Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 99, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF INFINITELY MANY PERIODIC SOLUTIONS
FOR SECOND-ORDER NONAUTONOMOUS HAMILTONIAN
SYSTEMS
WEN GUAN, DA-BIN WANG
Abstract. By using minimax methods and critical point theory, we obtain
infinitely many periodic solutions for a second-order nonautonomous Hamiltonian systems, when the gradient of potential energy does not exceed linear
growth.
1. Introduction and main results
Consider the second-order Hamiltonian system
u
¨(t) + ∇F (t, u(t)) = 0, a.e. t ∈ [0, T ],
u(0) − u(T ) = u(0)
˙
− u(T
˙ ) = 0.
(1.1)
Where T > 0 and F : [0, T ] × RN → R satisfies the following assumption:
(A1) 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 ∈ L1 ([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 existence of periodic solutions for problem (1.1) was obtained in [1, 3, 5, 6, 8, 9,
10, 11, 12, 13, 14, 15, 16, 18, 19, 21, 22, 23] with many solvability conditions by using
the least action principle and the minimax methods, such as the coercive type
potential condition [3], the convex type potential condition [8], the periodic type
potential conditions [18], the even type potential condition [6], the subquadratic
potential condition in Rabinowitz’s sense [11], the bounded nonlinearity condition
(see [9]), the subadditive condition (see [12]), the sublinear nonlinearity condition
(see [5, 14]), and the linear nonlinearity condition (see [10, 16, 22, 23]).
In particular, when the nonlinearity ∇F (t, x) is bounded; that is, there exists
g(t) ∈ L1 ([0, T ], R+ ) such that |∇F (t, x)| ≤ g(t) for all x ∈ RN and a.e. t ∈ [0, T ],
and that
Z
T
F (t, x)dt → ±∞ as |x| → ∞,
0
2000 Mathematics Subject Classification. 34C25, 58E50.
Key words and phrases. Periodic solutions; Minimax methods; linear; Hamiltonian system;
critical point.
c
2015
Texas State University - San Marcos.
Submitted November 11, 2014. Published April 14, 2015.
1
2
WEN GUAN, DA-BIN WANG
EJDE-2015/99
Mawhin and Willem [9] proved that problem (1.1) has at least one periodic solution.
Han and Tang [5, 14] generalized these results to the sublinear case:
|∇F (t, x)| ≤ f (t)|x|α + g(t)
for all x ∈ RN and a.e. t ∈ [0, T ]
(1.2)
with
|x|−2α
T
Z
F (t, x)dt → ±∞ as |x| → ∞,
(1.3)
0
where f (t), g(t) ∈ L1 ([0, T ], R+ ) and α ∈ [0, 1).
Subsequently, when α = 1 Zhao and Wu [22, 23], and Meng and Tang [10, 16]
proved the existence of periodic solutions for problem (1.1), i.e. ∇F (t, x) does not
exceed linear growth:
|∇F (t, x)| ≤ f (t)|x| + g(t)
1
for all x ∈ RN and a.e. t ∈ [0, T ],
(1.4)
+
where f (t), g(t) ∈ L ([0, T ], R ).
On the other hand, there are large number of papers that deals with multiplicity results for this problem. In particular, infinitely many solutions for (1.1) are
obtained in [2, 20, 24] when the nonlinearity F (t, x) have symmetry. Since the symmetry assumption on the nonlinearity F has play an important role in [2, 21, 24],
many authors have paid much attention to weak the symmetry condition and some
existence results on periodic solutions have been obtained without any symmetry
condition [4,7,17,25]. Especially, Zhang and Tang [25] obtained infinitely many periodic solutions for (1.1) when (1.2) holds and F has a suitable oscillating behaviour
at infinity:
Z T
lim sup
inf
|x|−2α
F (t, x)dt = +∞,
r→+∞ x∈RN ,|x|=r
lim inf
sup
R→+∞ x∈RN ,|x|=R
0
|x|−2α
Z
T
F (t, x)dt = −∞,
0
where α ∈ [0, 1).
Motivated by the results mentioned above, especially by ideas in [10,16,22,23,25],
in this article, by using the minimax methods in critical point theory, we obtain
infinitely many periodic solutions for(1.1).
Let HT1 be a Hilbert space HT1 = u : [0, T ] → RN : u is absolutely continuous,
u(0) = u(T ) and u˙ ∈ L2 ([0, T ], R) , with the norm
Z T
Z T
1/2
2
2
kuk =
|u(t)| dt +
|u(t)|
˙
dt
,
(1.5)
0
for u ∈
HT1 .
0
Let
Z
Z T
1 T
2
J(u) =
|u(t)|
˙
dt −
F (t, u(t))dt.
