Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 77, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF PERIODIC SOLUTIONS FOR
NON-AUTONOMOUS SECOND-ORDER
HAMILTONIAN SYSTEMS
YUE WU, TIANQING AN
Abstract. The purpose of this paper is to study the existence of periodic
solutions for a class of non-autonomous second order Hamiltonian systems.
New results are obtained by using the least action principle and the minimax
methods, without the so-called Ahmad-Lazer-Paul type condition.
1. Introduction and main results
Consider the second-order Hamiltonian system
ü(t) = ∇F t, u(t) ,
u(T ) − u(0) = u̇(T ) − u̇(0) = 0,
(1.1)
where T > 0 and F : [0, T ] × RN → R satisfies the following assumption:
(A) F (t, x) is measurable in t for every x ∈ RN , continuously differentiable in
x for a.e. t ∈ [0, T ], and there exist a ∈ C(R+ , R+ ), b ∈ L 1 (0, T ; R+ ) such
that
|F (t, x)| ≤ a(|x|)b(t), |∇F (t, x)| ≤ a(|x|)b(t)
for all x ∈ RN and a.e. t ∈ [0, T ].
The corresponding functional ϕ : HT1 → R,
Z
Z T
1 T
ϕ(u) =
|u̇(t)|2 dt +
F t, u(t) dt
2 0
0
is continuously differentiable and weakly lower semi-continuous on HT1 (see [4]),
where HT1 is the usual Sobolev space with the norm
Z T
hZ T
i1/2
2
kuk =
|u(t)| dt +
|u̇(t)|2 dt
.
0
0
It is well know that the solutions of problem (1.1) correspond to the critical points
of ϕ.
Problem (1.1) has been extensively studied in the past thirty years; see for
example the references in this article. Under some suitable solvability conditions,
2000 Mathematics Subject Classification. 34C25, 58F05.
Key words and phrases. Periodic solution; Hamiltonian systems; critical point;
variational method.
c
2013
Texas State University - San Marcos.
Submitted January 11, 2013. Published March 19, 2013.
1
2
Y. WU, T. AN
EJDE-2013/77
such as the coercivity condition (cf. [2]), the periodicity condition (cf. [5]), the
convexity condition (cf. [6]), the subadditive condition (cf. [10]), the existence and
multiplicity results are obtained. We note that in many contributions (for example,
see [1, 3, 9, 12, 13, 14, 15]), the following condition was assumed:
Z T
F (t, x)dt = ∞ or − ∞,
(1.2)
lim |x|−2α
|x|→∞
0
where α is a constant. In this article, instead of (1.2), we discuss the existence of peRT
riodic solutions of (1.1) under a weak condition that lim inf |x|→∞ |x|−2α 0 F (t, x)dt
RT
or lim sup|x|→∞ |x|−2α 0 F (t, x)dt has appropriate lower or upper bound.
Our main results are as follows:
Theorem 1.1. Suppose that F (t, x) = F1 (t, x) + F2 (x), where F1 and F2 satisfy
assumption (A) and the following conditions:
(F1) there exist f, g ∈ L 1 (0, T ; R+ ) and γ ∈ [0, 1) such that
|∇F1 (t, x)| ≤ f (t)|x|γ + g(t),
for all x ∈ RN and a.e. t ∈ [0, T ];
(F2) there exist constants r > 0 and α ∈ [0, 2) such that
(∇F2 (x) − ∇F2 (y), x − y) ≥ −r|x − y|α ,
for all x, y ∈ RN ;
item[(F3)]
lim inf |x|−2γ
T
Z
F (t, x) dt ≥
|x|→∞
0
T2
8π 2
T
Z
f 2 (t) dt.
0
Then problem (1.1) has at least one periodic solution which minimizes ϕ on HT1 .
Theorem 1.2. Suppose that F (t, x) = F1 (t, x) + F2 (x), where F1 and F2 satisfy
assumptions (A), (F1), (F2) and the following conditions:
(F4) there exist δ ∈ [0, 2) and C > 0 such that
(∇F2 (x) − ∇F2 (y), x − y) ≤ C|x − y|δ ,
for all x, y ∈ RN ;
(F5)
lim sup |x|
−2γ
|x|→∞
Z
0
T
3T 2
F (t, x) dt ≤ − 2
8π
Z
T
f 2 (t) dt.
0
Then problem (1.1) has at least one periodic solution which minimizes ϕ on HT1 .
