设为首页
加入收藏
期刊导航
网站地图
首页
期刊
数学与物理
地球与环境
信息通讯
经济与管理
生命科学
工程技术
医药卫生
人文社科
化学与材料
会议
合作
新闻
我们
招聘
千人智库
我要投稿
办刊
期刊菜单
●领域
●编委
●投稿须知
●最新文章
●检索
●投稿
文章导航
●Abstract
●Full-Text PDF
●Full-Text HTML
●Full-Text ePUB
●Linked References
●How to Cite this Article
AdvancesinAppliedMathematics
A^
ê
Æ
?
Ð
,2023,12(4),1704-1712
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.124177
˜
a
©
ê
Å
½
™
õ
Ñ
t
X
Ú
š
²
…
)
•
3
5
ŠŠŠ
ïïï
___
=
²
n
ó
Œ
Æ
n
Æ
§
[
‹
=
²
Â
v
F
Ï
µ
2023
c
3
24
F
¶
¹
^
F
Ï
µ
2023
c
4
18
F
¶
u
Ù
F
Ï
µ
2023
c
4
27
F
Á
‡
©
ï
Ä
˜
a
ä
k
C
Ò
©
ê
Å
½
™
õ
Ñ
t
X
Ú
−
(∆)
s
u
+
u
+
k
(
x
)
φu
=
a
(
x
)
|
u
|
p
−
1
u,x
∈
R
3
,
−
(∆)
t
φ
=
k
(
x
)
u
2
,x
∈
R
3
,
š
²
…
)
•
3
5
,
Ù
¥
3
s
+4
t
s
+
t
<p<
3+2
s
3
−
2
s
,
s,t
∈
(0
,
1)
…
4
s
+ 2
t>
3
,
a
(
x
)
∈
C
(
R
3
)
C
Ò
…
lim
|
x
|→∞
a
(
x
)=
a
∞
<
0
,
k
(
x
)
∈
C
(
R
3
)
∩
L
6
4
s
+2
t
−
3
(
R
3
)
.
A^
ì
´
Ú
n
,
©
T
X
Ú
–
•
3
˜
‡
š
²
…
)
.
'
…
c
©
ê
Å
½
™
õ
Ñ
t
X
Ú
§
C
Ò
§
š
²
…
)
ExistenceofNontrivialSolutionforaClass
ofFractionalSchr¨odinger-PoissonSystem
JuanxiaMeng
©
Ù
Ú
^
:
Š
ï
_
.
˜
a
©
ê
Å
½
™
-
Ñ
t
X
Ú
š
²
…
)
•
3
5
[J].
A^
ê
Æ
?
Ð
,2023,12(4):1704-1712.
DOI:10.12677/aam.2023.124177
Š
ï
_
CollegeofScience,LanzhouUniversityofTechnology,LanzhouGansu
Received:Mar.24
th
,2023;accepted:Apr.18
th
,2023;published:Apr.27
th
,2023
Abstract
Inthispaper,weareconcernedwiththeexistenceofnontrivialsolutionforaclassof
fractionalSchr¨odinger-Poissonsystem:
−
(∆)
s
u
+
u
+
k
(
x
)
φu
=
a
(
x
)
|
u
|
p
−
1
u,x
∈
R
3
,
−
(∆)
t
φ
=
k
(
x
)
u
2
,x
∈
R
3
,
where
3
s
+4
t
s
+
t
<p<
3+2
s
3
−
2
s
,
s,t
∈
(0
,
1)
and
4
s
+2
t>
3
,
a
(
x
)
∈
C
(
R
3
)
isasign-changingfunction
with
lim
|
x
|→∞
a
(
x
) =
a
∞
<
0
,
k
(
x
)
∈
C
(
R
3
)
∩
L
6
4
s
+2
t
−
3
(
R
3
)
.Byusingmountainpasstheorem,
weobtainthatthissystemhasatleastonenontrivialsolution.
Keywords
FractionalSchr¨odinger-PoissonSystem,Sign-ChangingWeight,NontrivialSolution
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.
