- Email: [email protected]
On the periodic nonlinearity and the multiplicity of solutions
:~on,mror Prmtrd
A~J,,w. Thuory. ,n Grear Bnrain.
.Merhods
8 .4pplrrom~ns.
Vol.
13.
No.
5. PP.
527-537.
0362-546X/89 $3.03 + .OU C 1989 Pergamon Press plc
1989.
ON THE PERIODIC NONLINEARITY AND THE MULTIPLICITY OF SOLUTIONS \
KUNG-CHING
CHANG*
Center for the hlathematical Sciences and Department of Mathematics, University of Wisconsin-Madison, WI 53706, U.S.A. and Peking University, People’s Republic of China
Madison,
(Received 7 March 1988; received for publication 27 April 1988) Key words and phrases: Periodic nonlinearity,
multiple solutions, critical point, Hamiltonian systems,
Neumann problem, periodic solution.
1. INTRODUCTION
by the work of Conley and Zehnder [3] on the solution of the Arnold conjecture, the author presented a different proof of their statement, and noticed that the periodicity of the Hamiltonian function is the essence of the occurrence of multiple periodic solutions [l-2]. Several other authors have recently studied questions of periodic solutions to periodic equations. They use a variety of methods based on their particular problems. The main purpose of this paper is to show that the following abstract theorem obtained in [l] can be used to give a unified approach to this class of problems. Let H be a real Hilbert space, and let A be a bounded self-adjoint operator defined on H. According to its spectral decomposition, H = H+ @ Ho @ H_, where H, , Ho, and H_ are invariant subspaces corresponding to the positive, zero, and negative spectrum of A respectively. INSPIRED
THEOREM 1. Suppose that A satisfies the following assumptions (Hi) A, P Al, has a bounded inverse on H,, (H,) y p dim(H_ @ H,,) c 00. Let V” be a C2 compact n-manifold without boundary, and let g E C’(H x V”, R’) be a function having a bounded and compact differential dg(x). Assume that
gW,x, 4 + - 03 where PO is the orthogonal
projection
as IRxll + ~0
if dim H,, # 0
onto Ho. Then the function
f(x, v) = *(Ax, x) + g(x, v) possesses at least cuplength (v”) + 1 distinct critical points. If further, we assume that g E C2(H x V, R’), and thatfis nondegenerate, thenfhas at least Cy= ,, pi( V”) critical points, where pi( V”) is the ith Betti number of V”, i = 0, 1, . . . , n. In the statement of theorem 8.3 in [l], the function g was assumed to be C2. However, in the proof of the first conclusion, Cr is sufficient.
Remark.
*Supported
by the U.S. Army Research Office under Contract No. DAAL03-87-K-0043. 527
KLTG-CHING
528
CHANC
hlost recent studies were only concerned with the case where A is positive definite. We shall give more applications where A is semidefinite, i.e. the negative eigenspace as well as the null space are finite dimensional. They are used to study semilinear elliptic systems and periodic solution problems for 2nd order ODE. Our theorem 3 generalizes and unifies results due to Mawhin [7], Mawhin and Willem (81, Li [6], Jiang [5], Franks [4], Pucci and Serrin (9, lo] and Rabinowitz [ 111. In our setting periodic solution problems for Hamiltonian systems reduce to case where A is unbounded and indefinite. Theorem 4 is a generalization of theorem 3. It implies the early results due to Conley and Zehnder [3] as special cases. In particular, multiple periodic solutions of Hamiltonian systems with resonance are studied, where the Hamiltonian functions are only periodic in certain variables. 2. SEhII-DEFINITE
FUNCTIONALS
A direct consequence of theorem 1 is the following. 2. Suppose that A is a self-adjoint operator satisfying (H,) and (Hz), defined on a Hilbert space H. Suppose that @ E C’(H, R’) is a function having a bounded and compact differential d@, and satisfies the following periodicity condition. (P) 3 e,, . . . . e, E ker A, they are linearly independent, and 3 (T, , . . . , K) E R’ such that THEOREM
i
mj7jej
j=I
>=we,
vx~H,v(m~,...,m,)~Z’
and the resonance condition. (LL) O(x) -+ -co if //xi1 -+ co and x E ker(A)n[e,, Then the equation
. . . . e,)‘.
Ax + d@(x) = 0 possesses at least r + 1 distinct solutions. If further, Q>E C’(H, R’) and all solutions of (2.1) are nondegenerate, at least 2’ solutions.
(2.1) then (2.1) possesses
Proof. We consider the following functional J(x) =
i(Ax, x) + Q(x).
According to (P), =
JO,
V (m, , . . . , m,) E Z’.
>
However, we have an orthogonal
decomposition H = ker A @I (ker A)’ = Z @ Y 0
(ker A)’
where Z = span(e,, . . . . e,), and Y = Z* fl ker(A). If we restrict ourselves on the quotient
Periodic nonlinearity and multiplicity of solutions
529
space T’ x (Y @ (ker A)‘) where T’ = Z/Z’(T,, functionals
Z’(T,,
. . . . T,),
. . . . T,): = ((m, T,, . . . . m,T,)I(m,,
. . . . m,) E Z’),
the
f(u, v) = JW, and g(u, v) = Q(x), are well defined, where (v, U) E T’ x (Y @ (ker A)‘) and x = u + v. The critical point off is a solution of (2.1). Sincefand g satisfy all conditions in theorem 1 with H, = Y and V = T’, the conclusion follows directly. We present here an application. THEOREM 3. Let M be a compact
matrix
valued
manifold without boundary, function defined on M, and let
continuous
ker(where 0 < r < N are assumptions F X,U+ (
~
A 0 I,_,
integers.
