A bisection algorithm for the numerical Mountain Pass

arXiv:math.CA/0410284 v1 12 Oct 2004

A bise tion algorithm for the numeri al Mountain Pass Vivina Barutello and Susanna Terra ini Abstra t. We propose a onstru tive proof for the Ambrosetti-Rabinowitz

Mountain Pass Theorem providing an algorithm, based on a bise tion method, for its implementation. The eÆ ien y of our algorithm, parti ularly suitable for problems in high dimensions, onsists in the low number of ow lines to be omputed for its onvergen e; for this reason it improves the one urrently used and proposed by Y.S. Choi and P.J. M Kenna in [3℄.

Introdu tion This paper deals with onstru tive methods to seek riti al points whi h are not minimizers for fun tionals de ned on Hilbert spa es. The existen e of su h riti al points may be dete ted from the topologi al features of the sublevels, under some

ompa tness onditions. The purpose of this work is to give a onstru tive version of the Ambrosetti-Rabinowitz Mountain Pass Theorem (see [1℄), whi h is one of the most useful abstra t riti al point theorems and that has found relevant appli ations in solving nonlinear boundary value problems. Our onstru tive proof relies on an algorithm based on a bise tion method; this algorithm, elaborated in dierent versions depending on the topology of the sublevels (see Algorithms 2 and 4), an be implemented numeri ally. In spite of its simpli ity, the algorithm proposed is new and in terms of numeri al optimization, it improves the one urrently used and proposed by Y.S. Choi and P.J. M Kenna in [3℄ (see also [4℄). The eÆ ien y of our algorithm onsists in the low number of steepest des ent ow lines to be determined to obtain a good approximation of a riti al point of mountain pass type; this fa t makes the method parti ularly t for problems in high dimensions su h as those

oming from the dis retization of nonlinear boundary value problems in in nite dimensional spa es. Moreover, ompared with the usual Newton's method, our algorithm has two main advantages: it always onverges and it does not need an a priori good initial guess. For some numeri al appli ations of Algorithms 2 and 4 we refer the reader to Chapter 2 of [2℄, where su h algorithms are applied to the a tion fun tional asso iated to the n-body problem with simple or double horeography onstraint 2000 Mathemati s Subje t Classi ation. Primary 46T99, 58E05; Se ondary 65J15. Key words and phrases. Abstra t riti al points theory, Nonlinear fun tional analysis. This work is partially supported by M.I.U.R. proje t \Metodi Variazionali ed Equazioni Dierenziali Nonlineari". 1

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V. BARUTELLO AND S.TERRACINI

respe tively. The theory exposed in this paper, permits the determination of a new solution for the 3-body problem in a rotating frame with angular velo ity ! = 1:5.

1. An iterative algorithm for riti al points 1.1. The steepest des ent ow. We onsider a Hilbert spa e X and a fun tional f : X ! R of lass C 2. Fixed 2 R, we de ne the -sublevel of f as the open set f := fx 2 X : f (x) < g and the set of riti al points of f as Crit(f ) := fx 2 X : rf (x) = 0g: The point x0 2 Crit(f ) is a lo al minimizer for the fun tional f if there exists r > 0, su h that f (x)  f (x0 ), 8x 2 Br (x0 ); x0 is a stri t lo al minimizer if there exists r0 > 0 su h that for every r < r0, inf f (x) > f (x0 ). x2Br (x0 ) Let  : R+  X ! X be the steepest des ent ow asso iated with the fun tional f de ned as the solution of the Cau hy problem 8 rf ((t; x)) < d (t; x) = (1.1) dt 1 + krf ((t; x))k : (0; x) = x We say that a subset X0  X is positively invariant for the ow  if f(t; x0); t  0g  X0 , for every x0 2 X0 . We term !-limit of x 2 X for the ow , the losed positively invariant set 



lim (tn ; x) : (tn )n  R+ tn !+1 We now state two useful preliminary Lemmata on erning some properties of the steepest des ent ow de ned in (1.1); for their simple proofs we refer, if ne essary, to [2℄. Lemma 1.1. Let  be the steepest des ent ow de ned in (1.1); let x 2 X ,

2 (0; 1℄ and T > 0, then jft 2 [0; T ℄ : krf ((t; x))k  gj  f (x) f2 (=2(T; x)) : Lemma 1.2. Let  be the steepest des ent ow de ned in (1.1); let x 2 X , then (i) !x  Crit(f ); moreover if x := limt!+1 f ((t; x)) > 1, then (ii) 9(tn )n ; tn ! +1 su h that lim f ((tn ; x)) = x and n! lim rf ((tn ; x)) = 0: n!+1 +1