Theorem 1.3. Suppose that F (t, x) = F1 (t, x) + F2 (x), where F1 and F2 satisfy
assumptions (A), (F1), and the following conditions:
(F2’) there exists a constant 0 < r < 4π 2 /T 2 , such that
(∇F2 (x) − ∇F2 (y), x − y) ≥ −r|x − y|2 ;
for all x, y ∈ RN ;
(F3’)
Z T
T2
f 2 (t) dt.
2(4π 2 − rT 2 ) 0
|x|→∞
0
Then problem (1.1) has at least one periodic solution which minimizes ϕ on HT1 .
lim inf |x|−2γ
Z
T
F (t, x) dt ≥
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PERIODIC SOLUTIONS OF HAMILTONIAN SYSTEMS
3
Theorem 1.4. Suppose that F = F1 + F2 , where F1 and F2 satisfy assumptions
(A), (F1) and the following conditions:
(F6) there exist k ∈ L 1 (0, T ; R+ ) and (λ, µ)-subconvex potential G : RN → R
with λ > 1/2 and 0 < µ < 2λ2 , such that
(∇F2 (t, x), y) ≥ −k(t)G(x − y),
for all x, y ∈ RN and a.e. t ∈ [0, T ];
(F7)
lim sup |x|
−2γ
T
Z
|x|→∞
0
lim sup |x|−β
Z
T
f 2 (t) dt,
0
T
Z
|x|→∞
3T 2
F1 (t, x) dt ≤ − 2
8π
Z
T
F2 (t, x) dt ≤ −8µ max G(s)
|s|≤1
0
k(t) dt,
0
where β = log2λ (2µ).
Then problem (1.1) has at least one periodic solution which minimizes ϕ on HT1 .
Remark 1.5. Theorems 1.1–1.3 extend some existing results. On the one hand,
we decomposed the potential F into F1 and F2 . On the other hand, we weaken
the so-called Ahmad-Lazer-Paul type condition (1.2) as conditions (F3), (F5) and
(F3’). Note that [13, Theorem 2] and [3, Theorem 1] are the direct corollaries of
Theorem 1.1 and Theorem 1.3 respectively. If F2 = 0, [11, Theorems 1 and 2] are
special cases of Theorem 1.1 and Theorem 1.2 respectively. Some examples of F are
given in section 3, which are not covered in the references. Moreover, our Theorem
1.4 is a new result.
2. Proof of Theorems
For u ∈
HT1 ,
let
ū =
1
T
Z
T
ũ(t) = u(t) − ū.
u(t) dt,
0
The following inequalities are well known (cf. [4]):
T
ku̇k2L2
12
T2
≤
ku̇k2L2
4π 2
kũk2∞ ≤
kũk2L2
For convenience, we denote
Z T
1/2
M1 =
f 2 (t) dt
,
(Sobolev’s inequality),
(Wirtinger’s inequality)
Z
M2 =
0
T
Z
f (t) dt,
0
M3 =
T
g(t) dt.
0
Now we give the proofs of the main results.
Proof of Theorem 1.1. By (F3), we can choose an a1 > T 2 /(4π 2 ) such that
Z T
a1
−2γ
F (t, x) dt > M12 .
lim inf |x|
2
|x|→∞
0
(2.1)
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Y. WU, T. AN
EJDE-2013/77
By (F1) and the Sobolev’s inequality, for any u ∈ HT1 ,
T
Z
[F1 (t, u(t)) − F1 (t, ū)] dt
0
=
Z
Z
T
0
T
1
Z
Z
∇F1 t, ū + sũ(t) , ũ(t) ds dt
0
1
≤
T
Z
γ
Z
f (t)|ū + sũ(t)| |ũ(t)| ds dt +
0
≤ |ū|γ
0
g(t)|ũ(t)| ds dt
0
T
Z
T
1/2 Z
f 2 (t) dt
0
1
0
1/2
|ũ(t)|2 dt
(2.2)
0
+ kũkγ+1
∞
T
Z
T
Z
f (t) dt + kũk∞
g(t) dt
0
0
≤
1
a1
2γ
kũk2L2 + M12 |ū| + M2 kũkγ+1
∞ + M3 kũk∞
2a1
2
≤
T2
T
a1
2γ
ku̇k2L2 + M12 |ū| +
2
8π a1
2
12
γ+1
2
M2 ku̇kγ+1
L2 +
T 1/2
M3 ku̇kL2
12
Similarly, by (F2) and the Sobolev’s inequality, for any u ∈ HT1 ,
Z
T
T
Z
Z
0
T Z 1
Z
[F2 (u(t)) − F2 (ū)]dt =
0
0
1
1
(∇F2 (ū + sũ(t)) − ∇F2 (ū), sũ(t)) ds dt
s
≥−
0
α
rsα−1 |ũ(t)| ds dt
0
rT
≥ − kũkα
∞
α
rT T α/2
≥−
ku̇kα
L2
α 12
(2.3)
It follows from (2.2) and (2.3) that
Z T
1
2
ϕ(u) = ku̇kL2 +
[F1 (t, u(t)) − F1 (t, ū)] dt
2
0
Z T
Z T
+
[F2 (u(t)) − F2 (ū)] dt +
F (t, ū)dt
0
≥
0
γ+1
2
T
T
T 1/2
γ+1
2
−
k
u̇k
M
k
u̇k
M3 ku̇kL2
2 −
2
2
L
L
2 8π 2 a1
12
12
Z
T
a1
rT T α/2
2γ
−2γ
ku̇kα
|ū|
F (t, ū)dt − M12
−
L2 + |ū|
α 12
2
0
1
2
−
for all u ∈ HT1 , which implies that ϕ(u) → ∞ as kuk → ∞, due to (2.1) and γ < 1.
