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PureMathematicsnØêÆ,2023,13(3),625-635
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.133067
‘k6Äéê.Kirchhoff
•§)•35
)))ooo
þ°nóŒÆnÆ§þ°
ÂvFϵ2023c222F¶¹^Fϵ2023c323F¶uÙFϵ2023c330F
Á‡
?ؘa‘k6Äéê.Kirchhoff•§)•35Úõ-5¯K"é6Ä‘¼êJÑÜ·
^‡§$^C©•{Úì´½n§3ëêœ¹e§©O•§)•35Úõ)5(
J"
'…c
Kirchhoff•§§éêš‚5‘§6Ä
ExistenceofSolutionsto
LogarithmicKirchhoff
EquationwithaSmall
Perturbation
JingTang
CollegeofScience,UniversityofShanghaiforScienceandTechnology,Shanghai
Received:Feb.22
nd
,2023;accepted:Mar.23
rd
,2023;published:Mar.30
th
,2023
©ÙÚ^:)o.‘k6Äéê.Kirchhoff•§)•35[J].nØêÆ,2023,13(3):625-635.
DOI:10.12677/pm.2023.133067
)o
Abstract
ThispaperconsiderstheexistenceandmultiplicityofsolutionsforlogarithmicKirch-
hoffequationswithasmallperturbation.Undersomeappropriateconditionsfor
perturbation,usingconstrainedvariationalmethodandMountain PassTheorem,the
existence and multiplicityof solutions areobtained respectivelywhen parameter small
enough.
Keywords
KirchhoffEquations,LogarithmicNonlinearity,SmallPerturbation
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
•Äe‘k6Äéê.Kirchhoff•§
(
−(a+b
R
Ω
|∇u|
2
dx)∆u= |u|
p−2
ulnu
2
+λf(x,u),inΩ,
u= 0,on∂Ω,
Ù¥Ω⊂R
3
´˜‡äk1w>.k.«•,a,b>0´½~ê,4<p<6,¼êf(x,u)÷
v±e^‡µ
(A)f(x,u) ∈C(Ω);
(B)f(x,u) ≥0,…f(x,0) 6= 0.
Kirchhoff•§•@Š•˜‡½.dKirchhoff[1]JÑ,35nØ!>^Æ!šÚîåÆ
ÔnÆ+•ké2•A^.CAc,NõÆöïÄL±e•§
(
−(a+b
R
Ω
|∇u|
2
dx)∆u= f(x,u),inΩ,
u= 0,on∂Ω,
(1)
XMa[2]ÏL¦^C©•{ïÄ•§(1))•35,Alves[3]|^.:nØaq(J.
DOI:10.12677/pm.2023.133067626nØêÆ
)o
•C,kNõÆö|^Œ‰ŒC©•{,Xµé¡ì´½n!½nïÄ•§(1))•3
59)ˆ«/,·‚Œ±ë„©z[4–11].AO/,š‚5‘f(x,u)•|Üœ/ž,7[
²[12]•§(1)õ)•35,Wen3[13]¥|^ÿÀÝnر9'uéê‘#O,
•§CÒ)•35.
Éþã©z 9ƒ'¤Jéu,©ïÄ‘kéêš‚5‘±96ÄKirchhoff•§)
•35.duš‚5‘´déêš‚5‘Ú=äkëY5Ÿ¼êf(x,u)|Ü¤,¦yì´
AÛ(Ú(PS)Sk.5C›©(J,ù•´©M#:ÚI‡ŽÑJ:.
e¡‰Ñ©̇(J.
½½½nnn1.1bΩ⊂R
3
´˜‡>.1wk.«•,a,b>0´½~ê,4 <p<6,¼
êf(x,u)÷v^‡(A),K•3~êλ
∗
>0,¦λ∈(0,λ
∗
)ž•§(1)•3˜‡).
½½½nnn1.2bΩ⊂R
3
´˜‡>.1wk.«•,a,b>0´½~ê,4 <p<6,¼
êf(x,u)÷v^‡(A)Ú(B),K|λ|vž,•§(1)•3,˜‡)v
λ
,¿…3H
1
0
(Ω)˜m
þ,λ→0ž,v
λ
→0.
2.ý•£
Äk`²©¥˜ÎÒ.˜mH
1
0
(Ω)L«ÊÏSobolev˜m,§‰ê´
kuk=

