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AdvancesinAppliedMathematics
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,2021,10(12),4262-4271
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1012453
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StudyonExistenceofSolutions
fora
p
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on
R
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LihuaLiu
YanchengNormalUniversity,Yancheng Jiangsu
Received:Nov.13
th
,2021;accepted:Dec.10
th
,2021;published:Dec.17
th
,2021
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DOI:10.12677/aam.2021.1012453
4
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Abstract
Inthispaper,westudy theexistenceofsolutionsforthefollowing
p
-Kirchhoffelliptic
equation
−
a
+
b
(
Z
R
N
|∇
u
|
p
)
τ
∆
p
u
+
V
(
x
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p
−
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u
=
|
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∈
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(1)
with
a,b,τ>
0
,
1
<p<N,p<q<p
(
τ
+ 1)
<p
∗
.Bythevariationalmethods,weprove
thatproblem(1)admitsatleastonenontrivialsolution.Themaindifficultyistoget
abounded
(
PS
)
sequenceandextractastrongconvergentsubsequencefromit.
Keywords
p
-KirchhoffEllipticEquation,VariationalMethod,
(
PS
)
Sequence,Pohoz
TTT
evIdentity
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2021.10124534265
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|
)
,
if
R<
|
x
|≤
R
+1
,
0
,
if
|
x
|
>R
+1
.
(14)
u
´
n
→∞
ž
§
k
l
R
(
x
)
∈
W
1
,p
(
R
N
)
,
k
l
R
(
x
)
k→∞
±
9
k∇
l
R
k
p
p
=
κ
p
0
((
R
+1)
N
−
R
N
)
σ
N
/N,
Z
R
N
G
µ
(
l
R
(
x
))
≥
G
1
/
2
(
κ
0
)
σ
N
R
N
/N,
(15)
Ù
¥
σ
N
•
S
N
−
1
þ
ü
¥
¡
.
d
(15)
Ú
G
1
/
2
(
κ
0
)
>
0
§
•
3
¿
©
Œ
R
0
¦
R
R
N
G
1
/
2
(
l
R
0
)
≥
1.
e
5
,
P
l
R
0
,β
=
l
R
0
(
x/β
).
u
´
k
k∇
l
R
0
,β
k
p
p
=
β
N
−
p
k
l
R
0
k
p
p
,
Z
R
N
G
1
/
2
(
l
R
0
,β
) =
β
N
Z
R
N
G
1
/
2
(
l
R
0
)
≥
β
N
.
l
,
I
µ
(
l
R
0
,β
) =
a
p
k∇
l
R
0
,β
k
p
p
+
b
p
(
τ
+1)
k∇
l
R
0
,β
k
p
(
τ
+1)
p
−
G
µ
(
l
R
0
,β
)
≤
a
p
β
N
−
p
k∇
l
R
0
k
p
p
+
b
p
(
τ
+1)
β
(
N
−
p
)(
τ
+1)
k∇
l
R
0
k
p
(
τ
+1)
p
−
β
N
(16)
du
p
(
τ
+1)
<p
∗
=
pN
N
−
p
,
u
´
(
N
−
p
)(
τ
+1)
<N
.
?
β
→∞
ž
§
k
I
µ
(
l
R
0
,β
)
→−∞
§
u
´
·
K
y
.
Ú
n
2.3
b
(
H
1
)
−
(
H
2
)
÷
v
.
K
•
3
~
ê
c>
0
§
¦
é
u
¤
k
µ
∈
[
1
2
,
1]
k
c
µ
≥
c>
0
.
y
²
:
Š
â
(2)
Ú
(5)
§
é
u
¤
k
u
∈
W
1
,p
(
R
N
)
Ú
µ
∈
[1
/
2
,
1]
k
I
µ
(
u
) =
a
p
k
u
k
p
+
b
p
(
τ
+1)
k∇
u
k
p
(
τ
+1)
−
µ
q
Z
R
N
|
u
|
q
,
≥
a
p
k
u
k
p
−
C
k
u
k
q
,
(17)
DOI:10.12677/aam.2021.10124534266
A^
ê
Æ
?
