PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON
DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS
May 24 – 27, 2002, Wilmington, NC, USA
pp. 1–8
MULTIPLE SOLUTIONS OF SUPER-QUADRATIC
SECOND ORDER DYNAMICAL SYSTEMS
XIANGJIN XU
DEPARTMENT OF MATHEMATICS
JOHNS HOPKINS UNIVERSITY
BALTIMORE, MD, 21218 USA
Abstract. In this paper the existence of periodic solutions of large norm for
the super-quadratic second order dynamical systems Aẍ = −∇V (x) is proved.
And some results for perturbed systems are also gained.
1. Introduction. This paper consider the existence of periodic solutions of the
second order dynamical systems
Aẍ = −∇V (x),
(1)
N
where x ∈ R , A is a nonsingular N ×N symmetric matrix, but not necessarily positive definite. Without loss of generality, we assume A = diag(1, · · · , 1, −1, · · · , −1)
with the number of 1 equal to N1 > 0 and the number of −1 equal to N2 = N − N1 .
In [2], Han stated that Professor L.Nirenberg asked the following question: whether
under suitable conditions, infinitely many periodic solutions of (1) with prescribed
period exist - as is known in the case A = Id. We have the following results:
Theorem 1. Suppose V ∈ C 1 (RN ) satisfies:
(V1) there exists µ > 2 such that 0 < µV (x) ≤ (x, ∇V (x)) for |x| large.
(V2) there exist a, b > 0, such that
|∇V (x)| ≤ c(∇V (x), x) + d,
∀x ∈ RN .
Then for any T > 0 and R > 0, (1) has a T -period solution x(t) with maxt∈[0,T ] |x(t)| ≥
R. If A is positive definite, (V2) does not need.
Theorem 2. Suppose V ∈ C 1 (RN ) satisfies (V1) and
(V3) let Π be the projection operator of RN onto the negative eigenspace of A,
assume (Πx, ∇V (x)) ≥ −αV (x) for some α > 0 and for |x| large.
then the results of Theorem 1 also holds.
Remark 1. Theorem 1.1 and Theorem 1.2 treat the case of a dynamical system
which has attraction in certain directions and repulsion in the other directions. The
condition (V 2) means that the direction difference between ∇V (x) and x is not too
large. The condition (V 3) introduced in [2], says that the repulsion is not to be
too strong. When the matrix A is positive definite, the results without condition
(V 2) and (V 3) is well known, cf. [4], [7]. Our theorems improve the main result
1991 Mathematics Subject Classification. 37J45,(34C25,47J30).
Key words and phrases. periodic solution, second order dynamical systems.
1
2
XIANGJIN XU
in [2], where the author further required V (x) ≥ 0 for all x ∈ RN and proved the
existence of one nonconstant solution essentially.
Following the idea of [6], where study the first order autonomous Hamiltonian
systems using S 1 index theory (i(·), B) introduced in [1], we firstly introduce a simple
system and study the structure of its solutions and the increasing estimates of the
critical values of this new system using S 1 index theory, then we prove Theorem
1.1 and Theorem 1.2 by comparing the critical values between the original system
and the new system. And we also obtain some results on perturbed systems by the
same argument and the increasing estimates gained in [3].
This paper grow from part of my master degree thesis [8]. It is a pleasure to
thank my advisor, Professor Yiming Long, for his constant encouragement and
helpful suggestions when I studied in Nankai Institute of Mathematics.
2. Proofs. For any given T > 0, let X = W 1,2 (ST , RN ). Define
Z T
Z T
I(x) =
(Aẋ, ẋ)dt −
V (x)dt,
∀x ∈ X.
0
0
It is well known that I ∈ C 1 (X, R). We define
X
X + = {x ∈ X|x =
cj eijt , cj ∈ CN1 , c−j = c̄j }
|j|6=0
X
X
0
−
= {x ∈ X|x = constants}
X
= {x ∈ X|x =
cj eijt , cj ∈ CN2 , c−j = c̄j }
|j|6=0
We have X = X + ⊕ X 0 ⊕ X − .