Z
Ω
|∇u|
2
dx

1/2
,
˜mL
s
(Ω)(2 ≤s≤6)‰ê½Â•
|u|
s
=

Z
Ω
|u|
r
dx

1
s
.
´••§(1)éAUþ•¼•
I(u) =
a
2
Z
Ω
|∇u|
2
dx+
b
4

Z
Ω
|∇u|
2
dx

2
+
2
p
2
Z
Ω
|u|
p
dx−
1
p
Z
Ω
|u|
p
lnu
2
dx−λ
Z
Ω
F(x,u)dx.
ÏL{üOŽ,éup∈(4,6)Úq∈(p,6),Œ±
lim
t→0
|t|
p−1
ln|t|
t
= 0,lim
t→∞
|t|
p−1
ln|t|
|t|
q−1
= 0,
=éu>0,•3C

>0,¦é?¿t∈R\{0},k
|t|
p−1
ln|t|≤|t|+C

|t|
q−1
.(2)
ddŒš‚ 5‘
R
Ω
|u|
p
lnu
2
dx3˜mH
1
0
(Ω)þ´k½Â,I(u)•˜‘3˜mH
1
0
(Ω)
þŒU´Ã½Â.Ïd,éI(u)?1±e?lŽÑd(J.
DOI:10.12677/pm.2023.133067627nØêÆ
)o
ëY¼êβ
k
(t)÷vXe^‡µ
β
k
(t) =
(
1,|t|≤k,
0,|t|≥k+1;
k<|t|<k+1ž,0 <β
k
(t) <1.½Â
f
k
(x,t) = β
k
(t)f(x,t),F
k
(x,t) =
Z
t
0
f
k
(x,s)ds,(3)
K•3~êC(k),k
|f
k
(x,t)|≤C(k),|F
k
(x,t)|≤C(k).(4)
À
¯
λ(k) >0,¦éu?¿x∈Ω,t∈R,k∈N,k
¯
λ(k)|f
k
(x,t)|≤1,
¯
λ(k)|F
k
(x,t)|≤1,
¯
λ(k)tf
k
(x,t) ≤1.(5)
dd,éuk∈N,λ∈

0,
¯
λ(k)

,½Â?•¼•
I
k
(u) =
a
2
Z
Ω
|∇u|
2
dx+
b
4

Z
Ω
|∇u|
2
dx

2
+
2
p
2
Z
Ω
|u|
p
dx
−
1
p
Z
Ω
|u|
p
lnu
2
dx−λ
Z
Ω
F
k
(x,u)dx.(6)
|^[11]aqy²,d(2)ªÚ(6)ª,·‚Œ±I
k
(u)∈C
1
(H
1
0
(Ω),R),±9éu?¿
u,v∈H
1
0
(Ω),¤á
(I
0
k
(u),v) =a
Z
Ω
|∇u∇v|dx+b
Z
Ω
|∇u|
2
dx
Z
Ω
|∇u∇v|dx
−
Z
Ω
|u|
p−2
uvlnu
2
dx−λ
Z
Ω
f
k
(x,u)vdx.(7)
3.½n1.1y²
ÚÚÚnnn3.1ba,b>0´½~ê,4 <p<6,p<q<6,K•3λ
∗
(k) >0,¦é?¿
λ∈(0,λ
∗
),k
(1)•3u
0
∈H
1
0
(Ω)Ú~êr>0,ku
0
k>rž,I
k
(u
0
) <0;
(2)•3~êρ>0,|u|= rž,I
k
(u) ≥ρ.
yyy²²²(1)|^Sobolevi\½n,(2)Ú(5),l(6)Œ
I
k
(u) >

a
2
−

p

kuk
2
−
C

p
kuk
q
−λC(k)|Ω|.
DOI:10.12677/pm.2023.133067628nØêÆ
)o
Àλ
∗
(k) >09r,ρ>0,Ké?¿λ∈(0,λ
∗
(k))Ñk
I
k
(u) ≥ρ,Ù¥kuk= r.
(2)duéu?¿s∈(0,+∞),¤á
2(1−s
p
)+ps
p
lns
2
>0,(8)
Ïdé?¿t>0,u∈EÑk
I
k
(tu) ≤
at
2
2
Z
Ω
|∇u|
2
dx+
bt
4
4

