Multiplicity result for periodic solutions of semilinear dissipative hyperbolic equations with coercive growth nonlinearity

MULTIPLICITY RESULT FOR PERIODIC SOLUTIONS OF SEMILINEAR DISSIPATIVE

HYPERBOLIC EQUATIONS WITH COERCIVE GROWTH NONLINEARITY

WAN SE KIM

ABSTRACT. The multiplicity of periodic solutions of semilinear dis- sipative hyperbolic equations is treated

1. Introduction

Let R be the set of all reals and n ~ Rn, n 2: 1, be a bounded domain with smoothbo~dary an which is assumed to be of class^C2. Let Q = (0,21r) x nand L2(Q) be the space of measurable and Lebesgue square integrable real-valued functions onQ with usual inner product < ',. > and corresponding norm 11·112.

By HJ(n) we mean the completion of cJ(n) with respect to the norm 11·111 defined by

IlcPlli =

1 ^L

Q1al9

H²(n) stands for the usual Sovolev space; i.e., the completion ofC²(O) with respect to the norm 11 . 112 defined by

IlcPll~ ⁼

1 ^L

Q 1£>19

Let9 : R ^{- 7} R be a continuous function. Moreover, we assume that there exist constants ao and bo^{such that}

Ig(u)1 ~ aolul+bo ^{for all} uE R.

Received October 27, 1999.

2000 Mathematics Subject Classification: 35S20, 35L70.

Key words and phrases: multiplicity, semilinear, dissipative, hyperbolic equation.

Supported by Hanyang University Research Grant 1999.

The purpose of this work is to investigate the multiplicity for periodic solutions of the semilinear dissipative hyperbolic equations

(E)

u(t, x) =0 on (0,21T) x an,

u(O,x) =U(21T,X) on n

where ^),1 and ^),2 denotes the first and second eigenvalues of-f::. with zero DiricWet boundary data and cl>1 is the positive normalized eigen- function corresponding to ),1 and h E L²(Q).

The purpose of this paper is to give a multiplicity result for semi- linear dissipative hyperbilic equations. Orginally, the linear dissipa- tive hyperbolic equations are derived from physical principle(see [4]).

The existence and asymptotic theory of dissipative hyperbolic equa- tions have been developed by several authors for initial value prob- lems, boundary value problems, or mixed problems. For information on dissipative hyperbolic equations, we refer to [24]. On the existence of doubly-periodic solutions of semilinear dissipative hyperbolic equa- tions have been done by Mawhin [22], Fucik and Mawhin [7]. Mawhin treat the existence of double-periodic solutions for semilinear dissipa- tive hyperbolic equations of several types ofg(u) with at most linear growth in connection with the set I; = {k^{2 -} j²lk,j integers}. Fucik and Mawhin consider also the existence double-periodic solutions of seminear dissipative hyperbolic equations with nonlinear term of the form g(u) = f.£u+ - vu- - cI>(u), where cl> is a continuous and bounded function, and f.£,v are real numbers related to the set I;. In [9, 15], the existence of solutions for DiricWet-periodic problem for semilin- ear dissipative hyperbolic equations at resonsnce, in [13, 14], the ex- istence of Dirichlet-periodic solutions for semilinear dissipative hyper- bolic problems with superlinear growth, in [16], the existence of double- periodic solutions for semilinear dissipative hyperbolic equations with non-decreasing type of non-linear term, in [19, 20], the multiple exis- tence of double-periodic and Dirichlet-periodic problem, respectably,

for semilinear dissipative hyperbolic equations and, in [17], the asymp- totic behavior of Dirichlet-initial problem of semi-linear dissipative hy- perbolic equations are disscused. Our result is related to the results in [19, 20] which are so called the Ambrosetti-prodi type multiplicity re- sult which has been initiated by Ambrosetti-Prodi [1] in the study of a Dirichlet problem to elliptic equations and developed in various direc- tions by several authors to ordinary and partial differential equations.

For more information on this problem for semilinear elliptic, parabolic and ordinary equations, we refer to [3, 5, 6, 10, 11, 12, 18, and 21] and their references.

In our result, we will trerat a multiplicity result for Dirichlet-periodic solutions of semilinear dissipative hyperbolic equations in n-dimensional space. we assume the coercive growth on9with restriction on the left- hand and our proof based on Mawhin's continuation theorem in [8].

2. Preliminary results

Let us define the linear operator

by

2 8u 8²u ²

DomL ={u E L²((0,27T),H (0) nH6(O»!8t E L²(Q), 8t₂ EL (Q), u(O, x) = U(27T,x),XE O}

and 8u 8²u

Lu⁼ (3at + 8t2 - ~u- ^Alu.

