Multiplicity results for a class of semilinear elliptic equations on <span class="mathjax-formula">$\mathbb {R}^m$</span>

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

P ^IERO M ONTECCHIARI

Multiplicity results for a class of semilinear elliptic equations on R

^m

Rendiconti del Seminario Matematico della Università di Padova, tome 95 (1996), p. 217-252

<http://www.numdam.org/item?id=RSMUP_1996__95__217_0>

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of Semilinear Elliptic Equations

^on

^Rm.

PIERO

MONTECCHIARI (*)

ABSTRACT - We _prove the existence of an uncountable set of bounded solutions for a class of semilinear

elliptic equations

^{on R~.}

1. - Introduction.

In this work we are concerned with the

study

of existence and

multiplicity

of solutions to the

problem

where m ~ 1 and

f

^{satisfies the}

assumptions

(f3) ~b1, b2 &#x3E; 0

^{such that}

V(X,Z)ER7,x R, where s e ( 1 , 2* - 1)with2*=2m/(m-2)ifm &#x3E; 2andsisnot

restricted if m ⁼

1,

^2.

The

hypotheses (f 1)-(f 3)

^are the ones studied in

[10] assuming

^also

that f(x, ^z)

^is

periodic

^{in x and}

superquadratic

^{in z. Here we consider}

the

following

^more

general

^case.

We _say that a set A c is

large

^at

infinity

^{if VR} &#x3E; 0 ~x ^E A such that x

I R ~

^{c A.}

Clearly

any ^cone in ~.m is lar- ge ^at

infinity.

^Another

example

^{is a cone} minus the union of the annuli centered in zero and with radii

( 2 n )2 , ( 2 n + 1 )2 .

(*) Indirizzo dell’A.: S.LS.S.A., Via Beirut

2/4,

^{34013 Trieste.}

We assume that there exists a function

f. :

^{x R - R}

verifying (f 1)-(f 3),

^{and a set}

large

^at

infinity

^{such that}

such that

and

Putting

^{we define on the}

functionals

and we look for sol- utions of

(P)

^as critical

points

^of 99.

As we will ^see the

assumptions (f 1)-(f 5)

^are sufficient to

guarantee

the existence of at least one non zero critical

point

^{of the}

«periodic»

functional _{cp 00 .}

By (f 5),

^{if u is a critical}

point of cp 00 ,

^{then also} p ^{* u =}

=

u( ~ - p)

^{is a critical}

point of 99

^for any p ^E Zm . If we translate this u in a

region

^where

f

^and

foo

^are very close ^{one to} the

other,

^{one could}

expect

^that

nearby

this translate of u there is a critical

point of q.

^In

general

^{this is not true as the}

following example

^shows.

Let f(x, ^z)

^{= with a E}

C1 (~, R), a(t) ~

ao &#x3E;

0, a(t)

&#x3E; 0 Vt E

R,

^{and assume that u is a solution of}

(P). By

^standard

bootstrap argument

^we

get

^{that therefore}

cp ’ ( u ) al u = 0 . But,

^if

e1 ⁼

(1, 0, ..., 0),

^{we have}

which

implies u

^{= 0}

(see [12]).

To avoid this situations we make a discreteness

assumption

^{on the}

set of the critical

points

^of the functional at

infinity.

We note first of all that _{cp 00} satisfies the

geometrical hypotheses

^of

the mountain pass theorem.

Letting

Setting j

^{we assume}

(*)

~c * &#x3E; c such that

n 199.

^is denumerable.

We note that the

hypothesis (*)

^{excludes the}

asymptotically

^autono-

mous cases, because

if foo

^{does not}

depends

^{on x and u E}

Kcoo*

^then p ^{* u E}

E

K~*

^for any

In this

setting

^{we are able} to prove the

following

THEOREM 1.1.

If (f 1)-(f 6)

and

(*)

hold then

(P)

^admits

infinitely

many distinct solutions.

Precisely

there exists u E

X,

^{solution to the}

equation -

^{du + u =}

(x, u) for

^{which we} have that Vr &#x3E; 0 there exist M

= M(r)

^{E 1~T and} R ⁼

R(r)

&#x3E; 0 such that

for

any

finite Pk I

^{c zm that}

verifies

there exists a solution v to

(P)

such that

if

^we

put

then

where

if

^{A c is} measurable then

In

particular,

^{for k = 1 we}

get

^{that verifies}

c

ABBR ^{( 0 )}

then there is a solution v to

(P)

^{which is near}

u( ~ - p ).

^Moreo-

ver for k &#x3E; 1 if we choose _{any set} of k

disjoint

^{annuli centered in zero,}

each of which intersects the set

ABBR ^{( 0)}

^{in a ball} of radius M centered in a

point

^of then there is a solution to

(P)

^{which is near a translate}

of u in each of this balls. We call this

type

of solution

k-bump

solution.

The first

proof

via variational methods of the existence of

2-bump

solutions was

given by

^{E. S6r6 in}

^[22] solving

^the homoclinic existence

problem

^{for first order}

periodic

^{and convex} Hamiltonian

systems.