Ð
4
á
u
Ï
•
q>p
,
¤
±
•
3
ρ>
0
¦
I
µ
(
u
)
>
0.
A
O
/
,
é
u
k
u
k
=
ρ
,
k
I
µ
(
u
)
≥
c>
0.
½
µ
∈
[
1
2
,
1]
Ú
γ
∈
Γ
µ
.
Š
â
Γ
µ
½
Â
Ú
Ù
ë
Y5
,
•
3
t
γ
∈
(0
,
1)
¦
k
γ
(
t
γ
)
k
E
=
ρ
and
k
γ
(1)
k
>ρ
.
Ï
d
,
é
u
?
Û
µ
∈
[
1
2
,
1],
c
µ
=inf
γ
∈
Γ
µ
max
t
∈
[0
,
1]
I
T
µ
(
γ
(
t
))
≥
inf
γ
∈
Γ
µ
I
T
µ
(
γ
(
t
γ
))
≥
c>
0
.
(18)
y
.
.
3.
(
PS
)
S
Â
ñ
5
Ú
k
.
5
d
Ú
n
2.1-2.3,
é
u
a.e.
µ
∈
[1
/
2
,
1],
•
3
k
.
S
{
u
n
}
in
X
,
÷
v
I
µ
(
u
n
)
→
c
µ
,
(
I
µ
(
u
n
))
0
→
0
,
sup
n
k
u
n
k
<T.
(19)
du
q
∈
(
p,p
∗
)
ž
§
i
\
X
→
L
q
(
R
N
)
´
ë
Y
,
•
d
·
‚
Œ
±
b
u
n
u,
in
X
;
u
n
→
u, inL
q
loc
(
R
N
);
u
n
→
u,a.e.
in
R
N
.
(20)
e
5
,
·
‚
ò
y
²
{
u
n
}
3
X
¥
Â
ñ
.
Ú
n
3.1
b
(
H
1
)
−
(
H
2
)
÷
v
.
é
u
?
Û
µ
∈
[
1
2
,
1]
,
X
J
S
{
u
n
}
k
.
…
÷
v
(20),
@
o
lim
n
→∞
Z
R
N
|
u
n
|
q
=
Z
R
N
|
u
|
q
,
lim
n
→∞
Z
R
N
|
u
n
−
u
|
q
dx
= 0
.
y
²
:
T
Ú
n
y
²
Œ
±
ë
„
©
z
[10].
Ú
n
3.2
(
H
1
)
−
(
H
2
)
÷
v
.
{
u
n
}
•
˜
(
PS
)
c
S
±
9
÷
v
(20),
K3
X
¥
§
u
n
→
u
,
=
,
é
u
¤
k
µ
∈
[1
/
2
,
1]
§
•
¼
I
µ
÷
v
(
PS
)
^
‡
.
y
²
:
d
(12),
(
I
µ
(
u
n
)
−
I
µ
(
u
))
0
(
u
n
−
u
) =
P
n
+
Q
n
+
K
n
,
(21)
Ù
¥
P
n
=
a
+
b
k∇
u
n
k
pτ
p
Z
R
N
(
|∇
u
n
|
p
−
2
∇
u
n
−|∇
u
|
p
−
2
∇
u
)
∇
(
u
n
−
u
)
+
Z
R
N
V
(
x
)(
|
u
n
|
p
−
2
u
n
−
u
p
−
2
u
)(
u
n
−
u
)
,
Q
n
=
b
(
k∇
u
n
k
pτ
−k∇
u
k
pτ
)
Z
R
N
|∇
u
|
p
−
2
∇
u
∇
(
u
n
−
u
)
,
K
n
=
µ
Z
R
N
(
|
u
n
|
q
−
2
u
n
−|
u
|
q
−
2
u
)(
u
n
−
u
)
.
DOI:10.12677/aam.2021.10124534267
A^
ê
Æ
?
Ð
4
á
u
w
,
(
I
µ
(
u
n
)
−
I
µ
(
u
))
0
(
u
n
−
u
)
→
0as
n
→∞
.