By the least action principle (see [4, Theorem 1.1 and Corollary 1.1]), the proof
is complete.
Proof of Theorem 1.2. Step 1. We firstly show that ϕ satisfies the (PS) condition.
Suppose that {un } is a (PS) sequence, that is, ϕ0 (un ) → 0 as n → 0 and {ϕ (un )}
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PERIODIC SOLUTIONS OF HAMILTONIAN SYSTEMS
is bounded. By (F5), we can choose an a2 > T 2 /(4π 2 ) such that
Z T
√
a
a2 T 2
2
−2γ
F (t, x) dt < −
lim sup |x|
M1 .
+
2
2π
|x|→∞
0
5
(2.4)
In a way similar to the proof of Theorem 1.1, one has
Z T
(∇F1 (t, un (t)), ũn (t)) dt
0
≤
T2
a2
T
2
2γ
ku̇n kL2 + M12 |ūn | +
2
8π a2
2
12
(γ+1)
2
T 1/2
M3 ku̇n kL2
12
(2.5)
γ+1
M2 ku̇n kL2 +
and
T
Z
(∇F2 (un (t)), ũn (t)) dt ≥ −
0
rT T α/2
α
ku̇n kL2
α 12
Z
T
r(t)dt
0
for all n. Hence one has
kũn k ≥ (ϕ0 (un ), ũn )
Z T
2
= ku̇n kL2 +
(∇F (t, un (t)), ũn (t))dt
0
(γ+1)
2
2
T T
a2
2γ
2
ku̇n kL2 − M12 |ūn | −
2
8π a2
2
12
1/2
α/2
T
rT T α
−
M3 ku̇n kL2 −
ku̇n kL2
12
α 12
for large n. It follows from Wirtinger’s inequality that
1/2
T 2 + 4π 2
kũn k ≤
ku̇n kL2 .
2π
By (2.6) and (2.7),
≥ 1−
T2 a2 2
T
2γ
2
M1 |ūn | ≥ 1 − 2
ku̇n kL2 −
2
8π a2
12
γ+1
2
γ+1
(2.6)
M2 ku̇n kL2
(2.7)
1
γ+1
M2 ku̇n kL2 −
α
T 2 + 4π 2
rT T 2
α
ku̇n kL2 −
−
α 12
2π
1
2
≥ ku̇n kL2 + C1 ,
2
T 2
M3 ku̇n kL2
12
1/2
ku̇n kL2
(2.8)
where
γ+1
n 4π 2 a − T 2
rT T α/2 α
T 2
2
2
s
−
M2 sγ+1 −
s
2
8π a2
12
α 12
s∈[0,+∞)
1/2 o
T 1/2
T 2 + 4π 2
−
M3 +
s .
12
2π
Note that a2 > T 2 /(4π 2 ) implies −∞ < C1 < 0. Hence, it follows from (2.8) that
C1 =
min
2
ku̇n kL2 ≤ a2 M12 |ūn |
and then
ku̇n kL2 ≤
√
2γ
− 2C1 ,
γ
a2 M1 |ūn | + C2 ,
(2.9)
(2.10)
6
Y. WU, T. AN
EJDE-2013/77
where 0 < C2 < +∞. In a way similar to the proof of Theorem 1.1, we have
Z
T
[F1 (t, u(t)) − F1 (t, ū)]dt
0
γ
≤ M1 |ū| kũkL2 + M2 kũkγ+1
∞ + M3 kũk∞
√
a2 T 2
π
2
2γ
γ+1
≤√
kũn kL2 +
M1 |ūn | + M2 kũn k∞ + M3 kũn k∞
a2 T
4π
γ+1
√
a2 T 2
T
T 2
2
2γ
γ+1
M1 |ūn | +
M2 ku̇n kL2
≤
√ ku̇n kL2 +
4π a2
4π
12
+
(2.11)
T 1/2
M3 ku̇n kL2 .