Ú
ó
C
c
5
,
N
õ
©
z
•
Ä
X
e
©
ê
Å
½
™
õ
Ñ
t
X
Ú
(
−
∆)
s
u
+
V
(
x
)
u
+
K
(
x
)
φu
=
f
(
x,u
)
,x
∈
R
3
,
(
−
∆)
t
φ
=
K
(
x
)
u
2
,x
∈
R
3
,
(1)
DOI:10.12677/aam.2023.1241771705
A^
ê
Æ
?
Ð
Š
ï
_
d?
s,t
∈
(0
,
1),
©
ê
Laplacian
Ž
f
(
−
∆)
s
½
Â
•
(
−
∆)
s
v
(
x
) =
C
N,s
P.V.
Z
R
N
v
(
x
)
−
v
(
y
)
|
x
−
y
|
N
+2
s
d
y,v
∈
S
(
R
N
)
,
d?
P.V.
L
«
…
Ü
ÌŠ
,
C
N,s
I
O
z
~
ê
,
S
(
R
N
)
´
d
¯
„P
~
¼
ê
|
¤
–
]
¼
ê
˜
m
.
5
¿
©
ê
Laplacian
Ž
f
(
−
∆)
s
´
3
©
z
[1,2]
Ä
g
Ú
\
.
•
õ
k
'
(
−
∆)
s
&
E
,
ž
ë
•
[3]
9
Ù
¥
ë
•
©
z
.
c
,
k
'
X
Ú
(1)
?
Ø
ý
Œ
õ
ê´
k
'
)
½
C
Ò
)
•
3
5
,
õ
)
5
(
J
,
X
[4–15].
Ù
¥
,
©
z
[5,6,12,14]
•
Ä
X
Ú
(1)
½
a
q
¯
K
C
Ò
)
•
3
5
;
©
z
[4,10]
ï
Ä
X
Ú
(1)
3
.
^
‡
e
p
U
þ
)
•
3
5
;
©
z
[7,13]
?
Ø
X
Ú
(1)
)
•
3
5
98
¥
5
;
©
z
[8,9,11,15]
•
X
Ú
(1)
Ä
)
•
3
5
½
õ
-
5
.
,
,
Ï
L
Î
n
ƒ
'
©
z
u
y
:
é
ä
k
C
Ò
©
ê
Å
½
™
-
Ñ
t
X
Ú
)
•
3
5
¯
K
ï
Ä
%
š
~
.
,
˜
•
¡
,
·
‚
5
¿
,
{
¡
Ÿ
3
[16]
¥
•
Ä
±
e
Å
½
™
-
Ñ
t
X
Ú
−
∆
u
+
u
+
φu
=
a
(
x
)
|
u
|
p
−
1
u,x
∈
R
3
,
−
∆
φ
=
k
(
x
)
u
2
,x
∈
R
3
,
(2)
Ù
¥
3
≤
p<
5,
a
(
x
)
∈
C
(
R
3
)
C
Ò
…
lim
|
x
|→∞
a
(
x
)=
a
∞
<
0,
k
(
x
)
∈
C
(
R
3
)
∩
L
2
(
R
3
).
|
^
ì
´
Ú
n
[17],
Š
ö
¼
Å
½
™
-
Ñ
t
X
Ú
(2)
–
k
˜
‡
š
²
…
)
•
3
5
(
J
.
É
±
þ
©
z
é
u
,
©
•
Ä
X
e
©
ê
Å
½
™
-
Ñ
t
X
Ú
−
(∆)
s
u
+
u
+
k
(
x
)
φu
=
a
(
x
)
|
u
|
p
−
1
u, x
∈
R
3
−
(∆)
t
φ
=
k
(
x
)
u
2
, x
∈
R
3
,
(3)
š
²
…
)
•
3
5
,
Ù
¥
3
s
+4
t
s
+
t
<p<
3+2
s
3
−
2
s
,
s,t
∈
(0
,
1)
…
4
s
+2
t>
3,
a
(
x
)
Ú
k
(
x
)
÷
v
µ
(A1)
a
(
x
)
∈
C
(
R
3
,
R
)
•
˜
a
C
Ò¼
ê
…
÷
v
a
∞
= lim
|
x
|→∞
a
(
x
)
<
0.
