hereafter
ci, c2,etc.,
denoteconstants)
there results~. Hence
(20) yields
This, jointly
with(Wi)
and(18), implies
that for all there resultsTherefore one infers there
exists p
> 0 such thatand
(b)
follows.Since
f(u)
=F(À(u)u)
andÀ(u)
satisfies(19),
then one findsthat
Thus,
if u e S is a criticalpoint of f I s
then there resultsfor some
Lagrange multiplier y
e R.Taking
the scalarproduct
with uand
using again (19)
it follows thatTherefore
(22) yields À(u)F’(À(u)u)
= 0 andfinally (21) implies F’(À(u)u)
= 0. This proves(c). m
4. A Lusternik-Schnirelman lemma.
This section is devoted to state in a
general
form a theoretical lem-ma which will furnish a basic tool for
proving
ourmultiplicity
results.We will use the Lusternik-Schnirelman
(L-S,
forshort) theory
of criti-cal
points.
See forexample [1]
for a short review.Let us consider a
C1
Riemannian manifold M and a functionalfe C1(M, R).
For c eR weset f’= lu
e M:f(u) c}.
LetK(f)
denotethe set of critical
points K( f ) _ fu
e M: =01.
We saysatisfies the condition
(C)
whenever for anye-neighbour-
hood U of
K(f)
there exists d > 0 such thatREMARK. In
using
variationaltools,
it isusually employed
the socalled
(PS)
condition introducedby
Palais and Smale[13]
rather thancondition
(C). Verifying (PS)
onf ~
amounts torequiring
that any se- quence unEfc
suchthat f (un )
is bounded(below)
and -0,
pos-sesses a
converging subsequence. Actually,
condition(C)
suffices toprove the main results in L-S critical
point theory.
Another main tool of the L-S
theory
is the L-Scategory.
Let T be atopological
space and let X c T. The L-Scategory
of X withrespect
toT, cat (X, T ),
is the leastinteger k
such that X c1
U
-
Xi, Xi being
1ik
closed subset of
T,
contractible in T.Among
theproperties
of the L-Scategory,
let us recall here for future references thefollowing
mono-tonicity properties:
In the
sequel
we will make use of thefollowing
result in the L-Stheory:
LEMMA 5.
Let feel (M, R)
be bounded below on M andsatisfy (C)
has at least
cat(fC,fl)
criticalpoints
inf~
The main result we will prove in this section is the
following
one.
LEMMA 6.
Suppose
that exist c, m E R and subsets X eYe M such that:2) 3q
EC( Y, X )
such that = xfor
all x EX;
3)
m andflM satisfies (C)
infc.
M
f
Then
II
M has al leastcat (X, X )
c r it icalpoints
inIC.
PROOF. First of
all, assumption 3)
allows us toapply
the L-Stheory
tof
onf c yielding
the existence of(at least)
cat(f c, f ~ )
criticalpoints
forflM
onf ~.
Next, using assumption 1) jointly
with(23)
and(24),
one infersWe claim
that,
if2) holds,
then one hasIndeed,
sinceX c Y, (23) yields
thatcat (X, Y) ~ cat (X, X).
Conver-sely,
let Then X cU
:!5:k Xi,
withXi
closed and con-tractible in Y.
Let hi
EC([O, 1]
xXi, Y)
denote thehomotopy
suchthat,
for all x E
Xi :
Let us set
Hi Then Hi
EC([0, 1]
xXi, X)
and there resultsThis means that
Xi
is contractible in X and therefore cat(X, X) ~
k == cat
(X, Y), proving (26).
As a consequence,(25)
becomesand an
application
of Lemma 5completes
theproof.
5. Condition
(C).
It is well known that the usual
(PS)
condition does not hold for Fon E because F is invariant under the
(non-compact)
action r -+ ut ----
u(. + z).
To overcome thisdifficulty,
thefollowing
result has beenproved in [8] and [9].
LetKo (F)
denote the setF’(u) = 01.
LEMMA 7.
Suppose
= inf F > 0 and e E be a se-quence such that UEKO(F)
Then there exist Tn E R and zo E E such
that, setting
zn(t)
= vn(t
+~n )
there results
(up
to asubsequence)
REMARK.
Actually,
theproof
in[8]
and[9]
isgiven
in the caseof non-autonomous
potentials,
but it can be carried over withoutchanges
in the autonomous case, too.Moreover,
let usexplicitly point
out
that,
in thepresent setting.
Lemma4 (b)-(c) readily implies
that1 = m(= inf f). 0
S
In
analogy
with Lemma7,
a condition similar to(27)
holds true forPrecisely
one hasLEMMA 8.