The minimax procedure that we will use in the proofs takes advantage of the
S 1 index theory (i(·), B) introduced in [1], where B denotes the family of closed S 1
invariant subset of X − 0 and i(·) is the S 1 cohomological index on the invariant
subset, since there is a natural S 1 invariance possessed by the functional I. One
can find the details of the index theory in [1].
For m ∈ N, we define
Hm = (span{eijt vk , e−ijt vk |j ≤ [m/N1 ], k ≤ m − N1 j} ∩ X + ) ⊕ X 0 ⊕ X − ,
where {vk } is the standard base of CN1 and [a] denotes the greatest integer not
greater than a. Then Hm is an invariant subspace of X. (V1) implies for certain
constants a1 and a2
V (x) ≥ a1 |x|µ − a2 .
By the Hölder inequality,
I(x) ≤ ||x+ ||2 − a1 ||x||µL2 + a2 ≤ ||x+ ||2 − a1 ||x+ ||µL2 + a2 .
(2)
+
Since Hm ∩ X is m dimensional and µ > 2,(2) shows that for any given constant
M (0) there is an Rm > 0 such that
I(x) ≤ −T M (0),
for all x ∈ Hm such that ||x|| ≥ Rm . Let Dm = BRm ∩ Hm . Let Gm denote the
class of mappings h ∈ C(Dm , X) which satisfy:
(g1 ) h is equivariant,
(g2 ) h(x) = x for x ∈ (∂BRm ∩ Hm ) ⊕ X 0 ,
MULTIPLE SOLUTIONS OF SUPER-QUADRATIC SYSTEMS
3
(g3 ) P − h(x) = α(x)x− + φ(x), where φ(x) is a compact map and α ∈ C(Dm , [1, ᾱ]),
constant ᾱ depending on h.
Since h(x) = x ∈ Gm for all m ∈ N, Gm 6= ∅. For j ∈ N, define
Γj = {h(Dm − Y )|m ≥ j, h ∈ Gm , Y ∈ B, and i(Y ) ≤ m − j}.
For Γj we have the following properties:
(i) (Monotonicity): Γj+1 ⊂ Γj ,
(ii) (Excision): If B ∈ Γj and Z ∈ B with i(Z) ≤ s < j, then B − Z ∈ Γj−s ,
(iii) (Invariance): If ϕ ∈ C(X, X) and satisfies (g1 ) − (g3 ) for m ≥ j, B ∈ Γj implies
ϕ(B) ∈ Γj .
The next result which was proved in [6] is crucial for later estimates.
⊥
Lemma 1. Let h ∈ Gm , j ≤ m, ρ < Rm , and Θ = {x ∈ Dm |h(x) ∈ ∂Bρ ∩ Vj−1
},
then Θ is compact and i(Θ) ≥ m − j + 1. If i(Y ) ≤ m − j,and W = Θ − Y , then
⊥
h(Dm − Y ) ∩ ∂Bρ ∩ Vj−1
⊃ h(w) 6= ∅.
And we will use the following calculus lemma:
Lemma 2. For m ≥ 1, x = (x1 , · · · , xN1 ) ∈ (Hm ∩ X + )⊥ , we have
[m/N1 ]||x||L2
||xl ||L∞
≤
||ẋ||L2
(3)
8
)1/2 ||ẋl ||L2 ,
≤ (
T ([m/N1 ] − 1
∀l = 1, · · · , N1 .
(4)
Proof. For x ∈ (Hm ∩ X + )⊥ , we have the Fourier series expansion,
X
x(t) =
cj eijt , cj ∈ CN1 , c−j = c̄j .
|j|≥[m/N1 ]
||ẋ||2L2 =
T
2
X
|cj |2 j 2 ≥
|j|≥[m/N1 ]
T
[m/N1 ]2
2
X
|cj |2 = [m/N1 ]2 ||x||L2 .
|j|≥[m/N1 ]
By the same computation as above, we get that for every l = 1, · · · , N1 ,
X
1
||xl ||L∞ ≤ 2
(|cj ||j|)
j
|j|≥[m/N1 ]
≤ 2(
X
|cj |2 j 2 )1/2 (
|j|≥[m/N1 ]
X
|j|≥[m/N1 ]
1 1/2
)
j2
8
)1/2 ||ẋl ||L2 .