(N - r)
+ (aij(X)) * ) = span(vt , . . ., rpkl,
Assume
= F(x, u)
mi~I;i
i=l
let (Uij(X)) be a symmetric
that
FE C’(M
v(x,u)~M
x RN, R’)
satisfies
x RN,v(ml
,...,
the
rn,)~Z’
following
(1)
>
where ei = (0, . . . , 1, , . . , 0), i = 1,2, . . . , r, and (T, , . . . , T,) E R’ is given,
(2) (3) and that h E C(M, RN), h = (h, , . . . , hN) satisfies hi(X)
dx = 0,
i = 1,2, . . . , r,
i ,‘+f and hi(x) = 0, j = r + 1, . . . , N. Then the elliptic -AU has at least r + 1 solutions,
Proof.
system
+ C(x) . u - F,(x, u) + h(x) = 0
on M
where
Let H = W’**(M, R “), A = Z, + (- A)-‘d(x), Q(u) =
and
- F(x, U(X)) + (h(x) - u(x)>,. M
530
KUNG-CHING CHANC
Obviously, ker A = spame, and @ E C’(H, R’), having a bounded
, . . . , e,, cp,, . . . , pkl,
and compact
differential,
satisfies
the conditions
(P) and
(LL). The conclusion
follows
immediately
from theorem
2.
Remark 2.1. In theorem 3, we may replace the compact manifold M by a smooth Sz in R”, in addition to the Neumann boundary value condition
bounded
domain
au 0, ay an = where v is the outward
pointing
normal
to the boundary,
a52.
Example 2.1. A4 = S’, r = N = 1. This is just the periodic
solution
problem
for the ODE
ii + F”(f, u) = h(f)
(2.2)
where F E C’(S’ x R’, R’) is periodic in u, and h E C(S’, R’) satisfies the zero mean condition j,&(t) dt = 0. Under these conditions, (2.2) has at least two solutions. It was shown by Mawhin and Willem [8], Li [6] and Franks [4].
Example 2.2. The case M = S’, and r = IV. The corresponding ODE system was studied by Jiang [5] and Rabinowitz [ll]. In this case, the following system possesses at least N + 1 solutions ii + F”(t, u) = h(t) (2.3) whereFEC’(S’
x
R”,R’)isperiodicinu=
(nt,...,
u,,,), and h E C(S’, RN), satisfies
j,&(t)
dt = 8. Example 2.3. The case M = S’, r < N, with (aji(t))~N_r~x~,v_r~ positive system was studied by Mawhin [7]. The system
definite.
The ODE
ii - a(t)24 + F,(t, u) = 0 possesses at least r + 1 solutions, provided that F E C’(S’ x RN, R’) is periodic variables (u, , . . . , U, and llF,(t, 4 IIP < 00.
Example 2.4. The case M = T”, r = N = 1. The problem IO]. The following equation Au + F,,(x, u) = 0 possesses
at least two solutions,
The Neumann Rabinowitz [ 111.
problem
for
provided the
was studied
in the first r
by Pucci and Serrin [9,
on T”
that F E C’(T” x R’, R’), and is periodic
elliptic
equation
(in
case
in u.
r = N = 1) was studied
by
Periodic nonlinearity and multiplicity of solutions
531
Remark 2.2. All the above examples deal only with functionals bounded from below, however, theorem 3 implies more than that. The improvements are in two directions. (1) The functional is semi-definite, i.e. it is bounded from below except on a finite dimensional subspace. (2) The resonance case is studied, it only happens when r < n. 3. INDEFINITE
FUNCTIONALS
In this section, we shall extend the results of Section 2 to indefinite functionals. The saddle point reduction argument will be applied. Let H be a Hilbert space, and let A be a self-adjoint operator with domain D(A) C H (unbounded). Assume that F is a potential operator with 0 E C’(H, RI), F = d@ and Q(0) = 0. The following assumptions are made. (A) 3 cx < 0 < p such that cr, /3 d a(A) and a(A) II [a, /3] consists of at most finitely many eigenvalues of finite multiplicities. (F) F is bounded and Gateaux differentiable, with
(D) For small E > 0, with --E d a(A),
let V = D(l(&Z + A)[“‘),
THEOREM 4. Suppose that (P) 3 e,, . . . . e, E ker A, they are linearly ~
Xf (
~
mj~ej
j= I
independent,
= @,(X1,
assume that 0 E C2( I’, R’).
and 3 (r, , . . . , T,) E R’, such that
, , .... m,) E z',
VW
>
(LL) Q(x) + &co if /Ix~/ + ~VXE Then the equation
kerAnspan(e,,...,e,JL.
Ax + W(x) = 0 has at least r + 1 distinct Proof.
A saddle point
solutions. reduction
procedure
where (EJ is the spectral
dJ%, I OL
p,
=
E,, 14
resolution
of A, and let
Ho = P,H, and for small E > 0, --E d a(A),
Let
+m
‘0 PO =
is applied.
H,
= P,H
let
I$ = [(&I + A)I-“‘Ho,
VI, = 1(&I + /I)!-“‘H,.
For each u E H, we have the decomposition u = u,
+ ug + u_
VXEH.