Using Fourier series and Parseval inequality, we get easily

8u 8u ²

<Lu, 8t >=(3118t11£2 for all u ^E DomL.

Hence kerL = ker(~ +^All) = kerL* since .6. +^All is self-adjoint and ker(~+All) is one space dimension generated by the eigenfunc- tion cPl. Therefore L is a closed, densely defined linear operator and

Im(L) = [kerL]1..; Le., L 2(Q) = kerLEBImL. Let's consider a conti- nous projection PI : L²(Q) ---+ L²(Q) such that ImPI = kerL. Then L²(Q) = kerLEBkerPI . We consider another continuous projection P2 : L2(Q) ---+ L 2(Q) defined by

(P₂h)(t, x) = ~

jr ^r

21r 1Q

Then we have L 2(Q) = ImPI E!1ImL, kerP2 = ImL, and L 2(Q)/ImL is isomorphism to ImP2.

Since dim[L2(Q)/ImL] = dim[ImP2 ] = dim[kerL] = 1, we have an isomorphism J : ImP2^---+kerL.

By the closed graph theorm, the generalized right inverse of L de- fined by

K = [LIDomLnlmL]-1 :ImL---+ ImL is continuous. Ifwe equip the space DomL with the norm

Then there exist a constant c> 0 independently ofhE ImL, U =Kh such that

IIKhllDomL ~ cllhllp·

Therefore K : ImL ---+ ImL is continuous and by the compact imbed- ding ofDomL in L 2(Q), we have that K: ImL ^---+ ImL is compact.

LEMMA 2.1. L is closed, densely defined linear operator such that kerL = [ImL]l.. and such that the right inverse K : ImL ---+ ImL is completely continuous.

Proof. See [2, 23].

3. Multiplicity result Let us consider the following

o

u(t, x) =0 on (0,21r) x an,

u(O, x) = u(21r, x) on n

where j.LE [0,1].

Let L : DomL _~ L 2(Q) ---> L2(Q) be defined as before. Ifwe define a substitution operator Nt :^{L2(Q) ---> L2}(Q) by

(Nt)(t, x) = j.Lg(u) - j.Lh(t, x)

for u E L²(Q) and (t,x) E Q, then Nt maps continuously into itself and take bounded sets into bounded set. Let G be any open bounded subset ofL2(Q), then P2Nt :G^---> L2(Q) is bounded and K(I-~P2) ^: G--->L²(Q) is compact and continuous. ThusNt is L-compact on G.

The coincidence degree D^L(L+Nt,G) is well defined and constant in j.L ifLu+Ntu ::/: 0 for j.L E [0,1] and u E DomLnaG. It is easy to check that (u, j.L) is a weak solution of (Et) if and only if u E DomL and

(3.1~) Lu+Ntu = o.

Here, we assume the following

lim infg(u) = +00, lul-too

. g(u)

hm sup1--1 < A2 - Al.

u-t-oo U

From (H2 ) and (H3 ), we may assume that

m = inf g(u) > 0

uER

and there exist aE (0,A2 - A1) and b2: 0 such that Ig(u)1 :::; alul +bfor all u:::; O.

For hE L²(Q), we write I\h= JJ_Q h(t, x)cjJ(x)dtdx.

LEMMA 3.1. If(HI), (H2) and (H_{s )} are satisfied, then, for each h* E L²(Q), there exists M(h*) >0 independentlyofJ.l such that

holds for each possible weak solution U= ac/>+^U, witha E Rand u E

ImL, of(E~)with J.l E [0.1], and with I\h _~ I\h* and IlhllL2 ~ Ilh*IIL2.

Proof. Suppose there exists h E L2(Q)withI\h ~ I\h* and Ilhll L2 ^~ Ilh* 11L2 and the corresponding sequence of solutions {(un, J.ln)}, with

J.l E [0,1], of _(3.1~n)such that

lim lIunllL2 =00, n---.+=

then clearly

lim IluⁿllL2 = ^00.

n---.+=

For each n 2: 1, we put Un(t,x) = anc/>(x)+un(t,x).

First, we are going to prove that

Ifit is not the case, we may assume that, by extracting subsequence if it is necessary,

tim IluⁿllL2 =o.

n---.+= lanl

Therefore, we may have a subsequence, say again, {un} such that we have easily

tim Iun(t,x)1 =00 a.e. onQ.

n---.+=

By taking the inner product with c/>on both sides of (3.1~),we have

110

110

On the other hand, by (H2 ) and Fatou's lemma, we have lim Je{g(un(t, x))c/>(x)dtdx=⁰⁰

n---.+= lQ

which leads to a contradiction. First, we assume that 0<^C <^00, then there exist no E N such that

For given ^E > 0, we may choose 8 > 0such that

for any measurable set A C Qwith IAI:S 8.