≤ (
T ([m/N1 ] − 1
Q.E.D.
As [6], we first study a simple system. There is M ∈ C 2 (R, R) satisfying:
(M1) M (|x|) ≥ V (x) for all x ∈ RN and M (s) − M (0) = o(s2 ) at s = 0,
(M2) M (s) is strictly monotonically increasing in s and tend to infinity as s → ∞,
(M3) N (s) = M 0 (s)/s is strictly monotone and tends to infinity as s → ∞,
(M4) 13 sM 0 (s)−M (s) is strictly monotone increasing in s, and sM 0 (s) ≥ µ(M (s)−
M (0)) for all s.
We refer to [6] for the construction of M . Define
Z T
Z T
J(x) =
(Aẋ, ẋ)dt −
M (|x|)dt,
∀x ∈ X.
0
0
4
XIANGJIN XU
We have J ∈ C 1 (X, R).
Now we show I, J satisfy (PS) condition.
Lemma 3. If V satisfies (V1) and (V2), I satisfies (PS) condition.
Proof. Let (xn ) be a sequence in X such that |I(xn )| < M and I 0 (xn ) → 0, as
n → ∞. Denote n = ||I 0 (xn )||W 1,2 . Let xn = (x1n , x2n ) ∈ RN1 ⊕ RN2 . We have
Z T
Z T
0
1
1 2
I (xn )xn =
|ẋn | dt −
∇V (xn )x1n dt,
0
I 0 (xn )x2n
0
T
Z
|ẋ2n |2 dt −
= −
Z
0
T
∇V (xn )x2n dt.
0
Since xn ∈ W 1,2 ,→ C 0 , we have
n ||x1n ||W 1,2
≥
||ẋ1n ||2L2
−
||x1n ||C 0
T
Z
|∇V (xn )|dt,
0
−n ||x2n ||W 1,2
≤
−||ẋ2n ||2L2 + ||x2n ||C 0
Z
T
|∇V (xn )|dt.
0
Since (V2), we have
||ẋn ||2L2 ≤ n ||xn ||W 1,2 + 2||xn ||C 0 [c
Z
T
(∇V (xn ), xn )dt + dT ]
0
On the other hand, we have
M + n ||xn ||W 1,2
1
≥ I(xn ) − (I 0 (xn ), xn )
2
Z
Z T
1 T
=
(∇V (xn ), xn )dt −
V (xn )dt
2 0
0
Z T
1
1
≥ ( − )
(∇V (xn ), xn )dt + c1
2 µ 0
Z T
µ
≥ ( − 1)
V (xn )dt + c2
2
0
≥ c3 ||xn ||µLµ + c4
(5)
Thus by the Hölder inequality and Sobolev embedding Theorem, we have
2/µ
||xn ||2L2 ≤ M1 (1 + ||xn ||W 1,2 )
||xn ||C 0 ≤ K||x||W 1,2
Z
T
(∇V (xn ), xn )dt ≤ M2 (1 + n ||xn ||W 1,2 )
0
Hence we have
2/µ
||xn ||2W 1,2 ≤ M3 (1 + ||xn ||W 1,2 + ||xn ||W 1,2 + n ||xn ||2W 1,2 )
Since n → 0, as n → ∞, we have ||xn ||W 1,2 bounded. From the form of I 0 we know
(PS) condition holds.
Q.E.D.
From (M3) and (M4), M (|x|) satisfies (V1) and (V2), we have
Lemma 4. J satisfies (PS) condition.
Lemma 5. If V satisfies (V1) and (V3), I satisfies (PS) condition.
