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Multiplicity Result for Semilinear Dissipative Hyperbolic Equations
Journal of Mathematical Analysis and Applications 231, 34]46 Ž1999. Article ID jmaa.1998.6207, available online at http:rrwww.idealibrary.com on
Multiplicity Result for Semilinear Dissipative Hyperbolic Equations* Wan Se Kim† Department of Mathematics, College of Natural Sciences, Hanyang Uni¨ ersity, Seoul 133-791, Korea Submitted by Jean Mawhin Received September 28, 1997
1. INTRODUCTION Let Zq, Z, R*, and R be the set of all positive integers, integers, nonnegative reals, and reals, respectively, and let V : R n, n G 1, be a bounded domain with smooth boundary V which is assumed to be of class C 2 . Let Q s Ž0, 2p . = V and L2 Ž Q . be the space of measurable Lebesgue square integrable real-valued functions on Q with usual inner product ² ? , ? : and corresponding norm 5 ? 5 2 . By H01 Ž V . we mean the completion of C01 Ž V . with respect to the norm 5 ? 5 1 defined by 5 f 5 12 s
HV Ý
< D af Ž x . < 2 dx.
< a
H 2 Ž V . stands for the usual Sovolev space; i.e.; the completion of C 2 Ž V . with respect to the norm 5 ? 5 2 defined by 5 f 5 22 s
HV Ý
< D af Ž x . < 2 dx.
< a
Let g: R ª R be a continuous function. Moreover, we assume that there exist constants a0 and b 0 such that < g Ž u . < F a0 < u < q b 0 , for all u g R. * This work was supported by BSRI N971429 Korea Ministry of Education. † E-mail: [email protected]. 34 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
Ž H1 .
DISSIPATIVE HYPERBOLIC EQUATIONS
35
The purpose of this work is to investigate the Ambrosetti]Prodi Žbriefly AP. type multiplicity result for the periodic solution of the semilinear hyperbolic equations,
b
u t
q
2u t
2
y D x u y l1 u q g Ž u . s
sf 1
'2p
q h Ž t , x . , in Q, Ž E .
u Ž t , x . s 0, on Ž 0, 2p . = V ,
Ž B1 .
u Ž 0, x . s u Ž 2p , x . , on V ,
Ž B2 .
where l1 denotes the first eigenvalue of yD with zero Dirichlet boundary condition and f 1 is the corresponding positive normalized eigenfunction; i.e., f 1Ž x . ) 0 on V and HU f 12 Ž x . dx s 1, and h g L2 Ž Q . with
HHQ h Ž t , x . f Ž x . dt dx s 0. 1
This type of result, so-called an Ambrosetti]Prodi type result was initiated by Ambrosetti]Prodi w1x in 1972 in the study of a Dirichlet problem to elliptic equations and was developed in various directions by several authors to ordinary and partial differential equations. A notable discussion for AP type results for periodic and Dirichlet boundary value problems has been done by Fabry, Mawhin, and Nkashama w5x and Chiappinelli, Mawhin, and Nugari w3x, respectively, for second-order ordinary differential equations. For AP type results for periodic solutions of higher order ordinary differential equations, we refer to the results of Ding and Mawhin in w4x. AP type results for Lienard systems have been done by Kim w9x, and Hirano and Kim w8x. Lazer and Mckenna treated AP type multiplicity results for elliptic and parabolic equations in w11x. In w10x, AP type results for doubly periodic solutions of dissipative hyperbolic equations in one space dimension have been treated by Kim. In our result, we treat an AP type multiplicity result for Dirichlet-periodic solutions of semilinear dissipative hyperbolic equations in n-dimensional space. We assume the coercive growth on g with a restriction on the left hand and our proof based on Mawhin’s continuation theorem.
2. PRELIMINARY RESULTS Let us define the linear operator L: Dom L : L2 Ž Q . ª L2 Ž Q . ,
36
WAN SE KIM
by
½
Dom L s u g L2 Ž Ž 0, 2p . , H 2 Ž V . l H01 Ž V . .
2u t2
u t
g L2 Ž Q . ,
g L2 Ž Q . ,u Ž 0, x . s u Ž 2p , x . , x g V
5
,
and Lu s b
u
q
t
2u t2
y D u y l1 u.
Using Fourier series and Parseval inequality, we get easily
¦
Lu,
u t
;
sb
u t
2 L2
, for all u g Dom L.
Hence ker L s kerŽ D q l1 I . s ker L* because D q l1 I is self-adjoint and kerŽ D q l1 I . is one-dimensional space generated by the eigenfunction f 1. Therefore L is a closed, densely defined linear operator and ImŽ L. s wker L x H ; i.e., L2 Ž Q . s ker L [ Im L. Let us consider a continuous projection P1: L2 Ž Q . ª L2 Ž Q . such that ker P1 s Im L. Then L2 Ž Q . s ker L [ ker P1. We consider another continuous projection P2 : L2 Ž Q . ª L2 Ž Q . defined by
Ž P2 h . Ž t , x . s
1 2p
HHQ h Ž t , x . f Ž x . dt dx f Ž x . . 1
1
Then we have L2 Ž Q . s Im P1 [ Im L, ker P2 s Im L, and L2 Ž Q .rIm L is an isomorphism to Im P2 . Because dimw L2 Ž Q .rIm L x s dimwIm P2 x s dimwker L x s 1, we have an isomorphism J: Im P2 ª ker L. By the closed graph theorem, the generalized right inverse of L defined by K s w L < Dom L l Im L x
y1
: Im L ª Im L
is continuous. If we equip the space Dom L with the norm, 5 u 5 Dom L s
HHQ
u q 2
u
2
q
2u
ž / ž / t
t2
2
q
Ý Ž Dxb u .
< b
2
dt dx,
DISSIPATIVE HYPERBOLIC EQUATIONS
37
then there exits a constant c ) 0 independent of h g Im L, u s Kh such that 5 Kh 5 Dom L F c 5 h 5 L2 . Therefore K: Im L ª Im L is continuous and by the compact imbedding of Dom L in L2 Ž Q ., we have that K: Im L ª Im L is compact. LEMMA 2.1. L is a closed, densely defined linear operator such that ker L s wIm L x H and such that the right in¨ erse K: Im L ª Im L is completely continuous.