(22)
d
Ú
n
3.1
±
9
H¨older
Ø
ª
,
k
|
K
n
|≤
µ
(
k
u
n
k
q
−
1
q
+
k
u
k
q
−
1
q
)
k
u
n
−
u
k
q
→
0
asn
→∞
.
(23)
½
Â
‚
5
•
¼
g
:
X
→
R
§
g
(
ω
) =
Z
R
N
|∇
u
|
p
−
2
∇
u
∇
ω.
5
¿
|
g
(
ω
)
|≤k∇
u
k
p
−
1
k
ω
k
,
Œ
±
w
g
3
X
þ
ë
Y
.
du
3
X
¥
§
u
n
u
,
g
(
u
n
−
u
) =
Z
R
N
|∇
u
|
p
−
2
∇
u
∇
(
u
n
−
u
)
→
0
,asn
→∞
.
(24)
q
Ï
k
u
n
k
3
X
þ
k
.
,
u
´
|
Q
n
|≤
Cg
(
u
n
−
u
)
→
0
,
as n
→∞
.
n
þ
?
Ø
,
·
‚
k
|
P
n
|→
0as
n
→∞
,
ù
·
‚
y
3
X
¥
§
u
n
→
u
.
y
.
.
Š
â
Ú
n
3.2,
•
3
S
{
µ
n
}⊂
[
1
2
,
1]
§
µ
n
→
1
±
9
n
→∞
ž
k
§
I
T
µ
n
(
u
n
) =
c
µ
n
,(
I
T
µ
n
)
0
(
u
n
) =
0
§
u
n
is
´
X
e
•
§
−
(
a
+
b
k∇
u
n
k
p
p
)∆
p
u
n
+
V
(
x
)
|
u
n
|
p
−
2
u
n
=
µ
k
u
n
k
q
−
2
u
n
.
(25)
š
²
…
)
.
e
5
,
·
‚
ò
/
Ï
u
Pohozav
ð
ª
y
²
k
u
n
k
<T
§
•
d
·
‚
Ä
k
ï
á
X
e
ð
ª
.
Ú
n
3.3
u
∈
X
´
X
e
•
§
−
(
a
+
b
k∇
u
k
pτ
p
)∆
p
u
+
V
(
x
)
|
u
|
p
−
2
u
=
µ
|
u
|
q
−
2
u.
(26)
f
)
§
K
÷
v
X
e
ð
ª
(
a
+
b
k∇
u
k
pτ
p
)
N
−
p
p
Z
R
N
|∇
u
|
p
+
1
p
Z
R
N
(
NV
(
x
)+
∇
V
(
x
)
.x
)
|
u
|
p
−
N
q
Z
R
N
|
u
|
q
= 0(27)
y
²
:
du
u
∈
X
´
•
§
(26)
f
)
,
Š
â
I
O
K
z
?
Ø
k
[9],
u
∈
C
2
loc
(
R
N
)
∩
W
1
,p
(
R
N
),
-
y
(
x,u
) =
µ
|
u
|
q
−
2
u
−
V
(
x
)
|
u
|
p
−
2
u
a
+
b
k∇
u
k
pτ
p
.
(28)
u
´
u
∈
X
•
´
X
e
•
§
−
∆
p
u
=
y
(
x,u
)
.
(29)
DOI:10.12677/aam.2021.10124534268
A^
ê
Æ
?
Ð
4
á
u
)
.
|
^
Pohozaev
ð
ª
[11],
N
−
p
p
Z
R
N
|∇
u
|
p
=
Z
R
N
(
NY
(
x,u
)+
Y
1
(
x,u
))(30)
Ù
¥
Y
(
x,u
) =
R
u
0
y
(
x,s
)
ds,Y
1
(
x,u
) =
N
P
i
=1
x
i
∂Y
(
x,u
)
∂x
i
§
(
Ø
¤
á
.
Ú
n
3.4
b
(
H
1
)
−
(
H
3
)
¤
á
.
{
µ
n
}⊂
[
1
2
,
1]
±
9
{
u
n
}⊂
X
¦
µ
n
%
1
,
I
µ
n
=
c
µ
n
Ú
I
0
µ
n
(
u
n
) =
0
.
u
´
S
{
u
n
}
3
X
¥
k
.
