Infinitely many periodic solutions for a superlinear forced wave equation

Prmrsd

Analysrr. Theory, .Werhodr & Appimaom. m Great Britam.

INFINITELY

Department

MANY

11. No.

1. pp. 85-104.

1987

0362-546.X87 S; 00 - “” Pergamon loumal~ L:d

PERIODIC SOLUTIONS FOR A SUPERLINEAR FORCED WAVE EQUATION

of Mathematics, (Received

Vol.

Faculty of Science, Nagoya University, Chikusa-ku, Nagoya, 164 Japan

1 November

1985; received for publication 20 January 1986)

Key words and phrases: Periodic solutions, nonlinear vibrating strings, perturbed symmetric functional.

INTRODUCTION IN THIS

AND

STATEMENT

OF RESULTS

article we shall study the nonlinear wave equation: Uff- u, + g(u) = f(x, 9, o(0, t) = u(n, t) = 0, u(x, f + 2X) = u(x, t),

(x7 t) E (0, ar) x R, fEW, (x9 r) E (0, d-r)x iw,

(0.1) (0.2) (0.3)

where g: iw+ 1wis a continuous function such that g(0) = 0, g(g)/E+ = as ]gl--, ;c and f(x, t) is a 2n-periodic function of 1. Many mathematicians are concerned with free uibrations; that is, assume f= 0 and find nontrivial solutions. Especially, Rabinowitz [14] proved that (O.l)-(0.3) _ possesses an unbounded sequence of weak solutions under the assumptions: g(j) is strictly increasing,

(g1)

and there exist p > 2 and 110 such that for Ifi L I, 0 < ,DG(~) = ,u ’ g(r) dt 5 Egg(:). I0

(&)

(Working within a restricted class of functions satisfying some symmetry properties. hypothesis (gr) can be eliminated. But when g(x, u) depends also on x, we can not eliminate (gt).) See also the earlier works by Rabinowitz [12], Brezis, Coron and Nirenberg [6] and Coron [8]. But forced vibrations for superlinear equations are not well studied. (For the existence of forced vibrations for asymptotically linear equations, many results are obtained. See Brezis [5] and its references.) However, in the case g(E) = ‘]@-‘E where s > 1 is a constant, we have proved in Tanaka [16] the following theorem (see also Ollivry [ll]). THEOREM

UL([OY 4 unbounded

1. (Tanaka [16].) Assume thatg(Q = ?I$]“-‘zwheres E (1, d(2) +l) andf(x, r) E x w (4 = u/4 +1> is a 2n-periodic function of t. Then (O.l)-(0.3) possesses an sequence of weak solutions in Lf,$‘([O, Jr] x W). 85

The present paper is a continuation of [16] and deals with the existence of forced vibrations for more general equations (O.l)-(0.3). Our main result is as follows: THEOREM

2. Suppose that g E C(Iw, W) and satisfies (gi), (gl) and there are constants s > 1 and C > 0 such that [g(5)] 5 C(]5]’ + 1)

Moreover,

for all

(93)

5 E W.

assume that f(x, t) E ~5.%([0, X] x R) is 2n-periodic 2 ->S--l

Then (O.l)-(0.3)

possesses an unbounded

and s satisfies

,u &U - 1’

(gl)

sequence of weak solutions in L”([O, ;t] x W).

As in [16], we use the dual variational method, that is, we convert the problem to a simpler one by a Legendre transformation which is used in Brezis, Coron and Nirenberg [6], and the method of Rabinowitz [13] which asserts the existence of infinitely many critical points of perturbed symmetric functionals. Rabinowitz [13] deals with the equation of elliptic type: xE D,

-An = g(n) + f(x), u = 0, where D C [W”’is a which ensures the Berestycki (31 and relman theory and

XEdD,

(0.4) (0.5)

bounded domain with a smooth boundary a D. [13] obtained the condition existence of infinitely many solutions of (0.4)-(0.5). (See also Bahri and Struwe [1.5]). Their arguments are based on restricted Lusternik-Schnienergy estimates for the functional on H;(D): u+

I/

lvul* dx -

D

1 G(u)

dx -

1 fu dx,

D

D

where G(E) = Jig(t) dt. In particular, in case N = 2 under the conditions (gZ)-(gJ g(-g)

= -g(e)

for all

and

j E W,

(gs)

Rabinowitz [13] obtained the desired existence result. Acting on S’-symmetry, Rabinowitz also obtained similar existence result for periodic solutions of perturbed second order Hamiltonian systems of ordinary differential equations. Acting on .I_?-symmetry, we also prove the existence of solutions without assumption (gj) for the wave equation (O.l)-(0.3). In case g(f) = +]@-‘g, we can verify the Palais-Smale compactness condition (P.S.) for the functional that corresponds to (O.l)-(0.3) with g(E) = t]t]‘-ig (see [16, lemma 31). But for general g(E) it is difficult to verify (P.S.). So we introduce a modified functional, in some sense, a truncation of g( Q, with a parameter E E (0,11.At first, with the aid of the arguments in Tanaka [16] and Rabinowitz [13], the existence of critical points of a modified functional will be established. At the same time a bound of critical values, which is uniform with respect

Infinitely many periodic solutions for a superlinear forced wave equation

87

to E, will be obtained. Using suitable estimates for critical points, we solve the original problem (O.l)-(0.3) via limit argument. This paper is organized as follows: 1. Dual variational formulation. 2. Modified functionals. 3. Construction of critical points. 4. Proof of theorem 2 and some remarks. 1. DUAL