Let un
E S be a sequence such thatThen there exist Tn E R and uo E E such
that, setting
wn(t)
= un(t
+2n)
there results
(up
to asubsequence)
PROOF. There results:
where (In - 0. From
(30)
it follows thatMultiplying by un
andtaking
into account(19),we get
0 = yn +(Jn un)
and hence
Since
À(un) *
p > 0(see (21))
then one infersThere results
and
hence, using again (21),
we deduce thatThis and
(29)
say that Lemma 7applies
to the sequence vn =À(un) un
with 1 = m, see the Remark after Lemma 7.
Therefore vn
satisfies(27)
as well as un verifies
(28), proving
the Lemma.As an immediate consequence of Lemma 8 we infer LEMMA 9. For all c 2m
satisfies
condition(C)
on6. Proofs.
The
proofs
of Theorems1,
2 and 3 willrely
on anapplication
of Lem-ma 6
to f,
with a suitable choice of X and Y.First,
somepreliminaries
are in order.
For K = a, b let us consider functionals
GK :
E -~ R definedby setting
Moreover,
the samearguments developed
in Section 3permit
to define(smooth)
functionals gK : S ~R,
From
assumption (W3)
it follows thatand hence
This,
inturn, implies
LEMMA 10.
I f
b a.2a-2/2, then gb
Cfor
all c and allF >
0,
small.PROOF. From the left-hand side
inequality
in(31)
weimmediately
infer
Furthermore,
for all U E S there results+ 00
Setting A(u) =
a direct calculationyields
- m
In the same way one finds
Using (7),
from(32)
and(33)
it followsThis and the
right-hand
sideinequality
in(31) yield
completing
theproof
of the lemma.PROOF OF THEOREM 1. The
proof
will be carried our in severalsteps.
(i)
If u E S is apossible
criticalpoint
of gb on,S,
then the argu- ments of Section 3yield
thatu)
= 0 and thusq (t)
=A(u) u(t)
is ahomoclinic orbit of
From the conservation of the
angular momentum,
it follows thatg(t) _ çr(t), where ç E SN-1
and r =r(t)
satisfies the scalarequation
Since
(34) has,
up to timetranslations,
aunique
solution it follows that gb has aunique
critical level on S.(ii) Let g
= inf gb .Clearly
Lemma 9applies
to gb with m = IA ands
therefore the L-S
theory
can be used to inferthat, actually, [A
= min gb .s
Let u* E ,S be such that gb
(u * )
=Then, according
topoint (i) above, IA
is the
only
critical levelof gb
andq * (t) _ ~(u * ) u * (t)
has the formq*(t) _ ~* r(t),
ESN-1
and rsatisfying (34).
(iii)
We willapply
Lemma 6 to M= S and f, by taking
c =2g -
E,with c > 0
small,
andFirst of
all,
let us notice that cat(X, X)
= 2. To seethis,
we recall thatX =
JU
= is a solution of(34)};
moreover, the sol-utions of
(34)
have the formr(t)
= ro(t + T),
for a fixed ro and any r E R.Thus X =
S N-1
x R and the claim follows.Next,
from Lemma 10(with
c =g)
it follows thatassumption 1)
of Lemma 6 is verified.Moreover,
frompoint (ii)
we deduce that X is a de- formation retract of Y.Indeed, gb
has no other criticalpoints
than thosein
X,
and Y can be deformed(actually, retracted)
on Xbecause, by
Lemma
9,
condition(C)
holds true forgb Is
on Inparticular,
sucha retraction
gives
rise to the map n EC(Y, X) satisfying assumption 2)
of Lemma 6.
From
(31)
it follows that =min f >
03BC. Hence2g - e
2m ands
therefore,
as a consequence of Lemma9, fls
verifies condition(C)
onf2m - ~
e. .(iv)
The discussion ofpoint (iii)
allows us toapply
Lemma 6yield- ing cat (X, X )
= 2 criticalpoints
for Moreprecisely, according
tothe L-S
theory, fs
has two critical levels C1 = m ~ c2 and if Cl = c2( = m)
thenfls
has at level minfinitely
many criticalpoints Km ,
such thatcat
( KM, f ~‘ - E)
= 2. It remains to showthat,
in any case,(1)
possesses at least 2geometrically
distinct homoclinic orbits. To seethis,
let ui besuch = ci and =
u(t)
= +r),
Ifthere exists
only
onegeometrically
distinct homoclinicorbit,
we wouldand therefore C1 = c2 = m. But it is immediate to see
that cat
( ~ (u), fC)
= 1 for all u can assume, without loss of gen-erality,
thatf ~
isconnected,
otherwise the same would be true in eachcomponent).