!x =

1.2. Dis onne ted sublevels. Let 2 R be su h that the sublevel f is dis onne ted, we term (Fi )i its disjoint onne ted omponents [ f = Fi ; Fi \ Fj = ;; 8i 6= j: i

For every index i, we onsider the basin of attra tion of the set Fi Fi := fx 2 X : !x  Fi g :

A BISECTION ALGORITHM FOR THE NUMERICAL MOUNTAIN PASS

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1.3. Let f be a dis onne ted sublevel for the fun tional f . Let be the disjoint onne ted omponents of f and Fi their basins of attra tion. Then, for every index i, the following assertions hold: (i) Fi is an open set; (ii)  Fi is a positively invariant set; (iii) inf f (x)  ; x2 Fi (iv) minimizers of f in  Fi are riti al points for f ; (v) riti al points for f in the set  Fi are not stri t lo al minimizers. Proof. We prove properties (i)-(v) when i = 1. (i) Let x 2 F1 , then !x  F1 and, sin e  is a gradient ow and F1 is a

onne ted omponent of the sublevel f , there exists T su h that (t; x) 2 F1 , for every t  T . The ontinuity of the ow  ensures that there exists Æ > 0 su h that (T; BÆ (x))  F1 and, hen e, BÆ (x)  F1 . (ii) By the sake of ontradi tion, suppose there exist x 2  F1 and T > 0 su h that xT := (T; x) 2=  F1 ; if xT 2 F1 , then there exists T su h that (T; x) 2 F1 , hen e x 2 F1 , but this ontradi ts (i). If xT 2 X nF1 , then there exists  > 0 su h that B (xT )  X nF1 ; hen e there exists Æ > 0 su h that (T; BÆ (x))  B (xT ) in

ontradi tion with the de nition of the set F1 . (iii) Follows from the de nition of the set F1 . (iv) Follows from (ii) and !x  Crit(f ), for every x 2  Fi . (v) Let x 2  F1 be a riti al point for the fun tional f ; by the sake of ontradi tion, suppose that x is a stri t lo al minimizer, hen e x 2= F1 . Let r > 0 be su h that Br (x) \ F1 = ; and r > 0 su h that f (x)  r + f (x); 8x 2 Br (x): By the ontinuity of f , the set U := fx 2 Br (x) : f (x) < r + f (x)g; is a neighborhood of x, hen e there exists a point xU 2 U \ F1 ; sin e the ow  is dissipative, and f (Br (x)) > f (xU ), we laim that !xU  Br (x), whi h ontradi ts the de nition of F1 .  We term path a ontinuous fun tion : [0; 1℄ ! X . Given a pair of points x1; x2 2 X , x1 6= x2 , we de ne the set of paths joining x1 to x2 as (1.2) x1 ;x2 := f 2 C ([0; 1℄; X ) : (0) = x1 ; (1) = x2 g :

Theorem 1.4. Let f be a dis onne ted sublevel for the fun tional f . Let Fi be the disjoint onne ted omponents of f and Fi their basins of attra tion. Let xi 2 Fi , i = 1; 2, and 2 x1 ;x2 ; then there exists x 2 ([0; 1℄) \  F1 . Proof. The rst step is the des ription of an algorithm that, given a path in the set x1;x2 , sele ts a point x 2 ([0; 1℄) \  F1 . Algorithm 1. s0 + s02 0 , x1 = x1, x02 = x2, x0m = (s0m ) Step 0. s01 = 0, s02 = 1, s0m = 1 2 si1 = sim 1 , si2 = si2 1 Step i. if !xim 1  F1 0 , else si1 = si1 1 , si2 = sim 1 si + si xi1 = (si1 ), xi2 = (si2 ), sim = 1 2 , xim = (sim ) 2

Fi

Proposition

4

and

V. BARUTELLO AND S.TERRACINI

We have then de ned two sequen es (si1 )i ; (si2 )i su h that 0 = s01  s11  : : :  si1 < si2  : : :  s12  s02 = 1 +1 0: jsi1 si2 j = 21i i!!

