A NNALES DE L ’I. H. P., SECTION C
K AZUNAGA T ANAKA
Homoclinic orbits for a singular second order hamiltonian system
Annales de l’I. H. P., section C, tome 7, n
o5 (1990), p. 427-438
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© Gauthier-Villars, 1990, tous droits réservés.
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
Homoclinic orbits for
asingular second order Hamiltonian system
Kazunaga TANAKA(*)
Department of Mathematics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya 464, Japan
Vol. 7, n° 5, 1990, p. 427-438. Analyse non linéaire
ABSTRACT. - We consider the second order Hamiltonian system:
where and is a
potential
with asingularity, i. e., ( V (q) I ~ oo
as q -~ e. We prove the existence of a homoclinic orbit of(HS)
under suitableassumptions.
Ourmain
assumptions
are the strong force condition of Gordon[8]
and theuniqueness
of aglobal
maximum of V.RESUME. - On considère le
système
hamiltonien du second ordreoù et est un
potentiel
singulier : | V(q)| ~
ooquand q ~
e. On montre alors 1’existence d’une orbitehomocline,
sousFhypothese
dite de « strong farce »(Gordon [8])
et a condition que le maximum de V soit
unique.
Mots clés : Homoclinic orbit, singular potential, critical point, minimax argument.
Classification A.M.S. : 58 E 05, 58 F 05.
(*) Current address: Department of Mathematics, College of General Education, Nagoya University, Chikusa-Ku, Nagoya 464, Japan.
Annales de l’Institut Henri Poincaré - Analyse non linéaire - 0294-1449 Vol. 7/90/OS/427/12/$3.20/© Gauthier-Villars
0. INTRODUCTION
The purpose of this paper is to
study
the existence of homoclinic orbits for asingular
Hamiltonian system:Here
q2, ... , qN), N~3,
and V issingular
onS,
i. e.,( V (q) ~ ~ oo
asq -~ S.
We assume that S is asingle point,
i. e.,S = ~ e ~ ,
e ~ 0.(Slight
modifications of our methodpermit
us to treatmore
general
compact sets S. See Remark3.3).
We also assumeV : RN B ~ e ~ --~ R
has aunique global
maximum 0 eRN,
and we considerthe existence of a homo clinic orbit which
begins
and ends at0,
i. e., asolution of
(HS)
which satisfiesMore
precisely,
ourassumptions
on V are as follows:(VI)
There is aneERN,
andR) ;
(V2) V (q) c 0
for allq E RN B ~ e ~
andV (q) = 0
if andonly
ifq = 0,
and lim
sup V (q) = V 0 ;
(V3)
There is aconstant 03B4~(0,1 2|e|)
such thatV(q)+1 2(V’(q),q)~0
for all q E
Bs (0),
whereBs (o) _ ~ x
ER" ; x ~ b ~ ;
(V4) -
V(q)
00 as q e; i(V5)
There is aneighbourhood
W of e inRN
and a functionsuch that
U(q)oo
as q --~ e and- V (q) % ~ U’ (q) ( 2 for q E W B ~ e ~ .
Now we can state our main result.
THEOREM 0 .1. -
If V satisfies (V 1)-(V5),
then(HS)
possesses at leastone
(nontrivial)
homoclinic orbit whichbegins
and ends at 0.Remark 0.2. - The
assumption (V5)
is the so-called strongforce
condition
(cf.
Gordon[8])
and it will be used toverify
the Palais-Smale compactness condition for the functionalcorresponding
to theapproxi-
mate
problem (HS : T) (see Proposition 1.1).
Forexample, (V5)
is satis-fied when V
(q) _ - I q - e I ~ °‘ (a
~2)
in aneighbourhood
of e. The assump- tion(V3)
is a kind ofconcavity
condition forV (q)
near 0. Inparticular (V3)
holds for small b > 0 whenV"(0)
isnegative
definite.This work is
largely
motivatedby
the work of Rabinowitz[12]
and theworks
[1-7]. [12]
studied via a variational method the existence and themultiplicity
of heteroclinic orbitsjoining global
maxima ofV (q)
for aperiodic
Hamiltonian system. On the other hand[1-7]
studied the existenceof time
periodic
solutions ofprescribed period
for the second ordersingular
Hamiltonian system(HS).