KUNG-CHING CHANG
532
wichu,EHo,u,EH,.Letx=x++x,+x_EV,where _r, = I(&1 + A)y*u,, We define
a function
x, = \(&I + /I)I-“*U*.
on the finite dimensional a(z) = $4x(z),
where x(z) = x+(z) + x_(z) + z, z=x, x,
= -(&I
space
V0 as follows
x(z)) + @(x(z))
E Vo, and x,(z) + A)-9*(&Z
are the solutions
of the equations
+ E-)(x+ + x_ + z).
We shall prove that 1” X*(Z+i,
VZE
Tej) =X+(Z),
V,.
In fact, P,(E/
+ F) X+ +X_
+ Z +
i
= PAW
Tej
j= I
(
f F)(x+
+ x_ + z)
>
therefore
2”
U(Z +j$,
qej)
=
Q(Z).
CLAIM.
Tej)) +@(x(z +j, 7;e,)) +@ i
+j,
a(Z+~,Tej)=i@x(Z+j$,?ej)~x(z
= + AX(Z),X(Z) + i Tjej ( j=l >
X(Z)+
7jej
j= I
(
>
= +(Ax(z),x(z)).+ @(x(z)) = a(z).
3” a satisfies spanle, , . . ., e,)‘. CLAIM. Suppose According
the
(PS)
condition
that (zk) is a sequence
to Chang
on
along
which (a(&)
17 V.
is bounded,
where
and
Y = N(A) fl
i/a’(8) 11= o(l).
[l, p. 1051, /IAx
Let Q be the orthogonal
T’ x (Y @ N(A)‘)
projection
+ F(x(zk)& onto
= o(l).
Y, which is considered
as a subspace
of the Hilbert
Periodic
nonlinearity
and multiplicity
of solutions
533
space X = Y @ N(A)*. Thus on the space 3~2, (I - Q)x(zk) = -A-‘(I since F is bounded,
- Q)F(x(?))
11 (I - Q)x(zk) ~1is bounded.
@(Q.4zkN = @,(x(z?) -
+ o(l),
Noticing
’ VMzkh .r0
V - QMzkN dt I
=
a(zk)
-
+(nx(zk),x(zk))
-
VMzkN,
(1
-
QMzkN dt,
1 0
where
x,(z) = ((1 - 01 + tQ)x(zh
and (Ax(zk), x(zk)) = (Ax(zk), (I -
QMz"N = (-FM.&) + o(l), U - QMzk
be bounded. According to the condition (LL), Qx(zk) is bounded. The compactness of zk now follows from the boundedness of x(zk) and the finiteness of the dimension of Vo. @(Qx(zk)) must
4” If we decompose
V. into span{e, , . . . , e,) @ (Y @ N(A)‘)
fl V,,
z = v + w, (v, w) E span(e, , . . ., e,] @ (Y @ fV(A)*)fl
g(w, v) =
Vo,
+(Ar(w + 9, <(w + 4) + cD(x(w + v))
where
r(z) = x+(z) + x-(z) then g is well defined on T’ x (Y @ N(A)‘)
n V,, and
dg(w, v) = P,F(x(w
+ v))
which is bounded and then is compact on finite dimensional manifold. The function a(z) now is written in the following form: a(w, v) = i(Aw, w) + g(w, v). Noticing that F is bounded, Y we have
dY, 4 = tb‘wY
/It(z) 11is always bounded.
If we denote y the projection of w onto
+ v), <(Y + VI) + WY) + way
+ v) + Y + 9 - WAI.
The first term and the third term are bounded, therefore g(Y* v) + f cQ
as IIYII -+ 00.
The function a(w, v) satisfies all assumptions of theorem 1. Theorem 4 is proved. Now we study the periodic solutions of the Hamiltonian functions are periodic in some of the variables.
systems, in which the Hamiltonian
531
I~UNG-CHINGCHANG
We use the following
notations:
p, q E RIV,
P = (P,, . ..>P!).,
(4,) . ..? qN)r
4 =
4 = (s,, . . . . qr),
P = (P,, . . ..P.),
P = (P,,
11
. . .
B = (4rt
PA
I
lz~r
17
q,),
. . .*
B = (Ps+,r . . ..PTh
B = (qs+,, . ..r (7Th
ii = (Pr+*, . ..rPN)r
4 = (q-r+,, .-.,qd.
We assume the following. (I) A(t), B(t), C(t) and D(t) are symmetric continuous matrix functions on S’, of order (S - r) x (S - r), (T - s) x (T - s), (N - T) x (N - T) and (N - T) x (N - T) respectively. Let A = j,&(t), and B = j,J?(t) be invertible. (II) fi E C’(S’ x R2N , R’) is periodic in the following variables p, q,$, 4, and @ is bounded. (III)
Let spanip,,
...,
-I$
- (C(t)
where
@ D(t)) >
J= -
(
0
IN-T
and vI , . . . , p,,, are linearly independent.
And
A(t,j_17jfj7j)-*m
as171=(7f+..-+7~)“2-+oo.
(IV)
c,d~C(S’,R~),withc=(c
,,___, c&d=(d ~S,ckG
i = 1, . . ..r. s + 1, . . . . T,j We define a Hamiltonian
,,...,
= j/ij(‘)
d,)and
= 0,
= I, . . ..s. function as follows
T +
THEOREM
5. Under
conditions
;C,
(C;(fk’i
(I)-(IV), -J$
has at least r + T + 1 periodic
solutions,
+
dAt)qi)
+
the Hamiltonian = H,(t, z),
fi(fv~,
4).
system
(HS)
t E s’
where z = (p, q) E RZN.