(A2)
k
(
x
)
∈
C
(
R
3
,
R
),
k
(
x
)
≥
0
…
k
(
x
)
∈
L
6
4
s
+2
t
−
3
(
R
3
).
©
Ì
‡
(
J
•
X
e
½
n
:
½
n
1
.
1
.
b
(A1),(A2)
¤
á
,
K
©
ê
Å
½
™
-
Ñ
t
X
Ú
(3)
–
•
3
˜
‡
š
²
…
)
.
2
½½½
nnn
1.1
yyy
²²²
é
u
½
u
∈
H
s
(
R
3
),
½
Â
D
t,
2
(
R
3
)
þ
‚
5
Ž
f
L
u
(
v
) =
Z
R
3
k
(
x
)
u
2
vdx,
DOI:10.12677/aam.2023.1241771706
A^
ê
Æ
?
Ð
Š
ï
_
@
o
Ò
k
|
L
u
(
v
)
|≤
Z
R
3
k
(
x
)
u
2
|
v
|
dx
≤
(
Z
R
3
k
(
x
)
6
4
s
+2
t
−
3
dx
)
4
s
+2
t
−
3
6
(
Z
R
3
u
6
3
−
2
s
)
3
−
2
s
3
(
Z
R
3
v
2
∗
t
)
1
2
∗
t
≤
C
k
k
(
x
)
k
L
6
4
s
+2
t
−
3
(
R
3
)
k
u
k
2
H
s
(
R
3
)
k
v
k
D
t,
2
(
R
3
)
.
Ï
d
d
Riesz
L
«
½
n
Œ
•
,
•
3
•
˜
φ
u
∈
D
t,
2
(
R
3
),
¦
h
φ
u
,v
i
D
t,
2
(
R
3
)
=
L
u
(
v
)
,
∀
v
∈
D
t,
2
(
R
3
)
,
=
Z
R
3
(
−
∆)
t
2
φ
u
(
−
∆)
t
2
vdx
=
Z
R
3
k
(
x
)
u
2
vdx,
∀
v
∈
D
t,
2
(
R
3
)
,
•
Ò
´
`
φ
u
´
(3)
¥
1
‡
•
§
f
)
.
ò
φ
u
‘
\
1
˜
‡
•
§
Ò
−
(∆)
s
u
+
u
+
k
(
x
)
φ
u
u
=
a
(
x
)
|
u
|
p
−
1
u.
(4)
Ï
d
¦
)
•
§
(3)
d
u
¦
)
•
§
(4).
•
§
(4)
)
é
A
U
þ
•
¼
F
(
u
) =
1
2
Z
R
3
|
(
−
∆)
s
2
u
|
2
+
u
2
dx
+
1
4
Z
R
3
k
(
x
)
φ
u
u
2
dx
−
1
p
+1
Z
R
3
a
(
x
)
|
u
|
p
+1
dx, u
∈
H
s
(
R
3
)
.
:
.
½
Â
Φ:
H
s
(
R
3
)
→
D
t,
2
(
R
3
)
•
Φ(
u
) =
φ
u
,
K
k
e
¡
Ú
n
.
Ú
n
2
.
1
(
i
)Φ
ë
Y
.
(
ii
)Φ
ò
k
.
8
N
k
.
8
.
(
iii
)
e
{
u
n
}
3
H
s
(
R
3
)
¥
k
.
…
u
n
u
,
@
o
k
Z
R
3
k
(
x
)
φ
u
n
u
2
n
dx
→
Z
R
3
k
(
x
)
φ
u
u
2
dx.
y
:(i )
é
¤
k
u
∈
H
s
(
R
3
),
k
|
L
u
|
=
k
φ
u
k
D
t,
2
(
R
3
)
=
k
Φ(
u
)
k
D
t,
2
(
R
3
)
DOI:10.12677/aam.2023.1241771707
A^
ê
Æ
?
Ð
Š
ï
_
¤
á
.
¤
±
,
•
y
²
Φ
ë
Y
,
•
I
y
:
u
7→
L
u
´
ë
Y
.
|
L
u
n
(
v
)
−
L
u
(
v
)
|≤
Z
R
3
k
(
x
)
|
v
||
u
2
n
−
u
2
|
dx
≤
C
k
k
k
L
6
4
s
+2
t
−
3
(
R
3
)
k
v
k
D
t,
2
(
R
3
)
k
u
2
n
−
u
2
k
L
6
3
−
2
s
(
R
3
)
Ï
•
v
´
?