The
proof
of Theorem 0.1 will begiven
in thefollowing
sections. Weconsider the
approximate problem:
Solutions of this
approximate problem
will be obtained as criticalpoints
of the functional
IT (q) (see
Section1).
We show the existence of criticalpoints
ofIT (q)
via a minimaxargument,
which isessentially
due to Bahriand Rabinowitz
[4] (see
alsoLyusternik
andFet [10] cf. Klingenberg [9]).
We also get some
estimates,
which are uniform with respect toT ~ 1,
for minimax values andcorresponding
criticalpoints q (t ; T).
These uniform estimatespermit
us to let T --~ oo ; for a suitable sequence 1 and asubsequence Tk
--~ ao, we see q(t
+Tk)
convergesweakly
to a homo clinic orbit of(HS)
as k -+ 00.1. APPROXIMATE PROBLEM
In this
section,
we solve theapproximate problem (HS : T)
via a minimax argument. LetHo (0, T ; RN)
denote the usual Sobolev space on(0, T)
with values inRN
under thenorm
=I q|2 dt)1/2 . Let
Clearly AT
is an open subset of(0, T ; RN).
ConsiderThen there is an one-to-one
correspondence
between criticalpoints
ofIT (q)
and classical solutions of(HS : T).
To obtain a critical
point
ofIT (q),
we use a minimax argument. To doso,
IT (q)
mustsatisfy
the Palais-Smale compactness condition(P.S.)
onAT.
(P.S.):
if(qm)m~ ~
cAT
is a sequence such thatIT (qm)
is bounded andIi (qm)
-+0,
then(qm)
possesses asubsequence converging
to someq
EAT,
PROPOSITION 1.1. -
If V (q) satisfies (V l), (V2), (V4), (V5),
thenIT (q)
satisfies
(P.S.).
Proof -
Let(qm)
cAT
be a sequence such thatIT (qm)
is bounded andIT (qm)
-+ 0. Thenby (V2)
and the definition ofIT (q), (qm)
is bounded inHQ (0, T ; RN).
Hence we can extract asubsequence
of(qm) -
still we denoteby (qm) - such
that qm converges toq E AT weakly
inH( (0, T ; RN).
OnVol. 7, n° 5-1990.
the other
hand, by
Greco([6],
Lemma2 .1 ),
if thenT
- i. e., Hence Therefore the form
of
IT (q) shows qm ~ q strongly
inHo (0, T ; RN). []
Now we introduce a minimax
procedure
forIT (q).
Letfor all and
For y E
rT
we observe y(x) (t)
= 0 for allSince
DN - 2 X [o, T]/a (DN - 2 X [o, T]) ^_~ SN -1,
we can associate for each03B3~0393T a map 03B3:SN-1 ~ SN-1
definedby
We denote
by deg y
the Brouwerdegree
of a mapy :
-~SN - ~ .
LetIt is clear that
rT ~ Q,~.
We define a minimax valueof IT (q) by
Then we have
PROPOSITION 1. 2. -
c (T) > 0
is a critical valueof IT (q).
Proof -
We will see later that c(T)
> 0 inProposition
1 . 4. Here we assume it and prove thatc (T)
is a critical value ofIT (q).
SinceIT (q)
satisfies
(P.S.),
we have thefollowing
deformation theorem(cf.
Rabinowitz
[ 11 ]).
LEMMA 1. 3. -
Suppose
that c is not a critical valueof IT (q).