Sin e (si1 )i ; (si2 )i are bounded monotone sequen es lim si = i!lim si = s 2 [0; 1℄: i!+1 1 +1 2

Let x = (s ), hen e

lim xi1 = lim xi2 = x; i!+1 and, ne essarily x lies on the positively invariant set  F1 . i!+1



1.5. In the same onditions of Theorem 1.4, let x 2 ([0; 1℄) \  F1 , then f (!x )  and there exists a sequen e (xn)n = (tn ; x)   F1 su h that lim rf (xn ) = 0; n!lim f (xn ) = f (!x ): n!+1 +1 Corollary

Sin e  F1 is a positively invariant losed set, !x   F1 and f (!x )  . We use Lemma 1.2, to de ne the sequen e (xn)n .  Proof.

Unfortunately, the proof of Corollary 1.5 is not onstru tive in the sense that it does not provide a method to determine the riti al set !x. The reasons why we an not have an implementable proof of this result are, rst, that we an not determine pre isely, in a nite number of steps, the point x, sin e it is the limit of the sequen e (xn1 )n in Algorithm 1. Se ond, !x is de ned as a limit for t ! +1 and we are not able to determine the value f (!x ) numeri ally. Although the following result is a onsequen e of Corollary 1.5, its relevan e

onsists in its onstru tive proof, that gives a method to determine a riti al point for the fun tional f at a level higher than . Corollary 1.6. In the same onditions of Theorem 1.4, let x  2 ([0; 1℄) \  F1 , n ~ then there exists a sequen e (~ yn )n  X , y~n := (Tn ; x1 ), su h that lim rf (~yn ) = 0; and  f (~yn )  f (x); 8n 2 N: n!+1 Let (xn1 )n be the sequen e de ned in Algorithm 1, (xn1 )n

([0; 1℄) and onverging to x 2  F1 . Sin e (xn1 )n  F1 , we an de ne Tn := inf ft  0 : f ((t; xn1 ))  g and

n := inf krf ((t; xn1 ))k: t2[0;Tn℄ Hen e, using Lemma 1.1 (we an suppose n < 1) we dedu e 2(f (xn1 ) ) Tn = jft 2 [0; Tn℄ : krf ((t; xn1 ))k  n gj 

n2 and we an on lude that s 2(f (xn1 ) )

n  : Tn Proof.

 F1 \

A BISECTION ALGORITHM FOR THE NUMERICAL MOUNTAIN PASS

5

Sin e (xn1 )n ! x as n ! +1 and f (!x )  , we have lim Tn = +1 and n! lim

= 0: n!+1 +1 n

We dedu e the existen e of a sequen e (T~n )n su h that (T~n )n 2 [0; Tn℄ and, de ning y~n := (T~n ; xn1 ), we have lim krf (~yn )k = 0: n!+1



The proof of Corollary 1.6 shows that Algorithm 1 an be improved in the following Algorithm 2 to obtain the sequen e (~yn )n . Algorithm 2. s0 + s02 0 , x1 = x1, x02 = x2, x0m = (s0m ) Step 0. s01 = 0, s02 = 1, s0m = 1 2 Step i. if !xim 1  F1 0 , si1 = sim 1 , si2 = si2 1 i 1 else si1 = s1 , si2 = sim 1 si + si xi1 = (si1 ), xi2 = (si2 ), sim = 1 2 , xim = (sim )  2 Ti := inf t  0 : f ((t; xi1 ))  T~i := t 2 [0; Ti℄ that minimizes krf ((t; xi1))k y~i := (T~i ; xi1): The sequen e (~yn )n de ned in Algorithm 2 will onverge, on e we impose some additional ompa tness onditions on the fun tional f . In this sense we give the following de nitions. Definition 1.7. A sequen e (xm )m  X is termed a Palais-Smale sequen e in the interval [a; b℄ for the fun tional f if +1 a  f (x )  b; 8m 2 N and rf (x ) m!! 0: m

m

The fun tional f satis es the Palais-Smale ondition in the interval [a; b℄ if every Palais-Smale sequen e in the interval [a; b℄ for the fun tional f , (xm )m , has a

onverging subsequen e xmk ! x0 2 X . Similarly, a sequen e (xm )m  X is a Palais-Smale sequen e at level for the fun tional f if +1 +1 f (xm ) m!!

