Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 64, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
PERIODIC SOLUTIONS FOR NON-AUTONOMOUS
SECOND-ORDER DIFFERENTIAL SYSTEMS WITH
(q, p)-LAPLACIAN
CHUN LI, ZENG-QI OU, CHUN-LEI TANG
Abstract. Some existence theorems are obtained for periodic solutions of
nonautonomous second-order differential systems with (q, p)-Laplacian by using the least action principle and the saddle point theorem.
1. Introduction
Consider the second-order system
d
|u̇1 (t)|q−2 u̇1 (t) = ∇u1 F (t, u1 (t), u2 (t)),
dt
d
(|u̇2 (t)|p−2 u̇2 (t)) = ∇u2 F (t, u1 (t), u2 (t)), a.e. t ∈ [0, T ],
dt
u1 (0) − u1 (T ) = u̇1 (0) − u̇1 (T ) = 0,
(1.1)
u2 (0) − u2 (T ) = u̇2 (0) − u̇2 (T ) = 0,
where 1 < p, q < ∞, T > 0 and | · | denotes the Euclidean norm in RN . F :
[0, T ] × RN × RN → R satisfies the following assumption
(A1)
– F is measurable in t for each (x1 , x2 ) ∈ RN × RN ;
– F is continuously differentiable in (x1 , x2 ) for a.e. t ∈ [0, T ];
– there exist a1 , a2 ∈ C(R+ , R+ ) and b ∈ L1 (0, T ; R+ ) such that
|F (t, x1 , x2 )|, |∇x1 F (t, x1 , x2 )|, |∇x2 F (t, x1 , x2 )| ≤ a1 (|x1 |) + a2 (|x2 |) b(t)
for all (x1 , x2 ) ∈ RN × RN and a.e. t ∈ [0, T ].
We denote by WT1,p the Sobolev space of functions u ∈ Lp (0, T ; RN ) having a
weak derivative u̇ ∈ Lp (0, T ; RN ). The norm in WT1,p is defined by
Z T
1/p
kukW 1,p =
|u(t)|p + |u̇(t)|p dt
.
T
0
The corresponding functional ϕ : W → R given is
Z
Z
Z T
1 T
1 T
ϕ(u1 , u2 ) =
|u̇1 (t)|q dt +
|u̇2 (t)|p dt +
F (t, u1 (t), u2 (t))dt,
q 0
p 0
0
2000 Mathematics Subject Classification. 34C25, 35B38, 47J30.
Key words and phrases. Periodic solution; differential systems; (q, p)-Laplacian;
least action principle; saddle point theorem.
c
2014
Texas State University - San Marcos.
Submitted March 18, 2012. Published March 5, 2014.
1
2
C. LI, Z.-Q. OU, C.-L. TANG
EJDE-2014/64
where W = WT1,q × WT1,p is a reflexive Banach space and endowed with the norm
k(u1 , u2 )kW = ku1 kW 1,q + ku2 kW 1,p .
T
T
It follows from assumption (A1) that the functional ϕ is continuously differentiable
and weakly lower semicontinuous on W . Moreover,
hϕ0 (u1 , u2 ), (v1 , v2 )i
Z T
=
[(|u̇1 (t)|q−2 u̇1 (t), v̇1 (t)) + (∇u1 F (t, u1 (t), u2 (t)), v1 (t))]dt
0
Z
T
[(|u̇2 (t)|p−2 u̇2 (t), v̇2 (t)) + (∇u2 F (t, u1 (t), u2 (t)), v2 (t))]dt
+
0
for all (u1 , u2 ), (v1 , v2 ) ∈ W .
For each u ∈ WT1,p can be written as u(t) = ū + ũ(t) with
Z
Z T
1 T
ū =
u(t)dt,
ũ(t)dt = 0.
T 0
0
We have the Sobolev’s inequality (for a proof and details see [4])
kũk∞ ≤ C1 ku̇kp , kṽk∞ ≤ C1 kv̇kq
for each u ∈ WT1,p , v ∈ WT1,q ,
and Wirtinger’s inequality (see [4])
kũkp ≤ C2 ku̇kp ,
kṽkq ≤ C2 kv̇kq
where
kukp =
T
Z
1/p
|u(t)|p dt
,
for each u ∈ WT1,p , v ∈ WT1,q ,
kuk∞ = max |u(t)|.
t∈[0,T ]
0
A function G : R
N
→ R is called to be (λ, µ)-subconvex if
G(λ(x + y)) ≤ µ(G(x) + G(y))
for some λ, µ > 0 and all x, y ∈ RN (see [19]).