Thenfor
all 1 > 0 there are an ~ E
(0, E) and ~
E C([0, 1] ] X AT, AT)
such that1°
2° IT (11 (T, q)) c IT (q) for
all(T, q)
E[0, 1 X AT;
3° ~ (l,
c where we use the notation :Arguing indirectly,
we supposec (T) > 0
is not a critical value.Applying
Lemma 1 . 3 to c = c
(T)
> 0 and s =c/2,
we have a deformationflow 11 (r, q)
with the
properties
1 °-3°.Moreover,
we haveIn
fact,
sincer~ ( 1, 0) = 0 (by 1 °),
we for Onthe other
hand,
we haveby
2°Hence 11 (r, y (x)) (t) ~ e
for all(x, t) E DN - 2
x[o, T]
andi E [o,1 ] .
Thus wehave
Therefore 11
for thatis,
we have( 1.1 ).
Choose
03B3~0393*T
such that max and consider x e DN - 2Then
by 3 °,
we haveThis contradicts with c =
c (T).
Therefore c(T)
> 0 is a critical value ofIT (R’) ~ .
PROPOSITION 1. 4. - There is a constant co > 0 which is
independent of
T > 1 such that
Proof. -
For anygiven
we define1) by
Then we can
easily
see thefollowing
1 °
deg yT
=deg 0,
thatis, yT
EI-’T
for all y EI-’ 1 ;
2°
IT (yT (x))
=I1 (y (x))
for all x E DN- 2 and03B3~0393*1.
Therefore we get
Next we prove the existence of a constant
co > 0
such that c(T) _>_
co for allT >_
1. For anygiven
we haveOtherwise,
we caneasily
see thatdeg y = 0.
Hence there is(xo, to) E DN - 2
x[0, T]
such thatSince y
(xo) (0)
=0,
there is an so E(0, to)
such thatVol. 7, n° 5-1990.
By
the Schwarzinequality,
we havefor q (t)
= y(xo) (t)
where
Thus we have
.
a. e.,
By ( 1. 2)
and( 1. 4)
we obtain the desired result..From
Proposition
1. 2 and1.4,
we deduce thefollowing.
PROPOSITION 1. 5. - For the
problem (HS : T)
has a solutionq
(t ; T)
such thatwhere co, c 1 > 0 are
independent of
T >_ 1 ..2. SOME ESTIMATES FOR SOLUTIONS
q (t ; T) Clearly
from the definition ofIT (q)
andProposition
1. 5, we haveLEMMA 2. 1. - There is a constant C > 0 which is
independent of
T >_ 1such that
v v
In what
follows,
we denoteby C, C’,
..., various constants which areindependent
of T >_ 1.PROPOSITION 2.2. -
Proof. - Suppose
thatq (t ; T)ft Bs (0)
for somet E (o, T).
We can findan interval
(s, t)
c(0, t)
such thatq (s ; T)
EaBg (0)
and q(i ; Bs (0)
forall
t).
ThenOn the other
hand,
where
ms > 0
is a constant defined in(1.3).
Combining
the above twoinequalities,
we getThus we have
Therefore we conclude
Since q (t ; T)
is a classical solution of(HS : T),
we can seeis constant in time t E
[0, T].
Moreover we haveLEMMA 2 . 3. -
ET
0 as T -~ oo . Inparticular,
Proof. - Integrate (2.1)
over(0, T),
we haveby
Lemma 2.1Hence we get
ET
0 as T -~ oo .Since q (0 ; T)
= q(T ; T)
=0,
we haveVol. 7, n° 5-1990.
The
following proposition gives
us an L oo-bound from belowfor q (t ; T).
This is an
only place
that the condition(V3) plays
an role.PROPOSITION 2 . 4. -
I) q ( . ; T) I IL~ ~o, T~ > ~ f~r a~l T >_ 1.
Proof - Using (2 . 1 ),
we getWe observe
ET =
2T) I2 > 0. Otherwise q(0; T) = 0
and then wehave
q (t ; T)
m0by
theuniqueness
of the solution of the initial valueproblem:
But this contradicts with
IT (q ( . ; T)) = c (T) > o.
Thereforeby (V3),
wehave
Suppose
that q (t ; T) I"
takes its maximum atT).
From the aboveinequality
we deduceq (to ; Bs (0).
Thus we haveBy
the aboveProposition 2 . 4,
we can find two numbers 0iT
_iT
Tsuch that q
(iT ; T),
q(iT ; T)
EaB~ (0)
and q(t ; T)
EBs (0) for
allt E
[0,
U[~T, T].