Periodic nonlinearity
Proof,
and multiplicity
535
of solutions
Let
and let (the subscripts on J coincide with those on p) Ct=
=
(-J$ -
A(f) >
(-1;) @(-I$- (a(f)o))0 (-J-g (” B(r)))
@(4
- (c(f) D(J).
We have
(a(‘)o)),
(p,Q)eker(-Jz-
with 4(2n) = 4(O),
(i.e. with A - C = 6). According to the assumption I, E = 0. We have
Similarly
Thus ker(@.) = R2' @ Rs-'@ RT-*@ spanIp,,...,( Let Q(z) =
fi(t,
Z(t))
+
i [Ci(f)Pi(f) i= I
+
di(th?i(OI
1
dt.
Then all the assumptions (A), (F), (D), (P) and (LL) are satisfied. The proof is complete.
KUNC-CHING CHANG
536
Example 3.1. If the Hamiltonian function H E C’(S’ x R ‘lv, R’) is periodic in each variable, then (HS) has at least 2N + 1 periodic solutions. This is the case r = s = T = N. This result related to the Arnold conjecture, was first obtained by Conley and Zehnder 131, see also Chang [2]. Example 3.2. If H E C2(S1 x RZN, R’), where His periodic in the components of q, and that there is an R > 0 such that for IPI 1 R, W,p,q)
=
$?P*P
+ asp
where a E RN, and M is a symmetric nonsingular time independent matrix, then the corresponding (HS) possesses at least N + I distinct periodic solutions. This is the case r = 0, S = T = N. This is a result obtained by Conley and Zehnder [3], see also P. H. Rabinowitz [ll]. Example 3.3. Let H E C2(S’ x R4, R’) be periodic in (p,, 4,). Assume that 3 R > 0 such that H(t, ~1,
~2 9 41,421
=
+(cP: + dq:) f Am
for m 2 R, where cd = kz 2 0 for some k E Z, and A > 0. Then the corresponding (HS) possesses at least 3 periodic solutions. In fact, ker( -J$
- (i
3)
= span((
-&sink,
coskt),
(Jcoskt,
sinki)]
it follows d (J.: + :) > - ( - ,J.,sinkt + A2coskt)’ + (1, coskt + A2 sinkt)2 2 min c
Therefore sinkt + 2, coskt), 0, (A, coskt + A2 sinkt)
A
) or -+ --co,
Remark. In the assumption (I), if the operators A and B are singular, then
Thus, ker(-J-$
- (a(l)
o))
= ((8,$ld~R~-‘)
0
[(E, ~~A(.s)ds?)[F~
herA].
Periodic
nonlinearity
and multiplicity
of solutions
537
Similarly
ker( 4; - (” B(I))) =((f, @I2
E
R-1
0
[(pwdd,d)ldE ker Bj.
In order to apply theorem 5, the assumption III is replaced by ~(r,c+
~~a(S)drC+
j:B(s)drd^+d+i~~r,s(r))-
foe,
as [cl + ld^l + 15) -+ CQ,where S E ker A, d^E ker 3, and 5 E R “‘. The same theorem holds. Example
3.4. Let H E C’(S’ x R’, R’) be periodic in (pl , ql, q2). Assume that 3 R > 0
such that H(t, ~1,
~2941,421
for lp21 2 R, where A > 0 is a constant, periodic solutions. In fact, &,p,,p,,q,,q,) Acknowledgement-We Sciences,
University
thank Professor of Wisconsin-Madison,
=
tcostp:
*Am
then the corresponding
= *Am
+ fm,
P. H. Rabinowitz for his invitation and for his very kind conversations
(HS) possesses at least 4
as ICI -+ m. to the Center for the Mathematical on his interesting preprint [I 11.
REFERENCES Morse theory and its applications, Univ. de Montreal (1985). 1. CHANC K. C., Indefinite dimensional of homology theory to differential equations, Proc. Symp. Pure Moth. AMS, 2. CHANC K. C., Applications Nonlinear Functional Analysis, (Edited by F. BROWDER) 1986. 3. CONLEY C. C. & ZEHNDER E., The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold, Invent. Marh. 73, 33-49 (1983). of the PoincarC-Birkhoff theorem, (preprint). 4. FRANKS J., Generalizations systems, Peking Univ. seminar report (1987). 5. JIANG M. Y., A report on the periodic solutions of Hamiltonian and some applications, ICTP, Tech. Rep. IC-86-191. 6. LI SHUJIE, Multiple critical points of periodic functional 7. MAWHIN J., Forced second order conservative systems with periodic nonlinearity, preprint (1987). equations, J. 8. MAWHIN J. & WILLEM M., Multiple solutions of the periodic BVP for some forced pendulum-type diff. Eqns 52, 264-287 (1984). 9. PUCCI 6. & SERRIN J., A mountain pass theorem, J. d$f. Eqns 60, 142-149 (1985). pass theorem, Univ. of Minnesota Math. Rep. 83-150. 10. Pur-CI P. & SERRIN J., Extensions of the mountain 11. RA~INOWI~Z P. H., On a class of functionals invariant under a Z” action, University of Wisconsin-Madison, Center for the Mathematical Sciences Technical Summary Report X88-1 (1987).
