# Infinitely Many Solutions for a Class of Fractional Impulsive Coupled Systems with(

Authors: Junping Xie; Xingyong Zhang
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

By using the symmetric mountain pass lemma, we investigate the problem of existence of infinitely many solutions for a class of fractional impulsive coupled systems with(
p
,
q
)-Laplacian, which possesses mixed type nonlinearities, and the nonlinearities do not need to satisfy the well-known Ambrosetti-Rabinowitz condition.


---

## Body

## 1. Introduction and Main Results

In this paper, we are concerned with existence of infinitely many solutions for the following fractional impulsive differential system with(
p
,
q
)-Laplacian:(1)
D
T
α
t
ρ
t
Φ
p
D
t
α
0
c
u
t
=
∇
u
W
t
,
u
t
,
v
t
,
a.e.
t
∈
0
,
T
,
D
T
β
t
γ
t
Φ
q
D
0
c
t
β
v
t
=
∇
v
W
t
,
u
t
,
v
t
,
a.e.
t
∈
0
,
T
,
Δ
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
i
=
∇
I
i
u
t
i
,
i
=
1,2
,
…
,
l
,
Δ
D
T
β
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
j
=
∇
H
j
v
s
j
,
j
=
1,2
,
…
,
m
,
u
0
=
u
T
=
0
,
v
0
=
v
T
=
0
,where T
>
0, α
∈
(
1
/
p
,
1
] with p
>
1, β
∈
(
1
/
q
,
1
] with q
>
1, Φ
s
(
x
)
=
x
s
-
2
x
(
s
>
1 and s
=
p
,
q
), and D
T
α
t (or D
T
β
t) denotes the right Riemann-Liouville fractional derivative of order α (or β), D
t
α
0
c (or D
0
c
t
β) is the left Caputo fractional derivative of order α (or β), ρ
,
γ
∈
L
∞
(
[
0
,
T
]
,
R
+
), I
i
,
H
j
:
R
N
(
N
≥
1
)
→
R are continuously differentiable, 0
=
t
0
<
t
1
<
⋯
<
t
l
+
1
=
T, 0
=
s
0
<
s
1
<
⋯
<
s
m
+
1
=
T, and (2)
Δ
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
i
=
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
i
+
-
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
i
-
,
Δ
D
T
β
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
j
=
D
T
β
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
j
+
-
D
T
β
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
j
-
,where (3)
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
i
+
=
lim
t
→
t
i
+
⁡
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
,
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
i
-
=
lim
t
→
t
i
-
⁡
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
,
D
T
β
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
j
+
=
lim
s
→
s
j
+
⁡
D
T
β
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
,
D
T
α
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
j
-
=
lim
s
→
s
j
-
⁡
D
T
β
-
1
t
γ
Φ
q
D
0
c
t
β
v
s
,
i
=
1,2
,
…
,
l, j
=
1,2
,
…
,
m, and W
:
[
0
,
T
]
×
R
N
×
R
N
→
R satisfies W
(
t
,
x
,
y
)
=
-
K
(
t
,
x
,
y
)
+
F
(
t
,
x
,
y
) and the following assumptions.(
W
0
)
W
(
t
,
x
,
y
) is measurable in t for each (
x
,
y
)
∈
R
N
×
R
N, continuously differentiable in (
x
,
y
)
∈
R
N
×
R
N for a.e. t
∈
[
0
,
T
], and there exist a
1
,
a
2
∈
C
(
R
+
,
R
+
) and b
∈
L
∞
(
[
0
,
T
]
;
R
+
) such that(4)
K
t
,
x
,
y
+
∇
K
t
,
x
,
y
+
F
t
,
x
,
y
+
∇
F
t
,
x
,
y
≤
a
1
x
+
a
2
y
b
t
,
I
i
x
+
∇
I
i
x
≤
a
1
x
,
i
=
1,2
,
…
,
l
,
H
j
y
+
∇
H
j
y
≤
a
2
y
,
j
=
1,2
,
…
,
mfor all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
]
.It is well known that critical point theory is a very important and effective tool to investigate the existence and multiplicity of various solutions for partial differential equations, ordinary differential equations, Hamiltonian systems, difference equations, and so on. Lots of important and interesting results have been established (see, e.g., [1–11] and reference therein). In 2011, Jiao and Zhou [12] first used critical point theory to investigate the existence of solutions for a class of fractional boundary value problems. Since then, critical point theory has also become an effective tool to obtain the existence and multiplicity results of solutions for various fractional differential equations (see, e.g., [13–17] and reference therein). Particularly, in [14], Zhao et al. considered existence of solutions for the following fractional coupled differential system with a parameter:(5)
D
T
α
t
a
t
D
t
α
0
u
t
=
λ
∇
u
F
t
,
u
t
,
v
t
,
t
∈
0
,
T
,
D
T
β
t
b
t
D
t
β
0
v
t
=
λ
∇
v
F
t
,
u
t
,
v
t
,
t
∈
0
,
T
,
u
0
=
u
T
=
0
,
v
0
=
v
T
=
0
,where λ
>
0 is a parameter, a
,
b
∈
L
∞
(
[
0
,
T
]
;
R
+
), and F
:
[
0
,
T
]
×
R
2
→
R. By using a critical point theorem in [18], they obtained system (5) which has at least three weak solutions. In [13], Li et al. investigated a class of fractional coupled differential systems with a parameter:(6)
D
T
α
t
a
t
D
t
α
0
u
t
=
λ
v
t
+
∇
u
F
t
,
u
t
,
v
t
,
t
∈
0
,
T
,
D
T
β
t
b
t
D
t
β
0
v
t
=
λ
u
t
+
∇
v
F
t
,
u
t
,
v
t
,
t
∈
0
,
T
,
u
0
=
u
T
=
0
,
v
0
=
v
T
=
0
,where λ
>
0 is a parameter, a
,
b
∈
L
∞
(
[
0
,
T
]
;
R
+
), and F
:
[
0
,
T
]
×
R
2
→
R. By using the least action principle and symmetric mountain pass theorem, they obtained system (6) which has at least one solution under asymptotically quadratic case and has infinitely many solutions under superquadratic case. For the superquadratic case, they assumed the following well-known Ambrosetti-Rabinowitz (AR) condition.(
A
R
) There are constants μ
>
2
,
M
>
0 such that (7)
0
<
μ
F
t
,
x
,
y
≤
x
F
x
t
,
x
,
y
+
y
F
y
t
,
x
,
yfor all t
∈
[
0
,
T
] and x
+
y
≥
M
.Over the past ten years, integer order impulsive differential equations with different boundary value conditions have been investigated deeply via variational methods (e.g., see [19–26] and reference therein). Recently, Bonanno et al. [27] and Rodrìguez-López and Tersian [28] were concerned with the following second-order impulsive fractional differential equation:(8)
D
T
α
t
D
t
α
0
c
u
t
+
a
t
u
t
=
λ
f
t
,
u
t
,
t
≠
t
i
,
a.e.
t
∈
0
,
T
,
Δ
D
T
α
-
1
t
D
t
α
0
c
u
t
i
=
μ
Q
i
u
t
i
,
i
=
1,2
,
…
,
l
,
u
0
=
u
T
=
0
,where α
∈
(
1
/
2
,
1
], λ
∈
(
0
,
+
∞
) and μ
∈
(
0
,
+
∞
) are two parameters, f
∈
C
(
[
0
,
T
]
×
R
,
R
), Q
i
∈
C
(
R
,
R
), and a
∈
C
(
[
0
,
T
]
,
R
). By using variational methods, they obtained some existence results about one or three solutions of (8). Subsequently, in [29], Nyamoradi and Rodríguez-López investigated the existence and multiplicity of solutions for (8) with λ
=
μ
=
1. They obtained some existence results about one or infinitely many solutions of (8) by using the least action principle, the mountain pass theorem, and the symmetric mountain pass theorem. In [30],Y. Zhao and Y. Zhao investigated the existence and multiplicity of solutions for a class of perturbed fractional differential system with impulsive effects and one parameter, and they obtained system that has at least one or two nontrivial solutions by using two abstract critical point theorems due to [31]. In [32], Heidarkhani et al. investigated the multiplicity of solutions for a class of perturbed fractional differential system with impulsive effects and two parameters, and they obtained that system has infinitely many solutions by using the smooth version of an abstract critical point theorem due to [33]. In [34], Zhao et al. investigated the existence of solution for (8) with λ
=
μ
=
1. By using the Morse theory and local linking argument, they obtained that equation has at least one nontrivial solution.In [35], Zhao and Tang investigated the following impulsive fractional differential equations with p-Laplacian:(9)
D
T
α
t
Φ
p
D
t
α
0
c
u
t
+
u
t
p
-
2
u
t
=
f
t
,
u
t
,
t
∈
0
,
T
,
Δ
D
T
α
-
1
t
Φ
p
D
t
α
0
c
u
t
i
=
Q
i
u
t
i
,
i
=
1,2
,
…
,
l
,
u
0
=
u
T
=
0
,where f
∈
C
(
[
0
,
T
]
×
R
,
R
) and Q
i
∈
C
(
R
,
R
). By using the mountain pass theorem, a critical point theorem in [36], and symmetric mountain pass theorem, they obtained two multiplicity results of solutions for (9). In detail, they obtained the following theorems.Theorem A (see [35]).