VARIATIONAL

FORMULATION

Let Q = (0, it) x (0,2x) and for p E [l, =) we denote functions of t whose pth powers are integrable, i.e.

by LP the space of 2;r-periodic

ll~ll$= 1 I@, t)lP < =. R (From now on, the measure dxdt is understood to be taken in all integrals over Q.) Consider the operator: Au = u, - u, acting on functions in L’ satisfying (0.2) and (0.3). The kernel N of A consists of functions of the form: N =

1

[ E L;o,(R)

[(t + x) - c(t - x); 5‘is 2n-periodic,

and

j--I.*c(r) d t = 0)

= span{sin j,r . sin jt, sin jx . cos jr; j E N} (N is closed in L’). Recall that gi_ven u E L’ such that Jnu$ = 0 for all $ E N n L”, there exists a unique function u E C(Q) such that Au = u and Jou+ = 0 for all @ E N n L”. Set u = Ko(=A-‘u). The following properties are well known:

II~ull~o.~z Cju/,

with

cy = 1 - i,

for all u, ur and u?. Set J5 4=(1_1

E (1,2),

(1.4)

1 r=-+1 s

(1.5)

We shall work in the space: E=(u,,q;IQ

u@ = 0

forall

r#~E N~I Lp

88

K. TANAKA

with norm ll.lj9, where (l/p) + (l/q) = 1. We denote by (. , .) the duality product between E* and E or between LP and L4, that is,

(u, u>= / uu for all u E LP

and

u E L9.

JR

Next we set h(c) = the inverse function of g(j), then we get from (gJ, (9s) and (g,): (i) there exists a constant I’ 2 0 such that for 14 2 I’, qH(E)

h(t) dt I @z(E) > 0

= q i;

(hd

0

where q is defined in (1.4); (ii) there exist constants Cr > 0 and C2 > 0 such that for all g E W, jh(;‘)l 2 c#j’-’

(hd

- c2

where r is defined in (1.5); 2(r -

(iii) In particular,

1)

2-r

W

‘4.

from (h,) we have Ih(

I C3(lE19-1 + 1)

0 s H(E) d

for

GM 4 + 1)

5E W,

CM

EE R.

(hd

for

For a given f E L” we define the functional Z(U) by Z(U) = -$(-Ku,

U) + / H(u +fl JR

for

K E E.

By (h6), Z(U) is well-defined and C’ on E, (Z’(u), w) = (Ku + h(u +f), w)

for

u, w E E.

If Z’(U) = 0, we have

Set u = h(u +f)(=

-Ku

+ $I), then we get Au + g(u) =f.

Hence we shall seek critical points of Z(U). This is so-called dual variational formulation of the problem (O.l)-(0.3). At the end of this section, we state some properties of H(e) which will be used in the proof of our theorem repeatedly. LEMMA 1. (i) There is a constant C5 > 0 such that for all 5 E R, lIr(Qj d c#Z(#4-l)‘q

+ 1).

U-6)

Infinitely many periodic solutions for a superlinear forced wave equation

89

(ii) There is a constant C6 = C,(]]f]ll) > 0 such that for all u E E, IH(r.4+j-) - H(u)1 5 C6 ((i,

H(u +jqq-l):*

+ 1).

From now on we denote by C various constants which are dependent of u E E.

(1.7)

on llfllx but independent

Proof. By (h6) we have Is’]E CH(Q’/q - C’

for all

f E R.

By (h2) we get for 151E I’,

IEIIWE) B (cH(5p - C’)lh(Ql.

@-GE) 2

Hence we have for all E E R, j/z(5)] 5 C(H(#q-“‘q

+ 1).

This proves (i). For p E [0, l] we have from (i)

IH(u+ H-I - H(u+ fll 5 1’ Ih(u+ trill lfl dt P 1 < =

Integrating

iP

C#Z(u

+ tf)(‘+)‘q

+ 1) If] dr.

over R, we get

1 IH(U +pfl-

H(U

+fll 5

1’\

Cj(H(u

+$)(q-l)‘q+ 1) IfI dt

R

c((/ H(u+ rf))(q-l)‘q+ 1) R

iilq’il P

We set M =

sup

PEP.11I Q

H(u+M),

then we have for any p,

1 H(u+pf)SjH(u+f)+j lH(u+pf)-H(u+f-j R

R

5

s-2

c

H(u +fl

+

c(lwq-l)‘q + 1)

dt.

(“@

K. TANAKA

90

Hence we get MS

i

H(z.4+f)

+ 1M +

c.

R

i.e.

n

From (1.8) with p = 0, we obtain the desired inequality. 2.

MODIFIED FUNCTIONXLS

To verify the Palais-Smale compactness condition, we replace Z(u) by a modified functional I( E; u). Choose a function q( +) E C”(R, R) such that: (9

for

q(t) = jr/q-%

jtlzl;

(171)

(ii) there exists a constant cq > 0 such that for

77(r) = 0

ItI 5 cq;

(II?)

I

(iii)

i0

q(s) dt = ;,

(173)

(iv) r~is odd and nondecreasing.

(113)

Set w(g) = J&(t) dt. By (7,) and (qs) we get

For E E [0, I] we define Z(E; u) E C’([O, l] x E, W) by I(&; u) = -1(-K&

u) +

IR

H(u +fl

+

IR

w( EU).

In what follows we denote by ’ the Frechet derivative with respect to u. Obviously we have for all 2.4,w E E, (I’(&; u), w) = (Ku + h(u +fl + Ed,

w).