A~J,,w. Thuory. ,n Grear Bnrain.
.Merhods
8 .4pplrrom~ns.
Vol.
13.
No.
5. PP.
527-537.
0362-546X/89 $3.03 + .OU C 1989 Pergamon Press plc
1989.
ON THE PERIODIC NONLINEARITY AND THE MULTIPLICITY OF SOLUTIONS \
KUNG-CHING
CHANG*
Center for the hlathematical Sciences and Department of Mathematics, University of Wisconsin-Madison, WI 53706, U.S.A. and Peking University, People’s Republic of China
Madison,
(Received 7 March 1988; received for publication 27 April 1988) Key words and phrases: Periodic nonlinearity,
multiple solutions, critical point, Hamiltonian systems,
Neumann problem, periodic solution.
1. INTRODUCTION
by the work of Conley and Zehnder [3] on the solution of the Arnold conjecture, the author presented a different proof of their statement, and noticed that the periodicity of the Hamiltonian function is the essence of the occurrence of multiple periodic solutions [l-2]. Several other authors have recently studied questions of periodic solutions to periodic equations. They use a variety of methods based on their particular problems. The main purpose of this paper is to show that the following abstract theorem obtained in [l] can be used to give a unified approach to this class of problems. Let H be a real Hilbert space, and let A be a bounded self-adjoint operator defined on H. According to its spectral decomposition, H = H+ @ Ho @ H_, where H, , Ho, and H_ are invariant subspaces corresponding to the positive, zero, and negative spectrum of A respectively. INSPIRED
THEOREM 1. Suppose that A satisfies the following assumptions (Hi) A, P Al, has a bounded inverse on H,, (H,) y p dim(H_ @ H,,) c 00. Let V” be a C2 compact n-manifold without boundary, and let g E C’(H x V”, R’) be a function having a bounded and compact differential dg(x). Assume that
gW,x, 4 + - 03 where PO is the orthogonal
projection
as IRxll + ~0
if dim H,, # 0
onto Ho. Then the function
f(x, v) = *(Ax, x) + g(x, v) possesses at least cuplength (v”) + 1 distinct critical points. If further, we assume that g E C2(H x V, R’), and thatfis nondegenerate, thenfhas at least Cy= ,, pi( V”) critical points, where pi( V”) is the ith Betti number of V”, i = 0, 1, . . . , n. In the statement of theorem 8.3 in [l], the function g was assumed to be C2. However, in the proof of the first conclusion, Cr is sufficient.
Remark.
*Supported
by the U.S. Army Research Office under Contract No. DAAL03-87-K-0043. 527
KLTG-CHING
528
CHANC
hlost recent studies were only concerned with the case where A is positive definite. We shall give more applications where A is semidefinite, i.e. the negative eigenspace as well as the null space are finite dimensional. They are used to study semilinear elliptic systems and periodic solution problems for 2nd order ODE. Our theorem 3 generalizes and unifies results due to Mawhin [7], Mawhin and Willem (81, Li [6], Jiang [5], Franks [4], Pucci and Serrin (9, lo] and Rabinowitz [ 111. In our setting periodic solution problems for Hamiltonian systems reduce to case where A is unbounded and indefinite. Theorem 4 is a generalization of theorem 3. It implies the early results due to Conley and Zehnder [3] as special cases. In particular, multiple periodic solutions of Hamiltonian systems with resonance are studied, where the Hamiltonian functions are only periodic in certain variables. 2. SEhII-DEFINITE
FUNCTIONALS
A direct consequence of theorem 1 is the following. 2. Suppose that A is a self-adjoint operator satisfying (H,) and (Hz), defined on a Hilbert space H. Suppose that @ E C’(H, R’) is a function having a bounded and compact differential d@, and satisfies the following periodicity condition. (P) 3 e,, . . . . e, E ker A, they are linearly independent, and 3 (T, , . . . , K) E R’ such that THEOREM
i
mj7jej
j=I
>=we,
vx~H,v(m~,...,m,)~Z’
and the resonance condition. (LL) O(x) -+ -co if //xi1 -+ co and x E ker(A)n[e,, Then the equation
. . . . e,)‘.
Ax + d@(x) = 0 possesses at least r + 1 distinct solutions. If further, Q>E C’(H, R’) and all solutions of (2.1) are nondegenerate, at least 2’ solutions.
(2.1) then (2.1) possesses
Proof. We consider the following functional J(x) =
i(Ax, x) + Q(x).
According to (P), =
JO,
V (m, , . . . , m,) E Z’.
>
However, we have an orthogonal
decomposition H = ker A @I (ker A)’ = Z @ Y 0
(ker A)’
where Z = span(e,, . . . . e,), and Y = Z* fl ker(A). If we restrict ourselves on the quotient
Periodic nonlinearity and multiplicity of solutions
529
space T’ x (Y @ (ker A)‘) where T’ = Z/Z’(T,, functionals
Z’(T,,
. . . . T,),
. . . . T,): = ((m, T,, . . . . m,T,)I(m,,
. . . . m,) E Z’),
the
f(u, v) = JW, and g(u, v) = Q(x), are well defined, where (v, U) E T’ x (Y @ (ker A)‘) and x = u + v. The critical point off is a solution of (2.1). Sincefand g satisfy all conditions in theorem 1 with H, = Y and V = T’, the conclusion follows directly. We present here an application. THEOREM 3. Let M be a compact
matrix
valued
manifold without boundary, function defined on M, and let
continuous
ker(where 0 < r < N are assumptions F X,U+ (
~
A 0 I,_,
integers.