Let 0 < 'Y < 114>11= and no = {x En: 4>(x) 2 'Y}. Choose Mo > 0 such that

8Mo-Im/

110

110

Then, since limu--+=g(u) =^00, we have that

mo=sup{lul :'Yg(u) < Mo} < ^00.

We put

Qn ={(t, x) E [0,27r] x no :lun(t, x)1 2 mo}.

Then we have IQnl :S8. Infact, ifIQnl > 8, then from the definition of mo we have

110

=I~ g(un)4>(x)dtdx+I~ g(un)4>(x)dtdx

Qn Q\Qn

> 8Mo - m

110

>

110

and this leads to a contradiction. Therefore, we have

On the other hand, 0=

110

=

11

11

~ ^(1/2)

11

11

Q\Qn Qn

From the definition of mo and the above facts, we have, for alln ~ no,

o~ (1/2)m~ - (1/2)(1 - E)(c/2)lIun lli2 + E(3c/2)lIun lli2

= (1/2)mg - (c/4)(1 + 5EC)l\un lli2.

Therefore, {llunllp} is bounded which leads to a contradiction.

Nex , we assume c -t - 0 th, en mnli ^--+oo IIuIlu_nⁿll11 - 1

L2 - .

Multiplying (3.1~) by ~~ and integrate over Q, we find from the peri- odicity of^U that

au ¹

11atlip ~ iI3Illhllp.

Again, taking the inner prodnct with Un on both sides of (3.1~), we have

(A2 - AI)llun lli2 -11a;n Ili2+ < g(un), Un >~ Ilhllpllunllp and hence

n~SUp(A2- AI-a )llun lli2 ~ ^[max{m,b}IQI^I/2 1

+ 11312I1h*lli2+llh*l\p].

Thus {llunl\p} isbounded which leads to another contradiction. 0

LEMMA 3.2. If(HI), (H2 ) and (H_{3 )} are satisfied, then, for each h* E L²(Q), there exists r =r(h*) > 0independently ofJ.L such that

lul ~ ^r

holds for each possible weak solution u = u+ U, with u ⁼ acP(x) , a E Rand UE ImL, of(3.1~) whereJ.L E [0,1]' and with /\h ~ /\h*

and IIhllp ~ Ilh*lIp·

Proof Suppose there existshE£2(Q) with I\h ::; I\h* and Ilhll£2 ::;

IIh*II£2, and the corresponding sequence of weak solutions {(un,,un)}

of (3.1~n) with {Iunl} is unbounded. Then (un, J..Ln) is a solution of

(3.1~n) where Un = Un +Un with Un = oncj>(x) and Un E Im£. We may choose a subsequence, say again {un} withUn =oncj>(x)such that

IOnl ^{-+ +00} as n -+ +00. Now, let M> M which is given InLemma 3.1. Let

Qo = {(t,x) EQllu(t,x)l2: 1~~}.

Then

M² 2:

lk

2:

lko

l+M ² 2: IQol[ IQI ].

Therefore IQol ::; [l:MFIQ\ and hence IQ" Qol = IHt,x) EQllu(t,x)1

< l+M}1 > [1- M ]21QI > O.

- IQI - l+M

Let W = (O.27f) x no. Then we have jOncj>(x)I^{-+ 00} for eachx E no

as n ^-+ 00. Hence, by Fatou's lemma and (H_{2 ),} we have

liminf

j-(

n-+oo lQ

=liminfjr ( g(oncj>(x) +u(t, x))cj>(x) dtdx

n-+oo lQ

2: Jr ( liminfg(oncj>(x) +u(t, x))cj>(x) dtdx lwnCQ'Qo) n-+oo

= 00.