(1.6)
2 0
0
It is well known that the function J is continuously differentiable and weakly lower
semicontinuous on HT1 and the solutions of (1.1) correspond to the critical points
of J (see [9]). Our main result is the following theorem.
RT
Theorem 1.1. Suppose that (A1) and (1.4) with 0 f (t)dt < T3 hold and
Z T
lim sup
inf
F (t, x)dt = +∞,
(1.7)
r→+∞ x∈RN ,|x|=r
0
EJDE-2015/99
lim inf
EXISTENCE OF INFINITELY MANY PERIODIC SOLUTIONS
sup
R→+∞ x∈RN ,|x|=R
|x|−2
Z
T
F (t, x)dt < −
0
3T 2
RT
2π 2 (12 − T 0 f (t)dt)
Z
3
T
f 2 (t)dt. (1.8)
0
Then
(i) There exists a sequence of periodic solutions {un } which are minimax type
critical points of functional J, and J(un ) → +∞ as n → ∞;
(ii) There exists another sequence of periodic solutions {u∗m } which are local
minimum points of functional J, and J(u∗m ) → −∞ as m → ∞.
Remark 1.2.
(i) As in [25], in this paper we do not assume any symmetry condition on nonlinearity;
(ii) Our main result in this paper extends main result in [25] corresponding to α = 1.
2. Proof of main results
For u ∈ HT1 , let
Z
1 T
u=
u(t)dt, u
e(t) = u(t) − u.
T 0
The following inequalities are well known (see [9]):
(2.1)
T
kuk
˙ 2L2 (Sobolev’s inequality),
12
T2
kuk
˙ 2L2 (Wirtinger’s inequality).
k˜
uk2L2 ≤
4π 2
For the sake of convenience, we denote
Z T
Z T
Z T
1/2
, M2 =
f (t)dt, M3 =
g(t)dt.
M1 =
f 2 (t)dt
k˜
uk2∞ ≤
0
0
Lemma 2.1. Suppose that
RT
0
f (t)dt < 3/T and (1.4) hold, then
e T1 ,
as kuk → ∞ in H
J(u) → +∞
where
e1
H
T
= {u ∈
HT1
0
| u = 0} be the subspace of
Proof. From (1.4) and
Z
1 T
J(u) =
2 0
Z
1 T
≥
2 0
Z
1 T
≥
2 0
(2.2)
HT1 .
e 1 we have
Sobolev’s inequality, for all u in H
T
Z T
2
|u(t)|
˙
dt −
F (t, u(t))dt
0
2
Z
|u(t)|
˙
dt −
T
2
f (t)|u(t)| dt −
0
2
|u(t)|
˙
dt −
Z
ke
uk2∞
T
g(t)|u(t)|dt
0
Z
T
Z
f (t)dt − ke
uk∞
0
T
g(t)dt
0
Z T
Z T
T
1
T 1/2
kuk
˙ 2L2 − kuk
˙ 2L2
f (t)dt −
kuk
˙ L2
g(t)dt
2
12
12
0
0
Z T
1
T
=
−
f (t)dt kuk
˙ 2L2 − C1 kuk
˙ L2 .
2 12 0
1/2
RT
2
By Wirtinger’s inequality, the norm kuk =
|u(t)|
˙
dt
is an equivalent
0
e 1 . So, J(u) → +∞ as kuk → ∞ in H
e1.
norm on H
T
T
≥
4
WEN GUAN, DA-BIN WANG
EJDE-2015/99
Lemma 2.2. Suppose that (1.7) holds. Then there exists positive real sequence
{an } such that
lim an = +∞,
n→∞
lim
lim bm = +∞,
where Hbm = {u ∈ RN : |u| = bm }
J(u) = −∞ .
n
The above lemma follows from (1.7).
RT
Lemma 2.3. Suppose that 0 f (t)dt <
positive real sequence {bm } such that
m→∞
sup
n→∞ u∈RN ,|u|=a
3
T
, (1.4) and (1.8) hold. Then there exists
lim
inf J(u) = +∞,
m→∞ u∈Hbm
L e1
HT .
Proof. By (1.8), we can choose an a > 3T 2 /(12π 2 − π 2 T M2 ) such that
Z T
a
|x|−2
F (t, x)dt < − M12 .