(N - r)
+ (aij(X)) * ) = span(vt , . . ., rpkl,
Assume
= F(x, u)
mi~I;i
i=l
let (Uij(X)) be a symmetric
that
FE C’(M
v(x,u)~M
x RN, R’)
satisfies
x RN,v(ml
,...,
the
rn,)~Z’
following
(1)
>
where ei = (0, . . . , 1, , . . , 0), i = 1,2, . . . , r, and (T, , . . . , T,) E R’ is given,
(2) (3) and that h E C(M, RN), h = (h, , . . . , hN) satisfies hi(X)
dx = 0,
i = 1,2, . . . , r,
i ,‘+f and hi(x) = 0, j = r + 1, . . . , N. Then the elliptic -AU has at least r + 1 solutions,
Proof.
system
+ C(x) . u - F,(x, u) + h(x) = 0
on M
where
Let H = W’**(M, R “), A = Z, + (- A)-‘d(x), Q(u) =
and
- F(x, U(X)) + (h(x) - u(x)>,. M
530
KUNG-CHING CHANC
Obviously, ker A = spame, and @ E C’(H, R’), having a bounded
, . . . , e,, cp,, . . . , pkl,
and compact
differential,
satisfies
the conditions
(P) and
(LL). The conclusion
follows
immediately
from theorem
2.
Remark 2.1. In theorem 3, we may replace the compact manifold M by a smooth Sz in R”, in addition to the Neumann boundary value condition
bounded
domain
au 0, ay an = where v is the outward
pointing
normal
to the boundary,
a52.
Example 2.1. A4 = S’, r = N = 1. This is just the periodic
solution
problem
for the ODE
ii + F”(f, u) = h(f)
(2.2)
where F E C’(S’ x R’, R’) is periodic in u, and h E C(S’, R’) satisfies the zero mean condition j,&(t) dt = 0. Under these conditions, (2.2) has at least two solutions. It was shown by Mawhin and Willem [8], Li [6] and Franks [4].
Example 2.2. The case M = S’, and r = IV. The corresponding ODE system was studied by Jiang [5] and Rabinowitz [ll]. In this case, the following system possesses at least N + 1 solutions ii + F”(t, u) = h(t) (2.3) whereFEC’(S’
x
R”,R’)isperiodicinu=
(nt,...,
u,,,), and h E C(S’, RN), satisfies
j,&(t)
dt = 8. Example 2.3. The case M = S’, r < N, with (aji(t))~N_r~x~,v_r~ positive system was studied by Mawhin [7]. The system
definite.
The ODE
ii - a(t)24 + F,(t, u) = 0 possesses at least r + 1 solutions, provided that F E C’(S’ x RN, R’) is periodic variables (u, , . . . , U, and llF,(t, 4 IIP < 00.
Example 2.4. The case M = T”, r = N = 1. The problem IO]. The following equation Au + F,,(x, u) = 0 possesses
at least two solutions,
The Neumann Rabinowitz [ 111.
problem
for
provided the
was studied
in the first r
by Pucci and Serrin [9,
on T”
that F E C’(T” x R’, R’), and is periodic
elliptic
equation
(in
case
in u.
r = N = 1) was studied
by
Periodic nonlinearity and multiplicity of solutions
531
Remark 2.2. All the above examples deal only with functionals bounded from below, however, theorem 3 implies more than that. The improvements are in two directions. (1) The functional is semi-definite, i.e. it is bounded from below except on a finite dimensional subspace. (2) The resonance case is studied, it only happens when r < n. 3. INDEFINITE
FUNCTIONALS
In this section, we shall extend the results of Section 2 to indefinite functionals. The saddle point reduction argument will be applied. Let H be a Hilbert space, and let A be a self-adjoint operator with domain D(A) C H (unbounded). Assume that F is a potential operator with 0 E C’(H, RI), F = d@ and Q(0) = 0. The following assumptions are made. (A) 3 cx < 0 < p such that cr, /3 d a(A) and a(A) II [a, /3] consists of at most finitely many eigenvalues of finite multiplicities. (F) F is bounded and Gateaux differentiable, with
(D) For small E > 0, with --E d a(A),
let V = D(l(&Z + A)[“‘),
THEOREM 4. Suppose that (P) 3 e,, . . . . e, E ker A, they are linearly ~
Xf (
~
mj~ej
j= I
independent,
= @,(X1,
assume that 0 E C2( I’, R’).
and 3 (r, , . . . , T,) E R’, such that
, , .... m,) E z',
VW
>
(LL) Q(x) + &co if /Ix~/ + ~VXE Then the equation
kerAnspan(e,,...,e,JL.
Ax + W(x) = 0 has at least r + 1 distinct Proof.
A saddle point
solutions. reduction
procedure
where (EJ is the spectral
dJ%, I OL
p,
=
E,, 14
resolution
of A, and let
Ho = P,H, and for small E > 0, --E d a(A),
Let
+m
‘0 PO =
is applied.
H,
= P,H
let
I$ = [(&I + A)I-“‘Ho,
VI, = 1(&I + /I)!-“‘H,.
For each u E H, we have the decomposition u = u,
+ ug + u_
VXEH.