The existence of periodic solutions for the second-order Hamiltonian system
ü(t) = ∇F (t, u),
a.e. t ∈ [0, T ],
u(0) − u(T ) = u̇(0) − u̇(T ) = 0,
(1.2)
has been extensively investigated in papers, such as [1, 2, 3, 4, 12, 13, 14, 15, 16,
18, 19] and the reference therein. Many solvability conditions are given, such as
the coercive condition (see [1]), the periodicity condition (see [18]), the convexity
condition (see [3]), the boundedness condition (see [4]), the subadditive condition
(see [12]), and the sublinear condition (see [14]). When the gradient ∇F (t, x) is
bounded; that is, there exists g ∈ L1 (0, T ; R+ ) such that
|∇F (t, x)| ≤ g(t)
N
for all x ∈ R and a.e. t ∈ [0, T ]. Mawhin and Willem [4] obtained the existence
of solutions for problem (1.2) under the condition
Z T
F (t, x)dt → +∞(or −∞), as |x| → ∞.
0
Tang [14] proved the existence of solutions for problem (1.2) when
|∇F (t, x)| ≤ f (t)|x|α + g(t)
(1.3)
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PERIODIC SOLUTIONS
3
for all x ∈ RN and a.e. t ∈ [0, T ], where f, g ∈ L1 (0, T ; R+ ) and α ∈ [0, 1). And, F
satisfies the condition
Z T
|x|−2α
F (t, x)dt → +∞ (or − ∞), as |x| → ∞.
0
Tang and Meng [16] studied the existence of solutions for problem (1.2) under the
conditions (1.3) or
|∇F (t, x)| ≤ f (t)|x| + g(t)
N
for all x ∈ R and a.e. t ∈ [0, T ], where f, g ∈ L1 (0, T ; R+ ). The results in [16]
complement those in [14, Theorem 1 and 2].
Recently, Paşca and Tang [9] established the existence results for problem (1.1)
which extend [14, Theorems 1 and 2]. By applying the least action principle, Paşca
[7] proved some existence theorems for problem (1.1) which generalize the corresponding Theorems of [19]. Using the Saddle Point Theorem, Paşca and Tang [10]
obtained some existence results for problem (1.1). Paşca [8] studied the existence of
periodic solutions for nonautonomous second-order differential inclusions systems
with (q, p)-Laplacian which extend the results of [5, 6, 9, 14].
In this paper, motivated by references [7, 9, 14, 16], we consider the existence
of periodic solutions for problem (1.1) by using the least action principle and the
Saddle Point Theorem. Our main results are the following theorems.
Theorem 1.1. Suppose that F = F1 + F2 , where F1 and F2 satisfy assumption
(A1) and the following conditions:
(H0) F1 (t, ·, ·) is (λ, µ)-subconvex with λ > 1/2 and 1/2 < µ < 2r−1 λr for a.e.t ∈
[0, T ], where r = min{p, q};
(H1) there exist fi , gi , hi ∈ L1 (0, T ; R+ ), i = 1, 2, α1 ∈ [0, q − 1), α2 ∈ [0, p − 1),
β1 ∈ [0, p/q 0 ), β2 ∈ [0, q/p0 ), q 0 = q/(q − 1) and p0 = p/(p − 1)
|∇x1 F2 (t, x1 , x2 )| ≤ f1 (t)|x1 |α1 + g1 (t)|x2 |β1 + h1 (t)
|∇x2 F2 (t, x1 , x2 )| ≤ f2 (t)|x2 |α2 + g2 (t)|x1 |β2 + h2 (t)
for all (x1 , x2 ) ∈ RN × RN and a.e. t ∈ [0, T ];
(H2)
Z T
1 Z T
1
F
(t,
λx
,
λx
)dt
+
F
(t,
x
,
x
)dt
> 2K,
1
1
2
2
1
2
|x|→∞ |x1 |γ1 + |x2 |γ2 µ 0
0
p
where |x| = |x1 |2 + |x2 |2 , γ1 = max{q 0 α1 , β2 p0 }, γ2 = max{p0 α2 , β1 q 0 }
and
n 4q0 /q (2q−1 kf k 1 C )q0 4p0 /p (2p−1 kf k 1 C )p0
1 L
1
2 L
1
K = max
,
,
q0
p0
0
0
0
0
4q /q (2β1 kg1 kL1 C1 )q 4p /p (2β2 kg2 kL1 C1 )p o
,
.
q0
p0
lim
Then problem (1.1) has at least one solution in W .
Corollary 1.2. Suppose that F = F1 + F2 , satisfies (H0), (H1) and
(H2’)
Z T
1 Z T
1
F
(t,
λx
,
λx
)dt
+
F
(t,
x
,
x
)dt
→∞
1
1
2
2
1
2
|x1 |γ1 + |x2 |γ2 µ 0
0
4
C. LI, Z.-Q. OU, C.-L. TANG
EJDE-2014/64
as |x| → +∞. Then problem (1.1) has at least one solution in W .
Remark 1.3. Corollary 1.2 generalizes Theorem 1 of [7]. In fact, it follows from
Corollary 1.2 by letting β1 = β2 = 0. There are functions satisfying the assumptions
of our Corollary 1.2 and not satisfying the assumptions in [7, 9]. For example, Let
α1 = α2 = 15/4, β1 = β2 = 11/4, p = q = 5, p0 = q 0 = 5/4, and
F1 (t, x1 ) = 5 + sin(|x1 |6 + |x2 |6 ),
2T
− t (|x1 |19/4 + |x2 |19/4 + |x1 |5/4 |x2 |5/4 ).
