# New Periodic Solutions for the Singular Hamiltonian System

Authors: Yi Liao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

By use of the Cerami-Palais-Smale condition, we generalize the classical Weierstrass minimizing theorem to the singular case by allowing functions which attain infinity at some values. As an application, we study certain singular second-order Hamiltonian systems with strong force potential at the origin and show the existence of new periodic solutions with fixed periods.


---

## Body

## 1. Introduction

We are mainly interested in the existence of periodic solutionst
↦
x
(
t
)
∈
Ω, with a prescribed period, of the second-order differential equation
(1)
x
¨
=
-
V
′
(
t
,
x
)
,
with Ω
=
R
N
-
{
0
} (N
∈
N, N
≥
2) and V
∈
C
1
(
R
×
Ω
,
R
) where V
′
(
t
,
·
) denotes the gradient of the function V
(
t
,
·
) defined on Ω. The study of the periodic solutions of such equations has a substantial literature with the works [1–21] of particular importance for our purpose.In the 1975 paper of Gordon [10], variational methods were used to study periodic solutions of planar 2-body type problems under assumptions which have come to be known as Gordon’s Strong Force condition (V
1):(
V
1
)
there exists a neighborhoodN of 0 and a function U
∈
C
1
(
Ω
,
R
) such that
(i)
lim
⁡
x
→
0
⁡
U
(
x
)
=
-
∞;
(ii)
-
V
(
t
,
x
)
≥
|
U
′
(
x
)
|
2 for every x
∈
N
-
{
0
} and t
∈
[
0
,
T
];
(iii)
lim
⁡
x
→
0
⁡
V
(
t
,
x
)
=
-
∞.For 2-body type problems inR
N
(
N
≥
2
), one can see the works of Ambrosetti-Coti Zelati, Bahri-Rabinowitz, Greco, and other mathematicians from [1–5, 18–21]. In this paper we wish to highlight two main results among them, that is, Theorems 1 and 2. Firstly, we must specify three separate conditions V
(
t
,
x
) may satisfy about its behavior at infinity. Suppose that V
(
t
,
x
) is T-periodic in t; then(
V
2
)
lim
⁡
|
x
|
→
∞
⁡
V
(
t
,
x
)
=
0, lim
⁡
|
x
|
→
∞
⁡
V
′
(
t
,
x
)
=
0 (uniformly for t) and V
(
t
,
x
)
<
0 for every t
∈
[
0
,
T
], x
∈
Ω;
(
V
3
)
there existc
1
,
M
1
,
R
1
,
ν
>
0 such that, for every t
∈
[
0
,
T
] and x
∈
R
N with |
x
|
≥
R
1,
(i)
|
V
′
(
t
,
x
)
|
≤
M
1;
(ii)
V
(
t
,
x
)
≥
c
1
|
x
|
ν;
(
V
4
)
there existc
1, R
1
>
0, θ
>
1
/
2, ν
>
1 such that, for every t
∈
[
0
,
T
], |
x
|
≥
R
1,
(i)
θ
V
′
(
t
,
x
)
x
≤
V
(
t
,
x
);
(ii)
V
(
t
,
x
)
≥
c
1
|
x
|
ν.SetK
=
{
x
∈
Ω
∣
V
′
(
t
,
x
)
=
0 for every t
∈
[
0
,
T
]
}; there hold the following results.Theorem 1 (Greco [11]).
IfK
=
∅ and (
V
1
) and one of (
V
2
)
–
(
V
4
) hold, then in (1) there is at least one nonconstant T-periodic C
2 solution.Theorem 2 (Bahri-Rabinowitz [3] and Greco [11]).
Suppose∂
V
/
∂
t
≡
0, so that V
(
t
,
x
)
≡
V
(
x
), and the following condition holds:
(
V
5
)
K is compact (or empty).
If(
V
1
) and one of (
V
2
)
–
(
V
4
) hold, then in (1) there exist infinitely many nonconstant T-periodic C
2 solutions.In this paper, we prove the following new theorem.Theorem 3.
SupposeV
∈
C
1
(
R
×
Ω
,
R
) satisfies condition (
V
1
) and conditions(
C
2
)
for the givenT
>
0, V
(
t
+
T
,
x
)
=
V
(
t
,
x
);
(
C
3
)
there existsA
>
0 such that, ∀
(
t
,
x
)
∈
R
×
Ω,
(2)
3
V
(
t
,
x
)
-
V
′
(
t
,
x
)
x
≤
A
;
(
C
4
)
V
(
t
,
x
)
→
+
∞ as |
x
|
→
+
∞ uniformly for 0
≤
t
≤
T.Then the system (1) has a T-periodic solution.Corollary 4.
Supposeα
≥
2, β
≥
3, c
1
>
0, c
2
>
0, V
∈
C
1
(
Ω
,
R
), and
(3)
V
(
x
)
=
-
c
1
x
-
α
,
f
o
r
0
<
x
≤
r
1
;
V
(
x
)
=
c
2
x
β
,
f
o
r
x
≥
r
2
>
r
1
;
then, ∀
T
>
0, the system (1) has a T-periodic solution. From the above example, we see that our potential does not satisfy any of conditions (
V
2
), (
V
3
), and (
V
4
).