KUNG-CHING CHANG
532
wichu,EHo,u,EH,.Letx=x++x,+x_EV,where _r, = I(&1 + A)y*u,, We define
a function
x, = \(&I + /I)I-“*U*.
on the finite dimensional a(z) = $4x(z),
where x(z) = x+(z) + x_(z) + z, z=x, x,
= -(&I
space
V0 as follows
x(z)) + @(x(z))
E Vo, and x,(z) + A)-9*(&Z
are the solutions
of the equations
+ E-)(x+ + x_ + z).
We shall prove that 1” X*(Z+i,
VZE
Tej) =X+(Z),
V,.
In fact, P,(E/
+ F) X+ +X_
+ Z +
i
= PAW
Tej
j= I
(
f F)(x+
+ x_ + z)
>
therefore
2”
U(Z +j$,
qej)
=
Q(Z).
CLAIM.
Tej)) +@(x(z +j, 7;e,)) +@ i
+j,
a(Z+~,Tej)=i@x(Z+j$,?ej)~x(z
= + AX(Z),X(Z) + i Tjej ( j=l >
X(Z)+
7jej
j= I
(
>
= +(Ax(z),x(z)).+ @(x(z)) = a(z).
3” a satisfies spanle, , . . ., e,)‘. CLAIM. Suppose According
the
(PS)
condition
that (zk) is a sequence
to Chang
on
along
which (a(&)
17 V.
is bounded,
where
and
Y = N(A) fl
i/a’(8) 11= o(l).
[l, p. 1051, /IAx
Let Q be the orthogonal
T’ x (Y @ N(A)‘)
projection
+ F(x(zk)& onto
= o(l).
Y, which is considered
as a subspace
of the Hilbert
Periodic
nonlinearity
and multiplicity
of solutions
533
space X = Y @ N(A)*. Thus on the space 3~2, (I - Q)x(zk) = -A-‘(I since F is bounded,
- Q)F(x(?))
11 (I - Q)x(zk) ~1is bounded.
@(Q.4zkN = @,(x(z?) -
+ o(l),
Noticing
’ VMzkh .r0
V - QMzkN dt I
=
a(zk)
-
+(nx(zk),x(zk))
-
VMzkN,
(1
-
QMzkN dt,
1 0
where
x,(z) = ((1 - 01 + tQ)x(zh
and (Ax(zk), x(zk)) = (Ax(zk), (I -
QMz"N = (-FM.&) + o(l), U - QMzk
be bounded. According to the condition (LL), Qx(zk) is bounded. The compactness of zk now follows from the boundedness of x(zk) and the finiteness of the dimension of Vo. @(Qx(zk)) must
4” If we decompose
V. into span{e, , . . . , e,) @ (Y @ N(A)‘)
fl V,,
z = v + w, (v, w) E span(e, , . . ., e,] @ (Y @ fV(A)*)fl
g(w, v) =
Vo,
+(Ar(w + 9, <(w + 4) + cD(x(w + v))
where
r(z) = x+(z) + x-(z) then g is well defined on T’ x (Y @ N(A)‘)
n V,, and
dg(w, v) = P,F(x(w
+ v))
which is bounded and then is compact on finite dimensional manifold. The function a(z) now is written in the following form: a(w, v) = i(Aw, w) + g(w, v). Noticing that F is bounded, Y we have
dY, 4 = tb‘wY
/It(z) 11is always bounded.
If we denote y the projection of w onto
+ v), <(Y + VI) + WY) + way
+ v) + Y + 9 - WAI.
The first term and the third term are bounded, therefore g(Y* v) + f cQ
as IIYII -+ 00.
The function a(w, v) satisfies all assumptions of theorem 1. Theorem 4 is proved. Now we study the periodic solutions of the Hamiltonian functions are periodic in some of the variables.
systems, in which the Hamiltonian
531
I~UNG-CHINGCHANG
We use the following
notations:
p, q E RIV,
P = (P,, . ..>P!).,
(4,) . ..? qN)r
4 =
4 = (s,, . . . . qr),
P = (P,, . . ..P.),
P = (P,,
11
. . .
B = (4rt
PA
I
lz~r
17
q,),
. . .*
B = (Ps+,r . . ..PTh
B = (qs+,, . ..r (7Th
ii = (Pr+*, . ..rPN)r
4 = (q-r+,, .-.,qd.
We assume the following. (I) A(t), B(t), C(t) and D(t) are symmetric continuous matrix functions on S’, of order (S - r) x (S - r), (T - s) x (T - s), (N - T) x (N - T) and (N - T) x (N - T) respectively. Let A = j,&(t), and B = j,J?(t) be invertible. (II) fi E C’(S’ x R2N , R’) is periodic in the following variables p, q,$, 4, and @ is bounded. (III)
Let spanip,,
...,
-I$
- (C(t)
where
@ D(t)) >
J= -
(
0
IN-T
and vI , . . . , p,,, are linearly independent.
And
A(t,j_17jfj7j)-*m
as171=(7f+..-+7~)“2-+oo.
(IV)
c,d~C(S’,R~),withc=(c
,,___, c&d=(d ~S,ckG
i = 1, . . ..r. s + 1, . . . . T,j We define a Hamiltonian
,,...,
= j/ij(‘)
d,)and
= 0,
= I, . . ..s. function as follows
T +
THEOREM
5. Under
conditions
;C,
(C;(fk’i
(I)-(IV), -J$
has at least r + T + 1 periodic
solutions,
+
dAt)qi)
+
the Hamiltonian = H,(t, z),
fi(fv~,
4).
system
(HS)
t E s’
where z = (p, q) E RZN.