¿
,
n
→∞
,
k
|
L
u
n
(
v
)
−
L
u
(
v
)
|→
0 .
(ii)
Ï
•
k
φ
u
k
D
t,
2
(
R
3
)
=
|
L
u
|≤
C
k
k
k
L
6
4
t
+2
s
−
3
(
R
3
)
k
u
k
2
H
s
(
R
3
)
.
¤
±Ú
n
¥
(ii)
¤
á
.
(iii)
3
©
z
[16]
¥
®
²
y
²
φ
u
n
φ
u
,
e
¡
y
²
Z
R
3
k
(
x
)
φ
u
n
u
2
n
dx
→
Z
R
3
k
(
x
)
φ
u
u
2
dx.
Ä
k
·
‚
5
¿
du
φ
u
n
φ
u
,
¤
±
·
‚
k
Z
R
3
k
(
x
)(
φ
u
n
−
φ
u
)
u
2
dx
→
0
.
(5)
e
¡
y
²
Z
R
3
k
(
x
)
φ
u
n
|
u
2
n
−
u
2
|
dx
→
0
.
(6)
Ï
•
k
(
x
)
∈
L
6
4
s
+2
t
−
3
(
R
3
)
…
ë
Y
,
Ï
d
é
∀
ε>
0,
∃
ρ
=
ρ
(
ε
)
>
0,
¦
|
x
|≥
ρ
ž
k
(
x
)
≤
ε
,
u
´
·
‚
Z
|
x
|≥
ρ
k
(
x
)
φ
u
n
|
u
2
n
−
u
2
|
dx
≤
ε
[
Z
|
x
|≥
ρ
φ
u
n
u
2
n
dx
+
Z
|
x
|≥
ρ
φ
u
n
u
2
dx
]
≤
εC
k
φ
u
n
k
D
t,
2
(
R
3
)
(
k
u
n
k
2
+
k
u
k
2
)
≤
Cε.
,
˜
•
¡
,
b
s,t
∈
(0
,
1),
*
X
J
4
s
+ 2
t>
3,
K
k
2
≤
12
3+2
t
<
6
3
−
2
s
,
Ï
d
H
s
(
R
3
)
→
L
12
3+2
t
(
R
3
).
k
:
Z
|
x
|≤
ρ
k
(
x
)
φ
u
n
|
u
2
n
−
u
2
|
dx
≤k
k
k
L
∞
(
R
3
)
[
Z
|
x
|≤
ρ
φ
u
n
|
u
2
n
−
u
2
|
dx
]
≤k
k
k
L
∞
(
R
3
)
C
k
φ
u
n
k
D
t,
2
(
R
3
)
[
Z
|
x
|≤
ρ
|
u
2
n
−
u
2
|
12
3+2
t
dx
]
3+2
t
6
→
0
.
ù
Ò
y
²
(6),
n
Ü
(5)
Ú
(6)
·
‚
R
R
3
k
(
x
)
φ
u
n
u
2
n
dx
→
R
R
3
k
(
x
)
φ
u
u
2
dx
.
Ú
n
y
.
.
DOI:10.12677/aam.2023.1241771708
A^
ê
Æ
?
Ð
Š
ï
_
Ú
n
2
.
2
.
•
¼
F
÷
v
(PS)
^
‡
.
y
:
u
n
⊂
H
s
(
R
3
)
÷
v
|
F
(
u
n
)
|≤
c
Ú
F
0
(
u
n
)
→
0 ,
@
o
k
Z
R
3
|
(
−
∆)
s
2
u
n
|
2
+
u
2
n
dx
+
Z
R
3
k
(
x
)
φ
u
n
u
2
n
dx
−
Z
R
3
a
(
x
)
|
u
n
|
p
+1
dx
=
o
(1)
k
u
n
k
(7)
1
2
Z
R
3
|
(
−
∆)
s
2
u
n
|
2
+
u
2
n
dx
+
1
4
Z
R
3
k
(
x
)
φ
u
n
u
2
n
dx
−
1
p
+1
Z
R
3
a
(
x
)
|
u
n
|
p
+1
dx
≤
c.
|
^
þ
¡
ü
ª
?
