Subharmonic oscillations of a semilinear wave equation

,Vonlrneor Anuirrlr. Thhmrv. Merhods & A,nplrcar~orn Vol Pnnted ,n Great Bncam

-SUBHARMONIC

9. No

5. pp. 503-514.

0362.546X/X5 $3 oO+ .oO Q 1985 Pergamon Press Ltd

1985.

OSCILLATIONS OF A SEMILINEAR EQUATION

WAVE

MICHEL WILLEM LInkersit

Catholique

de Louvain.

Institut

Mathimatique. 2. chemin Belgium

du Cyclotron,

B-1348 Louvain-la-Neuve.

(Received 25 February 1984; received for publication 9 October 1984) Key words and phrases: Subharmonics.

nonconvex duality, nonlinear strings equation. INTRODUCTION

CONSIDER

the following problem &I - u, + f(t, X, u) = 0

(1)

u(t, 0) = 0 = u(t, n)

where fis continuous on [wx IO, ~71 X [w and 2n-periodic in t. Using the dual action of Clarke and Ekeland [8] we prove the existence of infinitely many subharmonics of ( 1) when the potential F(t, x, u) =

I0

“f(r,x, s) ds

is either subquadratic of superquadratic in u. Moreover we deduce from the variational characterizations some estimates on the amplitude of the subharmonics. Our main results are: THEOREM

1.

If: F is convex in u:

(2)

F(t,x,u)-+

(3)

~0, Ju/-+ xc;

F(r, x, u)/u’*

0, Iu (+ ‘;c;

(4)

uniformly in r. then. for every k E N*, there exists a 2kn-periodic solution uk E L” of (1) such that I&= + x. k+ x. THEOREMS 2.

If: F is strictly convex in u;

there is q

>

(5)

2 such that qF(t, x, u) sf(t,

x. u)u;

there is p, y > 0 such thaf pi u19s F(t, x. u) c yju 14: then, for every

k E N*.

(6)

(7)

there exists a nonzero 2krr-periodic solution uk E L’ of (1) such that IUAL’---, 0, k-+ x. 503

M.

504

Example. Let g(t. x) be continuous

WILLEhl

on W x [O. X] . 2,7-periodic

in r and such that

PSg(t,x) for some /3 > 0. Then f(t.

X,

U)

=

g(t.

X)

iu 14-5

satisfies (2)-(3)-(4) if 1 < 4 < 2 and the amplitude of the subharmonics q = 2 problem (1) is linear and there is no subharmonic in general. (5)-(6)-(7) and the amplitude of the subharmonics tends to 0.

tends to infinity. When If 2 < 4 < x. f satisfies

Remarks. 1. The Hamiltonian version of theorem 1 was obtained in [12] (see also the expositions in [4] and [13]). In the autonomous case a related result is due to Brezis and Coron [5]. Our argument shows that the amplitude of the oscillations in [5] tends to infinity. 2. See [ll, 6, 71 for the autonomous superquadratic case. When 2 < 4 < 4, the existence of a nonzero 2.rc-periodic solution of (1) is proved in [3] under assumptions similar to (5)(6)-(7). 3. The argument of theorem 2 applies also to Hamiltonian systems. 4. If q(s) = F(t,x.su). assumption (6) implies that SW(S) 2 qy(s). Thus. s”v(l), i.e.

if s 3 1, v(s)

F(r. X. SU) 3 s9F(t. X, U). Then

there

is M, m > 0 such that

so that assumptions (6) and (7) are related. 5. The period 2n can be replaced by any rational multiple of X. 6. After the change of variable r = r/k, it suffices to find 2jr-periodic 1 k’ UU- u,, +

f(kr. x. u) = 0

Lc(t.

u(r,

0)

1. THE

Let Q = ]0,2,$

x

with periodic-Dirichlet

Let us define

=

0

=

LINEAR

solutions

of

2-c).