.
y
²
:
du
(
I
µ
n
)
0
(
u
n
) = 0,
u
´
Š
â
Ú
n
3.3,
u
n
÷
v
X
e
ª
:
0 =
P
µ
(
u
n
) = (
a
+
b
k∇
u
n
k
pτ
p
)
N
−
p
p
Z
R
N
|∇
u
n
|
p
+
N
p
Z
R
N
V
(
x
|
u
n
|
p
+
1
p
Z
R
N
∇
V
(
x
)
.x
|
u
n
|
p
−
N
q
Z
R
N
|
u
n
|
q
.
(31)
d
I
µ
n
(
u
n
) =
c
µ
n
,I
0
µ
n
(
u
n
)
u
n
= 0
,P
µ
(
u
n
) = 0and(
H
3
)
c
µ
n
=
I
µ
n
−
αP
µ
n
(
u
n
)
−
βI
0
µ
n
(
u
n
)
u
n
=
a
1
p
−
α
(
N
−
p
)
p
−
β
k∇
u
n
k
p
p
+
b
1
p
(
τ
+1)
−
α
(
N
−
p
)
p
−
β
k
u
n
k
pτ
+1
p
+
Z
R
N
1
p
−
Nα
p
−
β
)
V
(
x
)
−
α
p
(
∇
V.x
)
|
u
n
|
p
+
µ
1
q
−
Nα
q
−
β
k
u
n
k
q
q
≥
a
1
p
−
1
p
(
τ
+1)
k∇
u
n
k
p
p
+(
a
+
b
k∇
u
n
k
pτ
p
)
1
p
(
τ
+1)
−
α
(
N
−
p
)
p
−
β
k∇
u
n
k
p
p
+
1
p
−
α
(
N
+
λ
)
p
−
β
Z
R
N
V
(
x
)
|
u
n
|
p
+
µ
β
+
Nα
q
−
1
q
k
u
n
k
q
q
.
(32)
Ù
¥
α
=
(
p
(
τ
+1)
−
q
)
p
∗
N
(
p
∗
−
q
)
p
(
τ
+1)
,β
=
p
∗
−
p
(
τ
+1)
(
p
∗
−
q
)
p
(
τ
+1)
,λ
∈
(0
,
N
(
q
−
p
)(
p
∗
−
p
(
τ
+1))
p
∗
(
p
(
τ
+1)
−
q
)
]
.
u
´
{
ü
O
Ž
1
p
(
τ
+1)
−
α
(
N
−
p
)
p
−
β
= 0
,
1
p
−
α
(
N
+
λ
)
p
−
β
≥
0
,β
+
Nα
q
−
1
q
= 0
.
(33)
d
(32)and(33)
c
µ
n
≥
a
(
1
p
−
1
p
(
τ
+1)
)
k∇
u
n
k
p
p
.
(34)
ù
¿
›
X
∇
u
n
3
L
p
¥
k
.
.
,
˜
•
¡
,
du
c
µ
n
=
I
µ
n
u
n
)
−
1
q
I
0
µ
n
(
u
n
)
u
n
=
a
(
1
p
−
1
q
)+
b
(
1
p
(
τ
+1)
−
1
q
)
k∇
u
n
k
p
p
k∇
u
n
k
pτ
+
1
p
−
1
q
Z
R
N
V
(
x
)
|
u
n
|
p
.
(35)
DOI:10.12677/aam.2021.10124534269
A^
ê
Æ
?
Ð
4
á
u
ù
q
¿
›
X
u
n
3
L
p
¥
k
.
.
y
.
.
4.
š
²
…
)
•
3
5
½
n
1.1
y
²
d
Ú
n
3.4,
·
‚
b
½
k
u
n
k≤
T
,
¤
±
k
I
(
u
n
) =
I
µ
n
(
u
n
)+(
µ
n
−
1)
k
u
n
k
q
q
.
(36)
Ï
•
µ
n
→
1,
Œ
y
{
u
n
}
3
I
þ
÷
v
(
PS
)
^
‡
.
¯¢
þ
,
{
u
n
}
k
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DOI:10.12677/aam.2021.10124534271
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