and rf (xm ) m!! 0: The fun tional f satis es the Palais-Smale ondition at level , (PS) , if every Palais-Smale sequen e at level for f has a onverging subsequen e. Remark 1.8. Corollary 1.5 ensures the existen e of a Palais-Smale sequen e at level f (!x ) for the fun tional f . When the fun tional f satis es the (PS)f (!x ) , we on lude that there exists a riti al point x for f su h that f (x) = f (!x ). Corollary 1.6 implies the existen e of a Palais-Smale sequen e for f in the interval [ ; f (x)℄. When the fun tional veri es the Palais-Smale ondition in [ ; f (x)℄, it ensures the onvergen e of the sequen e (~yn )n onstru ted in Algorithm 2. Figure 1 shows the implementation of Algorithm 2. In the rst graph, ea h line represents the values of the a tion fun tional on the steepest des ent ow starting at the rst elements of the sequen e (xi1 )i . Remark that, as the index i in reases, the interval of time in whi h the a tion fun tion is approximately 5:2 in reases, too. This is due to the fa t that the line of the ow departing from xi1 passes ea h step

6

V. BARUTELLO AND S.TERRACINI 6.5 step1 step2 step3 step4 step6 step7

6

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A tion fun tional and norm of its gradient on the des ent ow from the rst elements of the sequen e (xj1 )j .

Figure 1.

loser to the desired riti al point, whose a tion level is approximately 5:2. In the se ond pi ture, we see the value of the norm of the gradient of the a tion fun tional on the same urves. This value de reases till the point y~i , then it in reases when the line (t; xi1) departs from a neighborhood of the mountain pass point, to rea h a lo al minimizer. Remark that both gures represents just the rst 1000 steps in the t-variable. We now point out some autions to be taken in the implementation of Algorithm 2. After a ertain number of steps, depending on the distan e between x1 and x2 , the points xi1 and xi2 may not be any more numeri ally distin t; moreover, taking into a

ount the numeri al errors in the integration method and a possible strong

A BISECTION ALGORITHM FOR THE NUMERICAL MOUNTAIN PASS

7

sharpness of the graph of f , the lines of the steepest des ent ow starting from xi1 and xi2 an be ome undistinguished. Hen e, ea h time we implement Algorithm 2 we a priori x a maximum number of iterations, Nmax , to avoid su h numeri al max ; xNmax )  , where  is a xed obstru tions and to make the distan e dist(xN 1 2 small positive number. Fixed ; > 0 we propose the following algorithm that allows us to approa h a lo ally optimal path joining the starting points x1 and x2 . Algorithm 3. 0 = Nmax ; Step 0 Nmax 0 ; y~ 0 ) Algorithm 2(x1; x2; Nmax Nmax Step k if krf (~ yNmax k 1 )k < , STOP n   o Nk 1 Nk 1 else Tdef := inf t > 0 : dist (t; x1 max ); (t; x2 max )  2k 1 ; k 1

xj := (Tdef ; xNj max ), j = 1; 2; k k 1 Nmax := Nmax 1; k ;y Algorithm 2(x1; x2; Nmax ~Nmax k ) It must be said that the hoi e of the se ond onne ted omponent where x2 lies is for ed by the problem itself. Nevertheless, we an onne t x1 to any given

onne ted omponent by juxtaposition of a nite number of lo ally optimal paths. 1.3. Sublevels with non trivial fundamental group. We re all that a path is a ontinuous fun tion : [0; 1℄ ! X . Given a path , we de ne the path

as

(s) := (1 s); 8s 2 [0; 1℄ and for every pair of paths 1 ; 2 su h that 1 (1) = 2 (0) we de ne the path

= 1 Æ 2 juxtaposition of 1 ; 2 as  (2s) s 2 [0; 21 ℄

(s) = ( 1 Æ 2 )(s) :=

1 (2 1 2 s 1) s 2 [ 2 ; 1℄ A path : [0; 1℄ ! X is a loop if (0) = (1). We say that a loop is ontra tible in X if there exist x 2 X and a ontinuous fun tion h : [0; 1℄  [0; 1℄ ! X su h that h(0; s) = (s), h(1; s) = x, 8s 2 [0; 1℄, and h(; 0) = h(; 1), 8 2 (0; 1). We observe that, for every  2 [0; 1℄, h(
) := h(;
) is a loop in X . Let Y be a subset of X , we say that Y is simply onne ted if every loop in Y is ontra tible. We onsider a fun tional f : X ! R, satisfying the following hypotheses (h1) f is bounded below, suppose f (x)  0; 8x 2 X ; (h2) there exists su h that the sublevel f 6= ; is not simply onne ted; (h3) f veri es the (PS) 0 for every 0  0 < . In the sequel, we will work in a onne ted omponent of the sublevel f ; we will still term it f . Let x 2 f be a stri t lo al minimizer for f and r > 0 su h that the ball Br (x) is

ontra tible in f , i.e. for every x 2 Br (x) the set f(1 )x + x :  2 [0; 1℄g  f . We onsider the sublevel f + , where  = f (x) and  > 0, su h that Fx +  Br (x), where Fx + is the onne ted omponent of f + ontaining x. We observe that Fx + is ontra tible in f . For every x1 2 f su h that !x1 = x we de ne the instant Tx1 := inf ft  0 : (t; x1) 2 Fx + g > 0:

