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AdvancesinAppliedMathematics
A^
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,2023,12(4),1567-1573
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.124161
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ux
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•
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lim
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ExistenceofSolutionfor
KirchhoffEquationwith
Sign-ChangingWeight
LipingChen
CollegeofScience,LanzhouUniversityofTechnology,LanzhouGansu
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A^
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Ð
,2023,12(4):1567-1573.
DOI:10.12677/aam.2023.124161
•
s
±
Received:Mar.19
th
,2023;accepted:Apr.15
th
,2023;published:Apr.24
th
,2023
Abstract
Inthispaper,wedealwiththeexistenceresultofKirchhoffequationwithsign-
changingweight
−
(
a
+
b
Z
R
3
|∇
u
|
2
dx
)∆
u
+
u
=
V
(
x
)
|
u
|
p
−
1
ux
∈
R
3
,
where
a,b>
0
,
3
<p<
5
,
V
(
x
)
isacontinuousandsign-changingfunctionsuchthat
lim
|
x
|→∞
V
(
x
) =
V
∞
<
0
.
Keywords
KirchhoffEquation,NonlocalTerm,VariationMethods,Sign-ChangingWeight
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.
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−
(
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dx
)∆
u
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u
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(
x
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|
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p
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ux
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∞
=lim
|
x
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V
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x
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<
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(1)
5
g
u
˜
„
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−
(
a
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b
Z
Ω
|∇
u
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dx
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u
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f
(
x,u
)
,
(2)
DOI:10.12677/aam.2023.1241611568
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3
[22]
¥
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Schr¨odinger-Poisson
X
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∆
u
+
u
+
φu
=
a
(
x
)
|
u
|
p
−
1
u,x
∈
R
3
,
−
∆
φ
=
k
(
x
)
u
2
,x
∈
R
3
,
(4)
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3
≤
p<
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a
(
x
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3
R
3
¥
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ë
Y
C
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lim
|
x
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a
(
x
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a
∞
<
0,
k
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k
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2.
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R
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e
(
u,v
) =
Z
R
3
∇
u
∇
v
+
uv,
k
u
k
= (
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1
/
2
.
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2
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3
|
u
|
2
dx
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4
(
Z
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3
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2
−
1
p
+1
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R
3
V
(
x
)
|
u
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p
+1
dx.
-
V
(
x
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V
+
(
x
)
−
V
−
(
x
)
,
(5)
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¥
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+
(
x
) =
V
(
x
)
,
X
J
V
(
x
)
≥
0
,
0
,
X
J
V
(
x
)
<
0
,
(6)
Ú
V
−
(
x
) =
0
,
X
J
V
(
x
)
≥
0
,
−
V
(
x
)
,
X
J
V
(
x
)
<
0
.
(7)
DOI:10.12677/aam.2023.1241611569
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k
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0
(
u
n
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→
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.
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y
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n
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(8),
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|
u
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(
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3
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u
n
|
2
dx
)
2
−
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R
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V
(
x
)
|
u
n
|
p
+1
dx
=
o
(1)
k
u
n
k
,
(9)
Ú
a
2
Z
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3
|∇
u
n
|
2
dx
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1
2
Z
R
3
|
u
n
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2
dx
+
b
4
(
Z
R
3
|∇
u
n
|
2
dx
)
2
−
1
p
+1
Z
R
3
V
(
x
)
|
u
n
|
p
+1
dx
≤
c.
(10)
|
^
(9)
Ú
(10),
k
(
1
2
−
1
p
+1
)[
Z
R
3
(
a
|∇
u
n
|
2
+
|
u
n
|
2
)
dx
]+
b
(
1
4
−
1
p
+1
)(
Z
R
3
|∇
u
n
|
2
dx
)
2
≤
c
+
o
(1)
k
u
n
k
.
(11)
du
p>
3,
Œ
±
(
1
2
−
1
p
+1
)
min
{
a,
1
}k
u
n
k
2
≤
c
+
o
(1)
k
u
n
k
.
(12)
¤
±
k
k
u
n
k≤
C
.
Ï
d
,
·
‚
b
3
H
1
(
R
3
)
¥
u
n
*u
Ú
n
→∞
ž
Z
R
3
|∇
u
n
|
2
dx
→
A
(13)
Š
â
(5),(9)
±
9
k
u
n
k≤
C
,
k
a
Z
R
3
|∇
u
n
|
2
dx
+
Z
R
3
|
u
n
|
2
dx
+
b
(
Z
R
3
|∇
u
n
|
2
dx
)
2
+
Z
R
3
V
−
(
x
)
|
u
n
|
p
+1
dx
=
Z
R
3
V
+
(
x
)
|
u
n
|
p
+1
dx
+
o
(1)
.
(14)
d
Γ
0
(
u
n
)
→
0
Œ
a
Z
R
3
∇
u
∇
vdx
+
Z
R
3
uvdx
+
bA
Z
R
3
∇
u
∇
vdx
+
Z
R
3
V
−
(
x
)
|
u
|
p
−
1
uvdx
=
Z
R
3
V
+
(
x
)
|
u
|
p
−
1
uvdx,
∀
v
∈
H
1
(
R
3
)
.