Z
Ω
|∇u|
2
dx

2
+
2t
p
p
2
Z
Ω
|u|
p
dx−
t
p
p
Z
Ω
|u|
p
lnu
2
dx
−
t
p
lnt
2
p
Z
Ω
|u|
p
dx+|Ω|
6
at
2
2
kuk
2
+
bt
4
4
kuk
4
+Ckuk
p
−
t
p
p
Z
Ω
|u|
p
lnu
2
dx+|Ω|.(9)
qÏ•p∈(4,6),¤±t→∞ž,I
k
(tu) →−∞.ùÒ¿›X,7½•3u
0
¦I
k
(u
0
) <0.Ún
3.1y..
ÚÚÚnnn3.2ba,b>0´½~ê,4 <p<6,p<q<6,Ké?¿λ∈(0,λ
∗
),I
k
3˜
mH
1
0
(Ω)¥÷v(PS)^‡,Ù¥λ
∗
(k)dÚn3.1‰Ñ.
yyy²²²b{u
n
}⊂H
1
0
(Ω)•I3H
1
0
(Ω)¥(PS)S,Kn→∞ž,•3C>0,¦
n→∞,¤á
|I
k
(u
n
) |<C,kI
0
k
(u
n
)k→0.(10)
d(5)−(7)ª,Œ±í
C>I
k
(u
n
)−
1
p
(I
0
k
(u
n
),u
n
)
>
(p−2)a
2p
ku
n
k
2
+
(p−4)b
4p
ku
n
k
4
+
2
p
2
Z
Ω
|u
n
|
p
dx
−λ
Z
Ω

F
k
(x,u
n
)−
1
p
f
k
(x
n
,u
n
)u
n

dx
>
(p−2)a
2p
ku
n
k
2
−
p+1
p
|Ω|.(11)
ùÒ¿›X{u
n
}3˜mH
1
0
(Ω)þk.,•3{u
n
},‡f£Ø”EP•{u
n
}¤9u∈
H
1
0
(Ω),n→∞žk









u
n
*u3H
1
0
(Ω)S,
u
n
→u3L
r
(Ω)S(1 ≤r<6),
u
n
→ua.e.x∈Ω.
(12)
DOI:10.12677/pm.2023.133067629nØêÆ
)o
|^(2),(5)ÚH¨olderØª




Z
Ω

|u
n
|
p−2
u
n
lnu
2
n
−|u|
p−2
ulnu
2

(u
n
−u)dx




≤
Z
Ω

|u
n
|+C

|u
n
|
q−1
+|u|+C

|u|
q−1

|u
n
−u|dx
≤(|u
n
|
2
+|u|
2
)|u
n
−u|
2
+C(|u
n
|
q−1
q
+|u|
q−1
q
)|u
n
−u|
q
= o
n
(1)(13)
Ú




λ
Z
Ω
(f
k
(x,u
n
)−f
k
(x,u))(u
n
−u)dx




≤λ
Z
Ω
(|f
k
(x,u
n
)|+|f
k
(x,u)|)|u
n
−u|dx
≤2|Ω|
1
2
|u
n
−u|
2
= o
n
(1).(14)
?(Ü(6)-(7)ªk
o
n
(1) = (I
0
k
(u
n
)−I
0
k
(u),u
n
−u)
= aku
n
−uk
2
+
b
2
(ku
n
k
2
Z
Ω
∇u
n
(∇u
n
−∇u)dx−kuk
2
Z
Ω
∇u(∇u
n
−∇u)dx)
−
Z
Ω

|u
n
|
p−2
u
n
lnu
2
n
−|u|
p−2
ulnu
2

(u
n
−u)dx
−λ
Z
Ω
(f
k
(x,u
n
)−f
k
(x,u))(u
n
−u)dx
≥Cku
n
−uk
2
,
Kn→∞ž,kku
n
−uk→0.Ún3.2y..
éuÚn3.1‰½u
0
,½Â
Γ :=