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V. BARUTELLO AND S.TERRACINI

We de ne the following paths (1.3)

x11 () := (Tx1 ; x1); x21 () := (1 )(Tx1 ; x1) + x; Definition 1.9. We term des ending path asso iated to x1 the path x1 := ( x11 Æ x21 ) where x11 ; x21 are de ned in (1.3).  := fx 2 X : !x = xg and X; Remark 1.10. We observe that, de ning X  x := f 2 x;x : x 2 X g, the maps X ! R+; x1 7! Tx1 and X ! X;  x ; x1 7! x1 are ontinuous on the set X .

Definition 1.11. Let be a path in f su h that ! (0) = ! (1) = x ; we de ne the loop  , the des ending loop asso iated to , as (1.4)

 := [(  (0) ) Æ ℄ Æ  (1) :

Definition 1.12. We say that a path in f is  - ontra tible in f if (a) ! (0) = ! (1) = x; (b) the loop  is ontra tible in f . Lemma 1.13. Let x  2 f be a stri t lo al minimizer for the fun tional f and

a path in f . If ! (s) = x, 8s 2 [0; 1℄, then is - ontra tible in f . Proof. To prove that the path  de ned in (1.4) is ontra tible, we de ne a

ontinuous fun tion .  1  h : [0; 1℄  [0; 1℄ ! f su h that h(0;
) =  (
) and h(1;
) = x For every  2 2 ; 1 we de ne the path  (s) :=  (1)(2(1 )s); s 2 [0; 1℄ ; hen e, the following fun tion h satis es our requirements     (2)) Æ ℄ Æ  (1);  2 0; 12  h := [( (  ) Æ  ;  2 12 ; 1 be a C 2 fun tional on a Hilbert spa e



Theorem 1.14. Let f X . Suppose that f satis es onditions (h1), (h2), (h3). Then one of the following situations o

urs: (i) f has a ontinuum of global minimizers; (ii) there exists x^ 2 Crit(f ) \ f that is not a stri t minimizer.

Proof. When the set Crit(f ) \ f is not a ontinuum of global minimizers, hypotheses (h1) and (h3) imply the existen e of at least one stri t minimizer x 2 f for the fun tional f . Let  f be a loop su h that (0) = (1) = x and suppose that is not ontra tible in f . Hen e, sin e  (s) = (s) for every s 2 [0; 1℄, is not - ontra tible in f . Using Lemma 1.13, we an on lude that there exists x 2 ([0; 1℄) su h that !x 6= x. The riti al point !x ould be a stri t minimizer or not. In the rst ase, we use Algorithm 2 to prove the existen e of a riti al point for f that is not a stri t lo al minimizer; otherwise !x is the sear hed riti al point for f . We now show an algorithm that determine a point x 2 ([0; 1℄) su h that !x 6= x. Let  f be a not ontra tible loop su h that (0) = (1) = x. We have already remarked that is - ontra tible.

A BISECTION ALGORITHM FOR THE NUMERICAL MOUNTAIN PASS

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4.
Step 0. 0 := , x0 = 0 21 if !x0 6= x,
STOP  
if 0 0; 21 is - ontra tible, 1 := 0 21 ; 1  1 
else 1 := 0 0; 2 Algorithm




xi = i 21 if !xi 6= x,
STOP 


i+1 := i 21 ; 1 if i 0; 21 is - ontra tible, 
else i+1 := i 0; 12 . Using Algorithm 4 we an dire tly nd a point whose !-limit is not x, in this

ase the i-loop is stopped. Otherwise, we de ne a sequen e of paths ( i )i , that are not - ontra tible. In this ase, let x0i = i (0) and x1i = i (0) be the initial and end points of these paths, then the sequen es (x0i )i ; (x1i )i  ([0; 1℄) and lim x0 = i!lim x1 = x: i!+1 i +1 i

Step i.