Then we have
LEMMA 2 . 5. - asToo.
Proof -
Let qa(t)
be a solution of thefollowing
initial valueproblem:
t~y tne continuous dependence of
qa (t)
on the initialdata a,
for any l > 0 there is an s > 0 such thatThus
by (2. 2),
for any l > 0 we can find 1 such thati. e., for
T >_ Tl.
Therefore we have03C41T ~
oo as T ~ ~.Similarly
wehave oo as T --~ 00..
3. LIMIT PROCESS AND PROOF OF THEOREM 0.1
In this
section,
we construct a homoclinic orbit of(HS)
as a limit ofq (t ; T)
as T -~ oo and wecomplete
theproof
of Theorem 0.1. An argu- ment similar to thefollowing
is usedby
Rabinowitz[12]
to show theexistence of heteroclinic orbits
joining global
maxima via a variational argument.For each
T >_ 1,
we defineq(t; T) E H 1 (R, RN) by
Then it
clearly
follows from Lemma 2.1 andProposition
2.2 that1°
q (t ; T)
is a solution of(HS)
in( - iT, T - iT) ;
2°
q (0 ; T)
E(0)
for all T >_1 ;
uniformly
bounded inT >_ 1.
By 3°,
we can extract asubsequence Tk
oo suchthat q (t ; Tk)
converges to somey (t)
E C(R, RN)
n L°°(R, RN) with y (t)
E L2(R, RN)
in the follow-ing
sense:Moreover we have
Similarly
as in[6],
Lemma2 .1,
we also havePROPOSITION 3 .1. - y
(t)
a nontrivial solutionof (HS)
on R.Proof. - Noting (3 . 4),
it is suffices to prove for any cp E~o (R, RN)
By
Lemma2. 5,
we can chooseko E N
such that supp ~p c( - iTk, Tk -
for all
k >_ ko. By
the property 1 °of q (t; T),
we have fork >_ ko
By (3 . 1 )
and(3. 2),
we can pass to the limit and we get(3. 5).
Nontriviality of y (t) clearly
follows from thefact y (0)
EaBs (0),
that is aconsequence of the property 2° of
q {t ; T)
and(3.1). N
Vol. 7, n° 5-1990.
As a last
step
of theproof
of Theorem0.1,
we prove PROPOSITION3 . 2. - y (t), y (t)
--~ ~ G!S t - + 00.Proof. -
First we provey(t) ~ 0
as t ~ ~.Arguing indirectly,
weassume y
(t)
+- 0. Then for some sequence tk --~ oo and for some E >0,
we haveOn the other
hand, by (V2)
and(3. 3),
Hence there is a
sequence tk ~
oo such that y(tk)
E(0).
Thus the curvey (t)
must intersect and~B~/2 (0) infinitely
often as t ~ ~. But thiscontradicts
RN)
and(3. 3).
Infact,
suppose(a, b)
c R isan interval such that
Then we have
By
Schwarzinequality,
we haveIf y (t)
intersects and(0) infinitely
often as t --~ oo, we can findinfinitely
manydisjoint
interval(ai’ b~)
with the property(3.6).
Thus wefind
This contradicts
with ~ (t)
e L~(R,
and(3.3).
Thus we obtain~)
-~ 0as
, ~
Since satisfies the
equation (HS),
isbounded on each compact interval
by
3° and(3.4).
Thus~(~;
convergesin
(R, RN)
to y(t).
HenceSince y (t) -~
0 as t - oo, we haveIn a similar way, we 0 as t -~ 2014 oo. N
Proof of
Theorem 0.1.2014Obviously by Propositions
3.1 and3.2,~ (t)
is a homoclinic orbit for
(HS). N
Remark 3.3.2014
Slight
modifications of our argumentpermit
us to treatmore
general
compact sets S. Moreprecisely,
we assume(VO)
S c= RN is a compact subset such that0~S
and 0belongs
to anunbounded component
We also assume the
following
instead of(V1)-(V5).