12
By (F4), we obtain
Z T
[F2 (un (t)) − F2 (ūn )] dt
0
Z
T
Z
T
Z
1
=
0
Z
0
≤
0
1
(∇F2 (ūn + sũn (t)) − ∇F2 (ūn ), sũn (t)) ds dt
s
1
δ
Csδ−1 |ũn (t)| ds dt ≤
0
CT
δ
kũn k∞
δ
CT T δ/2
δ
≤
ku̇n kL2 .
δ 12
It follows from the boundedness of {ϕ(un )} and (2.9)-2.11 that
C3 ≤ ϕ(un )
=
Z T
Z T
1
2
ku̇n kL2 +
[F1 (t, un (t)) − F1 (t, ūn )] dt +
[F2 (un (t)) − F2 (ūn )] dt
2
0
0
Z T
+
F (t, ūn )dt
0
√
γ+1
a2 T 2
T 2
2γ
γ+1
M1 |ūn | +
M2 ku̇n kL2
2 4π a2
4π
12
Z T
T 1/2
CT T δ/2
δ
+
M3 ku̇n kL2 +
ku̇n kL2 +
F (t, ūn )dt
12
δ 12
0
γ+1
1
√
T T 2
γ+1
2γ
γ
2
≤
+ √
a2 M1 |ūn | − 2C1 +
M2 ( a2 M1 |ūn | + C2 )
2 4π a2
12
≤
1
+
T
√
2
ku̇n kL2 +
√
T 1/2
CT T δ/2 √
δ
γ
γ
M3 ( a2 M1 |ūn | + C2 ) +
( a2 M1 |ūn | + C2 )
12
δ 12
Z T
√
a2 T 2
2γ
+
M1 |ūn | +
F (t, ūn )dt
4π
0
γ+1
√
a
√
a2 T 2
T 2
γ+1
2
2γ
γ(γ+1)
≤
+
M1 |ūn | +
M2 2γ ( a2 M1 )
|ūn |
+ 2γ C2γ+1
2
2π
12
δ/2 √
CT T
δ
γδ
+
2δ−1 ( a2 M1 ) |ūn | + 2δ−1 C2δ
δ 12
+
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7
Z T
√
T 1/2
T γ
F (t, ūn ) dt
+
C1 +
M3 ( a2 M1 |ūn | + C2 ) − 1 + √
12
2π a2
0
Z T
√
a
h
a2 T 2
2
2γ
−2γ
F (t, ūn )dt +
= |ūn | |ūn |
M1
+
2
2π
0
a T γ+1
a T 1/2
2
2
2
−γ
γ(γ−1)
+
2γ M1γ+1 M2 |ūn |
M1 M3 |ūn |
+
12
12
γ+1
i
CT a2 T δ/2 δ−1 δ
T 2 γ
γ(δ−2)
+
2 M2 C2γ+1
2 M1 |ūn |
+
δ
12
12
CT T δ/2 δ−1 δ
T 1/2
T +
C1
2 C2 +
M3 C2 − 1 + √
δ 12
12
2π a2
for large n. The above inequality and (2.4) imply that {|ū|} is bounded. Hence
{un } is bounded by (2.9). Arguing as in the proof of Proposition 4.1 of [4], we
conclude that (PS) condition is satisfied.
e1,
e 1 = {u ∈ H 1 : ū = 0}. We show that for u ∈ H
Step 2. Let H
T
T
T
ϕ(u) → +∞ (kuk → ∞).
(2.12)
In fact, by (F1) and Sobolev’s inequality, one has
Z
Z Z
T
T 1
[F1 (t, u(t)) − F1 (t, 0)]dt = (∇F1 (t, su(t)), u(t)) ds dt
0
0
T
Z
≤
0
f (t)|u(t)|
γ+1
g(t) |u(t)| dt
dt +
0
≤
T
Z
0
T
12
α+1
2
T 1/2
M3 ku̇kL2
12
M2 ku̇kα+1
L2 +
e 1 . It follows from (F2) that
for all u ∈ H
T
Z T
Z TZ 1
[F2 (u(t)) − F2 (0)]dt =
(∇F2 (su(t)) − ∇F2 (0) , u(t)) ds dt
0
0
Z
0
T Z 1
≥−
0
α
rsα−1 |u(t)| ds dt
0
rT
α
kuk∞
α
α/2
rT T
α
≥−
kukL2 .