˜
Ú
[
1
2
−
1
p
+1
]
Z
R
3
|
(
−
∆)
s
2
u
n
|
2
+
u
2
n
dx
+[
1
4
−
1
p
+1
]
Z
R
3
k
(
x
)
φ
u
n
u
2
n
≤
c
+
o
(1)
k
u
n
k
.
Ï
•
p>
3
s
+4
t
s
+
t
,
¤
±
k
[
1
2
−
1
p
+1
]
Z
R
3
|
(
−
∆)
s
2
u
n
|
2
+
u
2
n
dx
≤
c
+
o
(1)
k
u
n
k
.
d
þ
ª
•
u
n
k
.
.
Ï
d
·
‚
Œ
±
b
½
u
n
u
,
e
¡
y
²
u
n
→
u
.
•
I
y
²
k
u
n
k→k
u
k
.
d
•
§
(7)
·
‚
k
Z
R
3
|
(
−
∆)
s
2
u
n
|
2
+
u
2
n
dx
+
Z
R
3
k
(
x
)
φ
u
n
u
2
n
+
Z
R
3
a
−
(
x
)
|
u
n
|
p
+1
dx
=
Z
R
3
a
+
(
x
)
|
u
n
|
p
+1
dx
+
o
(1)
.
du
lim
|
x
|→∞
a
(
x
) =
a
∞
<
0 ,
Ï
d
a
+
(
x
)
k
;
|
8
,
|
^
¢
Ë
Å
i
\
½
n
•
Z
R
3
a
+
(
x
)
|
u
n
|
p
+1
dx
=
Z
R
3
a
+
(
x
)
|
u
|
p
+1
dx
+
o
(1)
.
¤
±
Z
R
3
|
(
−
∆)
s
2
u
n
|
2
+
u
2
n
dx
+
Z
R
3
k
(
x
)
φ
u
n
u
2
n
dx
+
Z
R
3
a
−
(
x
)
|
u
n
|
p
+1
dx
=
Z
R
3
|
(
−
∆)
s
2
u
|
2
+
u
2
dx
+
Z
R
3
k
(
x
)
φ
u
u
2
dx
+
Z
R
3
a
−
(
x
)
|
u
|
p
+1
dx
+
o
(1)
.
(8)
·
‚
ä
ó
Z
R
3
a
−
(
x
)
|
u
|
p
+1
dx
≤
liminf
n
→∞
Z
R
3
a
−
(
x
)
|
u
n
|
p
+1
dx.
DOI:10.12677/aam.2023.1241771709
A^
ê
Æ
?
Ð
Š
ï
_
¯¢
þ
d
Z
R
3
a
−
(
x
)
||
u
n
|
p
+1
−|
u
|
p
+1
−|
u
n
−
u
|
p
+1
|
dx
≤k
a
(
x
)
k
L
∞
(
R
3
)
Z
R
3
||
u
n
|
p
+1
−|
u
|
p
+1
−|
u
n
−
u
|
p
+1
|
dx
=
o
(1)
•
Z
R
3
a
−
(
x
)
|
u
n
|
p
+1
dx
=
Z
R
3
a
−
(
x
)
|
u
|
p
+1
dx
+
Z
R
3
a
−
(
x
)
|
u
n
−
u
|
p
+1
dx
+
o
(1)
,
•
Ò
´
ä
ó
¤
á
.
X
J
u
n
9
u
,
@
o
k
u
n
k
<
k
u
k
,
2
d
R
R
3
k
(
x
)
φ
u
n
u
2
n
dx
→
R
R
3
k
(
x
)
φ
u
u
2
dx
Ú
R
R
3
a
−
(
x
)
|
u
|
p
+1
dx
≤
liminf
n
→∞
R
R
3
a
−
(
x
)
|
u
n
|
p
+1
dx
.