^This

paper

inspired

^the work of V. Coti Zelati and P. H. Rabinowitz

[9]

^on second order

periodic

Hamiltonian

systems

^{where was}

proved

^{the exi-}

stence of

infinitely

many

k-bump

homoclinic solutions for any In

[10] they adapted

^these

techniques

^to find the existence of

infinitely

many

k-bump

solutions for

any k E N

^{for the}

problem ^(P)

^{in the case in}

which f(x, ^z)

^is

periodic

^{in x and}

superquadratic

^{in z .} S. Alama and Y.

Y. Lee in

[3]

studied the

problem ^(P) assuming f asymptotic

^as

I x I ~ 00

to a function

foo

^{of the}

type

considered in

[10].

In that paper

they

^were

able to prove that the

problem ^(P)

^admits

infinitely

many

k-bump

^sol-

utions. All this results are based on

assuming

^{that there exists c *} &#x3E; c

such that

K~*

^{is finite}

_(clearly,

^{in the}

_periodic

case, the functional is _cp

itself).

The existence of

k-bump

homoclinic solutions with minimum distan-

ce between the

bumps independent

^{from l~ was first}

proved by

^{E. S6r6}

in

[23]

for first order convex and

periodic

Hamiltonian

systems.

^{As a}

consequence,

considering

^the

C1loc-closure

^{of the set of the}

multibump

homoclinic

orbits, using

^{the Ascoli ArzelA}

Theorem,

he finds solutions with

possibly infinitely

^many

bumps.

Also in Theorem 1.1 the minimum distance between the

bumps

^of

any

k-bump

^solution

depends only

^{on r}

(being given by M(r)). Using

the Ascoli ArzelA theorem it is therefore

possible

to prove ^as in

[23]

^the existence of ^a class of bounded solutions of the

equation -

^{4u + u =}

= f (x, ^u). Precisely

^{we have:}

THEOREM 1.2. Under the same

assumptions of

^Theorem

1.1,

^{it hol-}

ds that

for

^{any r} &#x3E; 0 there exist M ⁼

M(r)

^{E =}

R(r)

&#x3E; 0 such that

for

any zm that

verifies

and

for

every sequence ^{Or =}

1IN

there exists _v, E

Bloc ^(Rm) satisfying - Llva

+ va

= f(x, Va)

^{such that}

The tools used in the

proof

^{of Theorem 1.1 are related to the ones de-}

veloped

ⁱⁿ

^[18]

^{and then}

improved

^{with P. Caldiroli in}

^[7]

^{and with S.}

Abenda and P. Caldiroli

in [1] studying

^the homoclinic existence pro- blem for second order Hamiltonian

systems.

^These

arguments, inspired by ^[23]

^and

[9], permits

^{us to}

strengthens

the results contained in

[3]

ⁱⁿ

a more

general setting.

^{In fact the}

superquadratic assumption ^{(f 4)}

^is

verified also

by

^functions

~oo

^which

change sign.

^{Moreover the assum-}

ption (*),

^{is satisfied if the}

functional 99 .

^{is for}

example

^a Morse functio- nal. In the one dimensional case

(m

⁼

1),

^{it is}

possible

^to

verify

^{this con-}

dition via the Melnikov

theory

^when

foo

^{is a}

periodic perturbation

^of

particular

autonomous

problems ^{(see [1]).}

Another differences with the work of S. Alama and Y. Y. Lee

[3]

^is the fact

that f is

^{not assumed to be}

asymptotic to f~

^as

x ~ I ~ 00

^but

only

^{on a set}

large

^at

infinity.

^This

permits

^{us to consider the}

problem (P) when f is

^{assumed to be}

asymptotic

^{in different sets}

large

^at

infinity

to different functions.

Precisely

^we consider the

hypothesis

we define

, the

^mountain pass

"-

level of _99, t and we assume

(*,) 3c,*

&#x3E; c, such that is denumerable.

then, by

^Theorem

1.2,

^we have 1 different sets of

multibump solutions,

each constructed with a suitable critical

point

^{of the}

functional q?,.

In

fact,

^we prove that there are also

multibump

^{solutions of}

^(P)

^of

mixed

type,

^{as said in the}

following

^theorem.

THEOREM 1.3. Assume that

(f 1)-(f 5), (f7)

^and

(*,)

^{hold. There}

exists _{ul ,} ..., ul ^E

X, satisfying - Au,

+ u,

= f (xg u‘) for

^{which we have}

that

for

any r &#x3E; 0 there exist M ⁼

M(r)

^{E N and R =}

R(r)

&#x3E; 0 such that

for

any

iii ii. N C119 ... , L ~N ,

^that

verify

and ,

for

every sequence ^{or =}

(ai)ieN E=- 10, 1IN

^{there exists v, E}

^{Hloc (Rm)}

satis, fying -

^{+ v, =}

f(x, v,)

^{such that}

If

ai ;e ⁰

only for

^a

finite

^number

of

indices then _va is

actually

^{a sol-}

ution to

(P).

As last remark we

point

^{out that an}

analogous

^{result was}

proved by

S.

Angenent

ⁱⁿ

^[2]

^{in a different}

setting (z - f(x, z)

is assumed to be

periodic

^{in x and bounded}

together

^{with its}

derivatives), using

^essen-

tially

^fixed

point arguments.