OPERATOR

IO, ~3 and let Lk be the operator

boundary

in L’(R).

conditions

for 1

u E Lp( 9):

UW

=otlW

The kernel

exponent EL”(R)

of Lk is given by

of p, n

KerLk 1

2

Subharmonic

oscillations

of a semilinear

wave equation

if it is easy to verify that u E ?& PROPOSITION 1. For every u E xk.p there

Moreover

there

is a constant

exists a unique

and

Lku = v

I$ TkU = 0.

IR

solution

u = ?&,cl E L” of

uw = 0. VW E Ker L k.

(9)

c(p) such that /&J1L=

c kc@)l&.

Proof. Let us write

If the Fourier

expansion

of u E xk., is given by f v mnelm’sin nx

the only solution

(10)

of (9) is U--Z

k’ ,&?

_

[j R

vmn eIm’sin

,,,?

nx

v(r. 5) e -m’rsinnedQ

lrnr 1

sin 11x e ,&,l _ &

(11)

Let us define

$2 pn2”_e’m’COSnx.

hk(t, x) =

m2

Then U =

;

[(hk

*

(12)

U)(t, X) - (hk * u)(l. -x)].

We obtain

It remains only to find a bound for /hklLq independant of k. Since 1 < p S 2, the Hausdorff-Young theorem [14] implies c(p) > 0 such that

the existence

of a constant

M.

506

Thus it suffices

to find a bound

WILLEM

for s(kil k’n’ - m’/ r independent

k

1

lk’n’-rn’~=ln

+m/k,/kn

-ml

<-

1

of k. But 1

n lkn -m;‘

so that we have

PROPOSITION

Proof.

2. For every

u E X,.,

It suffices to use (10). (11) and Parseval

2. THE

We assume

that F is convex

SUBQUADRATIC

in u. The Fenchel

n

identity.

CASE

transform

of F Lvith respect

to LI is defined

by

The dual action

is defined

on Xk = XL.2 by qn(o) = .i, [$Kru

where

11+ G(kt. s. u)] dR.

Kk = Kk.2.

LEMMA

1. If there

is my,/3 > 0 and

such that

(13) then there Proof.

For

is a 2rr-periodic We only outline

solution

II~ of (8) such that -L kuh minimizes

the argument

which follows

Brezis

and Coron

@i on XI, [5].

Subharmonic

oscillations

of a semilinear

SO7

wave equation

let us define ,

FJf.X.

E;

u) = F(r.x. u) +

and

&k(v) = where

G, is the Fenchel

transform

I, [i&u

of FE with respect

@(f. x, U) ?= -1” Bv proposition

2 there

. 11 + GXkr.

v2

1

2

E

is 6 > 0. independent

v)] dS2

to U. Assumption 1

- aZ-

)’ + h

(13) imply that

IJ: - 0.

of F. such that.

Since G, is convex in 11and. by (12), Kk:Xk+ X,. By the first order necessary condition KQI~ + $

x.

L’ is compact.

there

is l)F minimizing

@?.i on

G,(kr. x. u,) = w, E Ker Lk.

If u, = iv, - Ku,. then LkuE = - v, and. by duality. L ku, + f (kt, Moreover.

X.

J’,)

As in [5]. (u,) is bounded

in L”. Thus there UC,,L

=

c

$k(h).

in

u

ku c,, -

is a sequence

(13)

0.

then

Lku

to pass to the limit in (14) using

of theorem

1. Assumptions

(15) E,,--+ 0 such that

L”w*

f(kt, x. u,n) -f

Proof

M,

for every h E X,. &~(VE) C &k(h)

It suffices [lo]. n

+

Lxw*

in in

Lsw*.

Minty’s

device

(2) and (3) imply the existence piul

- NG F(r,x,

u).

as in [5] and in (15) using

of a, /3 > 0 such that (16)

It follows then from assumption (4) and lemma 1 that there is a 2rr periodic solution uh of (8) such that -L @k minimizes C$kon Xk. Let us estimate CL= &(-Lk&) from above. Using (16) and the definition of G. it is easy to verify that

/~~~/~~G(~.x,u)~cY.

