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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

o

5 (1990), p. 427-438

<http://www.numdam.org/item?id=AIHPC_1990__7_5_427_0>

© Gauthier-Villars, 1990, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

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Homoclinic orbits for

a

singular second order Hamiltonian system

Kazunaga TANAKA(*)

Department of Mathematics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya 464, Japan

Vol. 7, 5, 1990, p. 427-438. Analyse non linéaire

ABSTRACT. - We consider the second order Hamiltonian system:

where and is a

potential

with a

singularity, i. e., ( V (q) I ~ oo

as q -~ e. We prove the existence of a homoclinic orbit of

(HS)

under suitable

assumptions.

Our

main

assumptions

are the strong force condition of Gordon

[8]

and the

uniqueness

of a

global

maximum of V.

RESUME. - On considère le

système

hamiltonien du second ordre

et est un

potentiel

singulier : | V(q)| ~

oo

quand q ~

e. On montre alors 1’existence d’une orbite

homocline,

sous

Fhypothese

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

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0. INTRODUCTION

The purpose of this paper is to

study

the existence of homoclinic orbits for a

singular

Hamiltonian system:

Here

q2, ... , qN), N~3,

and V is

singular

on

S,

i. e.,

( V (q) ~ ~ oo

as

q -~ S.

We assume that S is a

single point,

i. e.,

S = ~ e ~ ,

e ~ 0.

(Slight

modifications of our method

permit

us to treat

more

general

compact sets S. See Remark

3.3).

We also assume

V : RN B ~ e ~ --~ R

has a

unique global

maximum 0 e

RN,

and we consider

the existence of a homo clinic orbit which

begins

and ends at

0,

i. e., a

solution of

(HS)

which satisfies

More

precisely,

our

assumptions

on V are as follows:

(VI)

There is an

eERN,

and

R) ;

(V2) V (q) c 0

for all

q E RN B ~ e ~

and

V (q) = 0

if and

only

if

q = 0,

and lim

sup V (q) = V 0 ;

(V3)

There is a

constant 03B4~(0,1 2|e|)

such that

V(q)+1 2(V’(q),q)~0

for all q E

Bs (0),

where

Bs (o) _ ~ x

E

R" ; x ~ b ~ ;

(V4) -

V

(q)

00 as q e; i

(V5)

There is a

neighbourhood

W of e in

RN

and a function

such 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 least

one

(nontrivial)

homoclinic orbit which

begins

and ends at 0.

Remark 0.2. - The

assumption (V5)

is the so-called strong

force

condition

(cf.

Gordon

[8])

and it will be used to

verify

the Palais-Smale compactness condition for the functional

corresponding

to the

approxi-

mate

problem (HS : T) (see Proposition 1.1).

For

example, (V5)

is satis-

fied when V

(q) _ - I q - e I ~ °‘ (a

~

2)

in a

neighbourhood

of e. The assump- tion

(V3)

is a kind of

concavity

condition for

V (q)

near 0. In

particular (V3)

holds for small b > 0 when

V"(0)

is

negative

definite.

This work is

largely

motivated

by

the work of Rabinowitz

[12]

and the

works

[1-7]. [12]

studied via a variational method the existence and the

multiplicity

of heteroclinic orbits

joining global

maxima of

V (q)

for a

periodic

Hamiltonian system. On the other hand

[1-7]

studied the existence

of time

periodic

solutions of

prescribed period

for the second order

singular

Hamiltonian system

(HS).

(4)

The

proof

of Theorem 0.1 will be

given

in the

following

sections. We

consider the

approximate problem:

Solutions of this

approximate problem

will be obtained as critical

points

of the functional

IT (q) (see

Section

1).

We show the existence of critical

points

of

IT (q)

via a minimax

argument,

which is

essentially

due to Bahri

and Rabinowitz

[4] (see

also

Lyusternik

and

Fet [10] cf. Klingenberg [9]).

We also get some

estimates,

which are uniform with respect to

T ~ 1,

for minimax values and

corresponding

critical

points q (t ; T).