^He

proved

his result under the assum-

ption

^{that the} solution u was such that the

operator -

^{L1 + I -}

u(x))

^{had a} bounded inverse. He was able to

verify

^this

hypothe-

sis for

periodic perturbation

^of

particular

autonomous

problem

^which

admits a

unique (up

^to

translations)

radial

solution, using

^a bifurcation theorem due to A.

Weinstein [24].

It is known that the

problem ^(P)

when

f(x, ^z)

^{= zP admits a}

unique positive

^solution

^{(see [13])}

^{and it}

should be

interesting

^to check if the

hypothesis (*)

^{holds for}

periodic perturbations

^{of this}

f.

The work is divided into four

parts.

^In the first section we

give

^some

preliminaries.

^{The second is devoted to the}

study

^{of some}

properties

^of

the

periodic problem

^{which we use} in section three to construct a

parti-

cular

pseudogradient

^{field of} 99.

Finally

in section four we prove the existence theorem.

2. - A local

compactness property.

In this section we

study

^some

properties

^{of the}

functional cp

^which

are

independent

^{on the}

assumptions

^{on the}

asymptotic

^behaviour

of f.

All the results contained here are true under the

hypotheses (f 1)-(f 3).

In the

proofs

^{that follow we shall}

always

consider the case m ;

3,

^the

proofs

^{for m = 1 or 2}

being

^{not more difficult.}

We have to note first of all that

(f 1)-(f 3) imply

and

obviously

^an

analogous

^estimate holds also for

F(x, z ).

^This

permits

us to say

that T

^is well defined on X because of the Sobolev Immersion Theorem.

Actually

^the

following

^holds:

PROPOSITION 2.2. ₉₉ E

C 1 (X, R).

PROOF. We _prove first

that cp

^{is Gateaux} differentiable. Given

h E-= X, by (2.1),

^we

get

^that

Being

^this last function in we can use the dominated conver-

gence theorem to

get

^that

We prove ^now

that p’G

is continuous. Let and

By

^{the Sobolev} Immersion theorem there and a

function such that a.e. in R7

and ~ I unkj ^{(x) I}

^~

^v(x)

^{a.e. in}

Using again

the dominated convergence theorem ^we

get cp G

^{X * . Since this can} be done for _any

subsequence

^the

proposition

^is

proved.

⁸

PROOF. If ^E &#x3E; 0

then, by ^(2.1),

which

Analogously

ii.

As a consequence ^we

get

^{a first}

compactness property

^of cp :

(2.4) 3g

&#x3E; 0 such that ~

2 Q and 99 ’ ( un )

^{- 0 then} Un ^- 0 . The

hypotheses (f 1 )-(f 3)

^are not sufficient to

guarantee

that the Pa- lais Smale _sequences ^are bounded in X. In the

following

^we

study

^the

behavior of the bounded Palais-Smale _sequences of _99. If M c Rm is mea-

surable then we

put ~ I

PROOF. We _prove first that

cp’ (u)

^{= 0. with}

Ilhll

^{= 1. Fi-}

xing E

&#x3E; 0 there exists R &#x3E; 0 such

that Ilhlllxl &#x3E; R

^{E .}

and be such that a.e. on

v( x )

^{a.e. on}

By ^(2.1)

^{and the dominated con-}

vergence theorem ^we

get

Since ^E is

arbitrary

the claim follows.

Let’s now prove that

cp’ (un - ^2c)

^{-~ 0. Since}

it is sufficient to show that

Given ^E &#x3E; 0 fix R &#x3E; 0 such

that ~ I

^Then

Consider the first addendum. Given

and ^a function such that

Since

we can use the dominated convergence theorem to

get

^that

Considering

^{that this can be done for} any

subsequence

^{of we ac-}

tually get

^that

For the second addendum we note

first, by

^{the choice of}

R,

Since

fx (x, ⁰⁾

^{= 0 we also infer that}

r . -. - -- .. - .

The

proof

^that

cp( Un - ^u)

^{2013~ b - is}

analogous.

^I

LEMMA 2.6. Let be a bounded

sequence in X

^and suppose that there exists _e &#x3E; 0 such that ^. Then

PROOF. We

give

^for

completeness

^the

proof given

ⁱⁿ

^{[10] (see}

^Lem-

ma

2.18).

Given q E (2, 2*) let 8

Holder

inequality

. So

where

By

^{the Sobolev} Immersion theorem there exists

Al

⁼ m,

~o)

^such

that

Assume now that

aq ;

^{2 that}

is q &#x3E; 4/m

^{+ 2. We have}

Choosing

^a

family

^of such that each

point

^{of R~ is}

contained in at least one and at most k of such

balls, summing

^{over this}

family,

^{we obtain}

Setting

^{in the above formula u} = un ^we

get

^{0 in for all}

q ^E

[ 4/m

⁺

2, ^2*).

^The

proof

^{now go on with another}

interpolation inequality.