α 12
≥−
Hence, we have
ϕ(u) =
1
ku̇k2L2 +
2
Z
T
Z
[F (t, u(t)) − F (t, 0)]dt +
0
T
F (t, 0)dt
0
α+1
1
T 2
T 1/2
M2 ku̇kα+1
−
≥ ku̇k2L2 −
M3 ku̇kL2
2
L
2
12
12
Z
T
rT T α/2
α
kukL2 +
F (t, 0)dt.
−
α 12
0
e 1 . Hence (2.12)
By Wirtinger’s inequality, kuk → ∞ if and only if ku̇kL2 → ∞ in H
T
is satisfied.
8
Y. WU, T. AN
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RT
Step 3. By (F5), we can easily see that 0 F (t, x)dt → −∞ as |x| → ∞ for all
e 1 )⊥ = RN ,
x ∈ RN . Thus, for all u ∈ (H
T
Z T
ϕ(u) =
F (t, u)dt → −∞ as |u| → ∞.
0
Now, the proof is completed by saddle point theorem (cf. [7, Theorem 4.6])
2
Proof of Theorem 1.3. By (F3’), we can choose an a3 > 4π2T−rT 2 such that
Z T
a3
F (t, x) dt > M12 .
lim inf |x|−2γ
(2.13)
2
|x|→∞
0
The condition (F2’) and the Sobolev’s inequality imply that
Z T
Z TZ 1
1
[F2 (u(t) − F2 (ū))] dt =
(∇F2 (ū + sũ(t)) − ∇F2 (ū), sũ(t)) ds dt
0 s
0
0
Z TZ 1
rT 2
2
≥−
rs|ũ(t)| ds dt − 2 ku̇k2L2 .
8π
0
0
It follows immediately from the similar method of the proof of Theorem 1.1 that
Z T
1
ϕ(u) = ku̇k2L2 +
F (t, u(t))dt
2
0
γ+1
1
T2
rT 2
T 2
2
≥
M2 ku̇kγ+1
−
− 2 ku̇kL2 −
L2
2 8π 2 a3
8π
12
Z T
T 1/2
a3
2γ
−2γ
−
M3 ku̇kL2 + |ū|
|ū|
F (t, ū)dt − M12
12
2
0
for all u ∈ HT1 , which implies that ϕ(u) → ∞ as kuk → ∞ by (2.13), due to the
2
facts that γ < 1, r < 4π
T 2 and kuk → ∞ if and only if
1/2
2
|ū| + ku̇k2L2
→ ∞.
By the least action principle, Theorem 1.3 holds.
Proof of Theorem 1.4. We firstly show that ϕ satisfies the (PS) condition. Suppose
that {un } satisfies ϕ0 (un ) → 0 as n → 0 and {ϕ (un )} is bounded. By (F7), we can
choose an a4 > T 2 /(4π 2 such that
Z T
√
a4 T 2
a4
−2γ
lim sup |x|
F1 (t, x) dt < −
+
M1 .
(2.14)
2
2π
|x|→∞
0
By the (λ,µ)-subconvexity of G(x), we have
β
G(x) ≤ 2µ|x| + 1 G0
(2.15)
for all x ∈ RN , and a.e. t ∈ [0, T ], where G0 = max|s|≤1 G(s), β = log2λ (2µ) < 2.
Then
Z T
Z T
(∇F2 (t, un (t)) , ũn (t)) dt ≥ −
k(t)G(ūn )dt
0
0
Z
≥−
T
β
k(t) 2µ|ūn | + 1 G0 dt
0
β
= −2µM4 |ūn | − M4 ,
(2.16)
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where M4 = G0
RT
0
9
k(t)dt. It follows from (2.5) and (2.16) that for large n,
kũn k ≥ (ϕ(un ), ũn )
Z
2
= ku̇n kL2 +
T
(∇F (t, un (t)), ũn (t))dt
0
T2 T
a4
2
2γ
≥ 1− 2
ku̇n kL2 − M12 |ūn | −
8π a4
2
12
1/2
T
β
−
M3 ku̇n kL2 − 2µM4 |ūn | − M4 .
