Time periodic solutions to a semilinear beam equation

NonlinearAnalysis, Theory, Methods & Applications, Vol. 12, No. 3, pp. 279-290, 1988.

0362-546X/88$3.00 + .00 ©1988 PergamonJournals Ltd.

Printed in Great Britain.

TIME PERIODIC

SOLUTIONS

TO A SEMILINEAR

BEAM

EQUATION E D U A R D FEIREISL Institute of Mathematics of the Czechoslovak A c a d e m y of Sciences, Zitn~ 25, Praha 1, 115 67, Czechoslovakia

(Received for publication 1 April 1987) Key words and phrases: Semilinear b e a m equation, periodic solution, superlinear function, Orlicz space.

LET {Pr} be the problem given by the equation Utt(x, t) + Uxxxx(X, t) + cu(x, t) + f(x, t, u(x, t)) = 0

(x,t)~(O,~)

(o.1)

× RI,c>-O

together with the homogeneous boundary conditions

u(O, t) = u(:r, t) = Uxx(O, t) = Uxx(~r, t) = O, t E R 1

(0.2)

as well as with the periodicity condition

u (x, t + 2-~) = u(x, t), (x, t) E (O, :r) x R ~,

TEN

(0.3)

where the symbol N stands for the positive integer set. Under appropriate hypotheses on f(x, t, v), in particular a superlinear growth in v, we find condition (R) (see Section 3), which is sufficient for the problem {Pr} to possess at least one solution. To begin with, we review some recent results related to this subject. The existence of at least one nonzero solution was proved in [1] provided that f(v) = o(v) for v ~ 0 and f(v) = O([vl p-zv) if v ~ -+ o%p > 2. In case f is odd, the abstract results achieved in [6] may be used in order to obtain an unbounded sequence of solutions to our problem. The general situation, however, when f depends actually on t and is superlinear, has not been treated satisfactorily in the literature to our knowledge. The method presented in [8] could be perhaps applied here to get similar results as for a wave equation. In this case, however, the function f has to be subjected to essential growth restrictions. The method we have used is based on the Rayleigh-Ritz approximation. The corresponding variational problem is solved on a finite dimensional space employing basic topological arguments only. The limit process consists of the combination of compactness arguments and slightly modified monotone operator theory. The latter makes use of some elements of the theory of Orlicz spaces. Our main existence result (stated in Section 3 and proved in Sections 4-6) covers some important special cases. For instance if f does not depend on t, we are in a position to prove the result analogous to that one achieved in [7] for a wave equation. 279

280

E. FEIREISL

1. THE LINEAR OPERATOR D Throughout the paper, the symbols ci, i E N will denote all strictly positive real constants. Z , R 1 are respectively the set of all integers, the real line. Eventually, we set

Qr = {(x, t)lx ~ [0, erl, Let us pay attention to the linear operator D, Dg = gtt + gxxxx defined, for the present, for all functions being both smooth on int(Qr) and satisfying (0.2), (0.3). The operator D is known to possess a system of eigenfunctions of the following form [ V ~ V 2 ;r -1 sin(kx) sin(jTt),

kEN, jEN kCN, j=O

ekj(x, t) = 1V"Tzt-1 sin(kx), I V 7 V ~ zr-1 sin(kx) cos(jTt),

kEN,-jEN

together with corresponding eigenvalues

Zkj = k4 - f i T2. We denote A = A ( T ) = max{2tkj + Cl)~kl + c < 0}

MA = span{ekj]Xkj + c = A}. Observe that M A is a nontrivial finite dimensional livear space. The spaces Lp of periodic functions can be defined as a closed real linear hull of the system {%}k,j according to norms

IIo11~ =

Io1~

,

p e [1, + 2 )

T

IIol[~ = sup Iol.

Qr

In this context, the system {eki}k,] represents a normed orthogonal basis to the space L 2. Consequently, the Fourier coefficients

akj(V) = fQr Vekj may be defined and, in view of the estimate

Ilekjll~ --< x @ v 2 :r -1 the definition above makes sense even for every function v E L 1.

(1.1)

Time periodic solutions to a semilinear beam equation

281

The operator D has a selfadjoint extension in L2 D o = ~ , ~.k]akj(v)ekj

k,j

~ ( O ) = (1)lV (=. L2, £1~kjakj(D)12 ~ q-a}. k,] For later purposes, the following denomination is convenient {D => z} = span{%]Xkj = z},

z ~ R 1.

We pause to list some technical results. LEMMA 1.1. lira A ( T ) = -

T-+ + oc

~.