If

ge(2,g) (q

⁼

4/m

⁺

2),

^{we have} q ⁼ 29 +

q( 1 - ⁰⁾

^{for some}

8 E

( o, 1). By

^{the Holder}

inequality

^{we have}

for _any natural n. The lemma follows from the fact that 0 in

LEMMA 2.7. Let 0

weakly

^{in X such that ~ 0.}

Then, for

any R &#x3E; 0 we have _{x -~} 0 and the

verifies

either

PROOF. Let R &#x3E; 0 and let gR ^E Coo

(R~, R)

^be such that 0

for any if

Clearly

We _prove first that

Since and since

ciao, it’s

enough

^{to show But this}

Rm

is a consequence of the

Lebesgue

^dominated convergence theorem sin-

ce

Therefore we have

Since

ficient to show that But this is true because

and because

u.,, 0

ⁱⁿ

L2 (B2R(O)BBR ^(0)).

The first

part

^of the lemma is ^so

proved.

^It’s easy to prove the alternative.

Assume that

(b)

^{does not hold. In that case}

for any &#x3E; 0. Since is

bounded, by

^{Lemma 2.6 we}

get

^that

) as we claimed.

Therefore ^if _un is ^a Palais-Smale sequence which converges

weakly

to a certain

point u,

^then un converges ^{to u} in

Hloc

Moreover if _un does not converge to u in

X,

then fixed

any R

&#x3E; 0 we have

lim sup llunll lxl &#x3E; x % r &#x3E;

^{0. This mass r cannot} be smaller than a certain

n - oo

positive

^{fixed value as the next lemma says.}

LEMMA 2.8. Let

weakly

ⁱⁿ

X, g~’ (un) -~ 0. If

^{there exists}

PROOF. Fix T &#x3E; 0 such that

Putting

^{M =}

= max{R, T }

^{we have}

by

^Lemmas

2.5, 2.7,

^that Therefore

from which we

get

^lim

u 11 ( 2 ~o . ^{u )}

^{~ 0 we de-}

rive from

(2.4)

that

This is a first local

compactness property

^of the functional which will be useful in the

following.

^{From it we derive}

easily

LEMMA 2.9.

If diam

o ^and

(un)

^{~ 0 then has an ac-} cumulation

point.

PROOF. Let ⁼ _go and T &#x3E; 0 such that

- In that case

Since

is bounded it has a

subsequence {unk}

^{which converges} weakly ^{in X to a} certain

point

^u.

Then, by

^Lemma

2.8,

_unk ^{- u.}

3. - The

periodic

^case.

Here we will

study

^some

properties

^{of the}

functional 99 Obviously

all the results

given

^{in the}

previous

section remain valid for _{cp 00 .} First of all we see how the further

hypothesis ^{(f 4)} implies

^that the functional cp 00 satisfies the

geometrical hypotheses

^{of the mountain} pass theo-

rem.

By

^{Lemma 2.3 we}

just

^{know that} there exists r &#x3E; 0 such that cp 00

(u ) &#x3E; ( 1 /4 ) r2 for

any u ^E Then we note that the

assumption (f 4) gives

information about the behavior of

F 00

at

infinity

^with

respect

to z. In

fact,

^{one can} infer that

given

^{0 then if 1 we}

have

LEMMA 3.2. There exists _ul E E such that _{cp 00}

(ul )

^0.

PROOF. Let

(xo ,

^{x R be}

given by (f 4),

^then

6 0

⁼

.

By continuity

^{there exists E} &#x3E; 0 such

Therefore 99 - (Auo)

^{~ - 00 and} the thesis follows.

This shows that the functional at

infinity

verifies the

geometrical hypotheses

of the mountain _pass theorem.

Then,

^{if we define r}

= ~ y

^E

~C([0,l],~):y(0)=0, ~(y(l))0}andc=inf

max cp 00

(y(s)),

^we

y e r8 e jo, 1]

infer that c is a

positive, asymptoticall

critical value for _{cp 00 .}

Now we use

again (f 4)

^to show that the Palais-Smale sequences of

are in fact bounded _sequences in X.

L E MMA 3.3. such that

~ ~ ( un ) ~ 0

^and

lim sup ( un )

+ ~ , bounded in E and lim inf

-&#x3E; 0. In

particular

any Palais-Smale _sequence

for

^{cp 00} is bounded in E.

PROOF. From

(f4),

^we

easily get

^that

Now, given

^a such that

cp ~ (un) -~ 0

^and

lim _{sup cp 00}

( un )

+ 00, ^from

(3.4)

^{we obtain} C for all n E

N,

C

being

^a

positive

^constant.

Consequently

^{we have}

that cp 00 (un) ~

I

^{and this}

implies

^that

Using

^the

periodicity

we can now prove that the

problem

^at

infinity always

^admits a non zero solution which is obtained as weak limit of a

suitable translated of the Palais-Smale _sequence

given by

the mountain pass theorem.

THEOREM 3.5. The

problem:

admits a non zero solution.

PROOF. be the Palais-Smale _sequence

given by

^{the moun-}

tain _pass theorem.

By

^{Lemma 3.3 we} can assume that

weakly

ⁱⁿ

X. If u ~ 0 then

by

Lemma 2.5 the theorem is

proved. Assume u.

⁰

weakly

^{in X. Since} 99 .

(un) -~

c &#x3E; 0 it cannot

be Un

2013&#x3E; 0 so the alternati-

ve

(b)

of Lemma 2.7 holds and

(up

^{to a}

subsequence)

c for which

(l yn, 11, ...,[~,J).