Thiscompletes
theproof.
Before
proving
Theorem2,
let us recall that when W is even, then F andf
are evenfunctionals,
and we can use theZ2-invariant
L-Stheory.
Let
y(A)
denote theZ2-genus
of theZ2-symmetric
closed setA,
withsee, for
example [1,
Section2].
Weanticipate that,
due to thespecific
features of theZ2-genus,
we do not need to use Lemma 6 anymore.
For k K
N,
let 7r* denote the k-dimensional linearsubspace
ofR N
such that for the ak defined in
(8)
there resultsDefine Ek
={xERN: Ixl [
= andSince
x R,
then= k,
see Lemma2.11-(vi) of [ 1 ].
Weclaim
PROOF. For all s E R and all x
E 2k
there resultsTherefore,
for any u =Xk
one hasand hence
The same calculation made in Lemma 10
yields
+ 00
where
A =
On the otherside,
as in Lemma10,
one also has -°°and
hence, using (35), (36)
andassumption (9)
Finally,
since u = solves(34),
there results and(37) implies 2g
for all u EXk .
0We are now in
position
to prove Theorem 2.PROOF OF THEOREM 2. From Lemma 11 and the
monotonicity property
of theZ2-genus (cfr.
Lemma 2.11of [1])
we infery ( f ~" - ~) >
~
y(Xk)
= k.Moreover, according
to Lemma satisfies(C)
because
2g -
c 2m. Thenfls
possesses at least k(pairs of)
criticalpoints,
whichgive rise,
as in theproof
of Theorem1,
to kgeometrically
distinct homoclinic orbits.
REMARK. As seen in the
preceding proof,
we did not need to findany set Y as claimed in Lemma 6. The reason is
because,
unlike thegenus, we do not
have,
ingeneral,
that cat(X, X) ~
cat(r, Y) provided
XcY.
In order to prove Theorem
3,
les usbegin modifying
thepotential
R. Let ro > 0 be such that 0
( 1 /2) r 2 implies
0 r ro andlet x
E C °°(R + , R)
be anonincreasing
function such thatDefine
R (q) by setting
and set
LEMMA 12. Let e > 0
and let q
be a horrcoclinic orbitof
Then q is a homoclinic orbit
of (1)
urith Vgiven by (10).
PROOF. From the conservation of the energy we infer
Hence it follows
Since e > 0 and
R(q) > 0,
then we find that1/2 |q(t)|2 > 1/a|q(t)|a.
Therefore ro and from the definition
of X
it follows that R = R.Hence
q(t)
solves(1),
withV(q) = -1/2IqI2
+PROOF OF THEOREM 3. First of
all,
let us check that for all e > 0 and smallenough, W,
verifiesassumptions (Wi)
and(W2).
Let us takeany
1/2[.
Then there resultsFrom
(W4)
it follows thatR’(q) ~ q
andR(q)
are as SinceR (q) = R(q)
forI q I small,
then there eidsts 6 > 0 such thatOn the other
hand,
sinceR (q)
= 0 for all ro +1,
then there existe1 > 0 such that
whenever This and
(39) imply
thatWe
verifies(Wi) provided 0ee1.
In a
quite
similar way, one shows that thereexists E2
> 0 such thatW~
verifies(W2) provided
0 ~ e ~ ê2.Finally,
let b be any real number such that 1 b2°‘ - 2/2 .
We claimthat,
for epossibly smaller, ( W3 )
holds true with~l (q) _ (1 /a)
and’P2
(q) = (b/a) I q I a.
To see
this,
we argue as before. We first remarkthat, being R(q) =
= as
q - 0,
then Ed’ > 0 such thatMoreover,
sinceR (q)
= 0 for ro +1,
then there exists E3 > 0 such thatTherefore
W~ (q) ~ ~2 (q) _ (b/«) ~ provided
0 ê ~e . Since, plain- ly, W~ (q) ~
VE >0,
the claim follows.The
preceding
discussion allows us toapply
Theorem 1(respect- ively,
Theorem2)
whenever 0 ~ ê ~ Eo =ê2, ê3}
and V has the form(10) (respectively,
V has the form(10)
and iseven), yielding
2(re- spectively, N)
homoclinic orbits forequation (38). According
to Lemma12,
these orbits are in fact solutions of(1),
and thiscompletes
theproof
of Theorem 3.
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