## 2. A Few Lemmas

In order to prove Theorem3, we will need to recall the following useful lemmas.Lemma 5 (Sobolev-Rellich-Kondrachov [15]).
It is well known that(4)
H
1
=
W
1,2
R
T
Z
,
R
N
⊂
C
R
T
Z
,
R
N
and the imbedding is compact.Lemma 6 (Eberlein-Shmulyan [15]).
A Banach spaceX is reflexive if and only if every bounded sequence in X has a weakly convergent subsequence.Lemma 7 (Ekeland [8]).
LetX be a Banach space, and suppose Φ defined on X is Gateaux-differentiable, lower semicontinuous, and bounded from below. Then there is a sequence {
x
n
} such that
(5)
Φ
x
n
⟶
inf
⁡
Φ
,
1
+
x
n
Φ
′
(
x
n
)
⟶
0
.Definition 8 (see [8]).
LetX be a Banach space and f
∈
C
1
(
X
,
R
). We say f satisfies the (
P
S
)
c condition if whenever {
x
n
}
⊂
X such that
(6)
f
x
n
⟶
c
with
f
′
x
n
⟶
0
,
then {
x
n
} has a strongly convergent subsequence.Interestingly, Cerami [22] considers a weaker compact condition on a Banach space than the classical (
P
S
)
c condition. Here we introduce a similar condition in an open subset of a Banach space.Definition 9 (see [8]).
LetX be a Banach space; Λ is an open subset; and suppose Φ defined on Λ is Gateaux-differentiable. We say that ϕ satisfies the (
C
P
S
)
c condition if whenever {
x
n
} is a sequence such that
(7)
Φ
(
x
n
)
⟶
c
with
(
1
+
x
n
)
Φ
′
(
x
n
)
⟶
0
,
then {
x
n
} has a strongly convergent subsequence in Λ.With this definition, we can deduce a minimizing result in an open subset of a Banach space, the proof of which is similar to the standard one.Lemma 10 (see Mawhin-Willem [15]).
LetX be a Banach space, Λ
⊂
X an open subset, and f
∈
C
1
(
Λ
,
R
). Assume f has a lower bound on the closure Λ
- of Λ, and let C
=
inf
⁡
x
∈
Λ
-
f
(
x
). If f satisfies (
C
P
S
)
C on Λ
⊂
X and f
(
x
n
)
→
+
∞ as x
n
→
∂
Λ, then C is a critical value for f.Lemma 11 (see [10]).
LetΩ
=
R
N
-
{
0
}
(
N
∈
N
,
N
≥
2
), and V
∈
C
1
(
R
×
Ω
,
R
) satisfies the Gordon’s Strong Force condition (
V
1
); let
(8)
H
1
=
q
:
R
⟶
R
N
∣
q
∈
L
2
,
q
˙
∈
L
2
,
q
t
+
T
=
q
t
,
Λ
=
q
∈
H
1
,
q
t
≠
0
,
∀
t
.
Define
(9)
f
(
q
)
=
1
2
∫
0
T
‍
q
˙
2
d
t
-
∫
0
T
‍
V
(
t
,
q
)
d
t
,
q
∈
Λ
.
Then f
(
x
n
)
→
-
∞ as x
n
→
∂
Λ.