(1.2)

The series Akj~O

I k]l

< +

(1.3)

is summable only if a~ E (a, + ~). For fixed fl ~ (a, 1) there is co(c, fl) satisfying IZkj + c I-~ <- co(c, fl) < + ~

(1.4)

2k]< - c

w does not depend on T. Proof. (i) Since c -> 0, we are able to estimate A ( T ) _= max{~.kj IXkj < 0} + c. Consequently seeing that Zkj < 0, we get

~'k] = k4 --] 2T2

= ( k2

- i T ) ( k2 + j T ) <-- - T.

Thus (1.2) holds. (ii) The estimate (1.3) was proved in [3] by means of merely elementary calculation. (iii) Since T is a positive integer, the following relation holds £ k 4 -j2T2<

Ik 4

-c

_]2T2 q_ cl-fl <=

£ ]k 4 _ ]2 "l- cI -fl ka-] 2< -c

It follows that co = co(c,fl) is independent of T. The fact co < + ~ is now an easy consequence of (ii). [] From here on, the value of fl will be kept fixed so that we are allowed to write co instead of

co(c, R e m a r k . After an easy computation we convince ourselves of the existence of fl E (], 1), co(0, fl) ~ V'2 z , for example.

282

E. FEIREISL

Keeping (1.3), (1.4) in mind together with the H61der inequality, it is not difficult to achieve the estimates (see [2] for details)

)'

(1.5)

el~a2j(v)

(1.6)

(

IIo11~-< c,V'f E [Zkjl~a~,j(o) \k,j

for all v E {D q: 0}, cr E (3, + oo);

IIo11~ -< V T K 1

IXkj +

for all v E {D < - c} where K1 = (2o)(c, fl))½ :r -1.

2. ORLICZ SPACES GENERATED BY A SUPERLINEAR FUNCTION

Definition 2.1. The function q) : R1---~ R 1 is said to be superlinear if the following conditions are satisfied.

(51)

tp is continuous on R 1. q9 is odd, i.e. t p ( - x ) = - q~(x),

x E R 1.

q~ is strictly increasing on R 1. Having denoted q~(x) =

-xq)(x)

P

-

($3)

~(s) ds, there exist

n u m b e r s p > 2, xo E R 1 such that 1

($2)

(S4)

¢(x) -> 0

holds for all

x>--Xo .

One observes easily (see [7]) that

a,(x)

lim

X---> + oo

x2 - + oo.

(2.1)

Passing to inversion, we set ~p = qg-1, ~ ( y ) = f~ ~p(s) ds. The Orlicz space L~, = L,~(Qr) is determined as the space of all measurable functions v on Qr satisfying

Ivw I < +

[Ivll¢ = s u p ~ w

J~

Y

where the supremum is taken over all functions w such that ~

~ ( w ) - 1. T

The space Ea, is the closure of all bounded measurable functions in L¢.

Time periodic solutions to a semilinear beam equation

283

We are about to recall some basic properties of the spaces having been just defined. The proofs are available in [5, Chapter 3], for instance. The spaces L~,, E~,, L~,, E~ are Banach spaces. Moreover, E~,, Eq, are separable and we may identify E~, = L~,,

E~, = L~.

(2.2)

Besides we have xcp(x) = ~(x) + tlJ(tp(x)), and, consequently, ~(qg(x))- (1-1)~p(qv(x))~(x)=

lxq~(x) - ~(x) _->0.

By means of ($3), (2.1) the existence of Y0--> 0 is ensured in such a way that

tit(y) >- 1 - P ~p(y)y

for all y _>-Y0.

The last inequality suffices the function W to belong to the A2-class (see [5]). As a consequence, we get E v = Lv. (2.3) 3. F O R M U L A T I O N

OF A M A I N R E S U L T

The first thing to state in our main result is the determination of conditions with which the function f is to agree. We will require the following. f = f(x, t, v) is continuous on [0, :r] x R 1 x R 1. (F1) fis ~

- periodic with respect to t.

(F2)

f i s nondecreasing in v for every x, t. There exists a superlinear function q9 such that / lim f(x,q g t,( v v) ~-

P(X,t),

0 < c 2 <=p(x, t) <- c3 We draw from the condition (F4)

uniformlyin

x,t

(F3)

(F4)

for all x, t.

lim f(x, t, v)v o-~+-= cp(v)o - p(x, t)

(3.1)

F(x, t, v) lim - - p(x, t)

(3.2)

having denoted

F(x, t, v) =

f0

f(x, t, s) ds.

Finally, after some computation, we derive

l f ( x , t, v)v - F(x, t, v) >=0 q where q is a fixed number, 2 < q < p, and Iv[ -> v0.

