一类 p-Laplace 方程基态解的存在性与集中性 Existence and Concentration ofGround States for a Class ofp-Laplace Equation

doi:10.12677/PM.2023.132016

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PureMathematicsnØêÆ,2023,13(2),131-148

PublishedOnlineFebruary2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.132016

˜ap−Laplace•§Ä)•35†8¥5

œœœKKK

úô“‰ŒÆêÆ†OŽÅ‰ÆÆ§úô7u

ÂvFÏµ2022c1231F¶¹^FÏµ2023c130F¶uÙFÏµ2023c26F

Á‡

©ïÄp−Laplace•§µ

−∆

u+|u|

p−2

u= Q

(x)|u|

q−2

u,x∈R

Ù¥µ∆

u=div(|∇u|

p−2

∇u)§1 <p<N§p<q<p

∗

N−p

"n→∞ž§k.¼êQ

(x)

gØsupp{Q

}Â •k•:8"·‚æ^å4Ú8¥;5ny²p−Laplace•§Ä

)•35Ú8¥5"

'…c

p−Laplace§Ä)§•35§8¥5

ExistenceandConcentrationof

GroundStatesforaClassof

p-LaplaceEquation

YingShi

CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang

Received:Dec.31

,2022;accepted:Jan.30

,2023;published:Feb.6

,2023

©ÙÚ^:œK.˜ap−Laplace•§Ä)•35†8¥5[J].nØêÆ,2023,13(2):131-148.

DOI:10.12677/pm.2023.132016

œK

Abstract

Inthispaper,westudythefollowingp−Laplaceequation:

−∆

u+|u|

p−2

u= Q

(x)|u|

q−2

u,x∈R

where∆

u=div(|∇u|

p−2

∇u),p>1,N≥1,p<q<p

∗

N−p

(1<p<∞,1≤N≤p).

areboundedfunctionswithself-focusingcoresuppQ

whichshrinkstoaﬁnite

setofpointsasn→∞.Viatheconstraintminimizingmethodandtheconcentration

compactnessprinciple,weprovetheexistenceandconcentrationforgroundstates.

Keywords

p-Laplace,GroundStates,Existence,Concentration

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

‘XêÆ!Ông,‰ÆuÐ,p−Lapace•§˜†´IS2Œ‰ïóŠöïÄ9:

¯K.3[1]¥,Bonanno-Livreay²p−Laplace•§n)•35.3[2]¥,Ferrero-Gazzola

y²g.O•½‡.O•š‚5p−Laplace•§»•)•35.3[3]¥§Liuy²

p-‡‚5g.p−Laplace •§Ä)•35.3[4]¥,Costa-Magalhesy²p−Laplace

Ä•§š²…)•35.

©•Äp−Laplace•§:

−∆

u+|u|

p−2

u= Q

(x)|u|

q−2

u,x∈R

,(E

)

Ù¥: N≥1, 1 <p<N,p<q<p

∗

N−p

.n→∞ž,k.¼êQ

(x)gØsupp{Q

}

Â •k•:8.

51éu•§(E

),¡{x|Q

(x)>0}Ú{x|Q

<0}¥:©O•gàÚà.3©

DOI:10.12677/pm.2023.132016132nØêÆ

œK

z[5]¥,Buryak-TrapaniïÄgàÚàœ/,¿y²fÚVf•35.éu

‘CÒ¼êš‚5 ‡©•§ƒ'(J§ë•[6–9].

p= 2ž,p−Laplace•§Œz•µ

−∆u+u= Q

(x)|u|

q−2

u,u∈H

(Ω),(1)

Ù¥:Q

(x)CÒ.3[10]¥,Ackermann-Szulkiny²µ

(i)n→∞,8Ü{x∈Ω|Q

>0}Â :x

ž,Šöæ^å4•{y²Laplace•§

(1)Ä)•35,¿?˜Úy²ùÄ)3x

?'uH

‰ê8¥.

(ii)n→∞,8Ü{x∈Ω|Q

>0}Â ü‡ ØÓ:x

6= x

ž,Š öæ^ä¼ê•{y

²n¿©ŒžÄ)•359ÙH

‰ê8¥5.dž,ùÄ)'uH

‰êØ

UÓž3x

Úx

?8¥.

•?˜Ú(½Ä)'uH

‰ê3x

?„´x

?8¥,Fang-Wang[11]ÒA½¼ê

(x)é•§(1)?1\ïÄ.e-

(x) =











1,|x|<ε

−1,|x|≥ε

)

Ù¥: ε

→0.Šöæ^å4•{y²(q

)Ä)•35¿ÏLÚ4••§?1'

Ä)'uH

‰ê3x

?8¥.e-

(x) =











,x∈B

−1,Ù§,

)

Ù¥: s

>0,x

6=x

,ε

→0.Šöæ^8¥;ny²Ä)'uH

‰ê3x

8¥.

É©zAckermann-Szulkin [10]ÚFang-Wang [11]éu,©‘3ïÄ•§(E

)Ä)

•35ÚH

‰ê8¥5"

b(q

)¤á,·‚kXe½n.