Periodic nonlinearity
Proof,
and multiplicity
535
of solutions
Let
and let (the subscripts on J coincide with those on p) Ct=
=
(-J$ -
A(f) >
(-1;) @(-I$- (a(f)o))0 (-J-g (” B(r)))
@(4
- (c(f) D(J).
We have
(a(‘)o)),
(p,Q)eker(-Jz-
with 4(2n) = 4(O),
(i.e. with A - C = 6). According to the assumption I, E = 0. We have
Similarly
Thus ker(@.) = R2' @ Rs-'@ RT-*@ spanIp,,...,( Let Q(z) =
fi(t,
Z(t))
+
i [Ci(f)Pi(f) i= I
+
di(th?i(OI
1
dt.
Then all the assumptions (A), (F), (D), (P) and (LL) are satisfied. The proof is complete.
KUNC-CHING CHANG
536
Example 3.1. If the Hamiltonian function H E C’(S’ x R ‘lv, R’) is periodic in each variable, then (HS) has at least 2N + 1 periodic solutions. This is the case r = s = T = N. This result related to the Arnold conjecture, was first obtained by Conley and Zehnder 131, see also Chang [2]. Example 3.2. If H E C2(S1 x RZN, R’), where His periodic in the components of q, and that there is an R > 0 such that for IPI 1 R, W,p,q)
=
$?P*P
+ asp
where a E RN, and M is a symmetric nonsingular time independent matrix, then the corresponding (HS) possesses at least N + I distinct periodic solutions. This is the case r = 0, S = T = N. This is a result obtained by Conley and Zehnder [3], see also P. H. Rabinowitz [ll]. Example 3.3. Let H E C2(S’ x R4, R’) be periodic in (p,, 4,). Assume that 3 R > 0 such that H(t, ~1,
~2 9 41,421
=
+(cP: + dq:) f Am
for m 2 R, where cd = kz 2 0 for some k E Z, and A > 0. Then the corresponding (HS) possesses at least 3 periodic solutions. In fact, ker( -J$
- (i
3)
= span((
-&sink,
coskt),
(Jcoskt,
sinki)]
it follows d (J.: + :) > - ( - ,J.,sinkt + A2coskt)’ + (1, coskt + A2 sinkt)2 2 min c
Therefore sinkt + 2, coskt), 0, (A, coskt + A2 sinkt)
A
) or -+ --co,
Remark. In the assumption (I), if the operators A and B are singular, then
Thus, ker(-J-$
- (a(l)
o))
= ((8,$ld~R~-‘)
0
[(E, ~~A(.s)ds?)[F~
herA].
Periodic
nonlinearity
and multiplicity
of solutions
537
Similarly
ker( 4; - (” B(I))) =((f, @I2
E
R-1
0
[(pwdd,d)ldE ker Bj.
In order to apply theorem 5, the assumption III is replaced by ~(r,c+
~~a(S)drC+
j:B(s)drd^+d+i~~r,s(r))-
foe,
as [cl + ld^l + 15) -+ CQ,where S E ker A, d^E ker 3, and 5 E R “‘. The same theorem holds. Example
3.4. Let H E C’(S’ x R’, R’) be periodic in (pl , ql, q2). Assume that 3 R > 0
such that H(t, ~1,
~2941,421
for lp21 2 R, where A > 0 is a constant, periodic solutions. In fact, &,p,,p,,q,,q,) Acknowledgement-We Sciences,
University
thank Professor of Wisconsin-Madison,
=
tcostp:
*Am
then the corresponding
= *Am
+ fm,
P. H. Rabinowitz for his invitation and for his very kind conversations
(HS) possesses at least 4
as ICI -+ m. to the Center for the Mathematical on his interesting preprint [I 11.
REFERENCES Morse theory and its applications, Univ. de Montreal (1985). 1. CHANC K. C., Indefinite dimensional of homology theory to differential equations, Proc. Symp. Pure Moth. AMS, 2. CHANC K. C., Applications Nonlinear Functional Analysis, (Edited by F. BROWDER) 1986. 3. CONLEY C. C. & ZEHNDER E., The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold, Invent. Marh. 73, 33-49 (1983). of the PoincarC-Birkhoff theorem, (preprint). 4. FRANKS J., Generalizations systems, Peking Univ. seminar report (1987). 5. JIANG M. Y., A report on the periodic solutions of Hamiltonian and some applications, ICTP, Tech. Rep. IC-86-191. 6. LI SHUJIE, Multiple critical points of periodic functional 7. MAWHIN J., Forced second order conservative systems with periodic nonlinearity, preprint (1987). equations, J. 8. MAWHIN J. & WILLEM M., Multiple solutions of the periodic BVP for some forced pendulum-type diff. Eqns 52, 264-287 (1984). 9. PUCCI 6. & SERRIN J., A mountain pass theorem, J. d$f. Eqns 60, 142-149 (1985). pass theorem, Univ. of Minnesota Math. Rep. 83-150. 10. Pur-CI P. & SERRIN J., Extensions of the mountain 11. RA~INOWI~Z P. H., On a class of functionals invariant under a Z” action, University of Wisconsin-Madison, Center for the Mathematical Sciences Technical Summary Report X88-1 (1987).