Suppose the following conditions hold.
(
A
1
) There exists a constant μ
>
p such that Q
i
(
u
)
u
≤
μ
∫
0
u
Q
i
(
s
)
d
s
<
0 for any u
∈
E
α
,
p
∖
{
0
}, i
=
1,2
,
…
,
l, where E
0
α
,
p is defined in Section 2.
(
A
2
) There exists a constant θ
∈
(
p
,
μ
] such that θ
F
(
t
,
u
)
≤
f
(
t
,
u
)
u for all u
∈
E
0
α
,
p and t
∈
[
0
,
T
], where F
(
t
,
u
)
=
∫
0
u
f
(
t
,
s
)
d
s.
(
A
3
) There exist constants δ
,
γ
>
0 such that F
0
≤
δ and F
∞
≥
γ, where (10)
F
0
=
l
i
m
s
u
p
u
→
0
F
t
,
u
u
θ
,
F
∞
=
l
i
m
i
n
f
u
→
∞
F
t
,
u
u
θ
.
(
A
4
) There exist constants δ
i
>
0 such that ∫
0
u
Q
i
(
s
)
d
s
≥
-
δ
i
u
μ for all u
∈
E
α
,
p
∖
{
0
}, i
=
1,2
,
…
,
l.
Then (9) has at least two weak solutions.Theorem B (see [35]).
Suppose(
A
1
)–(
A
4
) hold and f
(
t
,
u
) and I
i
(
u
) are odd about u, where i
=
1
,
…
,
l. Then (9) has infinitely many weak solutions.Motivated by [12–14, 35], in this paper, we investigate the existence of infinitely many solutions for system (1). Obviously, system (1) is more general and complex than system (5), system (6), and (9). We present some techniques in [35], which were applied to fractional p-Laplacian impulsive differential equation and can also be applied to fractional (
p
,
q
)-Laplacian impulsive differential system, and present some more relaxed superquadratic conditions for nonlinearities than those in [35]. It is remarkable that the fractional coupled (
p
,
q
)-Laplacian differential systems are different from the fractional p-Laplacian differential equations. One stark difference is that the solutions of system (1) are the combination of u
1 and u
2 but not of (5), which causes the fact that system (1) number is possibly more than that of (9) and, hence, it is impossible that system (1) reduces to system (9). Moreover, since, in general, p
≠
q and we present more relaxed superquadratic conditions, it is difficult to prove the boundness of Cerami sequence (see the definition in Section 2 below) and we have to develop some techniques on inequalities. When α
=
β
=
1, system (1) becomes the following integer order (
p
,
q
)-Laplacian impulsive differential system:(11)
d
d
t
ρ
t
Φ
p
u
˙
t
=
∇
u
W
t
,
u
t
,
v
t
,
a.e.
t
∈
0
,
T
,
d
d
t
γ
t
Φ
q
v
˙
t
=
∇
v
W
t
,
u
t
,
v
t
,
a.e.
t
∈
0
,
T
,
Δ
ρ
Φ
p
u
˙
t
t
i
=
∇
I
i
u
t
i
,
i
=
1,2
,
…
,
l
,
Δ
γ
Φ
q
u
˙
t
s
j
=
∇
H
j
v
s
j
,
j
=
1,2
,
…
,
m
,
u
0
=
u
T
=
0
,
v
0
=
v
T
=
0
.There have been some results on existence and multiplicity of solutions for integer order (
p
,
q
)-Laplacian impulsive differential systems with different boundary value conditions (see, e.g., [4, 37, 38]). However, system (11) which has Dirichlet boundary value is different from those systems in [4, 37, 38] and our assumptions on W are more relaxed than the well-known (AR) condition. Hence, our results are still new for integer order (
p
,
q
)-Laplacian impulsive differential systems. Next, we state our results.Theorem 1.
Suppose that(
W
0
) the following conditions hold.
(
A
)
ρ
-
≔
e
s
s
i
n
f
[
0
,
T
]
ρ
(
t
)
>
0, γ
-
≔
e
s
s
i
n
f
[
0
,
T
]
γ
(
t
)
>
0.
(
W
1
)
W
(
t
,
x
,
y
) is even in (
x
,
y
)
∈
R
N
×
R
N and W
(
t
,
0,0
)
≡
0 for a.e. t
∈
[
0
,
T
].
(
K
1
) There exist constants θ
1
∈
[
0
,
p
)
θ
2
∈
[
0
,
q
), d
1
,
d
2
>
0, M
1
>
0 such that (12)
K
t
,
x
,
y
≥
-
d
1
x
θ
1
-
d
2
y
θ
2for a.e. t
∈
[
0
,
T
] and all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≥
M
1.
(
K
2
) There exists a positive constant M
2 such that (13)
∇
x
K
t
,
x
,
y
,
x
+
∇
y
K
t
,
x
,
y
,
y
≤
max
⁡
p
,
q
K
t
,
x
,
yfor a.e. t
∈
[
0
,
T
] and all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≥
M
2.
(
K
3
) There exists positive constants c
1
,
c
2
,
M
3 such that (14)
K
t
,
x
,
y
≥
c
1
x
p
+
c
2
y
qfor a.e. t
∈
[
0
,
T
] and all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≤
M
3.
(
F
1
) There exist L
1
∈
(
0,1
], b
1
∈
0
,
min
⁡
ρ
-
/
p
,
c
1, b
2
∈
0
,
min
⁡
γ
-
/
q
,
c
2 such that (15)
F
t
,
x
,
y
≤
b
1
x
p
+
b
2
y
qfor a.e. t
∈
[
0
,
T
] and all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≤
L
1.
(
F
2
)
(16)
l
i
m
x
+
y
→
+
∞
F
t
,
x
,
y
x
p
+
y
q
=
+
∞
uniformly  for  a.e.
t
∈
0
,
T
.
(
F
3
) There exist constants ξ
,
η
1
,
η
2
,
L
2
∈
(
0
,
+
∞
) and ν
1
,
ν
2
∈
[
0
,
min
⁡
{
p
,
q
}
) such that (17)
max
⁡
p
,
q
+
1
ξ
+
η
1
x
ν
1
+
η
2
y
ν
2
F
t
,
x
,
y
≤
∇
x
F
t
,
x
,
y
,
x
+
∇
y
F
t
,
x
,
y
,
yfor a.e. t
∈
[
0
,
T
] and all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
>
L
2.
(
I
1
) There exist constants d
3
>
0
,
d
4
>
0
,
θ
3
∈
[
0
,
p
)
,
G
1
>
0 such that (18)
d
3
x
p
≥
I
i
x
≥
-
d
4
x
θ
3
,
∇
I
i
x
,
x
≤
max
⁡
p
,
q
I
i
xfor all x
∈
R
N with x
≥
G
1, i
=
1
,
…
,
l.
(
H
1
) There exist constants d
5
>
0
,
d
6
>
0
,
θ
4
∈
[
0
,
q
)
,
G
2
>
0 such that (19)
d
5
y
q
≥
H
j
y
≥
-
d
6
y
θ
4
,
∇
H
j
y
,
y
≤
max
⁡
p
,
q
H
j
yfor all y
∈
R
N with y
≥
G
2, j
=
1
,
…
,
m.
(
I
2
) There exist constants d
7
>
0
,
θ
5
∈
[
0
,
min
⁡
{
p
,
q
}
)
,
G
3
>
0 such that (20)
I
i
x
≥
-
d
7
x
θ
5for all x
∈
R
N with x
≤
G
3, i
=
1
,
…
,
l.
(
H
2
) There exist constants d
8
>
0
,
θ
6
∈
[
0
,
min
⁡
{
p
,
q
}
)
,
G
4
>
0 such that (21)
H
j
y
≥
-
d
8
y
θ
6for all y
∈
R
N with y
≤
G
4, j
=
1
,
…
,
m.
(
I
3
)
I
i
(
0
)
=
0, I
i
(
x
) is even in x
∈
R
N, i
=
1
,
…
,
l.
(
H
3
)
H
j
(
0
)
=
0, H
j
(
y
) is even in y
∈
R
N, j
=
1
,
…
,
m.
Then system (1) has an unbounded sequence of weak solutions.Theorem 2.
Suppose that(
W
0
), (
A
), (
W
1
), (
K
1
)–(
K
3
), (
F
1
)–(
F
3
), (
I
1
), (
H
1
), (
I
3
), (
H
3
), and the following conditions hold.