(15)
DOI:10.12677/aam.2023.1241611570
A^
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Ð
•
s
±
3
(15)
¥
v
=
u
,
Œ
±
a
Z
R
3
|∇
u
|
2
dx
+
Z
R
3
u
2
dx
+
bA
Z
R
3
|∇
u
|
2
dx
+
Z
R
3
V
−
(
x
)
|
u
|
p
+1
dx
=
Z
R
3
V
+
(
x
)
|
u
|
p
+1
dx.
(16)
Š
â
b
^
‡
(
V
),
•
V
+
(
x
)
k
˜
‡
;
|
8
,
l
Œ
Z
R
3
V
+
(
x
)
|
u
n
|
p
+1
dx
=
Z
R
3
V
+
(
x
)
|
u
|
p
+1
dx
+
o
(1)
.
(17)
(
Ü
(14),(16)
±
9
(17),
Œ
±
a
Z
R
3
|∇
u
n
|
2
dx
+
Z
R
3
u
2
n
dx
+
b
(
Z
R
3
|∇
u
n
|
2
dx
)
2
+
Z
R
3
V
−
(
x
)
|
u
n
|
p
+1
dx
=
a
Z
R
3
|∇
u
|
2
dx
+
Z
R
3
u
2
dx
+
bA
2
Z
R
3
|∇
u
|
2
dx
+
Z
R
3
V
−
(
x
)
|
u
|
p
+1
dx
+
o
(1)
.
(18)
b
u
n
9
u
,
K
k
k
u
k
+
o
(1)
<
k
u
n
k
.
?
k
u
k
L
2
+
o
(1)
<
k
u
n
k
L
2
Ú
k∇
u
k
L
2
+
o
(1)
<
k∇
u
n
k
L
2
–
k
˜
‡
´
¤
á
.
Ï
d
·
‚
Œ
±
í
ä
Ñ
a
Z
R
3
|∇
u
n
|
2
dx
+
Z
R
3
u
2
n
dx>a
Z
R
3
|∇
u
|
2
dx
+
Z
R
3
u
2
dx
+
o
(1)
.
(19)
Š
â
(13),(19)
Ú
Fatou
Ú
n
•
a
Z
R
3
|∇
u
n
|
2
dx
+
Z
R
3
u
2
n
dx
+
b
(
Z
R
3
|∇
u
n
|
2
dx
)
2
+
Z
R
3
V
−
(
x
)
|
u
n
|
p
+1
dx
>a
Z
R
3
|∇
u
|
2
dx
+
Z
R
3
u
2
dx
+
bA
2
Z
R
3
|∇
u
|
2
dx
+
Z
R
3
V
−
(
x
)
|
u
|
p
+1
dx
+
o
(1)
,
ù
†
(18)
g
ñ
.
¤
±
,
3
H
1
(
R
3
)
¥
u
n
→
u
.
½
n
1
.
1
y
²
:
˜
•
¡
,
d
Sobolev
Ø
ª
Ú
3
<p<
5,
·
‚
Œ
±
Γ(
u
)
≥
1
2
min
{
a,
1
}k
u
k
2
−k
V
k
L
∞
k
u
k
p
+1
,
(20)
l
•
3
~
ê
α,ρ>
0
¦
Γ
|
B
ρ
≥
α>
0.
,
˜
•
¡
,
À
J
ϕ
∈
H
1
(
R
3
)
¦
suppϕ
⊂
suppa
+
,
K
k
Γ(
tϕ
) =
t
2
2
Z
R
3
(
a
|∇
ϕ
|
2
dx
+
ϕ
2
)
dx
+
t
4
4
(
Z
R
3
|∇
ϕ
|
2
dx
)
2
−
t
p
+1
p
+1
Z
R
3
V
+
|
ϕ
|
p
+1
dx.
(21)
du
p>
3,
•
3
t
0
¦
Γ(
t
0
ϕ
)
<
0.
Ï
d
,
·
‚
y
²
¼
ê
Γ
ä
k
ì
´
A
Û
(
.
?
d
Ú
n
2
.
1
Œ
•
•
§
(1)
–
k
˜
‡
š
²
…
)
.
DOI:10.12677/aam.2023.1241611571
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±
3.
o
(
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²
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Kirchhoff
•
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)
•
3
5
.
ä
N
5
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5
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K
,
X
y
²
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(PS)
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©
z
[1]Kirchhoff,G.(1883)Mechanik.Teubner,Leipzig.
[2]Oplinger,D.W.(1960)FrequencyResponseofaNonlinearStretchedString.
TheJournalof
theAcousticalSocietyofAmerica
,
32
,1529-1538.https://doi.org/10.1121/1.1907948
[3]Lions,J.L.(1978)OnSomeQuestionsinBoundaryValueProblemsofMathematicalPhysics.
In:
North-HollandMathematicsStudies
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