MULTIPLE SOLUTIONS OF SUPER-QUADRATIC SYSTEMS
5
Proof. Let (xn ) be a sequence in X such that |I(xn )| < M and I 0 (xn ) → 0, as
n → ∞. Denote n = ||I 0 (xn )||W 1,2 . Let xn = (x1n , x2n ) ∈ RN1 ⊕ RN2 . We have
Z T
Z T
I 0 (xn )x1n = ||ẋ1n ||2L2 −
∇V (xn )xn dt +
∇V (xn )x2n dt,
0
Z
I 0 (xn )x2n
T
= −
|ẋ2n |2 dt −
0
0
Z
T
∇V (xn )x2n dt.
0
From (5) and (V3), we have
n ||x1n ||W 1,2
≥ ||ẋ1n ||2L2 − C1 (1 + n ||xn ||W 1,2 ) + C2 (1 + n ||xn ||W 1,2 ),
−n ||x2n ||W 1,2
≤
−||ẋ2n ||2L2 + C2 (1 + n ||xn ||W 1,2 ),
for some constants C1 and C2 . Thus we have
2/µ
||xn ||2W 1,2 ≤ C3 (1 + ||xn ||W 1,2 + ||xn ||W 1,2 + n ||xn ||2W 1,2 )
for some constant C3 . Since n → 0, as n → ∞, we have ||xn ||W 1,2 bounded. From
the form of I 0 we know (PS) condition holds.
Q.E.D.
Now we define two minimax sequences of I and J:
cj = inf sup I(x),
B∈Γj x∈B
dj = inf sup J(x).
B∈Γj x∈B
(6)
By (M1) we have cj ≥ dj , for all j ∈ N. By Lemma 2.1, we have cj+1 ≥ cj ,
dj+1 ≥ dj ≥ d1 . Set M̄ (s) = M (s) − M (0) and let
Z T
Z T
¯
M̄ (x)dt,
∀x ∈ X.
J(x)
=
(Aẋ, ẋ)dt −
0
0
we have
¯ − T M (0) = d¯j − T M (0).
dj = inf sup J(x)
B∈Γj x∈B
As Lemma 1.33 and Lemma 1.40 of [6] we have
Lemma 6. (1) d¯j+1 ≥ d¯j ≥ d¯1 > 0,
¯
(2) d¯j is a critical value of J,
¯ corresponding to d¯j lie in X − X 0 ,
(3) Any critical points of J,
¯ then i(K) ≥ l.
(4) If d¯j+l = · · · = d¯j+1 = d and K = (J¯0 )−1 (0) ∩ J¯−1 (d),
¯ Let x = (x1 , x2 ) ∈
Next we will make a closer study of the critical value d¯j of J.
N2
R ⊕ R be a corresponding critical point. Then x is a classical solution of
(
x1
∂
ẍ1
= − ∂x
M (|x|) = −M 0 (|x|) |x|
,
1
(7)
x2
∂
0
−ẍ2 = − ∂x2 M (|x|) = −M (|x|) |x|
.
N1
Condition (M1) guarantees that there are no problems with the right hand side
when x(t0 ) = 0. We first prove the following Lemma:
Lemma 7. For f ∈ C([0, T ], R), if f (t) ≥ 0 and f (t0 ) = 0 only when x(t0 ) = 0,
the boundary value problem
ẍ(t) = f (t)x(t),
∀t ∈ [0, T ]
(8)
x(0) = x(T ), ẋ(0) = ẋ(T ).
has only solution 0.
In the proof, we will use the following Theorem:
6
XIANGJIN XU
Theorem 3 (Theorem 3 of Chapter 1 in [5]). On interval (a, b), u(t) satisfies
ü + g(t)u̇ + h(t)u ≥ 0
where h(t) ≤ 0 and g and h are bounded on every closed subinterval of (a, b). If u
get the not negative maximum M at an inter point c, we have
∀t ∈ (a, b).
u(t) = M,
Proof. Let x(t) be a solution of (8). From the boundary values we have a
t0 ∈ [0, T ] such that ẋ(t0 ) = 0. Without loss generality, let t0 = 0. For b ∈ (0, T ],
we have
Z b
Z bZ t
Z bZ t
ẍ(s)dsdt =
f (s)x(s)dsdt.