First of all, we have the following LEMMA 2. For E E (0,11, I(&; u) satisfies (P.S.) condition. in E satisfies

That is, whenever

a sequence

{Uj};=l

IZ(&;Uj)] d iri Z’(&;Uj)--,O

there exists a subsequence

for some constant in

E*

M,

as ZA 2,

of {Uj}T=iwhich converges in E.

(2.1) (2.2)

Infinitely many periodic solutions for a superlinear forced wave equation

91

Proof. (Cf. Benci and Fortunato [4].) Assume that {Uj};=l in E satisfies (2.1) and (2.2). We may write KU, + h(K, + f) + E~(EKj) = Wj + $j with WjE LP, llw,/lp + 0 and @jE N fl Lp. We have (KKj,

CC,) + I

h(Kj

+nKj

+ ~

&Kjq(EKj)

= (Wj,

Kj).

R

R

On the other hand, I(&; Therefore

Kj)

~(KKj,

=

K,) +

I

H(Kj

+n

+

R

I

W(&K,)

5 M.

R

we have _/ (H(Kj $2

+_fl

-

Jh(Kj

+flKj)

+ 1

(O(EKj)

-

IEKjq(EKj))

5 ni + IlWjllp IIKjll4.

R

It follows from (h,), (h,), (r],), (71s) that

IIKjllqs C(E~A~ suP llwjllp) We extract a subsequence-still denote by u,-such that Uj converges weakly to u. in E. Set h(s; X, t, 5) = h(e + f(x, t)) + E~(EQ. This is a strictly increasing function of j such that for some constants Ci, C2, CiE, C,, > 0, ci,j$j~-’ By the compactness

of

K:

E-,

iR

- CZEd Ih(& X, t, E)j 5 Cil5’j@ + cz. (which follows from (1.3)), we get

E*

(h(E;X~ f~ uj>

-

h(E;X,

r~ uO))(K,

-

uO>

as Z+%.

=(--KKj+Wj-h(&;X,t,Ko),K,-Kg)-,0

For a subsequence -still

(2.3)

denoted by u,-we

may assume that

(h(&; X, t,

Kj) -

h(&; X, f, KO)) (Kj -

Ko) +

0

(h(E;X, f,

uj)-

h(E;X,f~

KO)s

ll(x,

uO))(Kj

-

a.e. in Q, f)~

where Z,(.K,t) E L’ is a fixed function. Thus we obtain from (2.3) Kj(x,

r>+

l"j(x,

where &, t) E L4 is a fixed function. Hence we have Uj’ u. strongly in

uO(x,

f>l s

l*(x9

E.

n

f>

a.e. in Q,

r>

a.e. in Q,

We shall find critical points of Z(u) by first obtaining critical points of I(&; u). These critical points satisfy Ku

+ h(K

+f)

+ Ed

E N.

K. TrLVAKA

92

With the aid of appropriate estimates (nt), we find these are critical points Next, as in [13] and [16], we Ci([O, l] X E, R). Let x E c”(W, R) such that x(t) = x(r) 5 1 for I E R. For E E [0, l] and

]]u]lr5 C wh’ic h are independent of E and the assumption of Z(u) for small E > 0. replace I(&; u) by the another functional J(E; u) E 1 for r 5 1, x(t) = 0 for t Z 2 and -2 5 x’(t) 5 0, 0 5 u E E we set

@(E; u) = a(l(&; U)’ + m)i!I, v(s; u) = X(@‘(E; u)-‘(-Ku, J(E; u) = -I(-Ku,

u) +

u)),

iR

H(u)

+ q(E; u) j- (H(u +n R

- HGO) + 1 I R

E C’([O, 11 x E, ‘JV, where 6q + 4

a=->1

2-q

and m > 1 is a constant. For large m > 1, J(E; u) possesses the following property: & J(E; u) 2 0 That is, J(E; u) is a nondecreasing &I(,;

for all

uE E

(Wu

+./I

-

H(4))

(2.4)

u))(-Ku,

X 1

u)Q(s; u)-sz(&; u)

d&u). R

R

Since Jouq(su)

E E [0, l]

function of E for fixed u. In fact, we have

u) = (1 - a2X’(iP(&; u)-‘(-Ku, X 1

and

E 0 for all u and E, it suffices to show that for large m > 1, la’~&.)(--Ku,

u)@(E; u)-~~(E; u) 1 (H(u +f) - H(u))1

5 4 for all

uE E

If u 4 supp ~_J(E;.), (2.5) clearly holds. Otherwise,

and’ E E [0, 11. we have the following lemma.

LEMMA3. If u E supp ij.~(E; . ), then /(-Ku,

I

R

u)j 5 2@(&; cc),

H(u f f) 5 ~Q(E; u).

(2.5)

Infinitely many periodic solutions for a superlinear forced wave equation

93

Proof. From the definition of I~/(E;. ), we have for u E supp V(E; . ), (-Ku,

u) 5 2@(E; u).

Hence for u E supp V(E; .), H(u

+j-)

= I(&; u) + g--Ku,

O(&U) z5

u) -

2@(&; u).

iR

IR

On the other hand, we have for all u E E, (--Ku, u) = -2Z(&; u) + 2

iR

H(u +J-) + 2

Thus we obtain the desired inequalities. Therefore,

O(ElO z -2Z(E; u) z -20(&; u).

n

we have by (1.7) I&‘( * *)(-Ku,

Up(E;

u)-v(&;

u) 1 (H(t4 +j-) - H(u))1 s CqE; R

up,

where C > 0 is a constant which is independent of E, m and u. Since @(.s; u) 2 am ‘j2, choosing m > 1 such that C(am”‘)-‘/q 5 l/2, we have (2.5). From now on we fix m > 1 so that (2.4) holds. Next we shall show that large critical values of J(E; u) are critical values of Z(E; u). PROPOSITION 1. There is a constant M > 0 independent of E E (0, l] such that for E E (0, 11, .Z(E;u) > M and ]]J’(E; u)]lE. < E imply that J(E; u) = Z(E; 10.