3. MULTIPLICITY RESULT Let us consider the following,
b
u t
q
2u t2
y D x u y l1 u q m g Ž u . s m sf q m h Ž t , x . , in Q,
Ž Esm .
u Ž t , x . s 0, on Ž 0, 2p . = V ,
Ž B1 .
u Ž 0, x . s u Ž 2p , x . , on V ,
Ž B2 .
where m g w0, 1x and f Ž x . s w f 1Ž x .r'2p x. Let L: Dom L : L2 Ž Q . ª L2 Ž Q . be defined as before. If we define a substitution operator Nsm : L2 Ž Q . ª L2 Ž Q . by
Ž Nsm . Ž t , x . s m g Ž u . y m sf y m h Ž t , x . , for u g L2 Ž Q . and Ž t, x . g Q, then Nsm maps continuously into itself and takes bounded sets into bounded sets. Let G be any open bounded subset of L2 Ž Q ., then P2 Nsm : G ª L2 Ž Q . is bounded and K Ž I y P2 .: G ª L2 Ž Q . is compact and continuous. Thus Nsm is L-compact on G. The coincidence degree DLŽ L q Nsm, G . is well defined and constant in m if Lu q Nsm / 0 for m g w0, 1x, s g R, and u g Dom L l G. It is easy to check that Ž u, m . is a weak solution of Ž Esm . if and only if u g Dom L and Lu q Nsm u s 0.
Ž 3.1 sm .
38
WAN SE KIM
Here we assume the following, lim inf g Ž u . s q`, < u <ª`
lim sup
g Ž u. u
uªy`
- l2 y l1 .
Ž H2 . Ž H3 .
From Ž H2 . and Ž H3 ., we may assume that m s inf g Ž u . ) y`, ugR
and there exist a g Ž0, l2 y l1 . and b G 0 such that < g Ž u . < F a < u < q b, for all u F 0. LEMMA 3.1. If Ž H1 . ] Ž H3 . are satisfied, then for any s g Rq, there exists M Ž s*. ) 0 such that 5 u 5 L2 F M Ž s* . holds for each possible weak solution u s af 1 q u, ˜ with a g R and u˜ g Im L, of Ž Esm . with m g w0, 1x and < s < F s*. Proof. Suppose there exists a constant s with < s < F s* and corresponding solutions Ž u n , m n . of Ž3.1 sm n . such that lim 5 u n 5 L2 s `.
nª`
For each n G 1, we put u nŽ t, x . s a n f Ž x . q u ˜nŽ t, x .. First, we are going to prove that lim
nª`
< an < 5u ˜n 5 L2
s c - `.
If it is not the case, then, if necessary we may take a subsequence, we have easily lim < u n Ž t , x . < s `, a.e. on Q.
nª`
By taking the inner product with f on both sides of Ž3.1 sm ., we have
HHQ g Ž u Ž t , x . . f Ž x . dt dx s s F s*. n
DISSIPATIVE HYPERBOLIC EQUATIONS
39
On the other hand, by H2 and Fatou’s lemma, we have lim
nª`
HHQ g Ž u Ž t , x . . f Ž x . dt dx s `, n
which leads to a contradiction. First, we assume that 0 - c - `, then there exists n 0 g N such that c 2
3c
5u ˜n 5 L2 F < a n < F
2
5u ˜n 5 L2 , for all n G n 0 .
For given e ) 0, we may choose d ) 0 such that
HHA < f <
dt dx - e 5 f 5 2L2 ,
2
for any measurable set A ; Q with < A < F d . Let 0 - g - 5 f 5 ` and V 0 s x g V: f Ž x . G g 4 . Choose M0 ) 0 such that
d M0 y < m <
HHQ f dt dx ) s*.
Then, because lim uª` g Ž u. s `, we have that m 0 s sup < u < : g g Ž u . - M0 4 - `. We put Q n s Ž t , x . g w 0, 2p x = V 0 : < u n Ž t , x . < G m 0 4 . Then we have < Q n < F d . In fact, if < Q n < ) d , then from the definition of m 0 we have
HHQ g Ž u Ž t , x . . f Ž x . dt dx n
s
HHQ
g Ž u n . f Ž x . dt dx q n
) d M0 y < m < ) s*,
HHQ_Q
HHQ f Ž x . dt dx
g Ž u n . f Ž x . dt dx n
40
WAN SE KIM
and this leads to a contradiction. Therefore, we have
HHQ_Q
< an f < 2 G Ž 1 y e . n
HHQ < a f < n
2
.
On the other hand, 0s s F
HHQ a f u˜ n
HHQ_Q 1 2
n
an f u ˜n q n
HHQ
HHQ_Q ž < a f q u˜ < n
2
n
an f u ˜n n
y < an f < 2 y < u ˜n < 2 q
/ HHQ
n
< an f < < u ˜n < . n
From the definition of m 0 and the preceding facts, we have, for all n G n 0 , 0F s
1 2 1 2
m20 y m20 y
1 2 c 4
c
3c
2
2
Ž 1 y e . 5 u˜n 5 2L2 q e
5u ˜n 5 2L2
Ž 1 q 5e c . 5 u˜n 5 2L2 .
Therefore, 5 u ˜n 5 L2 4 is bounded and hence 5 u n 5 L2 4 is bounded which leads to a contraction. Next, we assume c s 0, then lim nª`Ž5 u ˜n 5r5 u n 5 L2 . s 1. Multiplying Ž3.1 sm . by ur t and integrating over Q, we find from the periodicity of u that
u t
2
L
F
1
5 h 5 L2 .
Again, taking the inner product with u n on both sides of Ž3.1 sm ., we have
Ž l2 y l1 . 5 u˜n 5 2L2 y
un t
2 L2
q ² g Ž u n . , u n : F s* < a n < q 5 h 5 L2 5 u ˜n 5 L2 ,
and hence, lim sup Ž l 2 y l1 y a . 5 u ˜n 5 2L2 nª`
F max < m < , b 4 < Q < 1r2 q s* q
1 < b <2
5 h 5 2L2 q 5 h 5 L2 .