I
H
2
′ There exist positive constants d
7
′
,
d
8
′ with(22)
d
7
′
l
C
α
,
p
,
∞
min
⁡
p
,
q
+
d
8
′
m
C
β
,
q
,
∞
min
⁡
p
,
q
≤
min
⁡
min
⁡
ρ
-
p
,
c
1
-
b
1
,
min
⁡
γ
-
q
,
c
2
-
b
2
·
1
max
⁡
2
q
-
1
,
2
p
-
1
,and G
3
′
>
0 such that (23)
I
i
x
≥
-
d
7
′
x
min
⁡
p
,
qfor all x
∈
R
N with x
≤
G
3
′, i
=
1
,
…
,
l, and G
4
′
>
0 such that (24)
H
j
y
≥
-
d
8
′
y
min
⁡
p
,
qfor all y
∈
R
N with y
≤
G
4
′, j
=
1
,
…
,
m, where (25)
C
α
,
p
,
∞
=
T
α
-
1
/
p
Γ
α
α
p
-
p
′
+
1
1
/
p
′
,
p
′
=
p
p
-
1
,
C
β
,
q
,
∞
=
T
β
-
1
/
q
Γ
β
β
q
-
q
′
+
1
1
/
q
′
,
q
′
=
q
q
-
1
.Then system (1) has an unbounded sequence of weak solutions.It is easy to prove that the followingA
R
′ condition implies that (
F
2
) and (
F
3
) hold.A
R
′ there are constants μ
>
max
⁡
{
p
,
q
}, L
2
>
0 such that (26)
0
<
μ
F
t
,
x
,
y
≤
∇
x
F
t
,
x
,
y
,
x
+
∇
y
F
t
,
x
,
y
,
yfor a.e. t
∈
[
0
,
T
] and all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≥
L
2.Indeed, obviously, (AR)′ implies that(
F
3
) holds with ν
1
=
ν
2
=
0 and 1
/
ξ
=
μ
-
max
⁡
{
p
,
q
}. Moreover, by the proof of Theorem 1.2 in [4], (AR)′ and (
W
0
) imply that there exist positive constants B
i
,
i
=
1,2
,
3,4 such that (27)
F
t
,
x
,
y
≥
B
1
x
μ
+
B
2
y
μ
-
B
3
-
B
4
b
tfor all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
], and so it is easy to see that (
F
2
) holds. Then by Theorems 1 and 2, we have the following corollaries.Corollary 3.
Suppose that(
W
0
), (
A
), (
W
1
), (
K
1
)–(
K
3
), (
F
1
), (
I
1
)–(
I
3
), (
H
1
)
–
(
H
3
), and A
R
′ hold. Then system (1) has an unbounded sequence of weak solutions.Corollary 4.
Assume that(
W
0
), (
A
), (
W
1
), (
K
1
)–(
K
3
), (
F
1
), (
I
1
), (
I
3
), (
H
1
), (
H
3
), I
H
2
′, and A
R
′ hold. Then system (1) has an unbounded sequence of weak solutions.Remark 5.
There exist examples satisfying Theorems1 and 2. For example, let p
>
1
,
q
>
1, ρ
(
t
)
=
γ
(
t
)
=
t
2
+
1, I
i
(
x
)
=
x
p, i
=
1
,
…
,
l, H
j
(
y
)
=
y
p, j
=
1
,
…
,
m, and (28)
F
t
,
x
,
y
=
t
2
+
1
x
max
⁡
p
,
q
+
y
max
⁡
p
,
q
ln
⁡
1
+
x
p
+
y
q
,
K
t
,
x
,
y
=
t
2
+
1
θ
1
x
θ
1
+
t
2
+
1
θ
2
y
θ
2for all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
], where θ
1
∈
(
0
,
p
] and θ
2
∈
(
0
,
q
].With similar proofs of Theorems1 and 2, we can obtain the corresponding theorems for the following p-Laplacian system:(29)
D
T
α
t
ρ
t
Φ
p
D
t
α
0
c
u
t
=
∇
u
W
t
,
u
t
,
a.e.
t
∈
0
,
T
,
Δ
D
T
α
-
1
t
ρ
Φ
p
D
t
α
0
c
u
t
i
=
∇
I
i
u
t
i
,
i
=
1,2
,
…
,
l
,
u
0
=
u
T
=
0
,where W
(
t
,
x
)
=
-
K
(
t
,
x
)
+
F
(
t
,
x
) for all x
∈
R
N
(
N
≥
1
) and a.e. t
∈
[
0
,
T
].Theorem 6.
Suppose that the following conditions hold.
W
0
′
W
(
t
,
x
) is measurable in t for each x
∈
R
N, continuously differential in x
∈
R
N for a.e. t
∈
[
0
,
T
], and there exist a
∈
C
(
R
+
,
R
+
) and b
∈
L
1
(
[
0
,
T
]
;
R
+
) such that (30)
K
t
,
x
+
∇
K
t
,
x
+
F
t
,
x
+
∇
F
t
,
x
≤
a
x
b
t
,
I
i
x
+
∇
I
i
x
≤
a
x
,
i
=
1,2
,
…
,
lfor all x
∈
R
N and a.e. t
∈
[
0
,
T
].
A
′
ρ
-
≔
e
s
s
i
n
f
[
0
,
T
]
ρ
(
t
)
>
0.
W
1
′
W
(
t
,
x
) is even in x
∈
R
N and W
(
t
,
0
)
≡
0 for a.e. t
∈
[
0
,
T
].
K
1
′ There exist θ
1
∈
[
0
,
p
), d
1
>
0, M
1
>
0 such that (31)
K
t
,
x
≥
-
d
1
x
θ
1for a.e. t
∈
[
0
,
T
] and all x
∈
R
N with x
≥
M
1.
K
2
′ There exists a positive constant M
2 such that (32)
∇
x
K
t
,
x
,
x
≤
p
K
t
,
x
,for a.e. t
∈
[
0
,
T
] and all x
∈
R
N with x
≥
M
2.
K
3
′ There exist positive constants c
,
M
3 such that (33)
K
t
,
x
≥
c
x
pfor a.e. t
∈
[
0
,
T
] and all x
∈
R
N with x
≤
M
3.
F
1
′ There exist L
1
∈
(
0,1
], b
∈
0
,
min
⁡
ρ
-
/
p
,
c such that (34)
F
t
,
x
≤
b
x
pfor a.e. t
∈
[
0
,
T
] and all x
∈
R
N with x
≤
L
1.
F
2
′
(35)
l
i
m
x
→
+
∞
F
t
,
x
x
p
=
+
∞
uniformly  for  a.e.
t
∈
0
,
T
.
F
3
′ There exist constants ξ
,
η
,
L
2
∈
(
0
,
+
∞
) and ν
∈
[
0
,
p
) such that (36)
p
+
1
ξ
+
η
x
ν
F
t
,
x
,
y
≤
∇
x
F
t
,
x
,
xfor a.e. t
∈
[
0
,
T
] and all x
∈
R
N with x
>
L
2.
I
1
′ There exist constants d
2
>
0
,
θ
2
∈
[
0
,
p
)
,
G
1
>
0 such that (37)
I
i
x
≥
-
d
2
x
θ
2
,
∇
I
i
x
,
x
≤
p
I
i
xfor all x
∈
R
N with x
≥
G
1, i
=
1
,
…
,
l.
I
2
′ There exist constants d
3
>
0
,
θ
3
∈
[
0
,
p
)
,
G
2
>
0 such that (38)
I
i
x
≥
-
d
3
x
θ
3for all x
∈
R
N with x
≤
G
2, i
=
1
,
…
,
l.
I
3
′
I
i
(
0
)
=
0, I
i
(
x
) is even in x
∈
R
N, i
=
1
,
…
,
l.
Then system (29) has an unbounded sequence of weak solutions.Theorem 7.
Suppose thatW
0
′, A
′, W
1
′, K
1
′–K
3
′, F
1
′–F
3
′, I
1
′, I
3
′, and the following condition holds.
I
2
′
′ There exist positive constants d
2
′ with (39)
d
2
′
l
C
α
,
p
,
∞
p
≤
min
⁡
ρ
-
p
,
c
1
-
b
1
,and G
2
′
>
0 such that (40)
I
i
x
≥
-
d
2
′
x
pfor all x
∈
R
N with x
≤
G
2
′, i
=
1
,
…
,
l.
Then system (29) has an unbounded sequence of weak solutions.Corollary 8.
Suppose thatW
0
′, A
′, W
1
′, K
1
′–K
3
′, F
1
′, and I
1
′–I
3
′ and the following condition holds.
A
R
′
′ There are constants μ
>
p, L
2
>
0 such that (41)
0
<
μ
F
t
,
x
≤
∇
x
F
t
,
x
,
xfor a.e. t
∈
[
0
,
T
] and all x
∈
R
N with x
≥
L
2.
Then system (29) has an unbounded sequence of weak solutions.Corollary 9.