F2 (t, x1 , x2 ) =
3
Theorem 1.4. Suppose that F (t, x1 , x2 ) satisfies (H1) and
(H3)
Z T
1
F (t, x1 , x2 )dt < −(2q 0 + 2p0 + 1)2K.
lim
|x|→∞ |x1 |γ1 + |x2 |γ2 0
Then problem (1.1) has at least one solution in W .
Corollary 1.5. Suppose that F (t, x1 , x2 ) satisfies (H1) and
(H3’)
Z T
1
F (t, x1 , x2 )dt → −∞
|x1 |γ1 + |x2 |γ2 0
as |x| → ∞. Then problem (1.1) has at least one solution in W .
Remark 1.6. Corollary 1.5 extends [9, Theorem 2]. In fact, it follows from Corollary 1.5 by letting β1 = β2 = 0. There are functions satisfying the assumptions
of our Corollary 1.5 and not satisfying the assumptions in [9]. For example, Let
α1 = α2 = 15/4, β1 = β2 = 11/4, p = q = 5, p0 = q 0 = 5/4, and
T
F (t, x1 , x2 ) =
− t (|x1 |19/4 + |x2 |19/4 + |x1 |5/4 |x2 |5/4 ).
3
2. Proofs of main results
Tian and Ge [17] proved the following result which generalizes a very well known
result proved by Jean Mawhin and Michel Willem [4, Theorem 1.4].
Lemma 2.1 ([17]). Let L : [0, T ] × RN × RN × RN × RN → R, (t, x1 , x2 , y1 , y2 ) →
L(t, x1 , x2 , y1 , y2 ) be measurable in t for each (x1 , x2 , y1 , y2 ), and continuously differentiable in (x1 , x2 , y1 , y2 ) for a.e. t ∈ [0, T ]. If there exist ai ∈ C(R+ , R+ ),
i = 1, 2, b ∈ L1 (0, T ; R+ ), and c1 ∈ Lp (0, T ; R+ ), c2 ∈ Lq (0, T ; R+ ), 1 < p, q < ∞,
such that for a.e. t ∈ [0, T ] and every (x1 , x2 , y1 , y2 ) ∈ RN × RN × RN × RN , one
has
|L(t, x1 , x2 , y1 , y2 )| ≤ (a1 (|x1 |) + a2 (|x2 |))(b(t) + |y1 |q + |y2 |p ),
|Dx1 L(t, x1 , x2 , y1 , y2 )| ≤ (a1 (|x1 |) + a2 (|x2 |))(b(t) + |y2 |p ),
|Dx2 L(t, x1 , x2 , y1 , y2 )| ≤ (a1 (|x1 |) + a2 (|x2 |))(b(t) + |y1 |q ),
|Dy1 L(t, x1 , x2 , y1 , y2 )| ≤ (a1 (|x1 |) + a2 (|x2 |))(c1 (t) + |y1 |q−1 ),
|Dy2 L(t, x1 , x2 , y1 , y2 )| ≤ (a1 (|x1 |) + a2 (|x2 |))(c2 (t) + |y2 |p−1 ),
then the function ϕ : WT1,q × WT1,p → R defined by
Z T
ϕ(u1 , u2 ) =
L(t, u1 (t), u2 (t), u̇1 (t), u̇2 (t))dt
0
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PERIODIC SOLUTIONS
5
is continuously differentiable on WT1,q × WT1,p and
Z T
hϕ0 (u1 , u2 ), (v1 , v2 )i =
((Dx1 L(t, u1 (t), u2 (t), u̇1 (t), u̇2 (t)), v1 (t))
0
+ (Dy1 L(t, u1 (t), u2 (t), u̇1 (t), u̇2 (t)), v̇1 (t))
+ (Dx2 L(t, u1 (t), u2 (t), u̇1 (t), u̇2 (t)), v2 (t))
+ (Dy2 L(t, u1 (t), u2 (t), u̇1 (t), u̇2 (t)), v̇2 (t)))dt.
Corollary 2.2. Let L : [0, T ] × RN × RN × RN × RN → R be defined by
L(t, x1 , x2 , y1 , y2 ) =
1
1
|y1 |q + |y2 |p + F (t, x1 , x2 )
q
p
where F : [0, T ] × RN × RN → R satisfies condition (A1). If (u1 , u2 ) ∈ WT1,q × WT1,p
is a solution of the corresponding Euler equation ϕ0 (u1 , u2 ) = 0, then (u1 , u2 ) is a
solution of problem (1.1).
Remark 2.3. The function ϕ is weakly lower semi-continuous (w.l.s.c.) on W as
the sum of two convex continuous functions and of a weakly continuous one.
We will prove Theorem 1.1 by using the least action principle [4, Theorem 1.1],
and Theorem 1.4 by using the saddle point theorem [11, Theorem 4.6].
Proof of Theorem 1.1. Let β = log2λ (2µ). Then 0 < β < r. For |x| > 1, there
exists a positive integer n such that
n − 1 < log2λ |x| ≤ n.