Since Irk(t. s) = /3 sin(k

+ 1)f sin x E XL. we obtain

ckG&(h.)C_/

[n-&(sin(k+

l)~sin.r)‘]dR

R G c - c’k

(17)

for some c. c’ > 0. If. for some subsequence (k,,). lli,, ii' is bounded. bounded. Thus there is d > 0 such that

it follows from (8) that

L~,,llk,,~l.vis also

HA,,L=:L i,,ui,, I, 1 d d. The definition

of G implies

that G(r.s.

By using the preceding

inequalities. Ch,, =

contrary

to (17). Thus

we obtain

I [i(h,,. R

!L~~ILX-

LQQ~) + G(kt.x.

x. h-e

3. THE

0) 2 -d’.

11) 2 -F(r.s.

x.

CASE

so that the Fenchel

14 is

G(t. s. U) = 014 - F(t. s. H)

If p is the conjugate

exponent

where

and assumption

where

(7) imp]!, that

dQ

n

SUPERQUADRATIC

We assume that F satisfies (j)-(6)-(7) also the Legendre transform:

-L+x,,)]

of q. assumption

(6) imply that

transform

of F with respect

to

Subharmonic

The dual action

is defined

oscillations

on Xk = X,.,

of a semilinear

by

@k(u) = j-o [4&u. where

509

wave equation

u + G(kt,x,

u)] dQ,

Kk = Kk.p.

LEMMA 2. For every c E R the dual action @k satisfies the condition (PS),, i.e. if there sequence (u,) such that &(u,) + c and @;(u,,) + 0, then c is a critical value of @. Proof

Let (u,)

be such a sequence.

Then

Kkvn + g(kt, x, v,) - w, =fn 4: 0. n + x where

w E Ker Lk fl Lq. Since (&(u,)) E 3 @k(v,) =

is a

is bounded,

we obtain,

(20)

from (20), (18) and (19). that

I,[I(fn - g(kt, x, un>>u, +G(kr. x, on)] dQ

so that v,, is bounded in LP. Going if necessary compactness of & we can assume that

to a subsequence

and using

(20) and the

LP v,-

u L4

wn- w E KEr L,

(21)

L4

Kkv, + Ku. Since g is increasing

in v. we obtain 06

I

from (20) that, Vh E Lp.

R [(g(kt, x. v,) - g(kt, x. h))(v,

+ w, It follows

&$I,,

-

g(kt,x.

h))(v,,

- h)] dQ.

from (31) that 0 s

Using

-

- h)] dR

Minty’s

device

I

R [(w -

we conclude

KkV

point

of $k.

g(kt,x.

h))(v

that w -

i.e. u is a critical

-

&V

=

g(kt, x. v),

- h)] dR.

M.

5 10

It remains

only to prove

that &(u)

G(kt. x. u) dS2 3 =

iR f -R

WILLEM

= c. Since G is conv’ex in u. we have [G(kt.x.

u,,) +g(kt.x.

u,,)(v

-I:,,)]

[G(kt,x.

u,) + (f,, - K~c,,)(I’ - in)]

dQ dQ.

so that G(kt.

x, u) dR 2 lim -! ,1-x

*

G(kr.x.

u,,) dR.

But G(kr,x. Finally

u) dR cl&

G(kt,

x, u,,) d R.

G(kr,x,

u) dR.

we obtain G(kr. x. u,,) d Q -+

Moreover,

IR

by (31). we have KLU,,. u,, 151

!R

Kku

u.

so that &(u)

= lim qk([),,) = c. n- =

n

We shall construct a nonzero critical point of @k by using a variant of the AmbrosettiRabinowitz mountain pass theorem [I]. This theorem was applied to the dual action by. Ekeland [9] for Hamiltonian systems and by Brezis. Coron and Nirenberg [6] for the nonlinear wave equation. (see also [3.7]). MOUNTAIN PASS THEOREM. [6]

Let

X

be

a Banach

space,

$JE C’(X.