lim inf
sup
r→+∞ x∈RN ,|x|=r
2
0
e 1 . So, we have
For any u ∈ Hbm , let u = u + u
e, where |u| = bm , u
e∈H
T
Z T
F (t, u(t)) − F (t, u)dt
0
=
Z
0
T
T
Z
0
1
1
(∇F (t, u + se
u(t), u
e(t)) ds dt
Z TZ 1
f (t)|u + se
u(t)||e
u(t)| ds dt +
g(t)|e
u(t)| ds dt
0
0
0
0
Z T
Z T
1
≤
f (t) |u| + |e
u(t)| |e
u(t)|dt +
g(t)|e
u(t)|dt
2
0
0
Z
Z
Z T
Z T
T
1/2 T
1/2 1
2
2
2
≤ |u|
f (t)dt
|e
u(t)| dt
+ ke
uk∞
f (t)dt + ke
uk∞
g(t)dt
2
0
0
0
0
M2
ke
uk2∞ + M3 ke
uk∞
ukL2 +
= M1 |u|ke
2
M2 2
1
a
≤
ke
uk2L2 + M12 |u|2 +
ke
u k∞ + M3 ke
uk∞
2a
2
2
2
T 1/2
T
T M2 a
≤
+
kuk
˙ 2L2 + M12 |u|2 +
M3 kuk
˙ L2
2
8aπ
24
2
12
for all u ∈ Hbm . Hence we have
Z
Z T
Z T
1 T
2
J(u) =
|u(t)|
˙
dt −
[F (t, u(t)) − F (t, u)]dt −
F (t, u)dt
2 0
0
0
T2
T M2 1
T 1/2
−
−
kuk
˙ 2L2 −
M3 kuk
˙ L2
≥
2
2 8aπ
24
12
Z
T
a
− |u|2 |u|−2
F (t, u)dt + M12
2
0
1
for all u ∈ Hbm . As |u|2 + kuk
˙ L2 2 → ∞ if and only if kuk → ∞, then the Lemma
follows from (1.8) and the above inequality.
Z
Z
≤
Now prove our main result.
EJDE-2015/99
EXISTENCE OF INFINITELY MANY PERIODIC SOLUTIONS
5
Proof of Theorem 1.1. Let Ban be a ball in RN with radius an . Then we define a
family of maps
Γn = {γ ∈ C(Ban , HT1 ) : γ ∂B = Id∂B }
an
an
and corresponding minimax values
cn = inf max J(γ(x)).
γ∈Γn x∈Ban
e 1 , i.e., for any γ ∈ Γn ,
It is easy to see that each γ intersects the hyperplane H
T
1
e 6= ∅.
γ(Ban ) ∩ H
T
e 1 . So, there is a constant M
By Lemma 2.1, the functional J is coercive on H
T
such that
max J(γ(x)) ≥ inf J(u) ≥ M.
x∈Ban
e1
u∈H
T
Hence
cn ≥ inf J(u) ≥ M.
e1
u∈H
T
By Lemma 2.2, for all large value of n,
cn > max J(u).
u∈∂Ban
For such n, there exists a sequence {γk } in Γn such that
max J(γk (x)) → cn , k → ∞.
x∈Ban
Applying [9, Theorem 4.3 and Corollary 4.3], we know there exists a sequence {vk }
in HT1 such that
J(vk ) → cn , dist(vk , γk (Ban )) → 0, J 0 (vk ) → 0,
(2.3)
as k → ∞. If we can show {vk } is bounded, then there is a subsequence, which is
still be denote by {vk } such that
vk * un
weakly in HT1 ,
uniformly in C([0, T ], RN ).
vk → un
Hence
hJ 0 (vk ) − J 0 (un ), vk − un i → 0,
Z
T
(∇F (t, vk ) − ∇F (t, un ), vk − un )dt → 0
0
as k → ∞. Moreover, it is easy to see that
hJ 0 (vk ) − J 0 (un ), vk − un i
Z T
2
= kv˙k − u˙n kL2 −
(∇F (t, vk ) − ∇F (t, un ), vk − un )dt,
0
so kv˙k − u˙n k2L2 → 0 as k → ∞. Then, it is not difficult to obtain kvk − vn k → 0 as
k → ∞. So, we have
J 0 (un ) = lim J 0 (vk ) = 0,
k→∞
J(un ) = lim J(vk ) = cn .
k→∞
Thus, un is critical point and cn is critical value of functional J.
6
WEN GUAN, DA-BIN WANG
EJDE-2015/99
Now, let us show the sequence {vk } is bounded in HT1 . By (2.3), for any large
enough k, we have
cn ≤ max J(γk (x)) ≤ cn + 1,
(2.4)
x∈Ban
and we can find wk ∈ γk (Ban ) such that kvk − wk k ≤ 1.
Fix n, by Lemma 2.3, we can choose a large enough m such that
bm > an
and
inf
u∈Hbm
> cn + 1.