12
Then (2.7) and (2.17) imply that
γ+1
2
γ+1
(2.17)
M2 ku̇n kL2
γ+1
a4 2
T2 T 2
2γ
β
2
γ+1
M |ūn | + 2µM4 |ūn | ≥ 1 − 2
ku̇n kL2 −
M2 ku̇n kL2
2 1
8π a4
12
1/2 T 1/2
T 2 + 4π 2
−
M3 +
ku̇n kL2 − M4
12
2π
1
2
≥ ku̇n kL2 + C4 ,
2
(2.18)
where
γ+1
n 8π 2 a − T 2
T 2
4
2
C4 = min
s
−
M2 sγ+1 − M4
8π 2 a4
12
s∈[0,+∞)
1/2 i o
h T 1/2
T 2 + 4π 2
M3 +
s .
−
12
2π
Note that −∞ < C4 < 0 due to a4 >
2
T2
4π 2 .
ku̇n kL2 ≤ a4 M12 |ūn |
and then
2γ
By (2.18), one has
β
+ 4µM4 |ūn | − 2C4 ,
(2.19)
√
p
2a4
γ
β/2
M1 |ūn | + 2µM4 |ūn |
+ C5 ,
2
where C5 > 0. It follows from (F6) and (2.15) that
Z T
[F2 (t, u(t)) − F2 (t, ū)] dt
ku̇n kL2 ≤
(2.20)
0
T
Z
1
Z
=−
(∇F2 (t, ūn + sũn (t)) , −ũn (t)) ds dt
0
T
Z
0
Z
1
≤
k(t)G (ūn + (s + 1)ũn (t)) ds dt
0
0
T
Z
Z
≤
0
0
T
Z
≤ 4µ
≤2
≤
0
β+2
1
β
k(t) 2µ|ūn + (s + 1)ũn (t)| + 1 G0 ds dt
Z
β
β
k(t) |ūn | + 2β |ũn (t)| G0 dt + G0
0
β
β
µM4 kũn k∞ + 4µM4 |ūn | + M4
T β/2 β+2
β
β
2
µM4 ku̇n kL2 + 4µM4 |ūn | + M4
12
T
k(t)dt
(2.21)
10
Y. WU, T. AN
EJDE-2013/77
for all u ∈ HT1 . By the boundedness of {ϕ(un )} and the inequalities (2.11), (2.19)(2.21), one has
C6 ≤ ϕ(un )
=
Z T
1
2
ku̇n kL2 +
[F1 (t, un (t)) − F1 (t, ūn )] dt
2
0
Z T
Z T
+
[F2 (t, un (t)) − F2 (t, ūn )] dt +
F (t, ūn )dt
0
≤
1
+
2
T
√
0
√
4π a4
2
ku̇n kL2 +
a4 T 2
T
2γ
M1 |ūn | +
4π
12
γ+1
2
γ+1
M2 ku̇n kL2
T 1/2
T β/2 β+2
β
β
M3 ku̇n kL2 +
2
µM4 ku̇n kL2 + 4µM4 |ūn | + M4
12
12
+
T
Z
+
F (t, ūn )dt
0
√a T
4
2γ
2γ
β
M12 |ūn |
a4 M12 |ūn | + 4µM4 |ūn | − 2C4 +
2 4π a4
4π
γ+1
√
γ+1
p
T 2
γ
β/2
+
M2 a4 M1 |ūn | + 2 µM4 |ūn |
+ C5
12
√
p
T 1/2
γ
β/2
+ C5
+
pM3 a4 M1 |ūn | + 2 µM4 |ūn |
12
√
β
p
T β/2 β+2
γ
β/2
+
2
µM4 a4 M1 |ūn | + 2 µM4 |ūn |
+ C5
12
Z T
β
+ µM4 |ūn | + M4 +
F (t, ūn )dt
0
√
a
a4 T 2
T T 4
2γ
β
+
M1 |ūn | + 6 + √
µM4 |ūn | − 1 + √
C4
≤
2
2π
π a4
2π a4
γ+1
γ+1
β(γ+1)
γ+1
γ+1
T 2
γ(γ+1)
+
M2 2γ a4 2 M1 γ+1 |ūn |
+ 23γ+1 µ 2 M4 2 |ūn | 2
12
β
β2
β
β
β
T 2 β+2
γβ
+ 22γ C5γ+1 +
2
µM4 2β−1 a4 2 M1 β |ūn | + 23β−2 µ 2 M4 2 |ūn | 2
12
T 1/2
√
p
γ
β/2
+ 22(β−1) C5β +
M3
a4 M1 |ūn | + 2 µM4 |ūn |
+ C5
12
Z T
+ M4 +
F (t, ūn )dt
≤
1
T
√
+
0
2γ
= |ūn |
h
|ūn |
−2γ
Z
T
F1 (t, ūn )dt +
0
a
4
2
√
+
γ+1
a4 T 2
M1
2π
T 1/2 √
T 2 γ− 1 √
−γ
γ(γ−1)
+
a4 M1 M3 |ūn | +
2 2 a4 M1 M2 |ūn |
12
12
i
β
T β/2 2β+1