(9)
x(b) − x(0) =
ẋ(t)dt =
0
0
0
0
0
Case 1: If x(0) < 0, there exist a > 0 such that for all τ ∈ [0, a), x(τ ) < 0.
From (9) we know x(τ ) < x(0) for all τ ∈ [0, a), hence we know x(t) is a strictly
monotonically decreasing function. It is a contradiction.
Case 2: If x(0) ≥ 0, Since x(0) ≥ 0, we know x get the not negative maximum
M at an interior point c. Let g(t) = 0, and h(t) = f (t). We know the conditions of
above Theorem are satisfied, hence we have
x(t) = M,
∀t ∈ [0, T ].
Since f (t) = 0 only if x(t) = 0, we get M = 0.
Q.E.D.
0
(|x(t)|)
By the Lemma 7, each solution x(t) of (7) has the form (x1 (t), 0), since M |x(t)|
satisfies the conditions of Lemma 7. Hence we reduce (7) to the following second
order Hamiltonian systems
x1
ẍ1 = −M 0 (|x1 |)
,
(10)
|x1 |
As in [3], we have the following increasing estimates for the critical values {d¯j } of
(7):
Lemma 8. d¯j → ∞ as j → ∞.
Proof. Since the critical values {d¯j } of (7) are the same as those getting for
(10) by the minimax procedure on the corresponding invariant sets, we only need
to prove the Lemma for (10).
For A ∈ Γj , there are h ∈ Gm , Y ∈ B and i(Y ) ≤ m − j such that A =
h(Dm − Y ). By Lemma 2.1, there exist y ∈ Dm − Y such that
x = (x1 , · · · , xN1 ) = h(y) ∈ ∂BR0 (X + ) ∩ (Hj ∩ X + )⊥ .
By (3) of Lemma 2, we have
R02 = ||x||2X ≤ (1 + [j/N1 ]−2 )||ẋ||2L2 .
(11)
Hence we have
1 2
R and
2 0
By (4) of Lemma 2, we have
R02 ≥ ||ẋ||2L2 ≥
||xl ||L∞ ≤ (
||ẋl ||L2 ≤ R0 ,
f or
l = 1, · · · , N1 .
8
8
)1/2 ||ẋl ||L2 ≤ R0 (
)1/2 .
T ([j/N1 ] − 1)
T ([j/N1 ] − 1)
MULTIPLE SOLUTIONS OF SUPER-QUADRATIC SYSTEMS
7
Fix R0 = ( T ([j/N8 1 ]−1) )−1/2 , we have ||x||L∞ ≤ N1 . Hence from (11) we have
Z
Z T
1 T 2
¯
J(x)
=
|ẋ| dt −
M (|x|)dt
2 0
0
[j/N1 ]2
T
≥
([j/N1 ] − 1) − T max M (s)
2(1 + [j/N1 ]2 ) 8
|s|≤N1
T
≥
([j/N1 ] − 1) − T max M (s).
32
|s|≤N1
¯
By the definition of dj , we have
T
d¯j ≥
([j/N1 ] − 1) − T max M (s).
32
|s|≤N1
Hence the Lemma holds.
Q.E.D.
With the increasing estimates for d¯j and dj , now we study the minimax values
cj .
Lemma 9. If cj > T a2 ,
(i) cj is a critical value of I,
(ii) any corresponding critical point lies in X − X 0 ,
(iii) if cj+l = · · · = cj+1 = c > T a2 , i(I −1 (c) ∩ (I 0 )−1 (0)) ≥ l.