This implies γ(Ban ) cannot intersect the hyperplane Hbm for each k.
e 1 . Then we have |wk | < bm for
Let wk = wk + w
ek , where wk ∈ RN and w
ek ∈ H
T
each k. Also, by Sobolev’s inequality and (1.4), it is obvious that
cn + 1
≥ J(wk ) =
1
2
Z
1
≥
2
Z
1
2
Z
≥
≥
T
Z
1
2
|w˙ k (t)|2 dt −
F (t, wk (t))dt
0
0
T
|w˙ k (t)|2 dt −
T
Z
0
f (t)|wk (t)|2 dt −
0
T
2
0
2
g(t)|wk (t)|dt
Z
2
ek (t)| ]dt −
f (t)[|wk | + |w
T
|w˙ k (t)|2 dt − 2kw
ek k2∞
Z
T
T
Z
0
− kw
ek k∞
Z
0
|w˙ k (t)| dt − 2
0
T
Z
T
Z
Z
f (t)dt − 2|wk |2
0
T
g(t)dt − |wk |
0
T
Z
0
T
g(t)[|wk | + |w
ek (t)|]dt
T
f (t)dt
0
g(t)dt
0
Z T
Z T
1
T
kw˙ k (t)k2L2 − kw˙ k (t)k2L2
f (t)dt
f (t)dt − 2|wk |2
2
6
0
0
Z T
Z T
T 1/2
−
kw˙ k (t)kL2
g(t)dt − |wk |
g(t)dt
12
0
0
T 1/2
1 T
− M2 kw˙ k (t)k2L2 −
M3 kw˙ k (t)kL2 − C2
=
2
6
12
≥
˙ L2 )1/2 is an equivalent norm in HT1 , it follows that w
ek (t) is bounded.
As (|u|2 + kuk
Hence, wk is bounded. Also, {vk } is bounded in HT1 .
From the previous discussion we know that accumulation point un of {vk } is a
critical point and cn is critical value of J.
If we choose large enough n such that an > bm , then γ(Ban ) intersects the
hyperplane Hbm for any γ ∈ Γn .
It follows that
max J(γ(x)) ≥
x∈Ban
inf J(u).
u∈Hbm
From this inequality and Lemma 2.3 we obtain limn→∞ cn = +∞. Result (i) of
Theorem 1.1 is obtained.
Next we prove (ii). For fixed m, define the subset Pm of HT1 by
e T1 }.
Pm = {u ∈ HT1 : u = u + u
e, |u| ≤ bm , u
e∈H
(2.5)
EJDE-2015/99
EXISTENCE OF INFINITELY MANY PERIODIC SOLUTIONS
7
For u ∈ Pm , we have
Z
Z T
1 T
2
J(u) =
|u(t)|
˙
dt −
F (t, u(t))dt
2 0
0
Z
Z T
Z T
1 T
2
≥
|u(t)|
˙
dt −
f (t)|u(t)|2 dt −
g(t)|u(t)|dt
2 0
0
0
Z T
Z
Z T
1 T
2
≥
f (t)[|u(t)|2 + |e
|u(t)|
˙
dt − 2
g(t)[|u(t)| + |e
u(t)|2 ]dt −
u(t)|]dt
2 0
0
0
Z T
Z T
1
T
2
2
2
f (t)dt
≥ ku(t)k
f (t)dt − 2|u(t)|
˙
ku(t)k
˙
L2 −
L2
2
6
0
0
Z T
Z T
T 1/2
g(t)dt − |u(t)|
−
ku(t)k
˙
g(t)dt
L2
12
0
0
1 T
T 1/2
2
=
− M2 ku(t)k
˙
M3 ku(t)k
˙
L2 − C3
L2 −
2
6
12
(2.6)
Then J is bounded below on Pm .
Let
µm = inf J(u),
u∈Pm
and {uk } be a minimizing sequence in Pm ; that is,
J(uk ) → µm
By (2.6), {uk } is bounded in
denoted by {uk }, such that
HT1 .
as k → ∞.
Then there is a subsequence, which is still be
uk * u∗m weakly in HT1 .
Since Pm is a convex closed subset of HT1 , u∗m ∈ Pm . As J is weakly lower semicontinuous, we have
µm = lim J(uk ) ≥ J(u∗m ).
k→∞
Since u∗m ∈ Pm , µm = J(u∗m ).
If we can show u∗m is in the interior of Pm , then u∗m is a local minimum of
functional J. In fact, let u∗m = u∗m + u
e∗m . From Lemmas 2.2 and 2.3, we see
∗
|um | =
6 bm for large m, which means that u∗m is in the interior of Pm .
Since u∗m is a minimum of J on Pm , we have
J(u∗m ) = inf J(u) ≤ sup J(u).
u∈Pm
It follows from Lemma 2.2 that
complete.
J(u∗m )
|u|=bm
→ −∞ as m → ∞. Therefore, the proof is
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Wen Guan
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou,
Gansu 730050, China
E-mail address: [email protected]
Da-Bin Wang (corresponding author)
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou,
Gansu 730050, China
E-mail address: [email protected]