^Then

~(~)~c, p.(~)~0

^{and C.}

Let vn

^2013~

(up

^{to a}

subsequence) weakly

^{in X.}

By

^{Lemma 2.7 we have}

therefore So v is a non

zero critical

point

^..

Therefore E

XN( 0 ) /q l ^(v)

⁼

0 ) # il

^and

using

^{2.4 we have}

also that

As we have

point

^{out in the}

introduction,

^{the fact that the set of criti-}

cal

points

of the functional at

infinity

^{is not}

empty

^{does not}

guarantee

that the

problem ^(P)

^{has non} trivial solutions. We will prove that if the set of the critical

points

^{of the} functional at

infinity

^{is numerable}

then this forces the functional _cp itself to have

infinitely

many critical

points.

To

study

^{better the} Palais-Smale sequences,

following [23], [7],

^we

introduce two sets of real numbers.

Letting

we define

the set of the

asymptotic

critical values lower than b and

-1.. L ,. - _. 1.. II 1 - 1 1 ,

the set of the

asymptotic

distances between two Palais-Smale _sequen-

ces under b .

The

sets cp band D b

are

actually

closed subsets of R

(see [7])

and we

have:

where,

^{if S c X and I}

Using

^Lemmas

2.5, 2.7,

^{3.3 and}

(3.6) together

^{with the}

periodicity assumption

^{it is}

possible

^to characterize the Palais-Smale sequences of cp 00. As in

[10]

^we prove the

following.

L E MMA 3.9. such

- 0. Then there ^are

Vo e Koo U 101,

^vl,...,

Vk E Koo ,

^a

subsequence

denoted and

corresponding

^{E Z"l}

such

that,

as n -~ ~ :

PROOF.

By

^Lemma

3.3 ~ un }

^{is a bounded} sequence in X and we can assume that 3 lim and that _un converges

weakly

^{to some} vo in X. If _Vo the lemma follows. Otherwise the case

(b)

of the alternati-

ve in Lemma 2.7 holds for the sequence

~~o, ~ &#x3E; 0, ~ yn ~

^c

Z-

⁹

such

that,

up to ^a

subsequence, Putting

ZGn

⁼

(Un - vo)( ’

⁺

Yn)

^we observe that there exists _vi ^e such that

that the

sequence {un -

vo -

v1 ( · - y1n)}

verifies the case

(b)

of the al- ternative in Lemma 2.7 and we can continue as above. After a number of

steps

^not

greater

^{than ,} the lemma follows.

This characterization reflects on the structure of the

sets 0’ and

D b .

In

fact,

^{as in}

[7] (see

^Lemma

3.10),

^{one can} prove ^now that

Using

^the

hypothesis ^(*) given

^{in the} introduction this

permits

^{us to}

bound from below the ^norm of the

gradient

ⁱⁿ

large regions

^{of X. In}

fact if we assume

(*)

there exists c * &#x3E; c such that

K~* is

^countable

then both the sets

D ~ *

are countable too.

Being D ~ * and 0 c *

also

closed we then have that ^.

(3.10) D ~ *

^{does not} contain any

neighborhood

^{of 0 in}

R + , (3.11) ]0, c * [B øc*

^{is open} and dense in

[0, c * ] .

By ^(3.7)

^and

^(3.10)

^we have that around

K~*

there is a sequence of annuli of radii smaller and smaller on which there ^are not Palais-Smale ^se- quences at ^a level less then or

equal

^{to c * .}

Analogously, by ^(3.8)

^and

(3.11),

^fixed any

E E ( 0, c * - c )

there exist two closed intervals

[al , b1]

^c

(c -

E,

c)

^and

[a2 , b2]

^c

(c,

^{c +}

e)

^{such that the sets}

~

g~ ~ ~

^{not contain} Palais-Smale _sequen-

ces.

Using (3.7), (3.10)

and the local

compactness property given by

Lemma 2.9 it is

possible

^{to show as in}

^[7]

^{that the}

functional 99 .

^admits

a local mountain _pass

type

^critical

point (see [19]).

DEFINITION 3.12.

Let f be

^a functional of class

C 1

^{on a} Banach spa-

ce X and let Q be a

nonempty

open subset of X.

We _say that two

points

^are connectible

there is a

path

p ^E

C([ 0, 1 ], X) joining

xo and _{xl ,} with range p c ^Q and such that

max f

^c.

A critical

point x

^{E X for}

f

^{is called} of local mountain

pass-type

^for

f

on Q if i ^E Q and for _any

neighborhood 31

^{of x subset of Q the set}

n N contains two

points

^not connectible

We refer to Section 4 of

[7]

for the

proof

^{of the}

following

L E MMA 3.13.

If

cp 00

verifies ^(*)

^{then it admits} a non zero critical

point of

mountain pass

type.

^In

particular

there exist c ^E

[ c, c * )

^and

r E

(0, e)

^{such that}

for

any sequences

(rn)

^c

R.+ BD * ,

^{rn - 0 there is a se-}

quence

(vn)

^c

K 00 (c),

vn --~ v E

K 00 (C) having

^this

for

any ^{n e} N

and

for

any h &#x3E; 0 there is a

path

Yn ^E

C([ 0, 1 ], ~ satisfying:

(ii)

_y,,

( 0 )

and _y,,

( 1 )

^are not connectible in

where

Rn

&#x3E; 0 is

independent of s, hn

^{= and}

drn and Jl Tn

^are

defined by ^(3.7).