Proof. Note that
sup I(x) = T sup (−V (x)) ≤ T sup (a2 − a1 |x|µ ).
x∈X 0
x∈X 0
x∈X 0
Thus if cj > T a2 , similar to the proof of Lemma 6, using standard equivariant
deformation lemma, where we need I satisfying (PS) condition, we get statement
(i)-(iii).
Q.E.D.
Proof of Theorem 1.1 and Theorem 1.2. Since cj ≥ dj → ∞ as j → ∞
and the definition of d¯j , the requirement that cj > T a2 is satisfied for all large j.
Then following the above Lemma, we prove Theorem 1 and Theorem 2.
Q.E.D.
Remark 2. From the study of (7), N1 > 0 is necessary for our Theorems, which
is missed in [2]. And it is natural to ask the following question: Do the results of
Theorem 1 and Theorem 2 hold with only (V1) as the case A = Id? And from the
proofs, we use condition (V 2) or (V 3) only to show I satisfies (P S) condition.
In the following part, we will deal with the following perturbation problems:
Aẍ + ∇V (x) = f (t).
We have the following result:
(12)
Theorem 4. Suppose V ∈ C 1 (RN , R) and satisfying (V1) and (V2) (or (V3)),
then for any given T, R > 0 and T -periodic function f ∈ L2 ([0, T ], RN1 ), (12)
possesses a T -periodic solution x(t) with maxt∈[0,T ] |x(t)| ≥ R.
Proof. We first consider the system
Aẍ + ∇M (|x|) = f (t).
(13)
where M (|x|) satisfies (M1)-(M4). As the proof for Theorem 1.1 and Theorem
1.2, studying the solutions of (13), by using Lemma 7 we know each solution of (13)
has the form x(t) = (x1 (t), 0). Using the setting of invariant sets similarly to [3],
8
XIANGJIN XU
as the proof of Theorem 1.2 in [3], we know the critical values of the variational
functional of (13) are unbounded. Using the same argument, we have the result of
the theorem by comparing the critical values between the original system (12) and
the new system (13).
Q.E.D.
As section 6 in [3], for the following general forced systems
Aẍ + Ux (t, x) = 0,
(14)
Using the result of increasing estimates in Theorem 6.2 in [3] and the argument
of the above proof, we have the following result:
Theorem 5. Let U ∈ C 1 (ST × RN , R), where ST = R/(T Z), satisfies
(UI) (Ux (t, x), Πx) is independent of t, where Π is the projection operator of RN
onto the negative eigenspace of A.
(UII) There exist V : RN → R satisfying (V1) and (V2) (or (V3)) and constants
C > 0, 1 ≤ σ ≤ µ/2, such that
|U (t, x) − V (x)| ≤ C(1 + |x|σ−1 ).
Then (14) possesses infinitely many distinct T -periodic solutions.
REFERENCES
[1] E.R. Fadell and P.H. Rabinowtz, Generalized cohomological index theories for Lie grovp
actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math.
45 (1978), 139-174.
[2] Z. Han, Periodic Solutions for a Class of Nonautonomovs Dynamical Systems of Second Order.
J. Diff. Eq, 90 (1991), 408-417.
[3] Y. Long, Multiple solutions of perturbed superquadratic second order Hamiltonian systems.
Trans. A.M.S. 311 (1989), 749-780.
[4] J. Mawhin and M.Willem, Critical point theory and Hamiltonian systems. Applied Mathematical Science. Vol. 74. Springer-Verlag, 1989.
[5] M.H. Protter and H.F. Weinberger, ”Maximum Principles in Differential Equations”,
Prentice-Hall, 1967.
[6] P.H.Rabinowitz, Periodic solutions of Large Norm of Hamiltonian systems, J. Diff. Eq. 50,
(1983)33-48.
[7] P.H.Rabinowitz, ”Minimax Methods in Critical Point Theory with Applications to Differential
Equations”, CBMS 65, A.M.S., Providence, R.I., 1986.
[8] Xiangjin Xu, Periodic solutions of Hamiltonian systems and differential systems. Thesis for
Master. Nankai University. April 1999.
email address: [email protected]