41
DISSIPATIVE HYPERBOLIC EQUATIONS
Thus 5 u ˜n 5 L2 4 is bounded and thus 5 u n 5 L2 4 is bounded which leads to another contradiction. Remark. By Lemma 3.1, we may have a priori bounds M1Ž s*. ) 0 and g 1Ž s*. ) 0 such that < a < F g 1 Ž s* . , 5u ˜5 L2 F M1 Ž s* . , for each possible weak solution u s af q u ˜ of Ž Esm . with < s < F s* and m g w0, 1x. LEMMA 3.2. If Ž H1 . ] Ž H3 . are satisfied, then, for each s* g Rq, we can find an open bounded set GŽ s*. in L2 Ž Q . such that, for each open bounded set G in L2 Ž Q . such that G = GŽ s*., we ha¨ e DL Ž L q Ns1 , G . s 0, for all < s < F s*.
˜ ) M1 which is given in the Remark of Lemma 3.1. Let Proof. Let M
½
Q0 s Ž t , x . g Q < u ˜Ž t , x . < G
˜ 1qM < Q<
5
.
Then
˜2 G M
HHQ < u˜Ž t , x . <
G
HHQ < u˜Ž t , x . <
2
dt dx
2
dt dx
0
G < Q0 <
˜ 1qM < Q<
2
.
˜ Ž1 q M .x2 < Q < and hence < Q _ Q0 < s < uŽ t, x . g Therefore < Q0 < F w Mr ˜ Ž1 q M .x2 < Q < ) 0. Q< < u ˜Ž t, x .< F Ž1 q M˜ .r< Q <4< G w1 y Mr Let M0 s min x g V , ug R g Ž u. f Ž x . and W s Ž0, 2p . = V 0 . Suppose that a g R and < a < ª `, then < af Ž x .< ª ` for each x g V 0 . Hence, by Fatou’s lemma and Ž H2 ., we have lim inf < a <ª`
HHQ g Ž af Ž x . q u˜Ž t , x . . f Ž x . dt dx
s lim inf < a <ª`
G
HHQ
g Ž af Ž x . q u ˜Ž t , x . . f Ž x . y M0 dt dx q M0 < Q <
inf HHWlŽ Q_Q . lim < a <ª` 0
q M0 < Q < s `.
g Ž af Ž x . q u ˜Ž t , x . . f Ž x . y M0 dt dx
42
WAN SE KIM
Hence, there exists r 2 Ž s*. ) 0 such that, for < a < ) r 2 Ž s*., we have
HHQ g Ž af Ž x . q u˜Ž t , x . . f Ž x . dt dx ) s*. Let G Ž s* . s u g L2 Ž Q . N yr˜Ž s* . f Ž x . - af Ž x . - ˜ r Ž s* . f Ž x . for x g V , 5 u ˜5 L 2 - M˜ 4 , where u s af Ž x . q u ˜ with ˜r Ž s*. ) max r1Ž s*., r 2 Ž s*.4 and M˜ ) M1 which are given in the Remark of Lemma 3.1. Let s0 s d min g Ž u . , ugR
where d s 2p HV f Ž x . dx. If Ž3.1 sm . has a solution u for some s g R and m g w0, 1x, then by taking the inner product with f on both sides of Eq. Ž3.1 sm ., we have s0 F
HHQ g Ž u Ž t , x . . f Ž x . dt dx s s.
Thus Ž3.1 sm . has no solution for s - s0 . Hence for each open bounded set G = GŽ s*., we have DL Ž L q Ns1 , G . s 0, for s - s0 . Choose s - s0 and define F : Ž D Ž L . l G . = w 0, 1 x ª L2 Ž V . , by F Ž u, m . s Lu q NŽ1y m . sq m s Ž u . , for < s < F s*. They by Lemma 3.1 and the remark, we have 0 f F Ž D Ž L . l G . = w 0, 1 x , for < s < F s*. By the homotopy invariance of degree, we have, for all < s < F s*, DL Ž L q Ns1 , G . s DL Ž F Ž ?, 1 . , G . s DL Ž F Ž ?, 0 . , G . s DL Ž L q Ns1 , G . s 0, and the proof is complete.
DISSIPATIVE HYPERBOLIC EQUATIONS
43
LEMMA 3.3. If Ž H1 . ] Ž H3 . are satisfied, then there exists s1 ) s0 such that, for each s* ) s1 , we can find an open bounded set DŽ GŽ s*.. in L2 Ž Q . on which DL Ž L q Ns1 , D Ž G Ž s* . . . s 1, for all s1 - s F s*. Proof. Let g Ž a0 f Ž x0 . q u ˜0 . s
By Ž H1 ., we may have f Ž x . dt dx <. Define
min xgV < a
g Ž af Ž x . q u ˜. .
s1 s max 5 u˜5 L 2 F M˜ < HHQ g Ž a 0 f Ž x . q u ˜Ž t, x ..
D Ž G Ž s* . . s u g L2 Ž Q . N a 0 f Ž x . - af Ž x . - g ˜ Ž s* . f Ž x . for x g V , 5 u ˜5 L 2 - M˜ 4 , where g ˜ Ž s*. and M˜ are given in Lemma 3.2. If s ) s1 , m g w0, 1x and Ž u, m . is a possible solution of Ž3.1 sm . such that u g DŽ GŽ s*.., then by Ž B1 ., the Remark of Lemma 3.1, we have necessarily u s a 0 f Ž x . q u ˜ or u s g˜ Ž s*. f Ž x . q u. ˜ If u s a 0 f Ž x . q u˜ with ˜ 5u ˜5 L2 - M, then, by taking the inner product with f on the both sides of Ž3.1 sm ., we have
HHQ g Ž a
0f
Ž x . q u˜Ž t , x . . f Ž x . dt dx s s.