Assume thatW
0
′, A
′, W
1
′, K
1
′–K
3
′, F
1
′, I
1
′, I
2
′
′, I
3
′, and A
R
′
′ hold. Then system (29) has an unbounded sequence of weak solutions.Remark 10.
Corollaries8 and 9 are still different from Theorem B. Indeed, if N
=
1, ρ
(
t
)
≡
1, and K
(
t
,
x
)
≡
(
1
/
p
)
x
p for a.e. t
∈
[
0
,
T
] and all x
∈
R, system (29) reduces to (9). However, it is easy to see that I
1
′ and I
2
′ (or I
2
′
′) are different from (
A
1
). There exist examples satisfying I
1
′ and I
2
′ but not satisfying (
A
1
) and (
A
4
). For example, let N
=
1 and I
i
(
x
)
=
x
p for all x
∈
R, i
=
1
,
…
,
l. Then I
i
′
(
x
)
=
p
x
p
-
2
x. It is easy to see that I
i, i
=
1
,
…
,
l satisfy I
1
′ and I
2
′. Set Q
i
(
x
)
=
I
i
′
(
x
), i
=
1
,
…
,
l. Obviously, Q
i
(
x
), i
=
1
,
…
,
l do not satisfy (
A
1
). Moreover, there exist examples satisfying F
1
′–F
3
′ but not satisfying (
A
2
). For example, let (42)
F
t
,
x
≡
F
x
=
x
p
ln
⁡
1
+
x
pfor a.e. t
∈
[
0
,
T
]. Finally, one can also establish some results which are similar to Theorem A for system (1) and system (29) by combining those assumptions and arguments of Theorems 1 and 2 with those ideas proving Theorem A.


## 2. Preliminaries

In this section, we recall some known definitions and lemmas about fractional derivatives. For more details, the readers can see [12, 39–42].Leta
,
b
∈
(
-
∞
,
+
∞
) and (43)
A
C
a
,
b
≔
u
:
a
,
b
⟶
R
N
∣
u
is  absolutely  continuous  on
a
,
b
.Definition 11 (see [40, 42]).
Letf
∈
A
C
[
a
,
b
] and ϱ
∈
(
0,1
). D
t
ϱ
a and D
b
ϱ
t denote the left and right Riemann-Liouville fractional derivatives of order α for function f, respectively, which are defined by (44)
D
t
ϱ
a
f
t
=
1
Γ
1
-
ϱ
d
d
t
∫
a
t
t
-
s
-
ϱ
f
s
d
s
,
t
>
a
,
D
b
ϱ
t
f
t
=
-
1
Γ
1
-
ϱ
d
d
t
∫
a
t
s
-
t
-
ϱ
f
s
d
s
,
t
<
b
.Definition 12 (see [40, 42]).
Letf
∈
A
C
[
a
,
b
] and ϱ
∈
(
0,1
). D
a
c
t
ϱ and D
t
c
b
ϱ denote the left and right Caputo fractional derivatives of order ϱ for function f, respectively, which are defined by (45)
D
a
c
t
ϱ
f
t
=
D
t
ϱ
-
1
a
f
′
t
=
1
Γ
1
-
ϱ
∫
a
t
t
-
s
-
ϱ
f
′
s
d
s
,
D
t
c
b
ϱ
f
t
=
D
b
ϱ
-
1
t
f
′
t
=
1
Γ
1
-
ϱ
∫
t
b
s
-
t
-
ϱ
f
′
s
d
s
.Remark 13 (see [40, 42]).
Whenϱ
=
1, D
t
ϱ
a
f
t
=
D
a
c
t
ϱ
f
(
t
)
=
f
′
(
t
) and D
b
ϱ
t
f
t
=
D
t
c
b
ϱ
f
(
t
)
=
-
f
′
(
t
).Let(46)
C
0
∞
0
,
T
,
R
N
≔
u
∣
u
∈
C
∞
0
,
T
,
R
N
,
u
0
=
u
T
=
0with the norm u
∞
=
m
a
x
[
0
,
T
]
u
t, and, for s
>
1, (47)
L
s
0
,
T
,
R
N
≔
u
∣
u
:
0
,
T
⟶
R
N
,
∫
0
T
u
t
s
d
t
<
∞with the norm u
L
s
=
∫
0
T
u
t
s
d
t
1
/
s.Forϱ
∈
(
0,1
] and s
>
1, we define E
0
ϱ
,
s
(
0
,
T
) as the closure of C
0
∞
(
[
0
,
T
]
,
R
N
), with respect to the norm: (48)
u
s
=
∫
0
T
D
0
c
t
ϱ
u
t
s
d
t
+
∫
0
T
u
t
s
d
t
1
/
s
,
∀
u
∈
E
0
ϱ
,
s
0
,
T
.Then by Proposition 3.1 in [12], E is separable and reflexive Banach space, and if u
∈
E
0
ϱ
,
s
(
0
,
T
), then u, D
0
c
t
ϱ
u
∈
L
s
(
[
0
,
T
]
,
R
) and u
(
0
)
=
u
(
T
)
=
0. Moreover, by Remark 3.1 in [12], D
0
c
t
ϱ
u
t
=
D
t
ϱ
0
u
(
t
), t
∈
[
0
,
T
].Proposition 14 (see [12]).
Assume thatϱ
∈
(
0,1
] and s
>
1. For all u
∈
E
0
ϱ
,
s
(
0
,
T
), (49)
u
L
s
≤
C
ϱ
D
0
c
t
ϱ
u
L
s
,where C
ϱ
=
T
ϱ
/
Γ
(
ϱ
+
1
). Moreover, if ϱ
>
1
/
s, then (50)
u
∞
≤
C
ϱ
,
s
,
∞
D
0
c
t
ϱ
u
L
s
,where C
ϱ
,
s
,
∞
≔
T
ϱ
-
1
/
s
/
Γ
(
ϱ
)
(
ϱ
s
-
s
′
+
1
)
1
/
s
′ and s
′
=
s
/
(
s
-
1
).By Proposition14, it is easy to obtain that(51)
u
∞
≤
C
ϱ
,
s
,
∞
D
0
c
t
ϱ
u
L
s
≤
C
ϱ
,
s
,
∞
u
s
.Proposition 15 (see [12]).
Assume that1
/
p
<
α
≤
1 and 1
<
p
<
∞, and the sequence {
u
k
} converges weakly to u in E
0
α
,
p. Then u
k
→
u in C
(
[
0
,
T
]
,
R
N
).Assume thatα
,
β
∈
(
0,1
]. Let E
=
E
0
α
,
p
(
0
,
T
)
×
E
0
β
,
q
(
0
,
T
). On E, define the norm: (52)
u
,
v
E
=
u
p
+
v
qfor all (
u
,
v
)
∈
E
0
α
,
p
(
0
,
T
)
×
E
0
β
,
q
(
0
,
T
)
=
E.Similar to Definitions 2.4 and 2.5 in [35], we also present the following two definitions.Definition 16.
Let(53)
u
,
v
∈
u
,
v
∈
A
C
0
,
T
×
A
C
0
,
T
:
∫
t
i
t
i
+
1
u
t
p
+
D
t
α
0
c
u
t
p
d
t
<
∞
,
∫
s
j
s
j
+
1
v
t
q
+
D
t
α
0
c
u
t
q
d
t
<
∞
,
i
=
0,1
,
…
,
l
,
j
=
0,1
,
…
,
m
.If (
u
,
v
) satisfies the first equation of (1) for a.e. t
∈
[
0
,
T
]
∖
{
t
1
,
…
,
t
l
} and the second equation of (1) for a.e. t
∈
[
0
,
T
]
∖
{
s
1
,
…
,
s
m
}, D
T
α
-
1
t
Φ
p
(
D
t
α
0
c
u
)
(
t
i
+
), D
T
α
-
1
t
Φ
p
(
D
t
α
0
c
u
)
(
t
i
-
), D
T
β
-
1
t
Φ
q
(
D
0
c
t
β
v
)
(
s
j
+
), and D
T
α
-
1
t
Φ
p
(
D
0
c
t
β
v
)
(
s
j
-
) exist and satisfy the impulsive conditions of (1), and boundary conditions u
(
0
)
=
u
(
T
)
=
0 and v
(
0
)
=
v
(
T
)
=
0, then we call (
u
,
v
) a classical solution of (1).Definition 17.