The proof of proposition 1 is rather technical and independent of further arguments, so we prove it in the Appendix. As immediate corollaries to proposition 1, we have the following. COROLLARY 1. Whenever u E E and E E (0, l] satisfy J’(E; u) = 0 and J(E; u) > M, then I(&; u) = J(E; u) and Z’(E; u) = 0. 2. For E E (0, 11, J(E; u) satisfies (P.S.) condition on where M is independent of E.

COROLLARY M},

L~,~(E)

Proof of corollary 2. Using lemma 2, the proof is as in [16, corollary

3. CONSTRUCTION

OF CRITICAL

In this section we construct critical points of J(E; u) for Note that K is a compact self-adjoint operator in {u E L2; (u, @) = 0 =

for all

POINTS E E

(0,11.

+ E iV fl L2}

span{sin jx * cos kt, sin jx * sin kt;

j = 1,2,.

..,

k=0,1,2,.

..,

j#k}.

= {u E

21.

W

E;

J(E;

u) z

94

K. TANAKA

Its eigenvalues are {1/(j2 - k’); j # k} and corresponding eigenfunctions are sin jx . cos kt and sin jx * sin kt. We observe that each negative eigenvalue possesses even multiplicity. We rearrange the negative eigenvalues in the following order, denoted by -pi

5 -,u, 4 -,u, 5. . . < 0.

Here for each n, there exists a one-to-one invariant space:

correspondence

between -p,, and a 2-dimensional

span{ei = sin jx *cos kt, e; = sin jx * sin kr}(‘j2 - k2 = -,p;‘). Next we shall define spaces E,,, E,I and the projector E,=span{er,e;,ei,e;

,...,

P,,: E+ E, by

e,‘,e;}, for

E,I = {u E E; (e:, u) = (e;, u) = 0 P,u = &

i = 1,2,.

for

$ .((e:, u)e: + (e;, u)e;)

. . , n},

u E E.

For all u E E,,, we have by (h6) J(e; u) s +&4:

+ C(ll4l; + l),

where C > 0 is independent of E E [0, 11. Since 11.II2 and 11.II4 are equivalent to j/a11,in E,, there exists a constant R, > 0 such that J(E; u) s 0

for all

u E E,

with

for all

E E [0, 11.

]]u]]~B R,

We may assume that R, c R, +1 for all n. For 0 E [0,2x) = S’, define To: E+ E by (TBu)(x, t) = u(x, f + 0)

for

u E E.

Let BR = {u E E; l/uII, 5 R}

for

R > 0,

D, = BR* nE,,

rn={YE

C(D,, E); y(T@u) = Toy(u)

for all

8 E [0,2n)

and

u E D,,

Y(U)= u if lbllr= R,I. Moreover,

set

u,=

I

u=

+ ‘+‘;t E [O,&+I], w E BRn+,

A, = {AE C(U,, E); AjD, E r,, or

UE (BR,,+~~BR,)~E~I.

and

A(u) = u

II E,

if

and

lb4 5 Rn+l),

]]u]ll= R,,+*

Infinitely many periodic solutions for a superlinear forced wave equation

95

Define for n E N and E E [0,11,

The above definitions are analogous to those of Rabinowitz [13], which are used to prove the existence of infinitely many periodic solutions of a perturbed second order Hamiltonian system. By the definitions it is clear that C,(E) 2 b,(~). Since corollaries 1, 2 hold, i.e. for E E (0, 11, J(E; u) satisfies (P.S.) and large critical values of J(E; u) are also critical values of I(&; u), we have the following result by Rabinowitz [13]. He proved this with the aid of the standard “deformation theorem”. 2. (Rabinowitz [13]). For E E (0, 11, suppose that C,(E) > b,(c) Z M. Let d E (0, C,(E) - b,(~)) and

PROPOSITION

A,(&; d) = {A E A,;./(&;

A) 5 b,(~)

+ d

on

D,}.

Define c,(E; d) = IEp;c,d) uy; Jh n 9

W)).

n

Then c,(E; d) is a critical value of I(&; a). Hence the existence of a subsequence of c,( E)‘Swhich satisfy C,(E) > b,(c) 2 A4guarantees the existence of critical values. The main result in this section is as follows. PROPOSITION 3.

There exists a subsequence

{nj};= 1 such that for some constants q E (0, l] and

dj > 0, C,,(E) - 2di Z&(E)

Moreover

2 M

for all

&E (0,

Sj].

there exist sequences {mj}i”,1 and {Mj}i’, 1 which are independent as

mj-*x

mj 5 From propositions PROPOSITION

4.

c,,(E; dj)

5 Mj

for

of E and

j-xc, &E (0,

Sj].

2, 3 we deduce the following.

For any j E N and any E E (0, g], there exists Uj(E) E E such that I’(&; U,(E)) = 0, I(&; Uj(E))

Proof of proposition

Step Step Step Step

= Cnj(E;

dj) E [Mj, Mj].