REMARK 3.14.

Clearly

^the

property given

in this sections are true for all the functionals

(c

⁼

1, ... , L)

^{if the}

assumptions (*~)

^{are veri-}

fied. In the

following

^{we write c, as} the mountain _pass level of _q?,,

c,

^as

the local mountain _pass level

near 5,

^etc.

4. - The construction of a

pseudogradient

^{vector field.}

In this section we will

study

^some consequences of the

assumption (f7)

with which we ask that there exist ...,

Al

^c

R~, large

^{at infini-}

ty

^{on where}

f

^is

asymptotic

to 1 different

periodic

^and

superquadratic

functions

fl, ..., f, as x ~ -~ ~ .

First of all we show that

by ^(f7)

^{if a} function u is translated in a re-

gion

^where

f

^and

f

^{are close one to} the other then

cp’ ^(u)

^{is near}

(u) (here ~’~ (x, u) dx

^is the functional associated to the

Rm

function

f ^(t

⁼

^1,

^...,

l).

LEMMA 4.1. For _any 3 &#x3E; 0 and C &#x3E; 0 there exists R &#x3E; 0 such that

if

^{C and}

suppucA~BBR(0) then 11

PROOF. For any ^E &#x3E; 0 we choose R &#x3E; 0 such

that,

if supp u ^c

we have

for almost _{every x} E Then

1

A,

and the lemma follows. m Given

k,

^{N E N we} say that

If _p e

P(k, N)

^we define the annuli

and

where

pk

^{+ 1 ~}

1

^{= +00.}

Since p

^E

P(k, N)

the thickness of the annulus

Mi

is

alway greater

^{than or}

equal

^{to 2N.}

t ^v -1 ... I .- ^._

J

It is _{easy to} see that if N is

sufficiently large (depending

^{on V and}

^r)

then _p;

J)

^{is a}

nonempty

open subset of X. Moreover the ele- ments of _{p ;}

J)

^are

multibump

functions of the mixed

type.

^In

fact E p ;

J)

then u is near the function

V~i ( ~ - ^{pi )}

^{on the}

annulus

Defining

we

investigate

^some

properties

^{of the}

functionals q?

^and _{cP l,} on the set

Br(V ; p ; J).

We note that for _any

given Ñ(V, r)

^{e N is such} that

&#x3E; N £ r

^for any t ^e

11,

^...,

11

^then

given kEN,

N &#x3E;

N,

p ^E

r= P(k, N)

^{and Je}

11,

^...,

llk

^{then Vu E}

ar (V-,

p;

J)

^and

11, - - ., /c}

there

exists j

^e

{N

⁺

2,

^{..., 2N +}

11

^{such that}

Therefore if u ^E _p;

J)

^{then for} any i

E ~ 1,

^{... ,}

k 1,

the annulus

‘U,i

^{contains two annular}

regions

of thickness

N, symmetric

^with

respect

to pi , ^over which the norm of u is small as we want if N is

sufficiently

large. Moreover, by construction, Mi

^never intersects any one of these annuli.

We the smallest index in

IN

⁺

^2,

^{..., 2N}

+ 1 ~

^{which verifies}

(4.2).

For any E ^E

(0, r)

^there exists

2Vc

^E

r), 21

^such

that

So if &#x3E;

N, and p

^E

P(k, N),

^{then Vu E} p;

J)

^{and Vi E}

E ~ 1, ..., k}

^we

get

^that

Then,

^{fixed u e} p;

J),

^we define the

following

^{subsets of R~:}

With this notation

(4.3)

^can be rewritten in the form

We

plainly recognize

^{also that}

By

^wntixdai

Mi C Eu,

^{i ,} therefore the thickness of

Eu,

^{i is}

greater

than or

equal

^{to ~V V i}

E ~ 1,

^{... ,}

l~ ~,

^{Vu E}

^J).

^{This is true also}

for the connected

parts

^{of the sets i and}

.9u,

For _{... ,}

k ~,

^{we define} the cut-off functions:

with _i continuous on R~ and linear if restricted on the connected

parts

^of

Eu ~Eu

intersected with _any

straight

^line

passing through

^the

origin.

^We

put

^also

fl., 0

^{= 0.}

Then,

^{for i E}

{ 1,

^{... ,}

k 1,

^{we set:}

If (3

is any one of the above cut-off functions

then ~~8(x) ~ ~ 1/N,

^for

a.e. x e therefore since N ~

2,

^{it is} easy to ^see

that,

^{if A is measura-}

ble c Rm Vu e Z. Moreover if u e p ;

J)

^and i

e {1,

^{... ,}

1~ ~,

^then

by (4.5),

^we

get

Now we

define,

^{for i}

E=- 11,

^{... ,}

k 1,

^{the functions}

and we set

finally

If we define the finite cone C

= I y

^E

^R7; y ~ I 1/2, 1/4

^y,

1/2}

^{then the}

embedding

^{constant relative to} the immersion

H 1 ( S~ ) ~

~ L s +

1 (0)

^{can be chosen to be}

independent

^{of Q if S~ is an} open set which verifies the cone

property

^with

respect

^{to C.}

This

implies, by (f2) (f3),

^{that we can fix} ro E

(0, min jr, ý2 - 1})

such that if _{u, w} E X then

for _{any open} set A c Rm which satisfies the cone

property

^with

respect

to C. We can assume that _ro is such that

(4.7)

^{holds also if we}

consider f (i

⁼

1, ..., 1)

^instead

of f.