So, we have |x|β > (2λ)(n−1)β = (2µ)n−1 and |x| ≤ (2λ)n . Then, by (A1) and (H0),
one has
F1 (t, x1 , x2 ) ≤ 2µF1 (t, x1 /(2λ), x2 /(2λ)) ≤ . . .
≤ (2µ)n F1 (t, x1 /(2λ), x2 /(2λ))
≤ 2µ|x|β (a10 + a20 )b(t)
for a.e. t ∈ [0, T ] and all |x| > 1, where ai0 = max0≤s≤1 ai (s), i = 1, 2. Therefore,
F1 (t, x1 , x2 ) ≤ (2β/2+1 µ(|x1 |β + |x2 |β ) + 1)(a10 + a20 )b(t)
N
(2.1)
N
for a.e. t ∈ [0, T ] and all (x1 , x2 ) ∈ R × R .
It follows from (H1), Sobolev’s inequality and Young’s inequality that
Z T
(F2 (t, u1 (t), ū2 ) − F2 (t, ū1 , ū2 ))dt
0
Z T Z 1
=
(∇x1 F2 (t, ū1 + sũ1 (t), ū2 ), ũ1 (t))dsdt
0
0
Z TZ 1
Z TZ 1
≤
f1 (t)|ū1 + sũ1 (t)|α1 |ũ1 (t)|dsdt +
g1 (t)|ū2 |β1 |ũ1 (t)|dsdt
0
Z
0
T Z 1
+
≤2
0
q−1
0
0
h1 (t)|ũ1 (t)|dsdt
0
β1
1
(|ū1 |α1 + kũ1 kα
∞ )kũ1 k∞ kf1 kL1 + |ū2 | kũ1 k∞ kg1 kL1 + kũ1 k∞ kh1 kL1
1 +1
≤ 2q−1 C1α1 +1 kf1 kL1 ku̇1 kα
+ 2q−1 kf1 kL1 C1 |ū1 |α1 ku̇1 kq
q
6
C. LI, Z.-Q. OU, C.-L. TANG
EJDE-2014/64
+ C1 kg1 kL1 |ū2 |β1 ku̇1 kq + C1 kh1 kL1 ku̇1 kq
0
0
≤ 2q−1 C1α1 +1 kf1 kL1 ku̇1 kqα1 +1 +
0
1
4q /q (2q−1 kf1 kL1 C1 )q
|ū1 |q α1
ku̇1 kqq +
0
4q
q
0
0
0
1
4q /q (kg1 kL1 C1 )q
|ū2 |q β1 + C1 kh1 kL1 ku̇1 kq
ku̇1 kqq +
0
4q
q
+
0
0
0
1
4q /q (2q−1 kf1 kL1 C1 )q
|ū1 |q α1
ku̇1 kqq + 2q−1 C1α1 +1 kf1 kL1 ku̇1 kqα1 +1 +
2q
q0
=
0
0
0
4q /q (kg1 kL1 C1 )q
+
|ū2 |q β1 + C1 kh1 kL1 ku̇1 kq
q0
and
Z
T
0
(F2 (t, u1 (t), u2 (t)) − F2 (t, u1 (t), ū2 ))dt
Z T Z 1
=
(∇x2 F2 (t, u1 (t), ū2 + sũ2 (t)), ũ2 (t))dsdt
0
0
Z TZ 1
Z TZ 1
α2
≤
f2 (t)|ū2 + sũ2 (t)| |ũ2 (t)|dsdt +
g2 (t)|u1 |β2 |ũ2 (t)|dsdt
0
0
T Z 1
Z
+
≤2
0
0
h2 (t)|ũ2 (t)|dsdt
0
p−1
0
β2
β2
2
(|ū2 |α2 + kũ2 kα
+ kũ1 kβ∞2 )kũ2 k∞ kg2 kL1
∞ )kũ2 k∞ kf2 kL1 + 2 (|ū1 |
+ kũ2 k∞ kh2 kL1
2 +1
≤ 2p−1 C1α2 +1 kf2 kL1 ku̇2 kα
+ 2p−1 C1 kf2 kL1 |ū2 |α2 ku̇2 kp + C1 kh2 kL1 ku̇2 kp
p
+ 2β2 kg2 kL1 C1β2 +1 ku̇2 kp ku̇1 kβq 2 + 2β2 kg2 kL1 C1 ku̇2 kp |ū1 |β2
0
≤ 2p−1 C1α2 +1 kf2 kL1 ku̇2 kpα2 +1 +
0
+
0
0
1
4p /p (2p−1 kf2 kL1 C1 )p
ku̇2 kpp +
|ū2 |p α2
0
4p
p
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
1
1
ku̇2 kpp +
ku̇1 kβq 2 p + ku̇2 kpp
4p
p0
4p
0
0
0
4p /p (2β2 kg2 kL1 C1 )p
+
|ū1 |β2 p + C1 kh2 kL1 ku̇2 kp
p0
0
=
0
0
3
4p /p (2p−1 kf2 kL1 C1 )p
2 +1
|ū2 |p α2
ku̇2 kpp + 2p−1 C1α2 +1 kf2 kL1 ku̇2 kα