But s1 G
HHQ g Ž a
0f
Ž x . q u˜Ž t , x . . f Ž x . dt dx s s,
˜ then, by the result in which is impossible. If a 0 f Ž x . q u ˜ with 5 u˜5 L2 - M, the proof of Lemma 3.1 we have ss
HHQ g Ž g˜ Ž s*. f Ž x . q u˜. f Ž x . dt dx ) s*,
which is also impossible. Thus for s ) s1 , and m g Ž0, 1x, DLŽ L q Nsm, DŽ GŽ s*... is well defined and DL Ž L q Nsm , D Ž G Ž s* . . . s DB Ž JP2 Nsm , D Ž G Ž s* . . l ker L, 0 . ,
44
WAN SE KIM
where P2 Nsm : L2 Ž V . ª ker L is an operator defined by
Ž P2 Nsm u . Ž t , x . s mHH g Ž u Ž t , x . . f Ž x . dt dx y s f Ž x . . Q
Now let T : ker L ª R be defined by T Ž af Ž x . . s a . Then, for m s 1, DL Ž L q Ns1 , D Ž G Ž s* . . . s DB Ž JP2 N21 , D Ž G Ž s* . . l ker L, 0 . s DB Ž T Ž JP2 Ns1 . Ty1 , T Ž D Ž G Ž s* . . l ker L . , 0 . . If we let J: Im P2 ª ker L be the identity map, then the operator F s T Ž JP2 Ns1 .Ty1 is defined by FŽ a . s
HHQ g Ž af Ž x . . f Ž x . dt dx y s.
Thus, for s1 - s F s*, we have F Ž a0 . s
HHQ g Ž a
0f
Ž x . . f Ž x . dt dx y s - s1 y s - 0,
and by the choice of g ˜ Ž s*., we have FŽg ˜ Ž s* . . s
HHQ
gŽg ˜ Ž s* . f Ž x . . f Ž x . dt dx y s
) s* y s G 0. Therefore < DLŽ L q Ns1, DŽ GŽ s*...< s 1 and the proof is completed. THEOREM. such that
Assume Ž H1 . ] Ž H3 ., then there exists a real number s0 F s1
Ži. Ž E . has no solution for s - s0 . Žii. Ž E . has at least one solution for s s s1. Žiii. Ž E . has at least two solutions for s ) s1. Proof. Let s0 and s1 be constants defined in Lemma 3.2 and the remark. Part Ži. has been proved in Lemma 3.3. For part Žiii., if s ) s1 then we can choose G q DŽ GŽ s .., where G and DŽ GŽ s .. are defined in Lemmas 3.2 and 3.3, respectively.
DISSIPATIVE HYPERBOLIC EQUATIONS
45
By the additivity of degree, we have 0 s DL Ž L q Ns1 , G . s DL Ž L q Ns1 ,D Ž G Ž s . . . q DL Ž L q Ns1 , G y D Ž G Ž s . . . and hence, by Lemma 3.3, DL Ž L q Ns1 , G y D Ž G Ž s . . . s 1. Therefore Ž3.11s . has one solution in DŽ GŽ s .. and one solution in G y D Ž G Ž s . . . For part Žii., let sŽ n.4 be a sequence in R with sŽ1. ) sŽ2. ) ??? ) s1 such that sŽ n. ª s1 and let u n4 be the corresponding sequence of solutions of Ž3.11s Ž n. .. Then u n s a n f Ž x . q u ˜n with a n g R and u˜n g Im L. By Lemma 3.1, we have a subsequence a n k 4 of a n4 which converges to some a in R. On the other hand, by Ž H1 ., Lemma 3.1, and the remark, we can see that Lu n k 4 is a bounded sequence in Im L ; L2 Ž Q .. Because K: Im L ª Im L is a compact operator, and ˜ yn k s K Ž Lu n k ., we have a subsequence, say again, u ˜n k 4 converging to u˜ in Dom L l Im L. Therefore, we have a subsequence u n k 4 of u n4 which converges to u s af q u ˜ with a g R and u˜ g Im L. Because L is closed operator, u g Dom L and u is a solution of Ž3.11s . for s s s1. This completes our proof. REFERENCES 1. A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach space, Ann. Mat. Pura Appl. 93 Ž1972., 231]247. 2. H. Brezis and L. Nirenberg, Characterization of range of some nonlinear operators and applications to boundary value problems, Ann Scu. Norm. Sup. Pisa Cl. Sci. 4 Ž1978., 225]323. 3. R. Chiappinelli, J. Mawhin, and R. Nugari, Generalized Ambrosetti]Prodi conditions for nonlinear two-point boundary value problems, J. Differential Equation 69Ž3. Ž1987., 422]434. 4. S. H. Ding and J. Mawhin, A multiplicity result for periodic solutions of higher order ordinary differential equations, Differential Integral Equations 1Ž1. Ž1988., 31]40. 5. C. Fabry, J. Mawhin, and M. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. Soc. 18 Ž1986., 173]180. 6. N. Hirano and W. S. Kim, Multiplicity and stability result for semilinear parabolic equations, Discrete Continuous Dynamical Syst. 2Ž2. Ž1996., 271]280. 7. N. Hirano and W. S. Kim, Existence of stable and unstable solutions for semilinear parabolic problems with a jumping nonlinearity, Nonlinear Anal. 26Ž6. Ž1996., 1143]1160. 8. N. Hirano and W. S. Kim, Multiple existence of periodic solutions for Lienard system, Differential Integral Equations 8Ž7. Ž1985., 1805]1811.
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9. W. S. Kim, Existence of periodic solution for nonlinear Lienard systems, Internat. J. Math. 18Ž2. Ž1995., 265]272. 10. W. S. Kim, Multiplicity results for doubly periodic solutions of nonlinear dissipative hyperbolic equations, J. Math. Anal. Appl. 197 Ž1996., 735]748. 11. A. C. Lazer and P. J. Mckenna, Multiplicity results for a class of semi-linear elliptic and parabolic boundary value problems, J. Math. Anal. 107 Ž1985., 371]395. 12. M. N. Nkashama and M. Willem, ‘‘Time Periodic Solutions of Boundary Value Problems for Nonlinear Heat, Telegraph and Beam Equations,’’ Seminaire de mathematique, Universite ´ Catholique de Louvain Rapport No. 54, 1984.