3. The proof of proposition 3 is divided into several steps. 1: the growth of the value b,(O). 2: existence of a sequence {nj}i”,, such that c,,~(O)> bnj(0). 3: continuity of the functions b,(a), c,( .) : [0, l] + W. 4. conclusion.

aw

[f uog!sodoJd

‘911 may alzeq

uo ‘05 (W - I>‘n@- z>x-) aAeq fn(a

- z) ‘n(0

-

z)x-)

aM axJay aht ‘{N + (Ha

‘Tug 3 no 3 .Llu~u!s} ‘nay--)

saqdq p&s

= (0

= (n ‘v

+‘g 3 n ‘pueq Jaqlo - 1) a&28

IPJO3

aqs

-) aheq

‘N3U

aM

‘jO01d

o+l/(I_1)~-u~3 5 un

ieyl qsns 0 < 93 ~ue~suo:, e qs!xa aJay$ 0 < 9 fhe Jo3 JafioaJow ‘:3 3 n

II~ Jo3

;IlnOll”n S (n ‘w-)

leql yns 0 < % sluelsuo~ aJe aJaqL (*[f uoysodoJd n

‘0 #

‘911 eyeue~

.3~) ‘g vlru~~

{d = ‘l/M~jl f I-%’ 34) U (“a)A ‘S! 1-q

la8 am ‘[6] u! uIaJoay1 ureln-ynsJoa aqi 30 uo!sJak,S ue %uyrClddv ‘(0) = l-“g u rS x!~ pun! deur lUC?~JeA!nba-,Su1? sl (I-“3 ‘ne)~ 3 A1-Ud = Aar-“d uaqJ_ ‘0 %~!uyuos (“a),-(@) 30 1uauoduro2 aqi aq A ia? ‘“a u! (J 30 pooqJoq@!au (0 [IF?~03 (dg),_(La) ‘sploq (Z’f) pue ‘n pue 0 IF = ((da),-(@))B~ .a.!> 1ue!JeAu!-,S papunoq e s! (“f&(~a) ~03 (+?J)@ = (YZ)xaeI ‘a.! ‘WE~JeApIba-,S s! @ asq, ‘“3 3 A ia? .b F 1 l&?qialoN $001d

‘0 # {d = ‘IIMajj t’-:Y

3 M}

U (“a)x

‘“J 3 x pue (qu ‘0) 3 d ‘N 3 U IlE?JOA ‘p VLW3~ wxuural

(Z'f)

%1!~0~~03

‘(0)

aqi paau

= rS X!J u (a)

aM

‘g

uoysodoJd

30 3ooJd

ayl

JOJ

aguw 1q1

leap

S! 11

0

‘lp (1 ‘x)n

5 q

I

-

(J‘X)n = (J‘X)(na)

1

hq 3 ~3 .{N 3 .,:x/y} ‘i(uZ ‘O] 3 0

IF

'N 3 ~4 qXlS

IF

:a JOieJadO

Jo3

papunoq e auyap aM

ueds =

n = neJ tg 3

Jo3

:las lu!od

(T'f)

leq]

n} =

,s x!d

paxy e sassassod {(q

~-(‘-~)/(l-,)~~~3

‘01 3 0 !@L}lr?ql

aloN

2 (0)“4

0 < q3 lLIelSU0~ &?S! aJaq1

wv.w.L

‘0 < Q AlIE JOJ ‘C NOIlISOdOtId

96

‘)I

97

Infinitely many periodic solutions for a superlinear forced wave equation n

Thus we obtain the desired inequality.

Proof of proposition 5. Let y E I, and p E (0, R,). w E (xDn) rl {u E E,i_ i ; l/Qu/l, = p}, hence we have

4 there

is a

J(0; cl).

inf

sup J(0; y(u)) z J(0; W) z UED,

By lemma

u E E j+_L:‘QUjir = p

Let u E E,L_, with IIQul\, = p. It follows from (h,) that J(O;u)Z-A(--Ku,u)

+

IQ

H(n) - C (0,

S--1(--Ku,u)+1

iR

*(U))iq-l)jq

+ 1).

H(u)-C

= ’ - &(--Ku, U) + cl/ull: - C’. By lemma 5 and boundedness

of the operator

J(0; u) z - &2,-i

h -

la,_,p*

Q: L’+

L’, we get

IIQullS+ CllQull: - C’ + Cp’ - C’.

Hence we obtain 6,(O) 2 s;p (z

( 1 _ ;)

ka,_,p’ C2i(2-r)

+ Cp’ - C’) (y)

-r’(2-r)

-

C’.

W

Thus we obtain the desired inequality from lemma 5. Step 2. Existence

of a sequence {nj}i”,i such that c,,(O) > bnj(0).

Arguing indirectly, we have the following proposition, PROPOSITION 6. If c,(O) = 6,(O) for all n h no, then there is a constant

b,(O) S c,,nq To prove proposition

forall

C,, > 0 such that

nEN.

(3.3)

6, we need the following lemma.

LEMMA 6. There is a constant p > 0 such that for u E E and 8 E [0, 2n),

IJ(O; T@U)- J(0; u)l S P(IJ(O; u)](q-i)‘q + 1). Proof. By the definition of J(0; u), we have

KO; TBU) - J(0; u>I 5 V(O; U) 1 IH(u +f) R + V(0; Tf3u)

I R Ifmu

+f)

-

- H(u)/

w-@)I.