+
p
0
4p
p
0
0
0
0
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
4p /p (2β2 kg2 kL1 C1 )p
ku̇1 kqβ2 p +
|ū1 |β2 p
0
0
p
p
+ C1 kh2 kL1 ku̇2 kp
+
for all (u1 , u2 ) ∈ W . So, one has
Z
0
T
(F2 (t, u1 (t), u2 (t)) − F2 (t, ū1 , ū2 ))dt
Z
≤
0
T
(F2 (t, u1 (t), ū2 ) − F2 (t, ū1 , ū2 ))dt
EJDE-2014/64
PERIODIC SOLUTIONS
T
Z
+
0
7
(F2 (t, u1 (t), u2 (t)) − F2 (t, u1 (t), ū2 ))dt
1
3
≤
ku̇1 kqq + ku̇2 kpp + C1 kh1 kL1 ku̇1 kq + C1 kh2 kL1 ku̇2 kp
2q
4p
1 +1
2 +1
+ 2q−1 C1α1 +1 kf1 kL1 ku̇1 kα
+ 2p−1 C1α2 +1 kf2 kL1 ku̇2 kα
q
p
0
0
0
0
0
0
0
0
0
4q /q (kg1 kL1 C1 )q
4q /q (2q−1 kf1 kL1 C1 )q
|ū1 |q α1 +
|ū2 |q β1
+
0
q
q0
0
0
4p /p (2p−1 kf2 kL1 C1 )p
4p /p (2β2 kg2 kL1 C1 )p
p0 α2
+
+
|ū
|
|ū1 |p β2
2
p0
p0
0
+
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
ku̇1 kβq 2 p
0
p
for all (u1 , u2 ) ∈ W . Hence, we obtain from (H0), (2.1) and the above expression
that
1
ϕ(u1 , u2 ) =
q
Z
T
0
Z
1
|u̇1 (t)| dt +
p
q
T
Z
T
Z
p
|u̇2 (t)| dt +
F1 (t, u1 (t), u2 (t))dt
0
0
T
Z
F2 (t, ū1 , ū2 )dt
0
≥
T
(F2 (t, u1 (t), u2 (t)) − F2 (t, ū1 , ū2 ))dt +
+
0
1
1
ku̇1 kqq + ku̇2 kpp − C1 kh1 kL1 ku̇1 kq − C1 kh2 kL1 ku̇2 kp
2q
4p
− 2q−1 C1α1 +1 kf1 kL1 ku̇1 kqα1 +1 − 2p−1 C1α2 +1 kf2 kL1 ku̇2 kpα2 +1
0
−
0
0
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
4q /q (2q−1 kf1 kL1 C1 )q
p0 β2
k
u̇
k
−
|ū1 |q α1
1
q
0
0
p
q
0
0
0
0
0
4q /q (kg1 kL1 C1 )q
4p /p (2p−1 kf2 kL1 C1 )p
q 0 β1
|ū
|
|ū2 |p α2
−
2
0
0
q
p
Z T
0
p
p0 /p β2
0
4
(2 kg2 kL1 C1 )
−
|ū1 |β2 p +
F2 (t, ū1 , ū2 )dt
p0
0
Z
Z T
1 T
F1 (t, λū1 , λū2 )dt −
F1 (t, −ũ1 , −ũ2 )dt
+
µ 0
0
−
0
≥
0
0
1
4p /p (2β2 kg2 kL1 C1β2 +1 )p
1
ku̇1 kqq + ku̇2 kpp −
ku̇1 kqp β2
0
2q
4p
p
− 2q−1 C1α1 +1 kf1 kL1 ku̇1 kqα1 +1 − 2p−1 C1α2 +1 kf2 kL1 ku̇2 kpα2 +1
− C1 kh1 kL1 ku̇1 kq − C1 kh2 kL1 ku̇2 kp
− (2β/2+1 C1β µ(ku̇1 kβq + ku̇2 kβp ) + 1)(a10 + a20 )
Z
T
b(t)dt
0
+ (|ū1 |γ1 + |ū2 |γ2 )(
Z
|ū1
|γ1
1
1
(
+ |ū2 |γ2 µ
T
F2 (t, ū1 , ū2 )dt) − 2K) − K0
+
0
Z
T
F1 (t, λū1 , λū2 )dt
0
8
C. LI, Z.-Q. OU, C.-L. TANG
EJDE-2014/64
for all (u1 , u2 ) ∈ W and some positive constants K and K0 . It follows that
ϕ(u1 , u2 ) → +∞ as k(u1 , u2 )kW → ∞ due to (H2). By [4, Theorem 1.1] and
Corollary 2.2, The proof is complete.