For any(
h
,
w
)
∈
E, if the following two equalities (54)
∫
0
T
ρ
t
D
t
α
0
c
u
t
p
-
2
D
t
α
0
c
u
t
,
D
t
α
0
c
h
t
d
t
+
∑
i
=
1
l
∇
I
i
u
t
i
,
h
t
i
-
∫
0
T
∇
u
W
t
,
u
t
,
v
t
,
h
t
d
t
=
0
,
∫
0
T
γ
t
D
0
c
t
β
v
t
q
-
2
D
0
c
t
β
v
t
,
D
0
c
t
β
h
t
d
t
+
∑
j
=
1
m
H
j
u
t
i
,
w
s
j
-
∫
0
T
∇
v
W
t
,
u
t
,
v
t
,
h
t
d
t
=
0hold then the vector function (
u
,
v
)
∈
E is called a weak solution of (1).For(
u
,
v
)
∈
E, we define the functional J
:
E
→
R by(55)
J
u
,
v
=
1
p
∫
0
T
ρ
t
D
t
α
0
c
u
t
p
d
t
+
1
q
∫
0
T
γ
t
D
0
c
t
β
v
t
q
d
t
+
∑
i
=
1
l
I
i
u
t
j
+
∑
j
=
1
m
H
j
v
s
j
+
∫
0
T
K
t
,
u
t
,
v
t
d
t
-
∫
0
T
F
t
,
u
t
,
v
t
d
t
=
Φ
u
,
v
+
Ψ
u
,
v
,where(56)
Φ
u
,
v
=
1
p
∫
0
T
ρ
t
D
t
α
0
c
u
t
p
d
t
+
1
q
∫
0
T
γ
t
D
0
c
t
β
v
t
q
d
t
+
∫
0
T
K
t
,
u
t
,
v
t
d
t
-
∫
0
T
F
t
,
u
t
,
v
t
d
t
,
Ψ
u
,
v
=
∑
i
=
1
l
I
i
u
t
i
+
∑
j
=
1
m
H
j
v
s
j
.It follows from (
W
0
), the continuity of ∇
I
i and ∇
H
j and Theorem 5.41 in [42], that Φ and Ψ are continuously differentiable and so J
∈
C
1
(
E
,
R
) and(57)
J
′
u
,
v
,
h
,
w
=
∫
0
T
ρ
t
D
t
α
0
c
u
t
p
-
2
D
t
α
0
c
u
t
,
D
t
α
0
c
h
t
d
t
+
∫
0
T
γ
t
D
t
α
0
c
v
t
q
-
2
D
0
c
t
β
v
t
,
D
t
α
0
c
w
t
d
t
+
∫
0
T
∇
u
K
t
,
u
t
,
v
t
,
h
t
d
t
-
∫
0
T
∇
v
F
t
,
u
t
,
v
t
,
w
t
d
t
+
∑
i
=
1
l
∇
I
i
u
t
i
,
h
t
i
+
∑
j
=
1
m
∇
H
j
v
s
j
,
w
s
j
.Hence, the critical point of J is a weak solution of (1). Similar to the arguments of Propositions 2.5 and 2.6 in [35], it is easy to obtain that (
u
,
v
) is a classical solution of (1) if (
u
,
v
)
∈
E is a weak solution of (1).Assume thatE is a real Banach space and φ
∈
C
1
(
E
,
R
). For any sequence {
u
k
}
⊂
E, if φ
(
u
k
) is bounded and φ
′
(
u
k
)
→
0 as k
→
∞, then we call {
u
k
} a Palais-Smale sequence. If any Palais-Smale sequence {
u
k
} has a convergent subsequence, then we call φ which satisfies Palais-Smale condition.Similar to the proofs in [39], we will also use the following symmetric mountain pass theorem to prove our main results.Lemma 18 (see [2]).
LetE be an infinite dimensional Banach space and let φ
∈
C
1
(
E
,
R
) be even and satisfy Palais-Smale condition, and φ
(
0
)
=
0. If E
=
V
⊕
X, where V is finite dimensional, and φ satisfies the following, then φ possesses an unbounded sequence of critical values.(i)
There are constantsρ
,
ξ
>
0 such that φ
∂
B
ρ
∩
X
≥
ξ.
(ii)
For each finite dimensional subspaceE
~
⊂
E, there is R
=
R
E
~ such that φ
≤
0 on E
~
∖
B
R
E
~.Remark 19.
As shown in [43], a deformation lemma can be proved with replacing Palais-Smale condition with Cerami condition, which implies that Lemma 2.1 in [2] is true under Cerami condition. We say that φ satisfies Cerami condition; that is, for every sequence {
u
k
}
⊂
E, {
u
k
} has a convergent subsequence if φ
(
u
k
) is bounded and (
1
+
u
k
E
)
φ
′
u
k
E
∗
→
0 as k
→
∞, where E
∗ with the norm ·
E
∗ is the dual space of E.


## 3. Proofs of Theorems

Lemma 20.
Assume that(
A
), (
K
1
), (
K
2
), (
F
2
), (
F
3
), (
I
1
), and (
H
1
) hold. Then J satisfies Cerami condition.Proof.
For any sequence{
(
u
n
,
v
n
)
}
n
=
1
∞
⊂
E, suppose that there is a positive constant C
1
>
0 such that(58)
J
u
n
,
v
n
≤
C
1
,
1
+
u
n
,
v
n
E
J
′
u
n
,
v
n
E
∗
≤
C
1
.
∀
n
∈
N
,By (
F
2
) and assumption (
W
0
), there exist positive constants A, C
2, and C
3 such that(59)
F
t
,
x
,
y
≥
A
x
p
+
A
y
q
-
C
2
-
C
3
b
t
,
∀
x
,
y
∈
R
N
×
R
N
,
a.e.
t
∈
0
,
T
.It follows from (
F
3
) that(60)
∇
x
F
t
,
x
,
y
,
x
+
∇
y
F
t
,
x
,
y
,
y
-
max
⁡
p
,
q
F
t
,
x
,
y
ξ
+
η
1
x
ν
1
+
η
2
y
ν
2
≥
F
t
,
x
,
yfor all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
>
L
2 and a.e. t
∈
[
0
,
T
]. Assumption (
W
0
) and (60) imply that there exists a positive constant C
4 such that(61)
∇
x
F
t
,
x
,
y
,
x
+
∇
y
F
t
,
x
,
y
,
y
-
m
a
x
p
,
q
F
t
,
x
,
y
ξ
+
η
1
x
ν
1
+
η
2
y
ν
2
≥
F
t
,
x
,
y
-
C
4
b
tfor all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
].
Assume thatp
≥
q. Then, for any x
∈
R
N with x
≥
1 and all y
∈
R
N,(62)
x
+
y
min
⁡
p
,
q
=
x
+
y
q
≤
2
q
-
1
x
q
+
y
q
≤
2
q
-
1
x
p
+
y
qand, for any x
∈
R
N with x
≤
1 and all y
∈
R
N,(63)
x
+
y
min
⁡
p
,
q
=
x
+
y
q
≤
2
q
-
1
x
p
+
y
q
≤
2
q
-
1
x
p
+
y
q
+
1
.Then, (62) and (63) imply that(64)
x
+
y
min
⁡
p
,
q
≤
2
q
-
1
x
p
+
y
q
+
1for all (
x
,
y
)
∈
R
N
×
R
N. Similarly, if q
≥
p, we have(65)
x
+
y
min
⁡
p
,
q
≤
2
p
-
1
x
p
+
y
q
+
1for all (
x
,
y
)
∈
R
N
×
R
N. Combining (64) and (65), we have(66)
x
p
+
y
q
≥
x
+
y
min
⁡
p
,
q
max
⁡
2
q
-
1
,
2
p
-
1
-
1for all (
x
,
y
)
∈
R
N
×
R
N. Moreover, for ν
1
,
ν
2
∈
(
1
,
∞
) and η
1
,
η
2
∈
(
0
,
+
∞
), we have(67)
η
1