K. TANAKA

98

As in lemma 1, there is C > 0 independent

of E such that for all u E E and 8 E [0,2x),

IWou +f) - q7-,u)/ s c ((1 H(T@U

\

+f))Lq-l)lq +1),

H(fou +f))(q-l)/q +1). jk+f)-ii(.)~C((~ R

R

Hence, using lemma 3, the proof is essentially as in Tanaka [16, proposition Proof of proposition

6. As in Rabinowitz

11.

w

n

[13, lemma 2.311.

Comparing (3.1) and (3.3), we see that the inequalities are incompatible under the condition (h,). Therefore there is a sequence {nj},?,l such that forall

c,,(O) > b,,(O) 2 M

Step 3. Continuity of the functions b,( .), c,(a): [0, l] -

jEN.

;3.

Proposition 7. The functions b,( .), c,( .): [0, l] -+ $8 are right-continuous. are continuous at 0.

In particular,

they

Proof. We shall deal with b,( .) (the argument is the same for c,( .)). For any ;/ E T,, y(D,J is compact. Hence the function

E-

sup .I(&; y(u)); [O, 11+ 3 UED,

function of Thus b,,(s) = inf sup J(E; y(u)) is an upper-semicontinuous YEr, uED, E E [0, 11. On the other hand, it follows from (2.4) that b,(s) is a nondecreasing function of E. Hence we find that b,(.s) is a right-continuous function. n

is continuous.

Step 4. Conclusion.

We set dj = 4(cni(0) - b.,(O)) > 0.

From the definition of b,j(0),

there is /Ii E Anj such that

J(O; Aj(u)) ~ b,j(0) + Idj By proposition 7 and the continuity constant Sj E (O,l],

on

of the function:

E-

D,,.

sup J(E; Ai(U

we get for some

UED,, C,,(E)

-

J(E; A,(U)) S b.,(e)

b,,(e)

+ dj

2

2dj,

(3.4) on

for all E E (0, Sj]. These (nit dj, ~ji)jpossesses the desired property. t?Zj= bni(0), Mj =

SUP I(&; SUP O
(3.5)

D,,i,

Aj(ll)),

Set

Infinitely many periodic solutions for a superlinear forced wave equation

then “I, + m as j- x follows c,~(E), c,,(E; dj) and (2.4) that

from

(3.1).

We derive

from

99

the definitions

of b,, (E9,

T?IjG b,, (0) 5 b,,(E) ‘c c,, (E) 5 c,, (E; dj).

By (3.5) we have kj E A,,(&; dj) for all E E (0, S,]. Hence C,,(&;

dj)

~

SUP

J(E;

~j([c))

~

iMj.

l&u,,

Thus the proof of proposition

4. PROOF

3 is completed.

W

OF THEOREM

2 AND

SOLME REMARKS

With the aid of the argument which is essentially due to Brezis, Coron and Nirenberg [6] (see also Rabinowitz [12]), we obtain the following estimates for the critical points of I(&; u). PROPOSITION 8. .s E (0, l] satisfy

(Cf. Brezis, Coron and Nirenberg - 161, _ _ lemma 2.) Assume that u E E and I(&; U) 5 L.

and

I’(&; u) = 0

Then there exists a constant CL > 0 independent

of E E (0, l] such that

/IL& 5 CL. Now we can show that Uj(E) is a critical point of Z(U) for sufficiently small End of the proofof theorem 2. By proposition is a critical point U,(E) E E of I(&; u), that is,

Kuj(&) + h(Uj(E) +f)

E >

4, for any j E kJ and for any E E (0, Sj] there + E~(Eu,(E))

E N.

Moreover, I(&; Uj(E>)E It follows from proposition

where CMi is independent

Lrnj, Mj].

8 that

Il”j(E)ll= s c.Mj

0.

for

EE (0,

Sj],

of E. Choosing Ej = min{bj, c,C,;t}, we have by (qJ

V(&jUj(&j)) = O, i.e. Kuj(Ej) + h(Uj(Ej) +f) Hence uj(sj) is a critical point of Z(U). Thus pi = h(Uj(Ej) + ~)

E IV.

is

a weak solution of (O.l)-(0.3). G,(E,))

Since m, ---f 3~as j= Q!:

x, as j-2.

Q&j)) - =

In particular, (I[c,(Ej)(ix

+

as i-x.

x

Thus we obtain

Ii4 - x Therefore

as ;-

the proof of theorem 2 is completed.

x.

4

Remark

1. If 4 = r, i.e. !L = s + 1, we can verify that I(U) satisfies the Palais-Smale So we can prove the existence result without introducing I(&; u) (E E (0, 11).

condition.

Remark 2. In addition to the assumptions of theorem 2, if g and f are C”, then the solutions are also C’ (see B&is and Nirenberg [7, theorem 21). Remark

3. In the case that the nonlinear

term g(x, v) depends on x, we shall consider the

equation:

vrr

-

v.r.r+ g(x. u) = f(x, t),

v(0,

t) =

v(x,

t +

u(;T,

(x,

t) E (0,

,7)

x

w,

rr)

x

w,

fE8,

f) = 0,

2x) = u(x. t),

(x, t) E (0,

theorem 2 clearly holds under the similar conditions to (g,)-(gJ. But in the case that g(x, t, u) depends also on r, we must act on Z,-symmetry as in [16]. That is, assume that g(x, 1, v) is odd in v, then we can prove the existence of infinitely many solutions under the similar conditions to (gi)-(gJ. Remark 4. The question of whether or not the conditions (g2)-(gl) are essential for these existence results remains open. In case g(u) = ]u(s-‘u, (g,)-(g4) are equivalent to the restriction s < d(2) + 1. Brezis [5] conjectured that when g(E) = E3, (O.l)-(0.3) possesses a solutioneven infinitely many solutions-for every f (or at least for a dense set of f’s). See Bahri [2] for the similar question for elliptic equations and see also [17]. Acknowledgment-The

author would like to thank Professor Haruo Sunouchi for his advice and hearty encouragement.