Using (4.6), (4.7),

^{we can} prove ^now that:

LEMMA 4.8. Let

rE(0,(1/4)ro)

^{and Then Vu e}

e

8r (V-,

p;

J)

^we have

PROOF. We have that N

&#x3E; 2,

^{and the} thickness of the annuli and of the ones whose union is is

greater

^{than or}

equal

^{to N.}

Therefore these sets

satisfy

^{the cone}

property

^with

respect

^{to C. Mo-}

,.

Therefore, by (4.6),

^and

(4.7),

^we

get

The

computation

^is

perfectly analogous for q; t, i .

By

^{Lemma 4.8 we}

always

^{have that}

and

analogously

for all

Moreover if is

greater

^then

2c~

^{for a certain}

couple

^of

index

(i, j ),

^then

W~ indicates

^an

increasing

direction both

for q

^and

~p ~, i . ^{We note also that}

W~

^has

support in

^a

region

where each

Vii ^(.

^-

p,)

is small and it holds that

Let b,

^be any ^nonzero critical level of _gg,, and ~e

1

there exists a

locally lipschitz

continuous

function

Moreover

if ,

⁴ then there exists _{Jl k} &#x3E; 0 such that

by

Remark 3.9 we have that v &#x3E; 0.

We construct the vector field

Wu

^on

tBr3(V;

^p;

^{J), using}

^{Lemma 4.8}

with r ⁼ _r3. We will now define another vector field

analyzing

^{the diffe-}

rent cases.

Let i E J1 (u) and E1 =

We consider the two

possible

^subcases:

In the first one,

using

Lemma 4.8 and the fact that

(putting Eo = 0)

we

get

and

analogously

In the second subcase we

firstly

^note

that, arguing

^{as in}

(4.3),

^there

exists

... , N ~

^{such We}

put

and

Bu noting

^that

Eu ) ~

^{N and}

therefore that the set

1% B(Bu

^U

^Eu)

^{has the cone}

property

^with

respect

to C.

We define also the cut-off function _{t7 u} ^E

C( Rm , R)

^{such =}

(x)

and in such ^a is linear if restric- ted ^on the connected

parts

^of

Bu ~Bu

intersected with any line

passing through

pi .

Consider the

following

alternative:

If

i)

^{holds we}

put ’

Analogously

^we

get

We

finally

observe that

If

ii)

^{holds we} claim that 1 ⁷ In fact we

firstly

note that since

On the other hand since ^{I we}

get

To end the

proof

of the claim ^we note

that,

^since

we have that

Therefore

there exists 1 such that

Since ~ we have _supp Moreover

j,

^which

implies by

the choice of

As last

step

^{we note that}

and the same

argument gives

^also

From the two above

inequality

^it follows that

In this case we

put Wu,i

⁼

( 1 /2 ) yJ u Zu,

ⁱ

observing

^that

We now set

2,u

^{= and}

obtaining by

We also note

that 11 any i e { 1, ..., k}.

We

put

Fixing

^{also it can be either}

In the first

subcase, considering

^that

~i~~!/9,

^we

get

^{as above} that

For all

and ^i,

In the second subcase we

proceed

^{as in the case 1}

considering

^the

alternative

If

i)

holds then

putting

i =

( 1 - ~ u ) arguing

^{as in case 1}

we

get

’

If

ii)

holds then we claim that In fact

and

We

finally

derive that

which

implies

^{as we claimed. So} there exists

Zu,

^{i E}

Moreover, again

^{as in the case}

1), since E’I’ v/ 16

and we

put

^{in this case}

noting

^that

Let now’ and

obtaining,

^{as in the case}

1),

^that

As in the case

1)

^it is easy ^to prove that

there exists

V-, 3

E X such that i

and

We

put

In this case we

distinguish

^{between the two subcases:}

In the first _case, if we have Lemma 4.8 that

we

get using

and we set

’B?u,4

⁼

Wu .

In the second _case,

by

^{the local}

compactness property of gg ^(Lemma 2.8),

^{we obtain that if} p;

J) = 0

^then there exists

Vu

^E

X,

and there exists

/~~0, independent

^{of u, such that}

F or u E X we

put

i

noting

^{that 2. Then the}

proposition

follows with ^a classical

pseudogradient construction, by

using

^{a suitable}

partition

^of

unity

^{and suitable} cutoff functions.

5. -

Multiplicity

^result.

In this section we will state and _prove the main Theorem.

THEOREM 5.1. Assume that

(fl)-(f5), (f 7)

^and

(*,) ( c = 1, ..., l)

PROOF.

Suppose

^the

contrary,

then there exists

dent on s.

Fix 0 s

min{~i,(l/9)(c -c- ,),(1/2)~}.