γ(β−2)
+
2
µa4 2 M1 β M4 |ūn |
12
EJDE-2013/77
+ |ūn |
PERIODIC SOLUTIONS OF HAMILTONIAN SYSTEMS
β
Z
h
−β
|ūn |
T
0
11
T F2 (t, ūn )dt + 6 + √
µM4
π a4
1 2
β+2
T β/2 4β β+2
β −2
+
2 µ 2 M4 2 |ūn | 2
12
γ+1
i
p
β(γ−1)
γ+1
γ+1
T 2
T 1/2
−β/2
+
M2 23γ+1 µ 2 M2 M4 2 |ūn | 2 +
2M3 µM4 |ūn |
12
12
γ+1
2
T 1/2
T
T
C4 +
22γ M2 C5γ+1 +
M3 C5
− 1+ √
2π a4
12
12
T β/2 3β
2 µM4 C5β + M4
12
for large n. The above inequality and (2.14) imply that {|ū|} is bounded. Hence
{un } is bounded by (2.19). By using the usual method, the (PS) condition holds.
Similar to the proof of Theorem 1.2, we can verify that functional satisfies the
other conditions of the saddle point theorem. We omit the details.
+
3. Examples
In this section, we give some examples of F to illustrate that our results are new.
Example 3.1. Let F = F1 + F2 , with
2πt 7/4
|x| + (0.6T − t)|x|3/2 + (h(t), x) ,
F1 (t, x) = sin
T
3r
F2 (x) = C(x) − |x|4/3 ,
4
where h ∈ L 1 (0, T ; RN ), r > 0, C(x) =
3r
4
4 (|x1 |
+ |x2 |4/3 + · · · + |xN |4/3 ).
By Young’s inequality, it is easy to see that
2πt 7 3/4 3
|∇F1 (t, x)| ≤ sin
|x| + |0.6T − t||x|1/2 + |h(t)|
4
T
2
2πt 7 T3
≤
sin
+ ε |x|3/4 + 2 + |h(t)|
4
T
ε
for all x ∈ RN and a.e. t ∈ [0, T ], where ε > 0. And
(∇F2 (x) − ∇F2 (y), x − y) ≥ −r|x − y|4/3
for all x, y ∈ RN . Thus, (F1), (F2) hold with γ = 3/4, α = 4/3 and
7
2πt
(| sin(
)| + ε),
4
T
However, F does not satisfy (1.2). In fact
Z T
|x|−2γ
F (t, x)dt
f (t) =
g(t) =
T3
+ |h(t)|.
ε2
0
Z
T
3r
2πt 7/4
|x| + (0.6T − t)|x|3/2 + C(x) − |x|4/3 + (h(t), x) dt
T
4
0
Z T
3r
4/3
T (C(x) − 4 |x| )
= 0.1T 2 +
+
h(t)dt, |x|−3/2 x
|x|3/2
0
= |x|−3/2
sin
12
Y. WU, T. AN
EJDE-2013/77
On the other hand, we have
Z T
T2
49T 3 1 4ε
2
2
f
(t)dt
=
+
+
ε
8π 2 0
128π 2 2
π
If T <
128π 2
245 ,
we choose ε > 0 sufficient small such that
lim inf |x|−2γ
|x|→∞
Z
T
F (t, x)dt = 0.1T 2 >
0
T2
8π 2
Z
T
f 2 (t)dt
0
which implies that (F3) holds. Then F = F1 + F2 is not convex, not γ-subadditive,
not periodic, not a.e. uniformly coercive, and ∇F is not sublinear. Thus, F is not
covered by results in the references.
Example 3.2. Let F = F1 + F2 , with
F1 (t, x) = (0.5T − t)|x|7/4 + (0.4T − t)|x|3/2 + (h(t), x) ,
4r
F2 (x) = − |x|5/4 ,
5
where h ∈ L 1 (0, T ; RN ), r > 0.