Proof of Theorem 1.4. Firstly, we prove that ϕ satisfies the (P S) condition. Suppose that {(u1n , u2n )} is a (P S) sequence for ϕ, that is, ϕ0 (u1n , u2n ) → 0 as n → ∞
and {ϕ(u1n , u2n )} is bounded. In a way similar to the proof of Theorem 1.1, we
have
Z T
(∇x1 F (t, u1n (t), u2n (t)), ũ1n (t))dt
0
0
≤ 2q−1 C1α1 +1 kf1 kL1 ku̇1n kqα1 +1 +
0
0
0
0
3
4q /q (2q−1 kf1 kL1 C1 )q
|ū1n |q α1
ku̇1n kqq +
0
4q
q
0
0
0
0
4q /q (2β1 kg1 kL1 C1β1 +1 )q
4q /q (2β1 kg1 kL1 C1 )q
ku̇2n kpβ1 q +
|ū2n |q β1
0
q
q0
+ C1 kh1 kL1 ku̇1n kq
+
and
Z
T
0
(∇x2 F (t, u1n (t), u2n (t)), ũ2n (t))dt
0
p−1
≤2
C1α2 +1 kf2 kL1 ku̇2n kpα2 +1
0
0
0
0
3
4p /p (2p−1 kf2 kL1 C1 )p
+ ku̇2n kpp +
|ū2n |p α2
4p
p0
0
0
0
4p /p (2β2 kg2 kL1 C1 )p
4p /p (2β2 kg2 kL1 C1β2 +1 )p
β2 p0
k
u̇
k
+
|ū1n |p β2
1n
q
0
0
p
p
+ C1 kh2 kL1 ku̇2n kp
+
for all n. Hence, one has
k(ũ1n , ũ2n )kW
≥ hϕ0 (u1n , u2n ), (ũ1n , ũ2n )i
Z T
=
((∇x1 F (t, u1n (t), u2n (t)), ũ1n (t)) + (|u̇1n (t)|q−2 u̇1n (t), u̇1n (t))
0
+ (∇x2 F (t, u1n (t), u2n (t)), ũ2n (t)) + (|u̇2n (t)|p−2 u̇2n (t), u̇2n (t)))dt
4p − 3
4q − 3
1 +1
ku̇1n kqq +
ku̇2n kpp − 2q−1 C1α1 +1 kf1 kL1 ku̇1n kα
≥
q
4q
4p
0
0
0
0
−
0
4q /q (2β1 kg1 kL1 C1β1 +1 )q
4q /q (2q−1 kf1 kL1 C1 )q
β1 q 0
k
u̇
k
−
|ū1n |q α1
2n
p
0
0
q
q
−
0
4q /q (2β1 kg1 kL1 C1 )q
2 +1
|ū2n |q β1 − 2p−1 C1α2 +1 kf2 kL1 ku̇2n kα
p
0
q
0
0
0
0
0
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
4p /p (2p−1 kf2 kL1 C1 )p
β 2 p0
−
k
u̇
k
−
|ū2n |p α2
1n
q
p0
p0
0
0
0
4p /p (2β2 kg2 kL1 C1 )p
−
|ū1n |p β2 − C1 kh2 kL1 ku̇2n kp − C1 kh1 kL1 ku̇1n kq
p0
(2.2)
EJDE-2014/64
PERIODIC SOLUTIONS
9
for large n. It follows from Wirtinger’s inequality that
k(ũ1n , ũ2n )kW = kũ1n kW 1,q + kũ2n kW 1,p
T
≤
≤
T
(1 + C2q )1/q ku̇1n kq + (1 + C2p )1/p ku̇2n kp
max (1 + C2q )1/q , (1 + C2p )1/p ku̇1n kq +
(2.3)
ku̇2n kp
for all n. So, it follows from (2.2) and (2.3) that
0
0
0
0
K(|ū1n |p β2 + |ū2n |p α2 + |ū1n |q α1 + |ū2n |q β1 )
0
≥
0
0
0
0
0
4q /q (2q−1 kf1 kL1 C1 )q
4q /q (2β1 kg1 kL1 C1 )q
|ū1n |q α1 +
|ū2n |q β1
0
0
q
q
0
0
0
0
0
0
4p /p (2p−1 kf2 kL1 C1 )p
4p /p (2β2 kg2 kL1 C1 )p
|ū2n |p α2 +
|ū1n |p β2
0
p
p0
4q − 3
4p − 3
≥
ku̇1n kqq +
ku̇2n kpp
4q
4p
+
− 2q−1 C1α1 +1 kf1 kL1 ku̇1n kqα1 +1 −
0
0
0
0
0
0
0
0
0
4q /q (2β1 kg1 kL1 C1β1 +1 )q
ku̇2n kpβ1 q
0
q
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
ku̇1n kβq 2 p
−2
−
p0
− C1 kh2 kL1 ku̇2n kp − C1 kh1 kL1 ku̇1n kq − k(ũ1n , ũ2n )kW
4p − 3
4q − 3
ku̇1n kqq +
ku̇2n kpp − C1 kh2 kL1 ku̇2n kp − C1 kh1 kL1 ku̇1n kq
≥
4q
4p
p−1
C1α2 +1 kf2 kL1 ku̇2n kpα2 +1
− 2q−1 C1α1 +1 kf1 kL1 ku̇1n kqα1 +1 −
0