γ∈C

[0,1],H
1
0
(Ω)

: γ(0) = 0,γ(1) = u
0

,
d
k
:=inf
γ∈Γ
max
0≤t≤1
I
k
(γ(t)).
ÚÚÚnnn3.3ba,b>0´½~ê,4 <p<6,p<q<6,Ké?¿λ∈(0,λ
∗
),I
k
ä
k.Šd
k
,Ù¥λ
∗
(k)dÚn3.1‰Ñ.
yyy²²²dy²aqu[14],ùpÑ.
ÚÚÚnnn3.4ba,b>0´½~ê,4 <p<6,p<q<6,Ké?¿λ∈(0,λ
∗
),•3†
kÃ'~êM,¦ku
k
k<M,Ù¥u
k
´dÚn3.3½ÂI
k
.:,λ
∗
(k)dÚn3.1
‰Ñ.
DOI:10.12677/pm.2023.133067630nØêÆ
)o
yyy²²²½Â#•¼Xe
ˆ
I(u) =
a
2
Z
Ω
|∇u|
2
dx+
b
4
(
Z
Ω
|∇u|
2
dx)
2
+
2
p
2
Z
Ω
|u|
p
dx−
1
p
Z
Ω
|u|
p
lnu
2
dx+|Ω|.(15)
´I
k
(u) ≤
ˆ
I(u).$^aquy²Ún3.1•{Œ,
ˆ
I(u)•äkì´AÛ(.Ïd,e½Â
ˆ
d:=inf
γ∈Γ
max
0≤t≤1
ˆ
I(γ(t)).
Kk
d
k
≤
ˆ
d.
?d(11)ªŒ
ˆ
d≥d
k
≥

1
2
−
1
p

ku
k
k
2
−
p−1
p
|Ω|,
u´,ku
k
k≤M,Ù¥M>0…†kÃ'.Ún3.4y..
ÚÚÚnnn3.5ba,b>0´½~ê,4 <p<6,p<q<6,Ké?¿λ∈(0,λ
∗
),K•3
†kÚλÃ'~ê
˜
M,¦|u
k
|
∞
≤
˜
M,Ù¥u
k
´dÚn3.3½ÂI
k
.:,λ
∗
(k)dÚ
n3.1‰Ñ.
yyy²²²••Bå„,Ø”bu= u
k
.éu?¿m∈N,‰½β>1,½Â
B
m
=
n
x∈Ω;|u|
β−1
≤m
o
,D
m
= Ω\B
m
Ú
u
m
=
(
u|u|
2(β−1)
,x∈B
m
,
m
2
u,x∈D
m
.
´u
m
∈H
1
0
(Ω),u
m
≤|u|
2β−1
9
∇u
m
=
(
(2β−1)|u|
2(β−1)
∇u,x∈B
m
,
m
2
∇u,x∈D
m
,
ùÒ¿›Xu
m
Œ±Š•ÿÁ¼ê.Ødƒ,„Œ
Z
Ω
∇u∇u
m
dx= (2β−1)
Z
B
m
|u|
2(β−1)
|∇u|
2
dx+m
2
Z
D
m
|∇u|
2
dx.(16)
-
v
m
=
(
u|u|
β−1
,x∈B
m
,
mu,x∈D
m
,
lv
2
m
= uu
m
≤|u|
2β
±9
∇v
m
=
(
β|u|
β−1
∇u,x∈B
m
,
m∇u,x∈D
m
.
DOI:10.12677/pm.2023.133067631nØêÆ
)o
?k
Z
Ω
|∇v
m
|
2
dx= β
2
Z
B
m
|u|
2(β−1)
|∇u|
2
dx+m
2
Z
D
m
|∇u|
2
dx.(17)
2d(16)Ú(17)ªŒ
Z
Ω