(3.3)

284

E. FEIREISL

We are on the point of formulation of our main results. To begin with, let us determine a weak solution to the problem {Pr}.

Definition 3.1. The function u E L 1 is a weak solution of the problem {Pr} if f ( . , u) G L 1 and ~a (U(gtt + gxxxx + cg) + f(. , u)g) = 0

(3.4)

T

holds for all smooth functions g satisfying (0.2), (0.3). We proceed to the statement of condition (R) appearing in the existence theorem formulated below. To this end, we set F(v) = sup

F(x, t, v)

X,t

_F(v) = inf F(x,

t, v).

x~t

Condition (R). There are numbers r > 0, 6 > 0 such that -

Iml[x-eg2?

+ max(F(r), F ( - r ) ) = inf F(v) - 6 vER 1

where K2 = (21Qr[

TK2) -1

=

(4zr2K21)-1.

Our main goal is to prove the following theorem. THEOREM 3.1. Let f satisfy the conditions (F1) - (F4). Moreover, let condition (R) hold. Then there exists the solution u to {Pr} satisfying

26<=llfor

f(.,u)u<+~.

(3.5)

The proof of theorem 3.1 is carried out in Sections 4-6. Applications will be given in Section 7. 4. THE FINITE DIMENSIONAL APPROXIMATION In this section we deal with the spaces

E. = span{ekj[k < n, IJl --< n} endowed with the norm

Izkj + clea~j(v) + ~

IIIvll[ = .~,

C

a~j(v) .

~.kj>--c

On the space En, an energy functional In is considered,

In(v) = ½~ ()~k, + c)a~j(v) + ~ F(., v), v ~ En. ~QT We check easily that

In belongs to CI(En, R ~) with a gradient (I" (v), w) = g ('~k/ + C)akj(V)akj(W) + (

f(" , V)W T

where v, w ~ En.

Time periodic solutions to a semilinear beam equation

285

We are looking for appropriate critical points of In, i.e. solutions to the Euler equation

I',(u,) = O. In order to find such a solution, we take advantage of the following assertion proved in the Appendix. LEMMA. 4.1. Let H = V 1 ~ V 2 ~ V 3 be an orthogonal decomposition of a finite dimensional Hilbert space H, V2 4= {0}. Let the symbol S stand for a sphere in H with the centre in the point 0 @ H. Consider a functional J ~ CI(H, R 1) having the property lim

J ( h ) = + ~.

(4.1)

Ilhll--'+=

Finally, the following conditions are supposed to be satisfied. J =< b

on

S n (V~ • v2),

(4.2)

J>b

on

V3,

and

(4.3)

J > a

on

V 2 ~) V 3 ,

(4.4)

where a, b are some fixed real numbers. Then there exists a critical point h0 E H such that

J'(ho) = O,

J(ho) E [a, b].

(4.5)

We are going to apply lemma 4.1 to our situation. To begin with, some auxiliary estimates are needed. Taking v E En n {D -> z} for some z E R 1, we get in turn

In(v) >=½(z + c)Iloll~

+ (JQ

F ( . , v).

(4.6)

T

According to (3.2), (2.1), In satisfies (4.1) on E,. To keep the notation used in lemma 4.1, we set J=I,, V1 =

H=En,

{D < A - c} n E , ,

b = inf (_F(v) vER 1

V2=MANEn o3 = {D _-> - c} n E ,

6)leT].

Observe V2 ¢ {0} provided that n is sufficiently large. As to (4.2), let us take v E V 1 • v2, IIIvlll = s. After straightforward computation we get

In(O) = ½ E

(Zkj + c)a j(o) +

Xkj < - c

< -- ½]m[ l-flS2 + I O r l

F(., o) JQr

max(/~(X/TKls), F( -

X/TK,s)).

286

E. PEIREISL

Indeed, due to (1.6) we have - V ~ K l s we obtain

<-- o(x, t) <-- V'-TKls. Seeing that F is convex in v,

F(x, t, v) <=max(F(x, t, - N/-T KIs), F(x, t, V ~ K l S ) ) . Keeping condition (R) in mind, we pick out s = r(KIV'---T) -1 and get finally

I,(v) <--_- IAll-lJ(2TK21)-lr 2 + IQTI max(F(-r),/~(r)). In view of (R), the condition (4.2) follows. On the other hand if v ~ V3, we obtain

I,(v)>-_[ F ( . , v ) > b ~QT and (4.3) holds. By means of (4.6), we deduce the estimate I, > f2(A)

on

V 2 ~) V 3

where f2(A) < 0 is some fixed number independent on n E N. Setting a = f2, we are in a position to apply lemma 4.1. Thus we get the existence of a sequence {U,},eN satisfying