4q /q (2β1 kg1 kL1 C1β1 +1 )q
ku̇2n kpβ1 q
0
q
− 2p−1 C1α2 +1 kf2 kL1 ku̇2n kpα2 +1 −
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
ku̇1n kβq 2 p
p0
− (1 + C2p )1/p ku̇2n kp − (1 + C2q )1/q ku̇1n kq
p−1
q−1
ku̇1n kqq +
ku̇2n kpp − K1
≥
q
p
1
1
= 0 ku̇1n kqq + 0 ku̇2n kpp − K1
q
p
for large n and some positive constant K1 . Hence, by the above expression, we
obtain
1 1
2K(|ū1n |γ1 + |ū2n |γ2 ) ≥ min 0 , 0 ku̇1n kqq + ku̇2n kpp − K2
(2.4)
q p
for large n and some positive constant K2 . By the proof of Theorem 1.1, we have
Z T
(F (t, u1n (t), u2n (t)) − F (t, ū1n , ū2n ))dt
0
≤
3
1
ku̇1n kqq + ku̇2n kpp + C1 kh1 kL1 ku̇1n kq + C1 kh2 kL1 ku̇2n kp
2q
4p
1 +1
2 +1
+ 2q−1 C1α1 +1 kf1 kL1 ku̇1n kα
+ 2p−1 C1α2 +1 kf2 kL1 ku̇2n kα
q
p
0
0
0
0
0
4q /q (2q−1 kf1 kL1 C1 )q
4q /q (kg1 kL1 C1 )q
q 0 α1
+
|ū
|
+
|ū2n |q β1
1n
q0
q0
10
C. LI, Z.-Q. OU, C.-L. TANG
0
EJDE-2014/64
0
0
0
0
4p /p (2β2 kg2 kL1 C1 )p
4p /p (2p−1 kf2 kL1 C1 )p
p0 α2
+
+
|ū
|
|ū1n |p β2
2n
p0
p0
0
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
+
ku̇1n kβq 2 p
0
p
for all n. It follows from the boundedness of {ϕ(u1n , u2n )}, (2.4) and the above
inequality that
K3 ≤ ϕ(u1n , u2n )
Z
Z
1 T
1 T
|u̇1n (t)|q dt +
|u̇2n (t)|p dt
=
q 0
p 0
Z T
Z
+
F (t, u1n (t), u2n (t)) − F (t, ū1n , ū2n ) dt +
0
≤
T
F (t, ū1n , ū2n )dt
0
3
7
ku̇1n kqq + ku̇2n kpp + C1 kh1 kL1 ku̇1n kq + C1 kh2 kL1 ku̇2n kp
2q
4p
1 +1
2 +1
+ 2q−1 C1α1 +1 kf1 kL1 ku̇1n kα
+ 2p−1 C1α2 +1 kf2 kL1 ku̇2n kα
q
p
0
0
0
0
0
0
0
0
4q /q (kg1 kL1 C1 )q
4q /q (2q−1 kf1 kL1 C1 )q
q 0 α1
|ū
|
+
|ū2n |q β1
+
1n
q0
q0
0
0
4p /p (2p−1 kf2 kL1 C1 )p
4p /p (2β2 kg2 kL1 C1 )p
p0 α2
+
|ū
|
+
|ū1n |p β2
2n
p0
p0
Z T
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
β2 p0
k
u̇
k
+
F (t, ū1n , ū2n )dt
+
1n
q
p0
0
Z T
≤ 2(ku̇1n kqq + ku̇2n kpp ) + 2K(|ū1n |γ1 + |ū2n |γ2 ) +
F (t, ū1n , ū2n )dt + K4
0
0
0
≤ (2 max{q , p } + 1)2K(|ū1n |
γ1
Z
γ2
+ |ū2n | ) +
T
F (t, ū1n , ū2n )dt + K5
0
0
0
≤ (2q + 2p + 1)2K(|ū1n |
γ1
T
Z
γ2
+ |ū2n | ) +
F (t, ū1n , ū2n )dt + K5
0
≤ (|ū1n |γ1 + |ū2n |γ2 )(
|ū1n |γ1
0
0
+ (2q + 2p + 1)2K) + K5
1
+ |ū2n |γ2
Z
T
F (t, ū1n , ū2n )dt
0
for large n and some real constants K3 , K4 and K5 . The above inequality and (H3)
imply that (|ū1n |γ1 + |ū2n |γ2 ) is bounded. Hence, (u1n , u2n ) is bounded by (2.3)
and (2.4). By the compactness of the embedding WT1,p ( or WT1,q ) ⊂ C(0, T ; RN ),
the sequence {u1n } (or {u2n }) has a subsequence, still denoted by {u1n } (or {u2n }),
such that
u1n (or u2n ) * u1 (or u2 )
u1n (or u2n ) → u1 (or u2 )
weakly in WT1,p (or in WT1,q ),
N
strongly in C(0, T ; R ).