|∇v
m
|
2
−∇u∇u
m

dx= (β−1)
2
Z
B
m
|u|
2(β−1)
|∇u|
2
dx.(18)
òu
m
“\(7)ª,Kk
a
Z
Ω
∇u∇u
m
dx+b
Z
Ω
|∇u|
2
dx
Z
Ω
∇u∇u
m
dx=
Z
Ω
|u|
p−1
u
m
lnu
2
dx+λ
Z
Ω
f
k
(x,u)u
m
dx.
d(4),(16)Ú(18)ªk
a
Z
Ω
|∇v
m
|
2
dx+β
2
b
Z
Ω
|∇u|
2
dx
Z
Ω
uu
m
dx
= a(β−1)
2
Z
B
m
|u|
2(β−1)
|∇u|
2
dx+a
Z
Ω
∇u∇u
m
dx+β
2
b
Z
Ω
|∇u|
2
dx
Z
Ω
uu
m
dx
≤a

(β−1)
2
2β−1
+1

Z
B
m
∇u∇u
m
dx+β
2
b
Z
Ω
|∇u|
2
dx
Z
Ω
uu
m
dx
≤β
2
(a
Z
Ω
∇u∇u
m
dx+b
Z
Ω
|∇u|
2
dx
Z
Ω
uu
m
dx)
≤β
2

Z
Ω
|u|
p−1
|u
m
|lnu
2
dx+λ
Z
Ω
f
k
(x,u)|u
m
|dx

≤β
2


Z
Ω
|u||u
m
|dx+C

Z
Ω
|u|
q−1
|u
m
|dx+
Z
Ω
|u
m
|dx

≤β
2


Z
Ω
|u|
2β
dx+C

Z
Ω
|u|
q−2+2β
dx+
Z
Ω
|u|
2β−1
dx

≤β
2
C
1

Z
Ω
|u|
q−2+2β
dx

2β
q−2+2β
+β
2
C

Z
Ω
|u|
q−2+2β
dx
+β
2
C
2

Z
Ω
|u|
q−2+2β
dx

2β−1
q−2+2β
.(19)
Ø”b
R
Ω
|u|
q−2+2β
dx>1,¤á
Z
Ω
|∇v
m
|
2
dx≤β
2
C
Z
Ω
|u|
q−2+2β
dx.
ŠâSobolevØªŒ

Z
B
m
|v
m
|
6
dx

1
3
≤S
Z
Ω
|∇v
m
|
2
dx≤Sβ
2
C
Z
Ω
|u|
q−2+2β
dx.
DOI:10.12677/pm.2023.133067632nØêÆ
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?˜Ú,(ÜH¨olderØªÚÚn3.4k

Z
B
m
|v
m
|
6
dx

1
3
≤Sβ
2
C|u|
q−2
6
(
Z
Ω
|u|
2βr
1
dx)
1
r
1
≤Sβ
2
CM
q−2
(
Z
Ω
|u|
2βr
1
dx)
1
r
1
,(20)
Ù¥r
1
÷v
1
r
1
+
q−2
6
= 1.Ï•3B
m
¥,k|v
m
|= |u|
β
,¤±

Z
B
m
|u|
6β
dx

1
3
≤Sβ
2
CM
q−2
(
Z
Ω
|u|
2βr
1
dx)
1
r
1
.
-m→∞,ŠâüNÂñ½n,k
|u|
6β
≤β
1
β

SCM
q−2

1
2β
|u|
2βr
1
.(21)
σ=
3
r
1
,Kσ>1.-(21)ª¥β= σ,·‚k
|u|
6σ
≤σ
1
σ

SCM
q−2

1
2σ
|u|
6
.(22)
2ö,-(21)ª¥β= σ
2
,Kk
|u|
6σ
2
≤σ
2
σ
2

SCM
q−2

1
2σ
2
|u|
6σ
.(23)
Ïd,(Ü(22)Ú(23)ªŒ
|u|
6σ
2
≤σ
(
1
σ
+
2
σ
2
)