~k,j (Akj + c)akj(Un)akj(W) + fO f(',Un)W = O

(4.7)

T

1~ (~'k]+c)aEj(un) + f

F(',un) E [Q(A), IQTI ( i n f E ( v ) - 6 ) ] QT

(4.8)

\vU'R1

for all w E En. The functions un represent approximate solutions to our problem. 5. PASSING TO A LIMIT--COMPACTNESS ARGUMENTS Setting w = u, in (4.7), multiplying by -½, and adding to (4.8), we obtain

f(" , u,)u, T

fo

f(" , u,) < - f~. T

:

According to (3.3), we get f O r f ( ' , u,)u, --
fo q~(u.)u. <=c5.

(5.1)

T

We conclude that {u,},~N is bounded in L . (especially in L2 due to (2.1)), {q0(U,)}n~N is bounded in Lv. Thus (F4) results in boundedness of {f(., u,)},eu in Lv as well.

Time periodic solutions to a semilinear beam equation

287

As the first application of compactness, we obtain un ~ u

E w - weakly in L ¢

(5.2)

f ( . , u~) ~ h

E~, - weakly in Ly,

(5.3)

having considered sub-sequences as the case may be. The second compactness result is contained in the following lemma. LEMMA 5.1. For arbitrary e > 0 there is z > 0 such that

IA,~/la2:(u,,) < e

(5.4)

I~~jl>-z

holds for all n E N. Proof. Letting w = w,, =

~

sgn()~kj)ak/(u,~)ek:

IZkjl>=z

in (4.7), we get

I~k:laZj(u.) = ( [ZkYl>=z

- (cu. - f ( . , u . ) ) w

JQT ~-~ C61[WIIoo ~-~ C 7 IXk/l=z

(according to (1.5)) -< c7 Z(fl-1)/2 Seeing that/3 < 1, (5.4) follows.

[)tk/laZ/(u,

1~.~/ll3aZ/(Un) .

IZkjl=z •

6. T H E P R O O F O F T H E O R E M

3.1--MONOTONICITY

First of all, let us record an easy modification of known results on Lp-spaces. For the proof see [6]. LEMMA 6.1. Let Q be an open bounded set in R m, g = g ( y , v) continuous function on Q × R 1, g nondecreasing with respect to v. Suppose the existence of a sequence {On}hEN such that vn -'-> v

weakly in

L 1 (Q)

g(',Vn)--~h

weakly in

LI(Q)

limsup fQg(., v.)v. <-

hv E LI(Q).

E. FEIREISL

288

Then

h(y) =g(y, v(y))

fora.e,

y ~ Q.

Keeping w E En fixed, we pass to a limit in (4.7):

~,. (~.kj+c)akj(U)akj(W)+ I ~l

hw=O,

WEEn.

(6.1)

JO T

One observes easily that u is a solution to to lemma 6.1, we are verify but the relation lim e nsup - ~ =f

r

{Pr} as

soon as h = f ( . , u) is proved. According

f("u")u'<-fo

r hu.

(6.2)

To this end, we set w = un in (4.7) and get lim sup

I

n-.oo 3 Q

f(', u,)u, <- -

cllul[

- 3 : ~,k]a2](l~)

k,j

T

as a conclusion of lemma 5.1. Taking advantage of results achieved in Section 2, namely of (2.3), we have h E L v -- Ev. Letting w -- un in (6.1), we obtain

fQ hU = - ~, ~.kja~i(U) -- cIlulI~ T

k,]

which combinated with the inequality above is just (6.2). On the point of conclusion, we derive

½fo f ( ' , U n ) U o - - f o F(',Un)>----[QTl(inf\weR1E ( W ) - - 6 ) . T T By means of (6.2), the estimate (3.5) follows immediately. Theorem 3.1 has been proved.

7. E X A M P L E S

AND APPLICATIONS

We are about to demonstrate some important cases included in theorem 3.1. To begin with, we observe that condition (R) holds provided that f(u)= o(u) for u---~ 0. Thus the results achieved in [1] are generalized. COROLLARY 7.1. Let the function f satisfy (F1)-(F4). Moreover, let the condition lira

f(x, t, v)

v--~O

-

-

=

0

/)

hold uniformly in x, t. Then there is at least one nonzero weak solution to

{Pr}.