These uniform estimates

permit

us to let T --~ oo ; for a suitable sequence 1 and a

subsequence Tk

--~ ao, we see q

(t

+

Tk)

converges

weakly

to a homo clinic orbit of

(HS)

as k -+ 00.

1. APPROXIMATE PROBLEM

In this

section,

we solve the

approximate problem (HS : T)

via a minimax argument. Let

Ho (0, T ; RN)

denote the usual Sobolev space on

(0, T)

with values in

RN

under the

norm

=

I q|2 dt)1/2

. Let

Clearly AT

is an open subset of

(0, T ; RN).

Consider

Then there is an one-to-one

correspondence

between critical

points

of

IT (q)

and classical solutions of

(HS : T).

To obtain a critical

point

of

IT (q),

we use a minimax argument. To do

so,

IT (q)

must

satisfy

the Palais-Smale compactness condition

(P.S.)

on

AT.

(P.S.):

if

(qm)m~ ~

c

AT

is a sequence such that

IT (qm)

is bounded and

Ii (qm)

-+

0,

then

(qm)

possesses a

subsequence converging

to some

q

EAT,

PROPOSITION 1.1. -

If V (q) satisfies (V l), (V2), (V4), (V5),

then

IT (q)

satisfies

(P.S.).

Proof -

Let

(qm)

c

AT

be a sequence such that

IT (qm)

is bounded and

IT (qm)

-+ 0. Then

by (V2)

and the definition of

IT (q), (qm)

is bounded in

HQ (0, T ; RN).

Hence we can extract a

subsequence

of

(qm) -

still we denote

by (qm) - such

that qm converges to

q E AT weakly

in

H( (0, T ; RN).

On

Vol. 7, n° 5-1990.

(5)

the other

hand, by

Greco

([6],

Lemma

2 .1 ),

if then

T

- i. e., Hence Therefore the form

of

IT (q) shows qm ~ q strongly

in

Ho (0, T ; RN). []

Now we introduce a minimax

procedure

for

IT (q).

Let

for all and

For y E

rT

we observe y

(x) (t)

= 0 for all

Since

DN - 2 X [o, T]/a (DN - 2 X [o, T]) ^_~ SN -1,

we can associate for each

03B3~0393T a map 03B3:SN-1 ~ SN-1

defined

by

We denote

by deg y

the Brouwer

degree

of a map

y :

-~

SN - ~ .

Let

It is clear that

rT ~ Q,~.

We define a minimax value

of IT (q) by

Then we have

PROPOSITION 1. 2. -

c (T) > 0

is a critical value

of IT (q).

Proof -

We will see later that c

(T)

> 0 in

Proposition

1 . 4. Here we assume it and prove that

c (T)

is a critical value of

IT (q).

Since

IT (q)

satisfies

(P.S.),

we have the

following

deformation theorem

(cf.

Rabinowitz

[ 11 ]).

LEMMA 1. 3. -

Suppose

that c is not a critical value

of IT (q).

Then

for

all 1 > 0 there are an ~ E

(0, E) and ~

E C

([0, 1] ] X AT, AT)

such that

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 suppose

c (T) > 0

is not a critical value.

Applying

Lemma 1 . 3 to c = c

(T)

> 0 and s =

c/2,

we have a deformation

flow 11 (r, q)

with the

properties

1 °-3°.

Moreover,

we have

In

fact,

since

r~ ( 1, 0) = 0 (by 1 °),

we for On

the other

hand,

we have

by

(6)

Hence 11 (r, y (x)) (t) ~ e

for all

(x, t) E DN - 2

x

[o, T]

and

i E [o,1 ] .

Thus we

have

Therefore 11

for that

is,

we have

( 1.1 ).

Choose

03B3~0393*T

such that max and consider x e DN - 2

Then

by 3 °,

we have

This contradicts with c =

c (T).

Therefore c

(T)

> 0 is a critical value of

IT (R’) ~ .