(2.5)
(2.6)
EJDE-2014/64
PERIODIC SOLUTIONS
11
Note that
hϕ0 (u1n , u2n ), (u1 − u1n , 0)i
Z T
|u̇1n (t)|p−2 (u̇1n (t), u̇1 − u̇1n (t))dt
=
(2.7)
0
T
Z
−
(∇x1 F (t, u1n (t), u2n (t)), u1 (t) − u1n (t))dt → 0
0
as n → ∞. From (2.6), {u1n } is bounded in C(0, T ; RN ). Then we have
Z T
(∇x1 F (t, u1n (t), u2n (t)), u1 (t) − u1n (t))dt
0
T
Z
≤
|∇x1 F (t, u1n (t), u2n (t))| · |u1 (t) − u1n (t)|dt
0
T
Z
≤ K6
b(t)|u1 (t) − u1n (t)|dt
0
≤ K6 kbkL1 ku1 − u1n k∞
for some positive constant K6 , which combines with (2.6) implies that
Z T
(∇x1 F (t, u1n (t), u2n (t)), u1 (t) − u1n (t))dt → 0 as n → ∞.
0
Hence, by (2.7), one has
Z T
|u̇1n (t)|p−2 (u̇1n (t), u̇1 (t) − u̇1n (t))dt → 0
as n → ∞ .
0
Moreover, from (2.6) we obtain
Z T
|u1n (t)|p−2 (u1n (t), u1 (t) − u1n (t))dt → 0
as n → ∞ .
0
Setting
ψ(u1 , u2 ) =
1
p
Z
T
(|u1 (t)|p + |u̇1 (t)|p )dt +
0
1
q
Z
T
(|u2 (t)|q + |u̇2 (t)|q )dt,
0
one obtains
T
Z
hψ 0 (u1n , u2n ), (u1 − u1n , 0)i =
|u1n (t)|p−2 (u1n (t), u1 (t) − u1n (t))dt
0
Z
+
T
|u̇1n (t)|p−2 (u̇1n (t), u̇1 (t) − u̇1n (t))dt
0
and
hψ 0 (u1n , u2n ), (u1 − u1n , 0)i → 0 as n → ∞ .
(2.8)
By the Hölder’s inequality, we have
0 ≤ (ku1n kp−1 − ku1 kp−1 )(ku1n k − ku1 k) ≤ hψ 0 (u1n , u2n ) − ψ 0 (u1 , u2 ), (u1 − u1n , 0)i,
which together with (2.8) yields ku1n k → ku1 k. It follows that u1n → u1 strongly
in WT1,p by the uniform convexity of WT1,p . Similarly, we have u2n → u2 strongly in
WT1,q . Hence, the (P S) condition is satisfied.
12
C. LI, Z.-Q. OU, C.-L. TANG
EJDE-2014/64
f=W
f 1,q × W
f 1,p be the subspace of W given by
Let W
T
T
f = {(u1 , u2 ) ∈ W | (ū1 , ū2 ) = (0, 0)}.
W
Then
ϕ(u1 , u2 ) → +∞
(2.9)
f . In fact, by the proof of Theorem 1.1, one has
as k(u1 , u2 )kW → ∞ in W
Z T
Z
1 T
1
|u̇1 (t)|q dt +
|u̇2 (t)|p dt
ϕ(u1 , u2 ) =
q 0
p 0
Z T
Z T
+
(F (t, u1 (t), u2 (t)) − F (t, ū1 , ū2 ))dt +
F (t, ū1 , ū2 )dt
0
0
1
1
≥
ku̇1 kqq + ku̇2 kpp − 2q−1 C1α1 +1 kf1 kL1 ku̇1 kqα1 +1
2q
4p
0
0
0
4p /p (2β2 kg2 kL1 C1β2 +1 )p
ku̇1 kqβ2 p
p0
Z T
− C1 kh1 kL1 ku̇1 kq − C1 kh2 kL1 ku̇2 kp +
F (t, ū1 , ū2 )dt
2 +1
− 2p−1 C1α2 +1 kf2 kL1 ku̇2 kα
−
p
0
f . By Wirtinger’s inequality, the norm
for all (u1 , u2 ) ∈ W
k|(u1 , u2 )k| = k(u̇1 , u̇2 )kLq ×Lp = ku̇1 kq + ku̇2 kp
f . Hence, (2.9) follows from the above inequality.
is an equivalent norm on W
On the other hand, one has
ϕ(x1 , x2 ) → −∞
N
(2.10)
N
as |(x1 , x2 )| → ∞ in R × R , which follows from (H3). Now, Theorem 1.4 is
proved by (2.9), (2.10) and the Saddle Point Theorem (see [11, Theorem 4.6]). Acknowledgements. This research was supported by the National Natural Science Foundation of China (No. 11071198) and the Fundamental Research Funds
for the Central Universities (No. XDJK2010C055).
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Chun Li
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
E-mail address: Lch1999@swu.edu.cn
Zeng-Qi Ou
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
E-mail address: ouzengq707@sina.com
Chun-Lei Tang (corresponding author)
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Tel +86 23 68253135, fax +86 23 68253135
E-mail address: tangcl@swu.edu.cn