Hence, there exists ro(h*) > 0 such that, for IOnl > ro, we have (3.1)

110

J ¹⁰

On the other hand, by taking the inner product with<jJ(x) on the both sides of(3.1~n),we have

J

J k

which is impossible. The proof is complete. o

LEMMA 3.3. If(HI), (H_{2 )} and (H_{3 )} are satisfied, then, for each h* E L2(Q), we can find an open bounded set G(h*) in L 2(Q) such that, for each open bounded set G in L2(Q) such that G :2 G(h*), we have

DL(L+Nl,^{G) =0} for all hE L²(Q) with Ah ::; Ah* and Ilhll£2 ::; IIh*11£2'

Proof. By similar fashion as we did in the proof of Lemma 3.2 to get (3.1), there existsr(h*) >0 such that, for lal > r, we have

J k

J

Let

G(h*) ={u E L2(Q)I-r<jJ(x) <a<jJ(x) < r<jJ(x) forx E D,Ilull£2 < M}

whereu = a<jJ(x)+u withr(h*) >max{r(h*), ro(h*), r(h*)} and M>

M which are given in Lemma 3.1 and Lemma 3.2. If (3.1~) has a solution u for some h E L 2(Q) such that Ah < ^21fmIn<jJ(x)dx and

j1, E [0, 1], then by taking the inner product with <jJ on the both sides of the equation (3.1~),we have

21fm

l

Thus _(3.1~) has no solution for h E L²(Q) such that Ah < 21rm

In<jJ(x) dx.

Hence, for each open bounded set G 2 G(h*), we have DL(L+Nl,G) =0 for hEL²(Q)

such that I\h < 271"mIn^{rjJ(x)dx. Choose h E L2(Q) with I\h :; 271" m} In^{rjJ(x)dx and Ilhllp :; IIh*llp, and define}

F: (D(L) n^{G) x [0,1] -+L2(Q) by}

F(u,'\) =Lu+N(l->')h+>'h(U) for hE L²(Q)

withI\h:; I\h* and Ilhllp :; Ilh*llp. Then by Lemma 3.1 and Lemma 3.2, we have

ort. ^F(D(L) naG) x [0,1] for hE L²(Q)

with I\h :; I\h* and Ilhllp :; Ilh*lIp. By the homotopy invariance of degree, we have, for allhE L²(Q) withI\h:; I\h* and Ilhllp :; Ilh*llp,

DL(L+_N~,G) =DL(F(·,1), G)

=DL(F(·,0), G)

=DL(L+Nl,G)

=0

and the proof is completed. D

THEOREM. Assume (Hi), (H2 ) and (H3 ). Then there there exist a constantao such that the boundary value problem (E), (Bl ), and (B2 )

has at least two solutions for h such that

(3.2)

ffo

f ^fo

for everyu^E L²(D) having mean value zero on D, satisfying the con- ditions (Bd, and (B2 ) such that

(3.3)

where M is given Lemma 3.3.

Proof. Let

Define

g(ao<j>(xo)+uo) = mi!1 g(a<j>(x)+u).

xEn lal~r lii.I~M

.6.(G(h) ={u EL2(Q)lao4>(x) <a4>(x) <ro4>(x) for xEn, IIullL2 <M}

wherero(h) >r which is given in Lemma 3.3.

Ifu E 8/lG(h), then necessaryu =ao<j>(x) +uoru ==ro<j>(x) +u. If u =ao<j>(x) +u with Ilull£2 < M, then, by taking inner product with

<j>on the both sides of(3.1~),we have

J ¹⁰

Jl

which, from (3.2) and (3.3), is impossible. Ifu = ro<j>(x) +u with l1ull£2 < M, then, by the choice ofro > 0, we have

J10

which is also impossible. Thus for J.l ^E [0,1], DdL+Nt:,/lG(h)) is well defined and

DdL+Nt,/lG(h)) =DB (JP2Nt: ,/lG(h) n^kerL,O)

where DB ia Brouwer degree and P2Nt: : L2(Q) ^{- 7}kerL isan operator defined by

(P2Nt:u)(t, x)= J.l[J

10

l

Now letT :kerL ^{- 7} R be defined by T(a<j>(x)) =a.

Then, for 11=1,

DdL+Nt b.G(h)) =DB(JP2N~,b.G(h)nkerL,O)

=DB(T(JP2N~)T-1,T(b.G(h)) nkerL) ,0).

Ifwe let J : ImP2 ^{- 7} kerL be the identity map, then the operator

<I?=T(JP2N~)T-1 will be defined by

<I?(a) =

110

110

Thus, we have

<I?(ao) =

110

110

<I?(ro) =

110

110

Hence, the coincidence degree exists and the corresponding value

IDdL-N,b.G(h))1 = IDB[JP2N~,b.nKerL"OJI ^{= 1.}

Therefore, the equation (3,1_{1 )} has at least one solution in b.G(h) Choose G ;;;2 b.G(h), where G is defined in Lemma 3.3. By the additivity of degree, we have

and hence

IDdL+N~,G - b.G(h))1 = 1.

Therefore (3.1~) has another solution in G - b.G(h). This proves our

assertion. 0

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Department of Mathematics Hanyang University

Seoul 133-791, Korea

E-mail: [email protected]