Nores. The referee suggested that the author should refer to the papers [18, 191. In the recent paper [19], Bahri and Lions considered the problem: -Au = ]u]P-ru +f(x)

in

R,

u EHb(R).

(“) where R C w” is a bounded domain with a smooth boundary, p > 1 and f(x) E LLv’(.‘-Z)(R) if N 2 3, f(x) E Lq(R) with q > 1 if N = 2, q = 1 if N = 1. They obtained the following result: ifp < :V’/(,V- 2) (p < = if N z 2), there exist infinitely many solutions of (*)_ The author would like to thank the referee for suggesting it to him. The research for this paper was done at the Department of IMathematics, Waseda University, Tokyo, Japan.

Infinitely many periodic solutions for a superlinear forced wave equation

101

REFERENCES 1. AMBROSE~ A. & F~BINOWTZ P. H., Dual variational methods in critical point theory and applications, 1. funcr. Analysis 14. 349-381 (1973). 2. BAHRI A., Topological results on a certain class of functionals and application. /. funcr. Analysis 11, 397-427 (1981). 3. BAHRI A. & BERESTYCKIH., A perturbation method in critical point theory and applications, Trans. Am. marh. 3. Sot. 267, 1-32 (1981). 4. BEXCI V. & FORTUNATOD., The dual method in c%cal point theory. hlultiplicity results for indefinite functionals, Annali Mar.‘pura appl. 32, 215-242 (1982). 5. BR~ZIS H., Periodic solutions of nonlinear vibrating strings and duality principles, 8, 409-426 (1983).

Bull. Am. marh. Sec. (N.S.)

6. BRGZISH., CORONJ. M. & NIRENBERGL., Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Communs pure appl. Marh. 33. 667-689 (1980). 7. BRBZISH. & NIRENBERGL.. Forced vibrations for a nonlinear wave eauation. Commun.s qure aoo[. . . Mach. 31, 1-31 (1978). 8. CORONJ. hf.. Periodic solutions of a nonlinear wave equation without assumption of monotonicity. Math. Annln 262, 273-285 (1983). 9. FADELL E. R., HUSSEINIS. Y. & RABI~OWITZP. H., Borsuk-Ulam theorems for arbitrary S’ actions and applications, Trans. Am. math. Sot. 274, 345-360 (1982). 10. LOVICAROV.~H., Periodic solutions of a weakly nonlinear wave equation in one dimension, Czech. Malath.J. 19, 324-342 (1969). 11. OLLIVRYJ. P., Vibrations forcees pour une equation d’onde nonlineaire, Cr. hebd. stanc. Acad. Sci. Paris Ser. I, 297, 29-32 (1983). 12. RABINOWI~ZP. H., Free vibrations for a semilinear wave equation. Communs pure app’[. Marh. 31. 31-68 (1978). 13. RABISOWITZP. H., Multiple critical points of perturbed symmetric functionals, Trans. Am. math. Sot. 272, 753769 (1982). 14. R~\BINOWITZ P. H., Large amplitude time periodic solutions of a semilinear wave equation, Communspure appl. Marh. 37. 187-206 (1984).

15. STRUWEM.. Infinitkly many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta math. 32, 335-364 (1980). 16. TANAKAK., Infinitely many periodic solutions for the equation: u, - u, ? /u/‘-‘u = f(x, t), Communs part difi Eqns, 10, 1317-1345 (1985). 17. TANAKAK., Density of the range of a wave operator with nonmonotone superlinear nonlinearity. Proc. Japan Acad.,

62A, 129-132 (1986).

18. CLARKE F. & EKELANDI., Hamiltonian Marh. 33, 103-116

having prescribed minimal period, Commune pure appl.

trajectories

(1980).

19. B*R! A. & LIONSP. L., Remarques sur la thtorie variationnelle stanc. Acad. Sci. Paris 301, 145-147 (1985).

des points critiques et applications, C.r. hebd.

APPENDIX The purpose of this appendix is to prove proposition 1. To do so, we need the following lemma. LEM.M A.l. There is a constant M,, > 0 independent I(&; u) 2 (l/3)/( E; u).

of E E [0, l] such that J(E; u) Z M, and u E supp W(E; .) imply

Proof. As in Tanaka [16, lemma 21.

The proof of proposition 1 is essentially as in [16, proposition complicated estimates are needed. Proof of proposifion

21 and Rabinowitz [13, lemma 1.391. But more

1. It suffices to show that V(E; u) = 1, i.e., by the definition of I~(E; u), to show that (-Ku,

u) 5 a(&; u)

for u E E and E E (0, l] such that I(E; u) > M

and

~/J’(E;u)llE. < E.

(A.1)

K. TANAKA

102

If (-Ku, u) 5 0, then (A.l) is obvious. So we assume that (-Ku, Note that

u) > 0.

(l)(c; u), u) = - (-Ku, u) + j h(u)u + l/l(E; u) j a +

(h(u +f)

- h(u))u

R

(v’(c UL u> j* (ff(u +n - H(u)) + j W(EU), P

where (r/!‘(&;u),u) =~‘(@(&;U)-‘(-lCu,U))~(E;U)-’

x [Zq&;u)*(-Ku,U)

(I’(&; u), u) = - (-Ku,

u) + jo+

-a”(-Ku,U)z(E;U)(I)(&;U),U)1,

+ flu + jn EUn(&U).