x
ν
1
+
η
2
y
ν
2
≤
max
⁡
η
1
,
η
2
x
+
y
ν
1
+
x
+
y
ν
2
≤
2
max
⁡
η
1
,
η
2
x
+
y
max
⁡
ν
1
,
ν
2
+
1for all (
x
,
y
)
∈
R
N
×
R
N, and(68)
ξ
+
2
max
⁡
η
1
,
η
2
≤
ξ
+
2
max
⁡
η
1
,
η
2
x
+
y
max
⁡
ν
1
,
ν
2
.for all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≥
1. It follows from assumptions (59), (61), (66), (67), and (68) that there exist positive constants C
5
,
C
6
,
C
7 such that(69)
∇
x
F
t
,
x
,
y
,
x
+
∇
y
F
t
,
x
,
y
,
y
-
max
⁡
p
,
q
F
t
,
x
,
y
≥
F
t
,
x
,
y
-
C
4
b
t
ξ
+
η
1
x
ν
1
+
η
2
y
ν
2
≥
A
x
p
+
A
y
q
-
C
2
-
C
3
b
t
-
C
4
b
t
ξ
+
η
1
x
ν
1
+
η
2
y
ν
2
≥
A
/
max
⁡
2
q
-
1
,
2
p
-
1
x
+
y
min
⁡
p
,
q
-
A
-
C
2
-
C
3
b
t
-
C
4
b
t
ξ
+
2
max
⁡
η
1
,
η
2
x
+
y
max
⁡
ν
1
,
ν
2
+
2
max
⁡
η
1
,
η
2
≥
A
/
max
⁡
2
q
-
1
,
2
p
-
1
x
+
y
min
⁡
p
,
q
ξ
+
4
max
⁡
η
1
,
η
2
x
+
y
max
⁡
ν
1
,
ν
2
-
A
+
C
2
+
C
3
b
t
+
C
4
b
t
ξ
+
2
max
⁡
η
1
,
η
2
≥
C
5
x
+
y
min
⁡
p
,
q
-
max
⁡
ν
1
,
ν
2
-
C
6
b
t
-
C
7for all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≥
1 and a.e. t
∈
[
0
,
T
]. By (69) and assumption (
W
0
), there exist positive constants C
8
,
C
9 such that(70)
∇
x
F
t
,
x
,
y
,
x
+
∇
y
F
t
,
x
,
y
,
y
-
max
⁡
p
,
q
F
t
,
x
,
y
≥
C
5
x
+
y
min
⁡
p
,
q
-
max
⁡
ν
1
,
ν
2
-
C
8
b
t
-
C
9for all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
]. By (
K
1
), (
K
2
), and (
W
0
), there exists positive constant C
10 such that(71)
max
⁡
p
,
q
K
t
,
x
,
y
-
∇
x
K
t
,
x
,
y
,
x
-
∇
y
K
t
,
x
,
y
,
y
≥
-
C
10
b
t
,
(72)
K
t
,
x
,
y
≥
-
d
1
x
θ
1
-
d
2
y
θ
2
-
C
10
b
t
-
C
10for all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
]. Moreover, by (
I
1
), (
H
1
), and (
W
0
), there exist positive constants C
11 and C
12 such that(73)
max
⁡
p
,
q
I
i
x
-
∇
I
i
x
,
x
≥
-
C
11
,
I
i
x
≥
-
d
3
x
θ
3
-
C
11
,
i
=
1
,
…
,
l
,
max
⁡
p
,
q
H
j
y
-
∇
H
j
y
,
y
≥
-
C
12
,
H
j
x
≥
-
d
4
y
θ
4
-
C
12
,
j
=
1
,
…
,
mfor all (
x
,
y
)
∈
R
N
×
R
N. By (58) and (70), we have(74)
max
⁡
p
,
q
+
1
C
1
≥
max
⁡
p
,
q
J
u
n
,
v
n
-
J
′
u
n
,
v
n
,
u
n
,
v
n
≥
∫
0
T
-
∇
u
K
t
,
u
n
t
,
v
n
t
,
u
n
t
-
∇
v
K
t
,
u
n
t
,
v
n
t
,
v
n
t
+
max
⁡
p
,
q
K
t
,
u
n
t
,
v
n
t
d
t
+
∫
0
T
∇
u
F
t
,
u
n
t
,
v
n
t
,
u
n
t
+
∇
v
F
t
,
u
n
t
,
v
n
t
,
v
n
t
-
max
⁡
p
,
q
F
t
,
u
n
t
,
v
n
t
d
t
+
∑
i
=
1
l
max
⁡
p
,
q
I
i
u
n
t
i
-
∇
I
i
u
n
t
i
,
u
n
t
i
+
∑
j
=
1
m
max
⁡
p
,
q
K
j
v
n
s
j
-
∇
K
j
v
n
s
j
,
v
n
s
j
≥
∫
0
T
∇
u
F
t
,
u
n
t
,
v
n
t
,
u
n
t
+
∇
v
F
t
,
u
n
t
,
v
n
t
,
v
n
t
-
m
a
x
p
,
q
F
t
,
u
n
t
,
v
n
t
d
t
-
C
10
∫
0
T
b
t
d
t
-
C
11
l
-
C
12
m
≥
C
5
∫
0
T
u
n
t
+
v
n
t
min
⁡
p
,
q
-
max
⁡
ν
1
,
ν
2
d
t
-
C
8
+
C
10
∫
0
T
b
t
d
t
-
C
9
T
-
C
11
l
-
C
12
m
,which implies that there exists a positive constant C
13 such that ∫
0
T
(
u
n
t
+
v
n
t
)
min
⁡
{
p
,
q
}
-
max
⁡
{
ν
1
,
ν
2
}
d
t
<
C
13. Then ∫
0
T
u
n
t
m
i
n
{
p
,
q
}
-
m
a
x
{
ν
1
,
ν
2
}
d
t
<
C
13 and ∫
0
T
v
n
t
m
i
n
{
p
,
q
}
-
m
a
x
{
ν
1
,
ν
2
}
d
t
<
C
13. By (60) and (
F
2
), there exists C
14
>
0 such that(75)
∇
x
F
t
,
x
,
y
,
x
+
∇
y
F
t
,
x
,
y
,
y
-
max
⁡
p
,
q
F
t
,
x
,
y
ξ
+
η
1
x
ν
1
+
η
2
y
ν
2
≥
A
x
p
+
A
y
p
>
0for all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
>
L
2
+
C
14 and a.e. t
∈
[
0
,
T
]. Let F
~
(
t
,
x
,
y
)
≔
(
∇
x
F
(
t
,
x
,
y
)
,
x
)
+
(
∇
y
F
(
t
,
x
,
y
)
,
y
)
-
m
a
x
{
p
,
q
}
F
(
t
,
x
,
y
) for all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
], S
1
≔
{
t
∈
0
,
T
∣
u
n
t
+
v
n
t
>
L
2
+
C
14
}, S
2
≔
[
0
,
T
]
∖
S
1, and Λ
≔
min
⁡
{
p
,
q
}
-
max
⁡
{
ν
1
,
ν
2
}. By (58), (
W
0
), (
A
), (61), (72), (73), (74), (75), and (51), there exist positive constants C
15 and C
16 such that(76)
ρ
-
p
u
n
p
p
+
γ
-
q
v
n
q
q
≤
J
u
n
,
v
n
-
∫
0
T
K
t
,
u
n
t
,
v
n
t
d
t
+
∫
0
T
F
t
,
u
n
t
,
v
n
t
d
t
-
∑
i
=
1
l
I
i
u
t
j
-
∑
j
=
1
m
H
j
v
s
j
+
ρ
-
p
∫
0
T
u
n
t
p
d
t
+
γ
-
q
∫
0
T
v
n
t
q
d
t
≤
C
1
+
ρ
-
p
∫
0
T
u
n
t
p
d
t
+
γ
-
q
∫
0
T
v
n
t
q
d
t
+
d
1
∫
0
T
u
n
t
θ
1
d
t
+
d
2
∫
0
T
v
n
t
θ
2
d
t
+
∫
0
T
F
~
t
,
u
n
t
,
v
n
t
ξ
+
η
1
u
n
t
ν
1
+
η
2
v
n
t
ν
2
d
t
+
C
4
∫
0
T
b
t
d
t
+
C
10
∫
0
T
b
t
d
t
+
C
10
T
+
d
3
∫
0
T
u
n
t
θ
3
d
t
+
d
4
∫
0
T
v
n
t
θ
4
d
t
+
C
11
l
+
C
12
m
≤
C
1
+
ρ
-
p
∫
0
T
u
n
t
p
d
t
+
γ
-
q
∫
0
T
v
n
t
q
d
t
+
C
4
+
C
10
∫
0
T
b
t
d
t
+
∫
S
2
F
~
t
,
u
n
t
,
v
n
t
ξ
+
η
1
u
n
t
ν
1
+
η
2
v
n
t
ν
2
d
t
+
C
10
T
+
C
11
l
+
C
12
m
+
∫
S
1
F
~
t
,
u
n
t
,
v
n
t
ξ
+
η
1
u
n
t
ν
1
+
η
2
v
n
t
ν
2
d
t
+
d
1
∫
0
T
u
n
t
θ
1
d
t
+
d
2
∫
0
T
v
n
t
θ
2
d
t
+
d
3
∫
0
T
u
n
t
θ
3
d
t
+
d
4
∫
0
T
v
n
t
θ
4
d
t
≤
C
1
+
ρ
-
p
∫
0
T
u
n
t
p
d
t
+
γ
-
q
∫
0
T
v
n
t
q
d
t
+
C
4
+
C
10
∫
0
T
b
t
d
t
+
C
15
∫
0
T
b
t
d
t
+
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