(VF)
(V2’)
for and if andonly
if andlim
(V3’)
There is a constant8e(
such that(V4’) - V (q) -~ oo as q -~ S ;
(V5’)
There is aneighbourhood
W of S in RN and a functionU ~C1
(W B S, R)
such that U(q)
~ oo as q ~ S and Then we have thefollowing
theorem.THEOREM 3 . 4. -
If V satisfies (VO)
and(V 1’)-(VS’),
then(HS)
possesses at least one(nontrivialJ
homoclinic orbit whichbegins
and ends at 0. .Remark 3 . 5. - After
completing
thiswork,
the author learned from Professor Rabinowitz that Benci and Giannoni[13]
and Coti-Zelati and Ekeland[14]
have alsorecently
obtained results on homoclinic orbits of Hamiltonian systems.ACKNOWLEDGEMENTS
This work was done while the author was
visiting
Center for theMathematical
Sciences, University
of Wisconsin-Madison. He would likeVol. 7, n° 5-1990.
to thank Center for the Mathematical Sciences for kind
hospitality.
Hewould also like to thank Professor Paul H. Rabinowitz for his
helpful
advice.
REFERENCES
[1] A. AMBROSETTI and V. COTI-ZELATI, Critical points with lack of compactness and applications to singular dynamical system, Ann Mat. Pura Appl., Vol. 149, 1987, pp. 237-259.
[2] A. AMBROSETTI and V. COTI-ZELATI, Periodic solutions of singular dynamical systems, Periodic solutions of Hamiltonian systems and related topics, P. H. RABINOWITZ et al.
Eds., V209, NATO ASI Series, Reidel, 1987, pp. 1-10.
[3] A. AMBROSETTI and V. COTI-ZELATI, Noncollision orbits for a class of Keplerian-like potentials, Ann. Inst. Henri Poincaré, Analyse non linéaire, Vol. 5, 1988, pp. 287-295.
[4] A. BAHRI and P. H. RABINOWITZ, A minimax methods for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., Vol. 82, 1989, pp. 412-428.
[5] M. DEGIOVANNI, F. GIANNONI and A. MARINO, Periodic solutions of dynamical systems with Newtonian type potentials, Periodic solutions of Hamiltonian systems and related topics, P. H. RABINOWITZ et al. Eds., V209, NATO ASI Series, Reidel, 1987, pp. 111- 115.
[6] C. GRECO, Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Analysis ; T.M.A., Vol. 12, 1988, pp. 259-269.
[7] C. GRECO, Remarks on periodic solutions for some dynamical systems with singularities,
Periodic solutions of Hamiltonian systems and related topics, P. H. RABINOWITZ et al.
Eds., V209, NATO ASI Series, Reidel, 1987, pp. 169-173.
[8] W. B. GORDON, Conservative dynamical systems involving strong forces, Trans. Am.
Math. Soc., Vol. 204, 1975, pp. 113-135.
[9] W. KLINGENBERG, Lectures on closed geodesics, Grundlehren der Math. Wis.., Vol. 235, Springer-Verlag, Berlin-New York, 1978.
[10] L. A. LYUSTERNIK and A. I. FET, Variational problems on closed manifolds, Dokl.
Akad. Nauk. U.S.S.R. (N.S.), Vol. 81, 1951, pp. 17-18.
[11] P. H. RABINOWITZ, Minimax methods in critical point theory with applications to
differential equations, C.B.M.S. Reg. Conf. Ser. in Math. #65, Am. Math. Soc., Providence, RI, 1986.
[12] P. H. RABINOWITZ, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. Inst. Henri Poincaré, Analyse non linéaire, Vol. 6, 1989, pp. 331-346.
[13] V. BENCI and F. GIANNONI, Homoclinic orbits on compact manifolds, preprint, Università
di Pisa, 1989.
[14] V. COTI-ZELATI and I. EKELAND, A variational approach to homoclinic orbits in Hamil- tonian systems, preprint, S.I.S.S.A., 1988.
( Manuscript received May 16, 1989.)