Multiplicity Result for Semilinear Dissipative Hyperbolic Equations* Wan Se Kim† Department of Mathematics, College of Natural Sciences, Hanyang Uni¨ ersity, Seoul 133-791, Korea Submitted by Jean Mawhin Received September 28, 1997
1. INTRODUCTION Let Zq, Z, R*, and R be the set of all positive integers, integers, nonnegative reals, and reals, respectively, and let V : R n, n G 1, be a bounded domain with smooth boundary V which is assumed to be of class C 2 . Let Q s Ž0, 2p . = V and L2 Ž Q . be the space of measurable Lebesgue square integrable real-valued functions on Q with usual inner product ² ? , ? : and corresponding norm 5 ? 5 2 . By H01 Ž V . we mean the completion of C01 Ž V . with respect to the norm 5 ? 5 1 defined by 5 f 5 12 s
HV Ý
< D af Ž x . < 2 dx.
< a
H 2 Ž V . stands for the usual Sovolev space; i.e.; the completion of C 2 Ž V . with respect to the norm 5 ? 5 2 defined by 5 f 5 22 s
HV Ý
< D af Ž x . < 2 dx.
< a
Let g: R ª R be a continuous function. Moreover, we assume that there exist constants a0 and b 0 such that < g Ž u . < F a0 < u < q b 0 , for all u g R. * This work was supported by BSRI N971429 Korea Ministry of Education. † E-mail: [email protected]. 34 0022-247Xr99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
Ž H1 .
DISSIPATIVE HYPERBOLIC EQUATIONS
35
The purpose of this work is to investigate the Ambrosetti]Prodi Žbriefly AP. type multiplicity result for the periodic solution of the semilinear hyperbolic equations,
b
u t
q
2u t
2
y D x u y l1 u q g Ž u . s
sf 1
'2p
q h Ž t , x . , in Q, Ž E .
u Ž t , x . s 0, on Ž 0, 2p . = V ,
Ž B1 .
u Ž 0, x . s u Ž 2p , x . , on V ,
Ž B2 .
where l1 denotes the first eigenvalue of yD with zero Dirichlet boundary condition and f 1 is the corresponding positive normalized eigenfunction; i.e., f 1Ž x . ) 0 on V and HU f 12 Ž x . dx s 1, and h g L2 Ž Q . with
HHQ h Ž t , x . f Ž x . dt dx s 0. 1
This type of result, so-called an Ambrosetti]Prodi type result was initiated by Ambrosetti]Prodi w1x in 1972 in the study of a Dirichlet problem to elliptic equations and was developed in various directions by several authors to ordinary and partial differential equations. A notable discussion for AP type results for periodic and Dirichlet boundary value problems has been done by Fabry, Mawhin, and Nkashama w5x and Chiappinelli, Mawhin, and Nugari w3x, respectively, for second-order ordinary differential equations. For AP type results for periodic solutions of higher order ordinary differential equations, we refer to the results of Ding and Mawhin in w4x. AP type results for Lienard systems have been done by Kim w9x, and Hirano and Kim w8x. Lazer and Mckenna treated AP type multiplicity results for elliptic and parabolic equations in w11x. In w10x, AP type results for doubly periodic solutions of dissipative hyperbolic equations in one space dimension have been treated by Kim. In our result, we treat an AP type multiplicity result for Dirichlet-periodic solutions of semilinear dissipative hyperbolic equations in n-dimensional space. We assume the coercive growth on g with a restriction on the left hand and our proof based on Mawhin’s continuation theorem.
2. PRELIMINARY RESULTS Let us define the linear operator L: Dom L : L2 Ž Q . ª L2 Ž Q . ,
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WAN SE KIM
by
½
Dom L s u g L2 Ž Ž 0, 2p . , H 2 Ž V . l H01 Ž V . .
2u t2
u t
g L2 Ž Q . ,
g L2 Ž Q . ,u Ž 0, x . s u Ž 2p , x . , x g V
5
,
and Lu s b
u
q
t
2u t2
y D u y l1 u.
Using Fourier series and Parseval inequality, we get easily
¦
Lu,
u t
;
sb
u t
2 L2
, for all u g Dom L.
Hence ker L s kerŽ D q l1 I . s ker L* because D q l1 I is self-adjoint and kerŽ D q l1 I . is one-dimensional space generated by the eigenfunction f 1. Therefore L is a closed, densely defined linear operator and ImŽ L. s wker L x H ; i.e., L2 Ž Q . s ker L [ Im L. Let us consider a continuous projection P1: L2 Ž Q . ª L2 Ž Q . such that ker P1 s Im L. Then L2 Ž Q . s ker L [ ker P1. We consider another continuous projection P2 : L2 Ž Q . ª L2 Ž Q . defined by
Ž P2 h . Ž t , x . s
1 2p
HHQ h Ž t , x . f Ž x . dt dx f Ž x . . 1
1
Then we have L2 Ž Q . s Im P1 [ Im L, ker P2 s Im L, and L2 Ž Q .rIm L is an isomorphism to Im P2 . Because dimw L2 Ž Q .rIm L x s dimwIm P2 x s dimwker L x s 1, we have an isomorphism J: Im P2 ª ker L. By the closed graph theorem, the generalized right inverse of L defined by K s w L < Dom L l Im L x
y1
: Im L ª Im L
is continuous. If we equip the space Dom L with the norm, 5 u 5 Dom L s
HHQ
u q 2
u
2
q
2u
ž / ž / t
t2
2
q
Ý Ž Dxb u .
< b
2
dt dx,
DISSIPATIVE HYPERBOLIC EQUATIONS
37
then there exits a constant c ) 0 independent of h g Im L, u s Kh such that 5 Kh 5 Dom L F c 5 h 5 L2 . Therefore K: Im L ª Im L is continuous and by the compact imbedding of Dom L in L2 Ž Q ., we have that K: Im L ª Im L is compact. LEMMA 2.1. L is a closed, densely defined linear operator such that ker L s wIm L x H and such that the right in¨ erse K: Im L ª Im L is completely continuous.
