Addendum to closed orbits of fixed energy for a class of N-body problems

A NNALES DE L ’I. H. P., SECTION C

A. A MBROSETTI

V. C ^OTI Z ^ELATI

Addendum to closed orbits of fixed energy for a class of N-body problems

Annales de l’I. H. P., section C, tome 9, n

^o

3 (1992), p. 337-338

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© Gauthier-Villars, 1992, tous droits réservés.

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Addendum to

Closed orbits of fixed energy for

^a

class of N-body problems

by A. AMBROSETTI

V. COTI ZELATI Ann. Inst. Henri Poincaré,

Vol. 9, ^n° 3, 1992, p. 337-338. Analyse ^{non linéaire}

Scuola Normale Superiore, ^Pisa

and

University ^of Napoli, Faculty ^of Architecture, Napoli

In Lemma of 6 the paper Closed Orbits

of fixed

energy

for

^{a class}

of N-body problems, published

^on this Journal

9, No. 2, 1992,

pp. 187-

200)

^it is claimed that critical

points of f~

^{are found}

by applying

^the

Mountain-Pass Theorem.

Actually,

^{such a Theorem needs to be}

slightly changed,

^{in order to make it}

applicable

in Lemma 6. We indicate below the

required

modification.

Let

~p=~u~~o: ~~~II--’P~~ ~’p-~pEc~~O, ~~~ ~~~:l~p(Q~~I-~~ ^{~~~)=~1 ~~}

and

Hereafter, By

^{Lemma 2}

(i), Suppose

^{c is not a critical}

level for Since

(PS +) ^holds,

then there exist m > 0 and k

E ~

0,

j3,% 2 [

^such

for all

11 :

[0, 03C4u [ x 0 ~

^E denote the steepest descent flow

satisfying

338 A. AMBROSETTI AND V. COTI ZELATI

where X

is,

^as

usual,

^a

pseudo-gradient

^vector field for such that

(i) X (u)

^{= 0}

if f ’£ (u) _

^{c - 2 k}

or h (u) >_

^{c +}

^{2 k,} (ii ) X (u) = f £

^if

and

(s, M)) ~/, (M), (see

the Deform- ation lemma in

[2]).

Since,

^{as a} consequence of

(2 . 3), fE (u)

^{-~ + oo whenever}

u --~ v E aAo - ~ 0 ~,

^then

⁽ⁱⁱⁱ⁾

^above

readily implies that 11 (s, u) E Ao

^when-

ever

u E Ao and 11 (s,

^0.

Let us show that

p/2,

^{for all}

According

^{to the}

preceding remark,

^this follows in the usual _way

u) I > ^p/2,

^{V s iu}

^(indeed,

in such a case

~u = + oo ). Otherwise,

^let

[

^{be such that}

~ ~ ~ ^(S, u) ~ ~

⁼

^p/2,

^{for some u E}

Ep

^Then

proving

^{the claim.}

Note also that the same arguments ^{used to} prove Lemma 2

(i ) yield

that

( f E (u) ~ u) > o, ~ u E Ao, ~ ~ u ~ ~ = P-

^whenever

and

After these

preliminaries,

let k’ min

(k, p/4 m)

^{and let}

p~039303C1

^{be such}

that

(p (03BE))

^{c + k’. Consider the}

path

7?i

(03BE) = ~(03C1 2, p (03BE)). ^Using

03BE

the

properties

^{of the vector field}

X,

^one shows in the standard _way that

Moreover, whereas,

^{as noted}

before,

i

~ ~ p 1 (o) ~ ~ P.

^Let

^Setting

q (~) = p ~ (~o + ~ ( 1- ~o)),

^{it follows that as well as}

a contradiction which shows that ^c is a critical level

03BE

fory,.

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}