Regrouping terms shows that

(J’(E;u), u) = (1 - v(E; u,) j h(u)u + (ly(Eiu) - r,(E; u)) j h(u +f)u - (1 - T*(E;U))(-:u,

U) + (1 - T,(c; U)) I, su4(E:),

(A.21

where T,(E; u) = a’~‘(. .)(-Ku,

u)@(E; u)-~Z(E;u)

J(Wu+f) - H(U))? n

(H(u +f) - H(u)).

T*(&; u) = Tt(E; u) + 2x’(. .p(&; u)-’ j

R At iirst we shall show that

SU~~~T,(E;U)I,~TZ(E;U)/;EE(O, l],uE E, J(E;u) > M and (-Ku,u)> O}d 0 as

iM-+r.

Suppose that E E (0, l] and u E E satisfy I(&; u) > M and (-Ku,

u)> 0.By (1.7) we have

+ 1). IT,(&;u)/ 5 C~~'(+D(E;U)-~(-KU,U) j/j H(u+~))(~-"'~ n If u $ supp V(E; .), then T,(E; u) = 0. Otherwise, by lemma 3 and the definitions of V(E; u) and Q(E; u). we have

((j H(u+f))(q-‘)8 +1)

CqE;U)-’

Ir,(&;U)jd

0

(A.3) z CqE; u)-‘1’. On the other hand, by lemma A.1 we have @(E; U)

t

I(&;

U) ?Z d.!(E;

U) 2

f,M,

hence as

/T(I E,U)~~3~CM-'~~+O

M + =.

Similarly we have

I TAG u)l- cl

as

M-x.

Hence we can assume that ]T,(E; u)l 5 l/2, /TZ(&;u)] Z 1 and 1 - Tr(&; u) q(l-T,(e;u))

__‘,I 2-2

‘_I

(q

21 =b’o

Infinitely many periodic solutions for a superlinear forced wave equation

103.

for sufficiently large M > 0. By (A.2) we get I(&; u) 1 - Tz(E; u)

-1) (-Ku,u) 2

i

= q(1 - T,(c;u))

+

1 - 446 u) iR 4(1- TIC&iu,>

+

q(l

y&y;; u))

j

n

I

T,(E; u)

(qH(u + n - h(u)u)

(qfw +f) - h(u +f)u)

I (qH(u+ f) - wu+flu)

- 41 - TIC&;~1) n + ;

jQ(44EU) -

w(m)) (A.4)

= (I) + (II) + (III) + (IV) + (V). By (h2) and lemma 1 we have

1 (qH(u+f)

- Nub) = 1 (qH(u) - h(u)u) + J-(H(u +f) - H(u))

2:c-c((jn*(u+f)~(q-~~q+l),

R

f0

(qH(u+ f) -

wu+flu)

(qH(u+ f) - h(u + f)(u + n>+j Nu + flf

=j

R

R

z-c-c

I

(H(u +f)(q-‘)‘q + 1)/j

~-~(jn,u~~)~(q-l)iq-~. Since 1 - V(E; u) and V(E; u) are nonnegative,

we get

(II) + (III) 2 - c 0,

H(u +f)J”-“’

(A.5)

- c.

By (/I?) and lemma 1 we also get

j

jnh(u + flu j

5

I, h(u + f)(u

+ f) +

1I, h(u + flf 1

5 j (qH(u +f) - C) + / (H(u +fp-l)‘q R R

Hence we have

+ l)lfl

K. TANAKA

104 If u @ supp

~,!J(E: .). then T,(E; u) = 0. Otherwise I(IV)i S C!T,(E

by lemma

u)~(‘Q(E;

u)-’

5 m(&;

3 and (.\.3)

we get

u) + 1)

((\ H(u +fq-‘)’+1)(2qE;u) + 1) n

(A.61

-c!ij*H(uifl)(q-‘i’*+lj. By (vi) and (qj) we have I(V)/ 5 c. Hence

from (A.4).

(A.5)-(A.7)

(A.7)

we deduce 1

z(E;U) - q(1 - Ti(&;u)) (J’(c cl). II) ~b(-Ku.u)-C(~~*H(l‘+f))‘q-‘lq+I), By the assumption,

Adding

~(J’(E; u). u)i s ~/I’(E; u)il E‘ Iuilq 5 f/IuIlq, hence

bI(E; u) = - (b/2)(-G,

u) + b j

H(u +f)

+ b ,/ LO( EU) an d t h e above R

n b (1 + b)Z(e; u) e - (-Ku, 2 + [b j

we have

formula,

u)

H(u ff)

- Cij

H(u ,njiq-‘l’q

- C]

n +

b w(m) [J R

hand,

under

the assumption M
(-Ku,

u) > 0, we have

u)

< j

H(u) + VI(E; u) j

(H( u +f)

- H(u))

+ j

n

n S2

- c]

(-Ku, u) + (VI) + (VII)

= p On the other

-2jqjo:(&)~+

IR

H(u+f)

+ C+

w(m).

Hence jnH(u

+

f)+ ,/w(Eu)+~

as

R Thus we have (VI) + (VII) + 3: Now we may assume

as

M-*x.

that I(&; u) > M implies (VI) + (VII) E 0.

Thus (-Ku, Thus the proof

is completed.

W

4&u) n

u) d a[(&; u) 5 @(&; u).

M-+ =.

we get