Arguing as in the proof of the onvergen e of Algorithm 1, we dedu e that ne essarily !x 6= x; if !x is a stri t minimizer, we use Algorithm 2 to dedu e the existen e of a riti al point that is not a stri t lo al minimizer. 

2. The Mountain Pass Theorem Our goal now is to prove a version of the Mountain Pass Theorem (for a detailed theory on this subje t we refer to [1, 5℄). A Mountain Pass Theorem on erns itself with proving the existen e of riti al points whi h are not stri t lo al minimizers for the fun tional f ; using Corollary 1.5 and Proposition 1.3, we are now able to prove the following 2 Theorem 2.1 (Mountain Pass Theorem). Let f be a C fun tional on a Hilbert spa e X . Let x1; x2 2 X , let x1 ;x2 be the set of paths de ned in (1.2) and 0 the level

(2.1) su h that

(2.2)

0 :=

inf

sup f ( (s)):

2 x1 ;x2 s2[0;1℄

0 > maxff (x1 ); f (x2 )g:

If the fun tional f satis es the Palais-Smale ondition at level 0 then there exists a riti al point for the fun tional f at level 0, that is not a lo al minimizer. Proof. The de nition (2.1) of the level 0 and ondition (2.2), imply, rst, that the sublevel f 0 is dis onne ted, se ond that for every k 2 N, there exist

k 2 x1 ;x2 su h that 1 sup f ( k (s))  0 + : k s From Corollary 1.5, we dedu e the existen e of an element xk 2 k ([0; 1℄) \  F1 0 su h that 1

0  f (!xk )  0 + : k

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V. BARUTELLO AND S.TERRACINI

Let (xn1;k)n  F1 0 onverging to xk ; following the proof of Corollary 1.6, we dedu e the existen e of an index nk 2 N and an instant T~nk su h that, y~nk := (T~nk ; xn1 k ), where xn1 k 2 (xn1;k )n, veri es krf (~ynk )k < k1 and 0  f (~ynk )  0 + k1 : Sin e f satis es the Palais-Smale ondition at level 0 , we dedu e that there exists a riti al point, y~, for f at level 0 , limit of the sequen e (~ynk )k as k tends to +1. We on lude the proof showing that there exists a sequen e (yn k )k , onverging to a riti al point y that is not a minimizer and su h that f (yn k ) ! f (y ) = 0 , with f (yn k ) = 0 k1 < 0. Let   1 nk  Tnk := inf t  0 : f ((t; x1 ))  0 ; k hen e, de ning yn k := (Tnk ; xn1 k ), we have f (yn k ) = 0 k1 . If we an prove that lim rf (yn k ) = 0; k!+1 using the Palais-Smale ondition at level 0 for f , we on lude the proof. By the sake of ontradi tion suppose that there exists 0 > 0 su h that krf (yn k )k  0, for every k 2 N. Hen e, sin e the Palais-Smale ondition at level 0 holds, there exists Æ > 0 su h that jTnk T~nk j > Æ and krf ((Tnk t; xn1 k ))k  0 > 0, 8t 2 [0; Æ ). Then, using Lemma 1.1 r 2(f ((Tnk Æ; xn1 k )) f (yn k )) nk 0  0 <   inf krf ((Tnk t; x1 ))k  Æ2 t2[0;Æ) s

1  2 f (~ynk )Æ 2 0 + k and the right-hand side tends to 0 as k tends to +1.




 p1 ; kÆ



Referen es 1. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in riti al point theory and appli ations, J. Fun t. Anal. 14 (1973), 349{381. 2. V. Barutello, On the n-body problem, Ph.D thesis, Universita di Milano-Bi o

a, Milano 2004. 3. Y.S. Choi and P.J. M Kenna, A mountain pass method for the numeri al solution of semilinear ellipti problems, Nonlinear Anal. 20 4 (1993), 417{37. 4. Y.S. Choi, P.J. M Kenna and M. Romano, A mountain pass method for the numeri al solution of semilinear wave equations, Numer. Math. 64 (1993), 487{459. 5. P. Pu

i and J. Serrin, The stru ture of the riti al set in the Mountain Pass Theorem, Trans. Amer. Math. So . 299 1 (1987), 1115{132.  di Milano-Bi o

a, via Cozzi Dipartimento di Matemati a e Appli azioni, Universita 53, 20125 Milano, Italy. E-mail address : vivinamatapp.unimib.it, sustermatapp.unimib.it