3. MULTIPLICITY RESULT Let us consider the following,
b
u t
q
2u t2
y D x u y l1 u q m g Ž u . s m sf q m h Ž t , x . , in Q,
Ž Esm .
u Ž t , x . s 0, on Ž 0, 2p . = V ,
Ž B1 .
u Ž 0, x . s u Ž 2p , x . , on V ,
Ž B2 .
where m g w0, 1x and f Ž x . s w f 1Ž x .r'2p x. Let L: Dom L : L2 Ž Q . ª L2 Ž Q . be defined as before. If we define a substitution operator Nsm : L2 Ž Q . ª L2 Ž Q . by
Ž Nsm . Ž t , x . s m g Ž u . y m sf y m h Ž t , x . , for u g L2 Ž Q . and Ž t, x . g Q, then Nsm maps continuously into itself and takes bounded sets into bounded sets. Let G be any open bounded subset of L2 Ž Q ., then P2 Nsm : G ª L2 Ž Q . is bounded and K Ž I y P2 .: G ª L2 Ž Q . is compact and continuous. Thus Nsm is L-compact on G. The coincidence degree DLŽ L q Nsm, G . is well defined and constant in m if Lu q Nsm / 0 for m g w0, 1x, s g R, and u g Dom L l G. It is easy to check that Ž u, m . is a weak solution of Ž Esm . if and only if u g Dom L and Lu q Nsm u s 0.
Ž 3.1 sm .
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WAN SE KIM
Here we assume the following, lim inf g Ž u . s q`, < u <ª`
lim sup
g Ž u. u
uªy`
- l2 y l1 .
Ž H2 . Ž H3 .
From Ž H2 . and Ž H3 ., we may assume that m s inf g Ž u . ) y`, ugR
and there exist a g Ž0, l2 y l1 . and b G 0 such that < g Ž u . < F a < u < q b, for all u F 0. LEMMA 3.1. If Ž H1 . ] Ž H3 . are satisfied, then for any s g Rq, there exists M Ž s*. ) 0 such that 5 u 5 L2 F M Ž s* . holds for each possible weak solution u s af 1 q u, ˜ with a g R and u˜ g Im L, of Ž Esm . with m g w0, 1x and < s < F s*. Proof. Suppose there exists a constant s with < s < F s* and corresponding solutions Ž u n , m n . of Ž3.1 sm n . such that lim 5 u n 5 L2 s `.
nª`
For each n G 1, we put u nŽ t, x . s a n f Ž x . q u ˜nŽ t, x .. First, we are going to prove that lim
nª`
< an < 5u ˜n 5 L2
s c - `.
If it is not the case, then, if necessary we may take a subsequence, we have easily lim < u n Ž t , x . < s `, a.e. on Q.
nª`
By taking the inner product with f on both sides of Ž3.1 sm ., we have
HHQ g Ž u Ž t , x . . f Ž x . dt dx s s F s*. n
DISSIPATIVE HYPERBOLIC EQUATIONS
39
On the other hand, by H2 and Fatou’s lemma, we have lim
nª`
HHQ g Ž u Ž t , x . . f Ž x . dt dx s `, n
which leads to a contradiction. First, we assume that 0 - c - `, then there exists n 0 g N such that c 2
3c
5u ˜n 5 L2 F < a n < F
2
5u ˜n 5 L2 , for all n G n 0 .
For given e ) 0, we may choose d ) 0 such that
HHA < f <
dt dx - e 5 f 5 2L2 ,
2
for any measurable set A ; Q with < A < F d . Let 0 - g - 5 f 5 ` and V 0 s x g V: f Ž x . G g 4 . Choose M0 ) 0 such that
d M0 y < m <
HHQ f dt dx ) s*.
Then, because lim uª` g Ž u. s `, we have that m 0 s sup < u < : g g Ž u . - M0 4 - `. We put Q n s Ž t , x . g w 0, 2p x = V 0 : < u n Ž t , x . < G m 0 4 . Then we have < Q n < F d . In fact, if < Q n < ) d , then from the definition of m 0 we have
HHQ g Ž u Ž t , x . . f Ž x . dt dx n
s
HHQ
g Ž u n . f Ž x . dt dx q n
) d M0 y < m < ) s*,
HHQ_Q
HHQ f Ž x . dt dx
g Ž u n . f Ž x . dt dx n
40
WAN SE KIM
and this leads to a contradiction. Therefore, we have
HHQ_Q
< an f < 2 G Ž 1 y e . n
HHQ < a f < n
2
.
On the other hand, 0s s F
HHQ a f u˜ n
HHQ_Q 1 2
n
an f u ˜n q n
HHQ
HHQ_Q ž < a f q u˜ < n
2
n
an f u ˜n n
y < an f < 2 y < u ˜n < 2 q
/ HHQ
n
< an f < < u ˜n < . n
From the definition of m 0 and the preceding facts, we have, for all n G n 0 , 0F s
1 2 1 2
m20 y m20 y
1 2 c 4
c
3c
2
2
Ž 1 y e . 5 u˜n 5 2L2 q e
5u ˜n 5 2L2
Ž 1 q 5e c . 5 u˜n 5 2L2 .
Therefore, 5 u ˜n 5 L2 4 is bounded and hence 5 u n 5 L2 4 is bounded which leads to a contraction. Next, we assume c s 0, then lim nª`Ž5 u ˜n 5r5 u n 5 L2 . s 1. Multiplying Ž3.1 sm . by ur t and integrating over Q, we find from the periodicity of u that
u t
2
L
F
1
5 h 5 L2 .
Again, taking the inner product with u n on both sides of Ž3.1 sm ., we have
Ž l2 y l1 . 5 u˜n 5 2L2 y
un t
2 L2
q ² g Ž u n . , u n : F s* < a n < q 5 h 5 L2 5 u ˜n 5 L2 ,
and hence, lim sup Ž l 2 y l1 y a . 5 u ˜n 5 2L2 nª`
F max < m < , b 4 < Q < 1r2 q s* q
1 < b <2
5 h 5 2L2 q 5 h 5 L2 .