Sa), I),~E A’\.{O}.

I- = (s E %([O, 11, X): g(0) = 0. g( 1) = UU) and c = inf max @(g(s)).

qEr (15.7% I

If: (i) 4 satisfies

(PS),;

(ii) G(O), @(~a) s 0; (iii) there is 0 < r < I/uo// and d > 0 such that

then c 3 d is a critical Proof of theorem if u EXk,

value

of Q

2. Step 1. Construction

of a nontrivial

solution.

We deduce from (19) that.

Subharmonic

oscillations

of a semilinear

wave equation

511

and if u = p sin (k + l)t sin x, for some p 2 0. (22)

Thus,

for rk > 0 small enough,

is dk > 0 such that

there

b&~=rk+‘$k(U)~dk.

and. for R large enough, @k(R sin(k + l)t sinx) Since @k satisfies

(PS), by lemma

1, it follows

from the mountain

ck = inf max q&(S)) gEroc5s 1 is a critical

value

of &. i.e. there

G 0. pass theorem

that

> 0

is Uk E X, and wk E Ker Lk such that &Vk

+

g(kt,

X,

Vk)

=

wk.

(23)

As in lemma 1, uk = wk - Kkvk is a 2rr-periodic solution of (8). Moreover. since @(vk) = ck > 0 = @k(O). uk and uk are different from 0. From estimates for (uk) and (rvk) we shall prove that /&IL’+O. k-, x. Step 2. Estimates in the range. Let us first estimate ck from above as in [9]. We choose the path g(s) = sR sin(k + 1)t sin x, 0 G s G 1. Using

(22). we obtain ck

s

o~sa~l

@k(&))

S ?a;

&(P sin(k + 1)t

Sin x)

3

where

C is independent

of k. It follows from (23), (18) and (19) that X,

uk)

-

i&‘k(kt,x,

uk).

uk]

dQ

so that (24)

M.

512

We obtain

from proposition

h’ILLEh1

1 (25)

Step 3. .htimates in the kertlel. Since imply

uk = g(kt.x.

R G(kt.s. .I

In order to prove that /ukI~x -+o from ~uI, iLiZfollows from (8) that

(‘6)

vk) dQ

-+O,k--+

T,f(kt.x.

(6). (18). (19) and (‘1)

g(kt. X. vk) dR

V,!.

$

vk). inequditieS

x.

0. we use the Bahri-BrCzis

uk) = Tkf(kt,s.

M’k-Kpk)

=O.

method

[?I, It (‘7)

Let us write

g(u) = 4 (;;A”

t. x. u) - supf(t, x. 2) 11.1)

1

so that

We obtain

from (37)

or. after a change

of variable. i

SO

‘2.?!. ,, g(-/A

+ qk(T) - q&))

ds c 0.

that Mk = sup ess qk(7) < =. Moreover 2

h g(

-/3h

+

Mh

-

q,&))

ds 6 0.

(28)

Subharmonic

It

follows

from the definition

oscillations

of a semilinear

51.7

wave equation

of T, that (29)

where

@yk= ~u&I.

Let

then (29) implies

and we obtain

Assume

from (28)

Mk 2 fik, then

or, using assumptions

(6) and (7), /3(-/T& + Mk/2)‘-’

where ak+ 0 by (26) and pk-* 0 by (25). It is then easy to conclude argument shows that inf ess qk+ 0. n

d

0 that MA--* 0. A similar

Remark. After the completion of this work we learned from P. H. Rabinowitz that the existence of infinitely many distinct subharmonics was proved in [ 1.51under assumptions similar to (5)-(6)-(7). However, our approach is more direct. since [15] uses a perturbed linear operator. truncature and a Galerkin argument. Moreover we obtain an estimate in L” which is not contained in [15].

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M.

WILLEM

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10. 11. 12. 13.