Time periodic solutions to a semilinear beam equation

289

COROLLARY 7.2. Let h be a function satisfying (F1)-(F4), lim

h(x, t, v) -

v--~0

-

-

0

/)

uniformly in x, t. Let the function g agree with (F1)-(F3) and

Ig(x,t,v)l<-e Then there are positive numbers

e,

A

forallx, t,v. such that {Pr} possesses at least one solution satisfying

fQ f(.,u)u>-a T

whenever f = h + g. The number e depends on h only. If h, g are as in corollary 7.2 and g = g(x, t), some multiplicity results can be shown. Let us but sketch the way it could be done. Assume - c ~ a(D) ( a is the spectrum of D), h is of the class C 1 in v. We are in a situation to employ methods from [4] in order to obtain the existence of small solutions to {Pr} provided that g is small (we recall that the term cu in (0.1) has a regularization effect due to [1]). Consequently, we get the following corollary. COROLLARY 7.3. Let - c ~ or(D). Let f be as in corollary 7.2. Moreover suppose h E C 1 with respect to v and g = g(x, t). Then there is e > 0 such that the problem {Pr} has at least two solutions (geometrically distinct) provided that Ig(x, t) l < e for all x, t. In conclusion, we present the result analogous to that one achieved in [7] for a wave equation. Suppose f is independent on t. According to (1.2), condition (R) is valid for large 6 > 0 provided that T is sufficiently large. COROLLARY 7.4. Let f satisfy (F1)-(F4) and be independent on t. Then for arbitrary d > 0 there exists a solution u to {Pr} satisfying

fQ f(.,u)u>-d. 1

REFERENCES 1. CHANG K. C. & SANCHEZ L., Nontrivial periodic solutions of a nonlinear beam equation, Math. Meth. appl. Sci. 194-205 (1982). 2. FEIREISL E., Free vibrations for an equation of a rectangular thin plate, Aplik. mat. (to appear). 3. On periodic solutions of a beam equation, Dipl. Thesis, Faculty of Mathematics and Physics of the University Charles, Prague 4. McKENNA P. J., On the solutions of a nonlinear wave equation when the ratio of the period to the length of the interval is irrational, Proc. Am. math. Soc. 93, 59-64 (1985). 5. KUFNER A., JOHN O. & FU~K S., F u n c t i o n Spaces, Academia, Prague (1977). 6. LOVICARV., On infinitely many solutions to some nonlinear homogeneous equations, series of MI3 ~SAV, Prague (preprint). 7. RABINowrrz P. H., Large amplitude time periodic solutions of a semilinear wave equation, Communs pure appl. Math. 37, 189-206 (1984). 8. TANAKA K., Infinitely many periodic solutions for the equation uu - u~ - luls- lu = f(x, t), Proc. Japan Acad. 61, Ser. A, No. 3, 70-73 (1985). 9. PALAIS R. S., Critical point theory and the minimax principle, Proc. Symp. pure Math. 15, 185-212 (1970).

290

E. FEIREISL

APPENDIX--THE P R O O F O F L E M M A 4.1 To begin with, observe that a < b. We will assume the contrary to (4.5), i.e. J'(h) ~e 0 for all h ~ J-~ [a, b]. It is a matter of routine, proceeding similarly as in [9], to construct the homotopy ~/~ C([0, 1] x H, H) such that J ( r / ( . , h)) is a nonincreasing function on [0, 1], (8.1) r/(0, h) = h for every h ~ H r/(1, {hlJ(h ) <- b}) C {hlJ(h ) <=a}.

(8.2)

Let us only remark that the condition (4.1) stands for the usual Palais-Smale condition in view of dim(H) < + oo. Set p = S n (V~ ~ V2). From the topological point of view, p is a finite dimensional sphere (we assume a unit one for simplicity). Now, a new homotopy ~ E C([0, 1] x p, p) is defined as

~(t, h) = P~l(t, h)llPrl(t, h)l1-1 where P denotes an orthogonal projection on V1 ~) 112. This definition makes sense due to (4.2), (8.1) and (4.3). Obviously, we have ~(0, h) = h

for all h E p.

(8.3)

Seeing that V2 ~ {0}, there exists e0 E p n V2. We intend to show e0 ~ ~(1, p).

(8.4)

In such a way, keeping (8.3) in mind, the contradiction is obtained since the mappings ~(0, • ), ~(1, • ) belong no more to the same homotopy class. We are to demonstrate (8.4). If it were not true, we would have hi E p, ~(1, h 0 = e0. Consequently P0(1, hi) = Ze0 for some Z ~e 0. Thus we obtain r/(1, h 0 E V2 @ V3. According to (4.4), we get J(r/(1, h 0 ) > a. On the other hand using (4.2) together with (8.2), we deduce J ( r / ( 1 , h O ) =< a.

•