41
DISSIPATIVE HYPERBOLIC EQUATIONS
Thus 5 u ˜n 5 L2 4 is bounded and thus 5 u n 5 L2 4 is bounded which leads to another contradiction. Remark. By Lemma 3.1, we may have a priori bounds M1Ž s*. ) 0 and g 1Ž s*. ) 0 such that < a < F g 1 Ž s* . , 5u ˜5 L2 F M1 Ž s* . , for each possible weak solution u s af q u ˜ of Ž Esm . with < s < F s* and m g w0, 1x. LEMMA 3.2. If Ž H1 . ] Ž H3 . are satisfied, then, for each s* g Rq, we can find an open bounded set GŽ s*. in L2 Ž Q . such that, for each open bounded set G in L2 Ž Q . such that G = GŽ s*., we ha¨ e DL Ž L q Ns1 , G . s 0, for all < s < F s*.
˜ ) M1 which is given in the Remark of Lemma 3.1. Let Proof. Let M
½
Q0 s Ž t , x . g Q < u ˜Ž t , x . < G
˜ 1qM < Q<
5
.
Then
˜2 G M
HHQ < u˜Ž t , x . <
G
HHQ < u˜Ž t , x . <
2
dt dx
2
dt dx
0
G < Q0 <
˜ 1qM < Q<
2
.
˜ Ž1 q M .x2 < Q < and hence < Q _ Q0 < s < uŽ t, x . g Therefore < Q0 < F w Mr ˜ Ž1 q M .x2 < Q < ) 0. Q< < u ˜Ž t, x .< F Ž1 q M˜ .r< Q <4< G w1 y Mr Let M0 s min x g V , ug R g Ž u. f Ž x . and W s Ž0, 2p . = V 0 . Suppose that a g R and < a < ª `, then < af Ž x .< ª ` for each x g V 0 . Hence, by Fatou’s lemma and Ž H2 ., we have lim inf < a <ª`
HHQ g Ž af Ž x . q u˜Ž t , x . . f Ž x . dt dx
s lim inf < a <ª`
G
HHQ
g Ž af Ž x . q u ˜Ž t , x . . f Ž x . y M0 dt dx q M0 < Q <
inf HHWlŽ Q_Q . lim < a <ª` 0
q M0 < Q < s `.
g Ž af Ž x . q u ˜Ž t , x . . f Ž x . y M0 dt dx
42
WAN SE KIM
Hence, there exists r 2 Ž s*. ) 0 such that, for < a < ) r 2 Ž s*., we have
HHQ g Ž af Ž x . q u˜Ž t , x . . f Ž x . dt dx ) s*. Let G Ž s* . s u g L2 Ž Q . N yr˜Ž s* . f Ž x . - af Ž x . - ˜ r Ž s* . f Ž x . for x g V , 5 u ˜5 L 2 - M˜ 4 , where u s af Ž x . q u ˜ with ˜r Ž s*. ) max r1Ž s*., r 2 Ž s*.4 and M˜ ) M1 which are given in the Remark of Lemma 3.1. Let s0 s d min g Ž u . , ugR
where d s 2p HV f Ž x . dx. If Ž3.1 sm . has a solution u for some s g R and m g w0, 1x, then by taking the inner product with f on both sides of Eq. Ž3.1 sm ., we have s0 F
HHQ g Ž u Ž t , x . . f Ž x . dt dx s s.
Thus Ž3.1 sm . has no solution for s - s0 . Hence for each open bounded set G = GŽ s*., we have DL Ž L q Ns1 , G . s 0, for s - s0 . Choose s - s0 and define F : Ž D Ž L . l G . = w 0, 1 x ª L2 Ž V . , by F Ž u, m . s Lu q NŽ1y m . sq m s Ž u . , for < s < F s*. They by Lemma 3.1 and the remark, we have 0 f F Ž D Ž L . l G . = w 0, 1 x , for < s < F s*. By the homotopy invariance of degree, we have, for all < s < F s*, DL Ž L q Ns1 , G . s DL Ž F Ž ?, 1 . , G . s DL Ž F Ž ?, 0 . , G . s DL Ž L q Ns1 , G . s 0, and the proof is complete.
DISSIPATIVE HYPERBOLIC EQUATIONS
43
LEMMA 3.3. If Ž H1 . ] Ž H3 . are satisfied, then there exists s1 ) s0 such that, for each s* ) s1 , we can find an open bounded set DŽ GŽ s*.. in L2 Ž Q . on which DL Ž L q Ns1 , D Ž G Ž s* . . . s 1, for all s1 - s F s*. Proof. Let g Ž a0 f Ž x0 . q u ˜0 . s
By Ž H1 ., we may have f Ž x . dt dx <. Define
min xgV < a
g Ž af Ž x . q u ˜. .
s1 s max 5 u˜5 L 2 F M˜ < HHQ g Ž a 0 f Ž x . q u ˜Ž t, x ..
D Ž G Ž s* . . s u g L2 Ž Q . N a 0 f Ž x . - af Ž x . - g ˜ Ž s* . f Ž x . for x g V , 5 u ˜5 L 2 - M˜ 4 , where g ˜ Ž s*. and M˜ are given in Lemma 3.2. If s ) s1 , m g w0, 1x and Ž u, m . is a possible solution of Ž3.1 sm . such that u g DŽ GŽ s*.., then by Ž B1 ., the Remark of Lemma 3.1, we have necessarily u s a 0 f Ž x . q u ˜ or u s g˜ Ž s*. f Ž x . q u. ˜ If u s a 0 f Ž x . q u˜ with ˜ 5u ˜5 L2 - M, then, by taking the inner product with f on the both sides of Ž3.1 sm ., we have
HHQ g Ž a
0f
Ž x . q u˜Ž t , x . . f Ž x . dt dx s s.