## 3. The Proof of Theorem3

LetH
1
=
{
q
:
R
→
R
N
∣
q
∈
L
2
,
q
˙
∈
L
2
,
q
(
t
+
T
)
=
q
(
t
)
} and Λ
=
{
q
∈
H
1
,
q
(
t
)
≠
0
,
∀
t
}.Lemma 12 (see [2]).
SupposeV
∈
C
1
(
R
×
Ω
,
R
) satisfies condition (
C
2
) and define
(10)
f
(
q
)
=
1
2
∫
0
T
‍
q
˙
2
d
t
-
∫
0
T
‍
V
(
t
,
q
)
d
t
,
q
∈
Λ
.
Then the critical point of f
(
q
) is a T-periodic solution of (1).Lemma 13.
IfV satisfies (
V
1
) and (
C
2
)
–
(
C
4
) in Theorem 1, then f satisfies the Cerami-Palais-Smale condition for any c
≥
0; that is, for any {
x
n
}
⊂
Λ, if
(11)
f
(
x
n
)
⟶
c
,
(
1
+
x
n
)
f
′
(
x
n
)
⟶
0
,
then {
x
n
} has a strongly convergent subsequence and the limit is in Λ.Proof.
By condition(
V
1
) and Lemma 11, we must have f
(
x
n
)
→
+
∞ as x
n
→
∂
Λ. Since f
(
x
n
)
→
c, we know that, for any given ϵ
>
0, there exists N such that when n
>
N, there holds the inequality
(12)
1
2
∫
0
T
‍
x
˙
n
2
d
t
-
∫
0
T
‍
V
(
t
,
x
n
)
d
t
≤
c
+
ϵ
.
The limit (
1
+
x
n
)
f
′
(
x
n
)
→
0 implies
(13)
f
′
x
n
x
n
⟶
0
and so
(14)
2
f
x
n
+
∫
0
T
‍
2
V
t
,
x
n
-
V
′
t
,
x
n
x
n
d
t
=
f
′
(
x
n
)
x
n
⟶
0
.
Using condition (
C
3
) together with the limits and inequalities (11), (12), and (14), we can choose d
>
0 such that when n is large enough, there holds
(15)
∫
0
T
‍
x
˙
n
2
d
t
≤
d
,
which implies ∫
0
T
‍
|
x
˙
n
|
2
d
t is bounded.
In the following we prove that|
x
n
(
0
)
| is bounded; otherwise, there is a subsequence, still denoted by {
x
n
(
0
)
}, such that
(16)
x
n
0
⟶
+
∞
.
Then by Newton-Leibniz’s formula, we have
(17)
min
⁡
0
≤
t
≤
1
x
n
t
≥
x
n
0
-
x
˙
n
L
2
⟶
+
∞
.
Now, by (
C
3
) and (
C
4
), we have
(18)
2
V
t
,
x
n
t
-
V
′
t
,
x
n
t
·
x
n
t
⟶
-
∞
,
which contradicts the limit (14).
Hence,x
n
H
1
=
x
˙
n
L
2
+
x
n
0 is bounded; {
x
n
} has a weakly convergent subsequence; we still denote it by x
n, and let the limit be x. We can show in a standard fashion that this subsequence is strongly convergent in H
1. To complete the proof, we write it out.
Since the sequencex
n is bounded in H
1, so, by Sobolev’s embedding inequality, we know it is also bounded in maximum norm, and, by condition (
V
1
) and Lemma 11, we know that when n is large,
(19)
max
⁡
0
≤
t
≤
T
x
n
t
>
0
.
By V
∈
C
1
(
R
×
Ω
,
R
), when n is large, V
′
(
x
n
(
t
)
) is also uniformly bounded in maximum norm; we have
(20)
f
′
x
n
y
=
∫
0
T
‍
x
˙
n
y
˙
-
V
′
t
,
x
n
y
d
t
.
Taking y
=
x and y
=
x
n in the above equation, we get
(21)
f
′
x
n
x
=
∫
0
T
‍
x
˙
n
x
˙
-
V
′
t
,
x
n
x
d
t
,
f
′
x
n
x
n
=
∫
0
T
‍
x
˙
n
x
˙
n
-
V
′
t
,
x
n
x
n
d
t
.
Since f
′
(
x
n
)
→
0, hence f
′
(
x
n
)
x
→
0; furthermore, since x
n is bounded, so f
′
(
x
n
)
x
n
→
0. Hence, by (21) and the uniformly bounded property for V
′
(
x
n
(
t
)
), we have
(22)
lim
⁡
n
→
∞
∫
0
T
‍
x
˙
n
x
˙
d
t
=
lim
⁡
n
→
∞
∫
0
T
‍
x
˙
n
2
d
t
.
By x
n
⇀
x weakly, we have
(23)
∫
0
1
‍
x
˙
n
x
˙
d
t
+
x
n
0
·
x
0
=
∫
0
1
‍
x
˙
2
d
t
+
x
0
2
.
By Sobolev Embedding Theorem, {
x
n
} has a subsequence, still denoted by {
x
n
} subject to x
n
(
0
)
→
x
(
0
).
We notice(24)
x
n
-
x
=
∫
0
1
‍
x
˙
n
-
x
˙
2
d
t
1
/
2
-
x
n
0
-
x
0
=
∫
0
1
‍
x
˙
n
2
d
t
-
2
∫
0
1
‍
x
˙
n
x
˙
d
t
+
∫
0
1
‍
x
˙
2
d
t
1
/
2
-
x
n
0
-
x
0
⟶
x
˙
L
2
2
-
2
x
˙
L
2
2
+
x
˙
L
2
2
1
/
2
+
0
=
0
.
That is, x
n
→
x strongly in Λ. Then by Lemma 10 the proof of Theorem 3 is complete.



---

*Source: 703539-2014-11-09.xml*