^{We can also assume,}

enlarging

^T if necessary,

that ~ I and,

^{we fix an}

integer max ~ T , N, 4 }

^{where N is}

given by Proposition

^{4.9 for these value}

of r, b + , ~

⁼ c + , ~ ,

6-~

⁼ c _ , i ,

E/4

instead of E and

Vn

^{instead of V.}

If R &#x3E; 0 is

given by

^the

Proposition 4.9,

^{since x n}

(V;

p;

J) = 0

for

N, J),

there exists a

locally Lipschitz

continuous fun- ction W: X ~ X which satisfies the

properties (Wo)-(W4).

^{Let us consi-}

der the flow associated to the

following Cauchy problem

Plainly, by

for any u E X this

Cauchy problem

^{admits a}

unique

^sol-

ution

t7(-, ^u)

^{defined on}

^{R +}

^{and the}

function t7

is continuous on R + ^x E.

Moreover,

^the function s ^~--~

u))

^is

nonincreasing.

To _prove the

theorem,

^{we make the}

following

^claim.

-

CLAIM. There exists r &#x3E; 0 such that the continuous

function

G:

Q-~X given by G(O) = G(B)) satisfies:

(viii)

there is a

path ~ inside Q joining

^two

opposite faces F°

^and

Fi

^such

along ~,

^the

,function G

takes values under

C-,ji

^{+ E;}

namely: 3

^{i E}

{ 1,

^{... ,}

k } and ~ _ ( ~ 1, ... , ~ k )

^E

C( [ 0, 1], ~ )

^{such that}

Assume the claim holds and introduce a cut-off function _{x E}

e

C(Rm , R),

^{such that}

x(x)

^{= 0 if x ft}

’l4, x(t)

^{= 1 if t e U}

Mi _ 1 )

and linear ^on the connected

parts

^{of the} intersection of _any line

passing through

^the

origin

^{with the set}

’l4

ⁿ

(Mj

^U

Define g

^e

C([ 0, 1 ], X) by setting g(s)

^{= for s e}

[ 0, 1 ].

^We observe

that,

^{because of}

(vi)

^and

(viii)

^we have

g(0) = xG (1(0)) =

= =

~(0)(- - ^p1)

^and

^similarly

⁼

~(1)(- - Pt).

^{We have}

also that the

path g

^{is contained in the ball}

Pt). ^Indeed,

^first

of all

Secondly,

^since

E

(l/2)d~

^{and since}

Br+ ^{(Vn ,}

^p,

^J)

is invariant under _r¡, we

get

Translating by - pi-

^the

path

g, ^we

get

^a

curve joining

^with

y~i ( 1 )

ⁱⁿ

~3~. (v~i). ^Showing

^{that on g} the functional remains under the

level -

^{we will}

get

^a contradiction with the

property (ii)

_{of Y jt .}

To _prove

this,

^we notice that

and

Putting together

all these estimates in

(5.2),

^and

considering

^that

E

( 1 /9)(c - c _ ),

^we

finally get

^{that + 9 E}

[;- ,

^which

contradicts

(ii).

’ ~

To check the claim we

firstly

^{note that the}

properties ^(vi)

^and

^(vii)

are true for _any i &#x3E;

0,

and follows

easily

^from

(W2)

^and

(W3).

We divide the

proof

^of

(viii)

^{in some} lemmas.

LEMMA 5.3. There is 1 &#x3E; 0 such that

for

any u ^E

(Vn ;

^p;

^J)

ⁿ

there exists Z* e

{ 1,

^{... ,}

k } for

^which

PROOF. Set or ⁼ 2 diam

(Vn ;

p;

J)).

^Since

(Vn ;

p;

J))

^is

a bounded

set,

^{(7 +00. Put and let}

u E ~3r_

By (W2)

^{the curve s ~}

~(s, u)

^{remains in}

I

. Moreover it must goes out of at some

s E ]o, i[ , otherwise, by (W4),

in contrast with the definition of o.

Then,

there axe I

E {1, ... , k}

^{and an}

s ~ 0,

^we

get

But since for any s &#x3E; 0 we

get

^also

from which ⁱ

LEMMA 5.4. For _{any B} ^e

Q

^there

{ 1,

^...,

k}

^{such that}

PROOF. Assume first that e

83r- (V ;

p;

J). Then,

^since

by

^con-

struction ^we obtain the result

by

^{Lemma 5.3.}

In the other case there exists i

E {1,

^{... ,}

/c}

^{such that}

so

using

^the

properties ⁽ⁱⁱⁱ⁾

^and

^(iv)

^of yp ^we

get

By

^{Lemma 5.3 we then have that}

i7(s, ^{G( 8))}

^E

qtjfi

^{for any} s &#x3E; 0 and

the lemma follows. ^’~’

Now we can conclude the

proof

^of

property (viii).

^We

proceed by

contradiction

assuming

^the

contrary.

^That

is,

^for any i

E ~ 1, ... , k ~

^the

set

Di

⁼

(CPji,

^{i 0 + E, + ~}

[) separates F?

^from

Fi

ⁱⁿ

Q.

^Let

Ci

be the

component

^of

QBDi containing Fi

^{and let} ori:

Q ~ R

^{be the fun-}