But s1 G
HHQ g Ž a
0f
Ž x . q u˜Ž t , x . . f Ž x . dt dx s s,
˜ then, by the result in which is impossible. If a 0 f Ž x . q u ˜ with 5 u˜5 L2 - M, the proof of Lemma 3.1 we have ss
HHQ g Ž g˜ Ž s*. f Ž x . q u˜. f Ž x . dt dx ) s*,
which is also impossible. Thus for s ) s1 , and m g Ž0, 1x, DLŽ L q Nsm, DŽ GŽ s*... is well defined and DL Ž L q Nsm , D Ž G Ž s* . . . s DB Ž JP2 Nsm , D Ž G Ž s* . . l ker L, 0 . ,
44
WAN SE KIM
where P2 Nsm : L2 Ž V . ª ker L is an operator defined by
Ž P2 Nsm u . Ž t , x . s mHH g Ž u Ž t , x . . f Ž x . dt dx y s f Ž x . . Q
Now let T : ker L ª R be defined by T Ž af Ž x . . s a . Then, for m s 1, DL Ž L q Ns1 , D Ž G Ž s* . . . s DB Ž JP2 N21 , D Ž G Ž s* . . l ker L, 0 . s DB Ž T Ž JP2 Ns1 . Ty1 , T Ž D Ž G Ž s* . . l ker L . , 0 . . If we let J: Im P2 ª ker L be the identity map, then the operator F s T Ž JP2 Ns1 .Ty1 is defined by FŽ a . s
HHQ g Ž af Ž x . . f Ž x . dt dx y s.
Thus, for s1 - s F s*, we have F Ž a0 . s
HHQ g Ž a
0f
Ž x . . f Ž x . dt dx y s - s1 y s - 0,
and by the choice of g ˜ Ž s*., we have FŽg ˜ Ž s* . . s
HHQ
gŽg ˜ Ž s* . f Ž x . . f Ž x . dt dx y s
) s* y s G 0. Therefore < DLŽ L q Ns1, DŽ GŽ s*...< s 1 and the proof is completed. THEOREM. such that
Assume Ž H1 . ] Ž H3 ., then there exists a real number s0 F s1
Ži. Ž E . has no solution for s - s0 . Žii. Ž E . has at least one solution for s s s1. Žiii. Ž E . has at least two solutions for s ) s1. Proof. Let s0 and s1 be constants defined in Lemma 3.2 and the remark. Part Ži. has been proved in Lemma 3.3. For part Žiii., if s ) s1 then we can choose G q DŽ GŽ s .., where G and DŽ GŽ s .. are defined in Lemmas 3.2 and 3.3, respectively.
DISSIPATIVE HYPERBOLIC EQUATIONS
45
By the additivity of degree, we have 0 s DL Ž L q Ns1 , G . s DL Ž L q Ns1 ,D Ž G Ž s . . . q DL Ž L q Ns1 , G y D Ž G Ž s . . . and hence, by Lemma 3.3, DL Ž L q Ns1 , G y D Ž G Ž s . . . s 1. Therefore Ž3.11s . has one solution in DŽ GŽ s .. and one solution in G y D Ž G Ž s . . . For part Žii., let sŽ n.4 be a sequence in R with sŽ1. ) sŽ2. ) ??? ) s1 such that sŽ n. ª s1 and let u n4 be the corresponding sequence of solutions of Ž3.11s Ž n. .. Then u n s a n f Ž x . q u ˜n with a n g R and u˜n g Im L. By Lemma 3.1, we have a subsequence a n k 4 of a n4 which converges to some a in R. On the other hand, by Ž H1 ., Lemma 3.1, and the remark, we can see that Lu n k 4 is a bounded sequence in Im L ; L2 Ž Q .. Because K: Im L ª Im L is a compact operator, and ˜ yn k s K Ž Lu n k ., we have a subsequence, say again, u ˜n k 4 converging to u˜ in Dom L l Im L. Therefore, we have a subsequence u n k 4 of u n4 which converges to u s af q u ˜ with a g R and u˜ g Im L. Because L is closed operator, u g Dom L and u is a solution of Ž3.11s . for s s s1. This completes our proof. REFERENCES 1. A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach space, Ann. Mat. Pura Appl. 93 Ž1972., 231]247. 2. H. Brezis and L. Nirenberg, Characterization of range of some nonlinear operators and applications to boundary value problems, Ann Scu. Norm. Sup. Pisa Cl. Sci. 4 Ž1978., 225]323. 3. R. Chiappinelli, J. Mawhin, and R. Nugari, Generalized Ambrosetti]Prodi conditions for nonlinear two-point boundary value problems, J. Differential Equation 69Ž3. Ž1987., 422]434. 4. S. H. Ding and J. Mawhin, A multiplicity result for periodic solutions of higher order ordinary differential equations, Differential Integral Equations 1Ž1. Ž1988., 31]40. 5. C. Fabry, J. Mawhin, and M. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations, Bull. London Math. Soc. 18 Ž1986., 173]180. 6. N. Hirano and W. S. Kim, Multiplicity and stability result for semilinear parabolic equations, Discrete Continuous Dynamical Syst. 2Ž2. Ž1996., 271]280. 7. N. Hirano and W. S. Kim, Existence of stable and unstable solutions for semilinear parabolic problems with a jumping nonlinearity, Nonlinear Anal. 26Ž6. Ž1996., 1143]1160. 8. N. Hirano and W. S. Kim, Multiple existence of periodic solutions for Lienard system, Differential Integral Equations 8Ž7. Ž1985., 1805]1811.
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9. W. S. Kim, Existence of periodic solution for nonlinear Lienard systems, Internat. J. Math. 18Ž2. Ž1995., 265]272. 10. W. S. Kim, Multiplicity results for doubly periodic solutions of nonlinear dissipative hyperbolic equations, J. Math. Anal. Appl. 197 Ž1996., 735]748. 11. A. C. Lazer and P. J. Mckenna, Multiplicity results for a class of semi-linear elliptic and parabolic boundary value problems, J. Math. Anal. 107 Ž1985., 371]395. 12. M. N. Nkashama and M. Willem, ‘‘Time Periodic Solutions of Boundary Value Problems for Nonlinear Heat, Telegraph and Beam Equations,’’ Seminaire de mathematique, Universite ´ Catholique de Louvain Rapport No. 54, 1984.