∫
S
1
F
~
t
,
u
n
t
,
v
n
t
d
t
+
C
10
T
+
C
11
l
+
C
12
m
+
d
1
T
u
n
∞
θ
1
+
d
2
T
v
n
∞
θ
2
+
d
3
T
u
n
∞
θ
3
+
d
4
T
v
n
∞
θ
4
≤
C
1
+
ρ
-
p
∫
0
T
u
n
t
p
d
t
+
γ
-
q
∫
0
T
v
n
t
q
d
t
+
C
4
+
C
10
∫
0
T
b
t
d
t
+
C
15
∫
0
T
b
t
d
t
+
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
∫
0
T
F
~
t
,
u
n
t
,
v
n
t
d
t
-
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
∫
S
2
F
~
t
,
u
n
t
,
v
n
t
d
t
+
C
10
T
+
C
11
l
+
C
12
m
+
d
1
T
u
n
∞
θ
1
+
d
2
T
v
n
∞
θ
2
+
d
3
T
u
n
∞
θ
3
+
d
4
T
v
n
∞
θ
4
≤
C
1
+
ρ
-
p
∫
0
T
u
n
t
p
d
t
+
γ
-
q
∫
0
T
v
n
t
q
d
t
+
C
4
+
C
10
∫
0
T
b
t
d
t
+
C
15
∫
0
T
b
t
d
t
+
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
∫
0
T
F
~
t
,
u
n
t
,
v
n
t
d
t
+
C
16
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
∫
0
T
b
t
d
t
+
C
10
T
+
C
11
l
+
C
12
m
+
d
1
T
u
n
∞
θ
1
+
d
2
T
v
n
∞
θ
2
+
d
3
T
u
n
∞
θ
3
+
d
4
T
v
n
∞
θ
4
≤
C
1
+
ρ
-
p
∫
0
T
u
n
t
p
d
t
+
γ
-
q
∫
0
T
v
n
t
q
d
t
+
C
4
+
C
10
∫
0
T
b
t
d
t
+
C
15
∫
0
T
b
t
d
t
+
max
⁡
p
,
q
+
1
C
1
+
C
10
∫
0
T
b
t
d
t
+
C
11
l
+
C
12
m
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
+
C
16
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
∫
0
T
b
t
d
t
+
C
10
T
+
C
11
l
+
C
12
m
+
d
1
T
u
n
∞
θ
1
+
d
2
T
v
n
∞
θ
2
+
d
3
T
u
n
∞
θ
3
+
d
4
T
v
n
∞
θ
4
≤
C
1
+
ρ
-
p
u
n
∞
p
-
Λ
∫
0
T
u
n
t
Λ
d
t
+
γ
-
q
v
n
∞
q
-
Λ
∫
0
T
v
n
t
Λ
d
t
+
C
4
+
C
10
∫
0
T
b
t
d
t
+
C
15
∫
0
T
b
t
d
t
+
max
⁡
p
,
q
+
1
C
1
+
C
10
∫
0
T
b
t
d
t
+
C
11
l
+
C
12
m
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
+
C
16
ξ
+
η
1
u
n
∞
ν
1
+
η
2
v
n
∞
ν
2
∫
0
T
b
t
d
t
+
C
10
T
+
C
11
l
+
C
12
m
+
d
1
T
u
n
∞
θ
1
+
d
2
T
v
n
∞
θ
2
+
d
3
T
u
n
∞
θ
3
+
d
4
T
v
n
∞
θ
4
≤
C
1
+
C
α
,
p
,
∞
p
-
Λ
ρ
-
p
u
n
p
p
-
Λ
∫
0
T
u
n
t
Λ
d
t
+
C
β
,
q
,
∞
q
-
Λ
γ
-
q
v
n
q
q
-
Λ
∫
0
T
v
n
t
Λ
d
t
+
C
4
+
C
10
∫
0
T
b
t
d
t
+
C
15
∫
0
T
b
t
d
t
+
max
⁡
p
,
q
+
1
C
1
+
C
10
∫
0
T
b
t
d
t
+
C
11
l
+
C
12
m
ξ
+
η
1
C
α
,
p
,
∞
ν
1
u
n
p
ν
1
+
η
2
C
β
,
q
,
∞
ν
2
v
n
q
ν
2
+
C
16
ξ
+
η
1
C
α
,
p
,
∞
ν
1
u
n
p
ν
1
+
η
2
C
β
,
q
,
∞
ν
2
v
n
q
ν
2
∫
0
T
b
t
d
t
+
C
10
T
+
C
11
l
+
C
12
m
+
d
1
T
C
α
,
p
,
∞
θ
1
u
n
p
θ
1
+
d
2
T
C
β
,
q
,
∞
θ
2
v
n
q
θ
2
+
d
3
C
α
,
p
,
∞
θ
3
T
u
n
p
θ
3
+
d
4
T
C
β
,
q
,
∞
θ
4
v
n
q
θ
4
.Since max
⁡
{
ν
1
,
ν
2
}
<
min
⁡
{
p
,
q
}, θ
1
<
p, θ
2
<
q, θ
3
<
p, θ
4
<
q, and 0
<
Λ
<
m
i
n
{
p
,
q
}, the boundness of ∫
0
T
u
n
t
Λ
d
t and ∫
0
T
v
n
t
Λ
d
t implies that u
n
p and v
n
q are bounded. Going if necessary to a subsequence, assume that u
n
⇀
u in E
0
α
,
p
(
0
,
T
) and v
n
⇀
v in E
0
β
,
q
(
0
,
T
). Then, by Proposition 15, we obtain u
n
-
u
∞
→
0 and v
n
-
v
∞
→
0, and so u
n
-
u
L
p
→
0 and v
n
-
v
L
q
→
0. Similar to the arguments of Lemma 3.1 in [35], we can obtain that D
t
α
0
c
u
n
(
t
)
-
D
t
α
0
c
u
(
t
)
L
p
→
0 and D
t
α
0
c
v
n
(
t
)
-
D
t
α
0
c
v
(
t
)
L
q
→
0. Therefore, u
n
-
u
p
→
0 and v
n
-
v
q
→
0 as n
→
∞, which shows that (
u
n
,
v
n
)
-
(
u
,
v
)
E
→
0 as n
→
∞.Lemma 21.
Assume that(
A
), (
K
3
), (
F
1
), (
I
2
), and (
H
2
) hold. There are constants ρ
,
ξ
>
0 such that J
∂
B
ρ
∩
E
≥
ξ.Proof.
Choosingρ
=
min
⁡
{
G
3
,
G
4
,
M
3
,
L
1
}/max
⁡
{
C
α
,
p
,
∞, C
β
,
q
,
∞
}. Then, for all (
u
,
v
)
∈
E
∩
∂
B
ρ, we have u
p
+
v
q
=
ρ and then, by (51), u
∞
≤
C
α
,
p
,
∞
u
p
≤
C
α
,
p
,
∞
ρ
≤
min
⁡
{
G
3
,
G
4
,
M
3
,
L
1
} and, similarly, v
∞
≤
min
⁡
{
G
3
,
G
4
,
M
3
,
L
1
}. Note that b
1
<
min
⁡
ρ
-
/
p
,
c
1 and b
2
<
min
⁡
γ
-
/
q
,
c
2. Hence, it follows from (
K
3
), (
F
1
), (
I
2
), (
H
2
), and (66) that(77)
J
u
,
v
=
1
p
∫
0
T
ρ
t
D
t
α
0
c
u
t
p
d
t
+
1
q
∫
0
T
γ
t
D
0
c
t
β
v
t
q
d
t
+
∑
i
=
1
l
I
i
u
t
i
+
∑
j
=
1
m
H
j
v
s
j
+
∫
0
T
K
t
,
u
t
,
v
t
d
t
-
∫
0
T
F
t
,
u
t
,
v
t
d
t
≥
ρ
-
p
∫
0
T
D
t
α
0
c
u
t
p
d
t
+
γ
-
q
∫
0
T
D
t
α
0
c
v
t
q
d
t
-
d
7
∑
i
=
1
l
u
t
j
θ
5
-
d
8
∑
j
=
1
m
v
s
j
θ
6
+
c
1
∫
0
T
u
t
p
d
t
+
c
2
∫
0
T
v
t
q
d
t
-
b
1
∫
0
T
u
t
p
d
t
-
b
2
∫
0
T
v
t
q
d
t
≥
min
⁡
ρ
-
p
,
c
1
u
p
p
+
min
⁡
γ
-
q
,
c
2
v
q
q
-
d
7
l
u
∞
θ
5
-
d
8
m
v
∞
θ
6
-
b
1
u
p
p
-
b
2
v
q
q
≥
min
⁡
ρ
-
p
,
c
1
u
p
p
+
min
⁡
γ
-
q
,
c
2
v
q
q
-
d
7
l
C
α
,
p
,
∞
θ
5
u
p
θ
5
-
d
8
m
C
β
,
q
,
∞
θ
6
v
q
θ
6
-
b
1
u
p
p
-
b
2
v
q
q
≥
min
⁡
min
⁡
ρ
-
p
,
c
1
-
b
1
,
min
⁡
γ
-
q
,
c
2
-
b
2
1
max
⁡
2
q
-
1
,
2
p
-
1
u
p
+
v
q
min
⁡
p
,
q
-
min
⁡
min
⁡
ρ
-
p
,
c
1
-
b
1
,
min
⁡
γ
-
q
,
c
2
-
b
2
-
d
7
l
C
α
,
p
,
∞
θ
5
u
,
v
E
θ
5
-
d
8
m
C
β
,
q
,
∞
θ
6
u
,
v
E
θ
6
.Since θ
5
,
θ
6
∈
[
0
,
min
⁡
{
p
,
q
}
), (77) implies that there exists a sufficiently large constant ρ
>
0 such that J
(
u
,
v
)
>
0 for all (
u
,
v
)
∈
E with (
u
,
v
)
E
=
ρ.Lemma 22.