PROPOSITION 1. 4. - There is a constant co > 0 which is

independent of

T > 1 such that

Proof. -

For any

given

we define

1) by

Then we can

easily

see the

following

1 °

deg yT

=

deg 0,

that

is, yT

E

I-’T

for all y E

I-’ 1 ;

IT (yT (x))

=

I1 (y (x))

for all x E DN- 2 and

03B3~0393*1.

Therefore we get

Next we prove the existence of a constant

co > 0

such that c

(T) _>_

co for all

T >_

1. For any

given

we have

Otherwise,

we can

easily

see that

deg y = 0.

Hence there is

(xo, to) E DN - 2

x

[0, T]

such that

Since y

(xo) (0)

=

0,

there is an so E

(0, to)

such that

Vol. 7, 5-1990.

(7)

By

the Schwarz

inequality,

we have

for 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 and

1.4,

we deduce the

following.

PROPOSITION 1. 5. - For the

problem (HS : T)

has a solution

q

(t ; T)

such that

where co, c 1 > 0 are

independent of

T >_ 1 ..

2. SOME ESTIMATES FOR SOLUTIONS

q (t ; T) Clearly

from the definition of

IT (q)

and

Proposition

1. 5, we have

LEMMA 2. 1. - There is a constant C > 0 which is

independent of

T >_ 1

such that

v v

In what

follows,

we denote

by C, C’,

..., various constants which are

independent

of T >_ 1.

(8)

PROPOSITION 2.2. -

Proof. - Suppose

that

q (t ; T)ft Bs (0)

for some

t E (o, T).

We can find

an interval

(s, t)

c

(0, t)

such that

q (s ; T)

E

aBg (0)

and q

(i ; Bs (0)

for

all

t).

Then

On the other

hand,

where

ms > 0

is a constant defined in

(1.3).

Combining

the above two

inequalities,

we get

Thus we have

Therefore we conclude

Since q (t ; T)

is a classical solution of

(HS : T),

we can see

is constant in time t E

[0, T].

Moreover we have

LEMMA 2 . 3. -

ET

0 as T -~ oo . In

particular,

Proof. - Integrate (2.1)

over

(0, T),

we have

by

Lemma 2.1

Hence we get

ET

0 as T -~ oo .

Since q (0 ; T)

= q

(T ; T)

=

0,

we have

Vol. 7, n° 5-1990.

(9)

The

following proposition gives

us an L oo-bound from below

for 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 get

We observe

ET =

2

T) I2 > 0. Otherwise q(0; T) = 0

and then we

have

q (t ; T)

m0

by

the

uniqueness

of the solution of the initial value

problem:

But this contradicts with

IT (q ( . ; T)) = c (T) > o.

Therefore

by (V3),

we

have

Suppose

that q (t ; T) I"

takes its maximum at

T).

From the above

inequality

we deduce

q (to ; Bs (0).

Thus we have

By

the above

Proposition 2 . 4,

we can find two numbers 0

iT

_

iT

T

such that q

(iT ; T),

q

(iT ; T)

E

aB~ (0)

and q

(t ; T)

E

Bs (0) for

all

t E

[0,

U

[~T, T].

Then we have

LEMMA 2 . 5. - asToo.

Proof -

Let qa

(t)

be a solution of the

following

initial value

problem:

t~y tne continuous dependence of

qa (t)

on the initial

data a,

for any l > 0 there is an s > 0 such that

Thus

by (2. 2),

for any l > 0 we can find 1 such that

i. e., for

T >_ Tl.

Therefore we have

03C41T ~

oo as T ~ ~.

Similarly

we

have oo as T --~ 00..

(10)

3. LIMIT PROCESS AND PROOF OF THEOREM 0.1

In this

section,

we construct a homoclinic orbit of

(HS)

as a limit of

q (t ; T)

as T -~ oo and we

complete

the

proof

of Theorem 0.1. An argu- ment similar to the

following

is used

by

Rabinowitz

[12]

to show the

existence of heteroclinic orbits

joining global

maxima via a variational argument.