•
Z
R
3
|
(
−
∆)
s
2
u
n
|
2
+
u
2
n
dx
+
Z
R
3
k
(
x
)
φ
u
n
u
2
n
dx
+
Z
R
3
a
−
(
x
)
|
u
n
|
p
+1
dx
>
Z
R
3
|
(
−
∆)
s
2
u
|
2
+
u
2
dx
+
Z
R
3
k
(
x
)
φ
u
u
2
dx
+
Z
R
3
a
−
(
x
)
|
u
|
p
+1
dx
+
o
(1)
,
ù
†
(8)
ª
g
ñ
.
Ï
d
u
n
→
u
,
y
.
.
½
n
1
y
²
:
Ä
k
y
²
•
3
α,ρ>
0
¦
F
∂B
ρ
>α>
0.
¯¢
þ
d
¢
Ë
Å
i
\
½
n
k
F
(
u
)
≥
1
2
k
u
k
2
−
C
k
a
k
L
∞
k
u
k
p
+1
L
p
+1
(
R
3
)
,
d
þ
ª
Œ
•
•
3
ρ>
0 ,
¦
F
∂B
ρ
>α>
0 .
·
‚
2
y
²
•
3
η
∈
H
s
(
R
3
) ,
k
η
k
>ρ
,
¦
F
(
η
)
<
0 ,
¯¢
þ
,
À
¼
ê
ϕ
θ
=
θ
s
+
t
ϕ
(
θx
)
∈
H
s
(
R
3
),
θ
∈
R
+
,
ϕ
θ
6
= 0,
¦
suppϕ
θ
⊂
suppa
+
,
@
o
Ò
k
F
(
θ
s
+
t
ϕ
(
θx
))
≤
θ
4
s
+2
t
−
3
2
k
ϕ
k
2
+
k
k
k
∞
θ
4
s
+2
t
−
3
4
Z
R
3
φ
ϕ
ϕ
2
dx
−
θ
(
p
+1)(
s
+
t
)
−
3
p
+1
Z
R
3
a
+
|
ϕ
|
p
+1
dx.
Ï
•
p>
3
s
+4
t
s
+
t
,
¤
±
(
p
+1)(
s
+
t
)
−
3
>
4
s
+2
t
−
3.
θ
→
+
∞
ž
,
F
(
θ
s
+
t
ϕ
(
θx
))
→−∞
.
Ï
d
,
é
,
‡
θ
0
¿
©
Œ
ž
,
-
η
=
ϕ
θ
0
=
θ
s
+
t
0
ϕ
(
θ
0
x
),
¤
±
k
F
(
η
)
<
0.
½
Â
Γ =
{
γ
∈
C
([0
,
1]
,H
s
(
R
3
)) :
γ
(0) = 0
,γ
(1) =
η
}
,
c
=inf
γ
∈
Γ
sup
θ
∈
[0
,
1]
F
(
γ
(
θ
))
,
@
o
d
ì
´
Ú
n
•
c
´
F
˜
‡
š
²
…
.
:
,
=
©
ê
Å
½
™
-
Ñ
t
X
Ú
(3)
–
•
3
˜
‡
š
²
…
)
.
DOI:10.12677/aam.2023.1241771710
A^
ê
Æ
?
Ð
Š
ï
_
3
(((
ØØØ
©
Ï
L
|
^
ì
´
Ú
n
,
©
ê
Å
½
™
õ
Ñ
t
X
Ú
–
•
3
˜
‡
š
²
…
)
.
Ù
Ì
‡
g
´
´
é
˜
‡
k
.
(PS)
S
,
¦
T
S
÷
v
(PS)
^
‡
,
d
d
Œ
±
y
²
š
²
…
)
•
3
5
.
ë
•
©
z
[1]Laskin,N. (2000)Fractional Quantum Mechanics andL´evyPath Integrals.
PhysicsLettersA
,
268
,298-305.https://doi.org/10.1016/S0375-9601(00)00201-2
[2]Laskin,N.(2002)FractionalSchr¨odingerEquations.
PhysicalReview
,
66
,56-108.
https://doi.org/10.1103/PhysRevE.66.056108
[3]MolicaBisci,G.,R˘adulescu,V.D.andServadei,R.(2016)VariationalMethodsforNonlocal
FractionalProblems.In:
EncyclopediaofMathematicsandItsApplications
,
162
.Cambridge
UniversityPress,Cambridge.https://doi.org/10.1017/CBO9781316282397
[4]Chen, M.,Li, Q.andPeng,S.J.(2021)BoundStatesforFractionalSchr¨odinger-PoissonSystem
with CriticalExponent.