Assume that(
A
), (
K
3
), (
F
1
), and I
H
2
′ hold. There are constants ρ
,
ξ
>
0 such that J
∂
B
ρ
∩
E
≥
ξ.Proof.
By (77) and I
H
2
′, it is easy to obtain that(78)
J
u
,
v
≥
min
⁡
min
⁡
ρ
-
p
,
c
1
-
b
1
,
min
⁡
γ
-
q
,
c
2
-
b
2
1
max
⁡
2
q
-
1
,
2
p
-
1
u
,
v
E
min
⁡
p
,
q
-
min
⁡
min
⁡
ρ
-
p
,
c
1
-
b
1
,
min
⁡
γ
-
q
,
c
2
-
b
2
-
d
7
′
l
C
α
,
p
,
∞
min
⁡
p
,
q
u
,
v
E
min
⁡
p
,
q
-
d
8
′
m
C
β
,
q
,
∞
min
⁡
p
,
q
u
,
v
E
min
⁡
p
,
q
.Then (22) and (78) imply that there exists a sufficiently large constant ρ
>
0 such that J
(
u
,
v
)
>
0 for all (
u
,
v
)
∈
E with (
u
,
v
)
E
=
ρ.Lemma 23.
For each finite dimensional subspaceE
~
⊂
E, there is R
=
R
E
~ such that J
(
u
,
v
)
≤
0 on E
~
∖
B
R
E
~.Proof.
For each given finite dimensional spaceE
~
⊂
E, we claim that there exists R
>
0 such that φ
(
u
)
≤
0 on E
~
/
B
R. Indeed, obviously, for any (
u
,
v
)
∈
E
~, (
u
,
v
) can be rewritten by (
u
,
v
)
=
(
u
,
0
)
+
(
0
,
v
), where u
∈
E
1
~
⊂
E
0
α
,
p
(
0
,
T
), v
∈
E
2
~
⊂
E
0
β
,
q
(
0
,
T
), and E
1
~ and E
2
~ are finite dimensional ones. So there exist positive constants d
9
,
d
10
,
d
11
,
d
12
,
d
13
,
d
14
,
d
15
,
d
16 such that(79)
d
9
u
L
p
p
≤
∫
0
T
u
t
max
⁡
p
,
q
d
t
≤
d
10
u
L
p
p
,
d
11
v
L
q
q
≤
∫
0
T
v
t
max
⁡
p
,
q
d
t
≤
d
12
v
L
q
q
,
d
13
u
p
p
≤
∫
0
T
u
t
p
d
t
≤
d
14
u
p
p
,
d
15
v
q
q
≤
∫
0
T
v
t
q
d
t
≤
d
16
v
q
q
.Similar to the proof of Theorem 1.1 in [4], by (
K
2
) and assumption (
W
0
), there exists positive constant D
0 such that(80)
K
t
,
x
,
y
≤
D
0
b
t
x
max
⁡
p
,
q
+
D
0
b
t
y
max
⁡
p
,
q
+
D
0
b
tfor all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
]. By (
F
2
), there exist(81)
D
1
>
max
⁡
1
d
13
e
s
s
s
u
p
0
,
T
ρ
t
p
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
10
+
d
3
l
C
α
,
p
,
∞
p
,
1
d
15
e
s
s
s
u
p
0
,
T
γ
t
q
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
12
+
d
5
m
C
β
,
q
,
∞
qand positive constant M
4 such that(82)
F
t
,
x
,
y
≥
D
1
x
p
+
D
1
y
qfor all (
x
,
y
)
∈
R
N
×
R
N with x
+
y
≥
M
4 and a.e. t
∈
[
0
,
T
]. Then by (82) and assumption (
W
0
), there exist positive constants D
2 and D
3 such that(83)
F
t
,
x
,
y
≥
D
1
x
p
+
D
0
y
q
-
D
2
-
D
3
b
tfor all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
]. By (
I
1
) and (
H
1
), there exist positive constants D
4
,
D
5 such that(84)
I
i
x
≤
d
3
x
p
+
D
4
b
t
,
H
j
y
≤
d
5
y
q
+
D
5
b
tfor all (
x
,
y
)
∈
R
N
×
R
N and a.e. t
∈
[
0
,
T
]. It follows from (83), (80), (84), and (79) that (85)
J
u
,
v
=
1
p
∫
0
T
ρ
t
D
t
α
0
c
u
t
p
d
t
+
1
q
∫
0
T
γ
t
D
t
α
0
c
v
t
q
d
t
+
∑
i
=
1
l
I
i
u
t
j
+
∑
j
=
1
m
H
j
v
s
j
+
∫
0
T
K
t
,
u
t
,
v
t
d
t
-
∫
0
T
F
t
,
u
t
,
v
t
d
t
≤
e
s
s
s
u
p
0
,
T
ρ
t
p
∫
0
T
D
t
α
0
c
u
t
p
d
t
+
e
s
s
s
u
p
0
,
T
γ
t
q
∫
0
T
D
t
α
0
c
v
t
q
d
t
+
d
3
∑
i
=
1
l
u
t
j
p
+
d
5
∑
j
=
1
m
v
s
j
q
+
D
0
∫
0
T
b
t
u
t
max
⁡
p
,
q
d
t
+
D
0
∫
0
T
b
t
v
t
max
⁡
p
,
q
d
t
-
D
1
∫
0
T
u
t
p
d
t
-
D
1
∫
0
T
v
t
q
d
t
+
D
2
T
+
D
3
+
D
0
∫
0
T
b
t
d
t
+
D
4
l
+
D
5
m
≤
e
s
s
s
u
p
0
,
T
ρ
t
p
∫
0
T
D
t
α
0
c
u
t
p
d
t
+
e
s
s
s
u
p
0
,
T
γ
t
q
∫
0
T
D
t
α
0
c
v
t
q
d
t
+
d
3
l
u
∞
p
+
d
5
m
v
∞
q
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
10
∫
0
T
u
t
p
d
t
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
12
∫
0
T
v
t
q
d
t
-
D
1
d
13
u
p
p
-
D
1
d
15
v
q
q
+
D
2
T
+
D
3
+
D
0
∫
0
T
b
t
d
t
+
D
4
l
+
D
5
m
≤
e
s
s
s
u
p
0
,
T
ρ
t
p
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
10
u
p
p
+
e
s
s
s
u
p
0
,
T
γ
t
q
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
12
v
q
+
d
3
l
C
α
,
p
,
∞
p
u
p
p
+
d
5
m
C
β
,
q
,
∞
q
v
q
q
-
D
1
d
13
u
p
p
-
D
1
d
15
v
q
q
+
D
2
T
+
D
3
+
D
0
∫
0
T
b
t
d
t
+
D
4
l
+
D
5
m
=
e
s
s
s
u
p
0
,
T
ρ
t
p
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
10
+
d
3
l
C
α
,
p
,
∞
p
-
D
1
d
13
u
p
p
+
e
s
s
s
u
p
0
,
T
γ
t
q
+
D
0
·
e
s
s
s
u
p
0
,
T
b
t
·
d
12
+
d
5
m
C
β
,
q
,
∞
q
-
D
1
d
15
v
q
q
+
D
2
T
+
D
3
+
D
0
∫
0
T
b
t
d
t
+
D
4
l
+
D
5
m
.Thus, (81) and the above inequality imply that there exists a sufficiently large constant R
>
0 such that J
(
u
,
v
)
<
0 if u
p
=
v
q
=
R
/
2.Proof of Theorems1 and 2.
By(
W
1
), (
I
3
), and (
H
3
), we have J
(
0,0
)
=
0 and J is even. Similar to the proof of Lemma 4.4 in [44], we can choose e
j
∈
E
∖
{
0
}, j
=
1,2
,
…, and define V
=
s
p
a
n
{
e
1
} and X
=
s
p
a
n
{
e
j
,
j
=
2
,
…
}
¯. Then E
=
V
⊕
X. Note that ∂
B
ρ
∩
X
⊂
∂
B
ρ
∩
E. Thus by Lemma 21 (or Lemma 22) and Lemma 23 it is easy to see that (i) and (ii) of Lemma 18 hold. Lemma 20 implies that J satisfies Cerami condition. So, by Lemma 18 and Remark 19, there exists a sequence {
(
u
n
,
v
n
)
} such that J
(
u
n
,
v
n
)
→
+
∞ and then it is easy to see that (
u
n
,
v
n
)
→
∞ as n
→
∞.



---

*Source: 9256192-2018-05-08.xml*