For each

T >_ 1,

we define

q(t; T) E H 1 (R, RN) by

Then it

clearly

follows from Lemma 2.1 and

Proposition

2.2 that

q (t ; T)

is a solution of

(HS)

in

( - iT, T - iT) ;

q (0 ; T)

E

(0)

for all T >_

1 ;

uniformly

bounded in

T >_ 1.

By 3°,

we can extract a

subsequence Tk

oo such

that q (t ; Tk)

converges to some

y (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],

Lemma

2 .1,

we also have

PROPOSITION 3 .1. - y

(t)

a nontrivial solution

of (HS)

on R.

Proof. - Noting (3 . 4),

it is suffices to prove for any cp E

~o (R, RN)

By

Lemma

2. 5,

we can choose

ko E N

such that supp ~p c

( - iTk, Tk -

for all

k >_ ko. By

the property 1 °

of q (t; T),

we have for

k >_ 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 the

fact y (0)

E

aBs (0),

that is a

consequence of the property 2° of

q {t ; T)

and

(3.1). N

Vol. 7, n° 5-1990.

(11)

As a last

step

of the

proof

of Theorem

0.1,

we prove PROPOSITION

3 . 2. - y (t), y (t)

--~ ~ G!S t - + 00.

Proof. -

First we prove

y(t) ~ 0

as t ~ ~.

Arguing indirectly,

we

assume y

(t)

+- 0. Then for some sequence tk --~ oo and for some E >

0,

we have

On the other

hand, by (V2)

and

(3. 3),

Hence there is a

sequence tk ~

oo such that y

(tk)

E

(0).

Thus the curve

y (t)

must intersect and

~B~/2 (0) infinitely

often as t ~ ~. But this

contradicts

RN)

and

(3. 3).

In

fact,

suppose

(a, b)

c R is

an interval such that

Then we have

By

Schwarz

inequality,

we have

If y (t)

intersects and

(0) infinitely

often as t --~ oo, we can find

infinitely

many

disjoint

interval

(ai’ b~)

with the property

(3.6).

Thus we

find

This contradicts

with ~ (t)

e L~

(R,

and

(3.3).

Thus we obtain

~)

-~ 0

as

, ~

Since satisfies the

equation (HS),

is

bounded on each compact interval

by

3° and

(3.4).

Thus

~(~;

converges

(12)

in

(R, RN)

to y

(t).

Hence

Since y (t) -~

0 as t - oo, we have

In a similar way, we 0 as t -~ 2014 oo. N

Proof of

Theorem 0.1.2014

Obviously by Propositions

3.1 and

3.2,~ (t)

is a homoclinic orbit for

(HS). N

Remark 3.3.2014

Slight

modifications of our argument

permit

us to treat

more

general

compact sets S. More

precisely,

we assume

(VO)

S c= RN is a compact subset such that

0~S

and 0

belongs

to an

unbounded component

We also assume the

following

instead of

(V1)-(V5).

(VF)

(V2’)

for and if and

only

if and

lim

(V3’)

There is a constant

8e(

such that

(V4’) - V (q) -~ oo as q -~ S ;

(V5’)

There is a

neighbourhood

W of S in RN and a function

U ~C1

(W B S, R)

such that U

(q)

~ oo as q ~ S and Then we have the

following

theorem.

THEOREM 3 . 4. -

If V satisfies (VO)

and

(V 1’)-(VS’),

then

(HS)

possesses at least one

(nontrivialJ

homoclinic orbit which

begins

and ends at 0. .

Remark 3 . 5. - After

completing

this

work,

the author learned from Professor Rabinowitz that Benci and Giannoni

[13]

and Coti-Zelati and Ekeland

[14]

have also

recently

obtained results on homoclinic orbits of Hamiltonian systems.

ACKNOWLEDGEMENTS

This work was done while the author was

visiting

Center for the

Mathematical

Sciences, University

of Wisconsin-Madison. He would like

Vol. 7, n° 5-1990.

(13)

to thank Center for the Mathematical Sciences for kind

hospitality.

He

would 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.)

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