SCM
q−2

1
2
(
1
σ
+
1
σ
2
)
|u|
6
.
-(21)ª¥β= σ
i
,i= 1,2,...,¿-E±þÚ½Œ
|u|
6σ
i
≤σ

P
i
j=1
j
σ
j


SCM
q−2


1
2
P
i
j=1
1
σ
j

|u|
6
.
-þª¥i→∞,(ÜÚn3.4,·‚k
|u|
∞
≤C|u|
6
≤CM≤
˜
M,
Ù¥
˜
M>0…
˜
MŠ†λÚkÃ'.Ún3.5y..
½½½nnn1.1yyy²²²ÀJ~êk,¦k>
˜
M,Ù¥
˜
MdÚn2.5‰Ñ.-λ
∗
= λ
∗
(k),Ké?¿
λ∈(0,λ
∗
),·‚k|u
k
|
∞
<k.Ïd,Šâf
k
(x,t)ÚF
k
(x,t)½Â,k
f
k
(x,u
k
) = f(x,u
k
)a.e.x∈Ω.
u´u
k
´•§(1)).
DOI:10.12677/pm.2023.133067633nØêÆ
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4.½n1.2y²
5¿,3Ún3.1¥,XJ·‚òrÚρ©OO†¤•~ê,K•3λ
0
∈(0,λ
∗
),¦
λ∈(0,λ
0
)ž,I
k
Eäkì´AÛ(.=ke¡Ún.
ÚÚÚnnn4.1•3~ê0<r
0
<r,0 <ρ
0
<ρ,±90 <λ
0
(k) <λ
∗
(k),K•3u
0
∈H
1
0
(Ω),é
?¿λ∈(0,λ
0
)kI
k
(u
0
) <0ku
0
k>r
0
Ú
I
k
(u) ≥ρ
0
Ù¥kuk= r
0
,(24)
Ù¥r,ρ,λ
∗
(k)dÚn3.1‰½.
ybëY¼êf(x,u)3Ωþ÷vf(x,0) ≥0,f(x,0) 6= 0.dÚn3.1Œ
inf
kuk=r
I
k
(u) ≥ρ>0 ≤I
k
(0).(25)
-B
r
= {u∈H
1
0
(Ω)|kuk≤r},KI
k
3B
r
þŒ±4Š,4Š:•v
λ
,´•v
λ
∈B
r
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¢þ,ŠâEklandC©n,·‚Œ±ÀJ˜‡S{v
n
}⊂B
r
,¦3B
r
þkI
k
(v
n
)→I
k
(v
λ
),
kI
0
k
(v
n
)k→0.w,,•3{v
n
}˜‡fS(Ø”EP•{v
n
}),3B
r
þkv
n
*v
λ
.Š âI
k

feŒëY5,Œ±I
k
(v
λ
)≤liminf
n→∞
I
k
(v
n
),Kv
λ
´I
k
3B
r
þ4Š.2dI
k
(0)≤0,·
‚Œ±I
k
(v
λ
) <0 <I
k
(u
k
),Ù¥u
k
´½n1.1¥¤‰½I
k
.:.Ïdv
λ
6= u
k
.|^
†Ún3.5aqy²Œ±,|λ|<λ
∗
vž,·‚k|v
λ
|
∞
≤C,Ù¥C>0†kÃ'.
dd,Àk>C,Kf
k
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λ
) = f(x,v
λ
),lv
λ
´•§(1)˜‡).
,˜•¡,dÚn4.1Œ•,éz˜‡λ∈(0,λ
0
),v
λ
Ñ´•§(1)).Kdv
λ
∈B
r
,k
kv
λ
k<r.ÄKλ→0ž,kv
λ
k→α,Ù¥0<α<r.e¡,·‚ÀJS{λ
n
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0
),K
n→∞ž,λ
n
→0.ddŒ•,•3ρ
0
>0,¦n¿©Œž,kI
k
(v
λ
n
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I
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λ
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k→0.
ë•©z
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