DiscreteandContinuousDynamicalSystems-SeriesS
,
14
,1819-1835.
https://doi.org/10.3934/dcdss.2021038
[5]Guo,L.(2018)Sign-ChangingSolutionsforFractionalSchr¨odinger-PoissonSystemin
R
3
.
ApplicableAnalysis
,
98
,2085-2104.
[6]Ianni,I.(2013)Sign-ChangingRadialSolutionsfortheSchr¨odinger-Poisson-SlaterProblem.
TopologicalMethodsinNonlinearAnalysis
,
41
,365-385.
[7]Liu,Z.andZhang,J.(2017)MultiplicityandConcentrationofPositiveSolutionsforthe
FractionalSchr¨odinger-Poisson Systemswith CriticalGrowth.
ESAIM:Control,Optimisation
andCalculusofVariations
,
23
,1515-1542.https://doi.org/10.1051/cocv/2016063
[8]Luo,H.andTang,X.(2018)GroundStateandMultipleSolutionsfortheFractional
Schr¨odinger-PoissonSystemwithCriticalSobolevExponent.
NonlinearAnalysis:RealWorld
Applications
,
42
,24-52.https://doi.org/10.1016/j.nonrwa.2017.12.003
[9]Shen,L.andYao,X.(2018)LeastEnergySolutionsforaClassofFractionalSchr¨odinger-
PoissonSystems.
JournalofMathematicalPhysics
,
59
,ArticleID:081501.
https://doi.org/10.1063/1.5047663
[10]Sun,X.andTeng,K.M.(2020)PositiveBoundStatesforFractionalSchr¨odinger-Poisson
SystemwithCriticalExponent.
CommunicationsonPureandAppliedAnalysis
,
19
,3735-
3768.https://doi.org/10.3934/cpaa.2020165
DOI:10.12677/aam.2023.1241771711
A^
ê
Æ
?
Ð
Š
ï
_
[11]Teng,K.M.(2016)ExistenceofGroundStateSolutionsfortheNonlinearFractional
Schr¨odinger-PoissonSystemswithCriticalSobolevExponent.
JournalofDifferentialEqua-
tions
,
261
,3061-3106.https://doi.org/10.1016/j.jde.2016.05.022
[12]Wang,D.B.,Zhang,H.,Ma,Y.andGuan,W.(2019)GroundStateSign-ChangingSolutions
foraClassofNonlinearFractionalSchr¨odinger-PoissonSystemwithPotentialVanishingat
Infinity.
JournalofAppliedMathematicsandComputing
,
61
,611-634.
https://doi.org/10.1007/s12190-019-01265-y
[13]Yu,Y.,Zhao,F.andZhao,L.(2017)TheConcentrationBehaviorofGroundStateSolutions
foraFractionalSchr¨odinger-PoissonSystem.
CalculusofVariationsandPartialDifferential
Equations
,
56
,ArticleNo.116.https://doi.org/10.1007/s00526-017-1199-4
[14]Yu,Y.,Zhao,F.andZhao,L.(2018)PositiveandSign-ChangingLeastEnergySolutions
foraFractionalSchr¨odinger-PoissonSystemwithCriticalExponent.
ApplicableAnalysis
,
99
,
2229-2257.
[15]Zhang,J.,do
´
O,J.M.andSquassina,M.(2016)FractionalSchr¨odinger-PoissonSystemswith
aGeneralSubcriticalorCriticalNonlinearity.
AdvancedNonlinearStudies
,
16
,15-30.
https://doi.org/10.1515/ans-2015-5024
[16]
{
¡
Ÿ
.
˜
a
Å
½
™
-
Ñ
t
•
§
)
•
3
5
[J].
A^
ê
Æ
,2010,23(3):648-652.
[17]Willem,M.(1996)MinimaxTheorems.Birkh¨auser,Bosten.
DOI:10.12677/aam.2023.1241771712
A^
ê
Æ
?
Ð