Similar to Example 3.1, we can see that all conditions of Theorem 1.2 hold but
F is not covered by results in the references.
Example 3.3. Let F = F1 + F2 , with
F1 (t, x) = (0.5T − t)|x|7/4 + (0.6T − t)|x|3/2 + (h(t), x) ,
r
F2 (x) = C(x) − |x|2 ,
2
where h ∈ L 1 (0, T ; RN ), C(x) = 2r (|x1 |4 + |x2 |2 + · · · + |xN |2 ), 0 < r <
4π 2
T2 .
In a way similar to Example 3.1, it is easy to see that condition (F1) and (F2’)
are satisfied with γ = 3/4. However, F does not satisfies (1.2). In fact,
Z T
|x|−2γ
F (t, x)dt
0
Z
T
r
(0.5T − t)|x|7/4 + (0.6T − t)|x|3/2 + C(x) − |x|2 + (h(t), x) dt
2
0
Z T
r
2
T C(x) − 2 |x|
= 0.1T 2 +
+
h(t)dt, x|x|−3/2
|x|3/2
0
Z T
4
2
rT
(|x
|
−
|x
|
)
1
1
−3/2
= 0.1T 2 +
+
h(t)dt,
x|x|
.
2|x|3/2
0
= |x|−2/3
We can choose ε > 0 small enough and some suitable T such that
Z T
Z T
T2
lim inf |x|−2γ
F (t, x)dt = 0.1T 2 >
f 2 (t, x)dt,
2 − rT 2 )
2(4π
|x|→∞
0
0
which implies that (F3’) holds. F is also not covered by results in the references.
EJDE-2013/77
PERIODIC SOLUTIONS OF HAMILTONIAN SYSTEMS
13
References
[1] Nurbek Aizmahin, Tianqing An; The existence of periodic solutions of non-autonomous
second-order Hamiltonian systems. Nonlinear Analysis TMA 74, (2011), 4862–4867.
[2] M. S. Berger, M. Schechter; On the solvability of semilinear gradient operator equations.
Adv. Math. 25, (1977), 97–132.
[3] Jian Ma, Chunlei Tang; Periodic solutions for some nonautonomous second-order systems.
J. Math. Anal. Appl. 275, (2002), 482–494.
[4] J. Mawhin, M. Willem; Critical Point Theory and Hamiltonian Systems. Springer-Verlag,
New York, 1989.
[5] J. Mawhin; Oscillations forcees de systemes hamiltoniens. Semin. Analyse Non linaire, Univ.
Besancon, 1981.
[6] J. Mawhin; Semi-coercive monotone variational problems. Acad. Roy. Belg. Bull. Cl. Sci. 73,
(1987), 118–130.
[7] P. H. Rabinowitz; Minimax methods in critical point with applications to differential equations. CBMS 65, American Mathematical Society, Providence, RI, 1986.
[8] Chunlei Tang; Existence and multiplicity of periodic solutions for non-autonomous second
order systems. Nonlinear Analysis TMA 32, (1998), 299–304.
[9] Chunlei Tang; Periodic solutions for nonautonomous second order systems with sublinear
nonlinearity. Pro. Amer. Math. Soc. 126, (1998), 3263–3270.
[10] Chunlei Tang; Periodic solutions of nonautonomous second order systems with γquasisubadditive potential. J. Math. Anal. Appl. 186, (1995), 671–675.
[11] X.H. Tang, Qiong Meng; Solutions of a second-order Hamiltonian system with periodic
boundary conditions. Nonlinear Analysis RWA 11, (2010), 3722–3733.
[12] Xingping Wu, Chunlei Tang; Periodic solutions of a class of non-autonomous second-order
systems. J. Math. Anal. Appl. 236, (1999), 227–235.
[13] Rigao Yang; Periodic solutions of some non-autonomous second order Hamiltonian systems.
Nonlinear Analysis TMA 69, (2008), 2333–2338.
[14] Xingyong Zhang, Yinggao Zhou; Periodic solutions of non-autonomous second order Hamiltonian systems. J. Math. Anal. Appl. 345, (2008), 929–933.
[15] Fukun Zhao, Xian Wu; Periodic solutions for a class of non-autonomous second order systems. J. Math. Anal. Appl. 296, (2004), 422–434.
Yue Wu
College of Science, Hohai University, Nanjing 210098, China
E-mail address: wyue007@126.com
Tianqing An
College of Science, Hohai University, Nanjing 210098, China
E-mail address: antq@hhu.edu.cn