½n1(i)(E

) –•3˜‡Ä)¶

(ii)u

•(E

)˜‡Ä).Š C†

:= ε

q−p

(ε

x),

DOI:10.12677/pm.2023.132016133nØêÆ

œK

K•3ϕ

f,EP•ϕ

,¦ϕ

→ϕuD

1,p

),…ϕ´•§

−∆

u= Q(x)|u|

q−2

u,x∈R

˜‡Ä).ùp

Q(x) =











1,e|x|<1,

−1,Ù§.

(iii)é?¿δ>0,

lim

n→0

|x|≥δ

(|∇u

+|u

)dx

(|∇u

+|u

)dx

= 0,lim

n→0

|x|≥δ

|∇u

= 0.

52Sobolev˜mD

1,p

) := {u|∇u∈L

),u∈L

∗

)},Ù¥‰ê½Â•

kuk



|∇u|



b(q

) ¤á.Ø”˜„5,Ø”s

≤s

¿½Âβ=





q−p

−N





q−p

(

= s

β<1

(

β≤1,

·‚kXe½n.

½n2(i)(E

)–•3˜‡Ä)¶

(ii)u

•(E

)˜‡Ä).Š C†

(x) := ε

q−p

(ε

x),

K•3ϕ

f,EP•ϕ

,¦ϕ

(x+

) →ϕuD

1,p

),…ϕ´•§

−∆

ϕ= K

(x)|ϕ|

q−2

ϕ,x∈R

˜‡Ä).ùp











,x∈B

(0),

−1,Ù§;

DOI:10.12677/pm.2023.132016134nØêÆ

œK

(iii)éu?¿δ>0,

lim

n→0

|x−x

|≥δ

(|∇u

+|u

)dx

(|∇u

+|u

)dx

= 0,lim

n→0

|x−x

|≥δ

= 0.

©(Xe:1!æ^ å4y²½n1;1n!|^å4Ú8¥;•{y²½n

2.ÎÒ`²:C,C

,···“L~ê;B

(y) := {x∈R

||x−y|<r}L«±y•%r•Œ»

¥;|u|

L«L

)¥‰ê;Q

(x) = max{±Q(x),0}L«¼êQ(x)Ü†KÜ.

2.½n1y²

!æ^å4•{•§(E

)Ä)•35,¿ÏL†4••§'Œ±Ä

)8¥5§?¤½n1•ªy².

Äkòε

•ε,ŠCþO†

ϕ(x) := ε

q−p

u(εx),

Kd(E

)

−∆

ϕ+ε

|ϕ|

p−2

ϕ= Q(x)|ϕ|

q−2

ϕ,x∈R

.(2)

5¿

Q(x) =











1,|x|<1,

−1,|x|≥1.

/ªþ,•§(2)4••§•

−∆

ϕ= Q(x)|ϕ|

q−2

ϕ,x∈R

.(3)

•Äå4¯K:

= inf



|∇u|



u∈D

1,p

Q(x)|u|

dx= 1



·‚keãÚn.

Ún1•3¼êu

∈D

1,p

)¦I

?Œˆ,¿…ϕ

= I

q−p

••§(3)Ä

).

y².{u

}⊂D

1,p

)•I

˜4zS,@on→∞ž,

|∇u

dx→I

Q(x)|u

dx= 1,n= 1,2,···.

DOI:10.12677/pm.2023.132016135nØêÆ

œK

Äk,·‚y²I

>0.¯¢þ,dQ(x)½ÂŒ•

|x|<1

dx= 1+

|x|≥1

dx≥1.(4)

qdH¨olderØªÚSobolevØªŒ

|x|<1

dx≤C



|x|<1

∗



∗

≤C



|∇u



→CI

.(5)

Ïd,d(4) Ú(5)I

≥C

−p/q

>0.

½Â,{u

}3D

1,p

)¥k..fS,EP•{u

},Kn→∞ž

1,p

),u

→u

a.e.uR

→u

loc

),∀1 <r<p

∗

5¿|u

|•4zS,KØ”u

≥0.Ïdu

≥0 a.e.uR

.é(4)e4•,dFatou Ú

n

|x|<1

dx≥1+

|x|≥1

dx,

Q(x)u

dx≥1.

Q(x)u

dx>1,Kd‰êfeŒëY5

0 <I

≤

|∇u



Q(x)u



|∇u

dx≤liminf

n→∞

|∇u

dx= I

gñ.Ïd, I

≥0?ˆ,…u

6= 0.

-ϕ

= I

q−p

,Kϕ

∈D

1,p

)÷v

−∆

= Q(x)ϕ

q−1

≥0,

dý•§K5,

∈C

)∩C

(

)∩C(R

2dý•§r4ŒŠnϕ

>0.Ïd,•§(3)•3Ä).2

e5§•Äå4¯Kµ

= inf



(|∇u|

+ε

|u|

)dx



u∈W

1,p

Q(x)|u|

dx= 1



DOI:10.12677/pm.2023.132016136nØêÆ

œK

éu0 <ε≤1,aquÚn1y²,ØJeã·K.

·K1•3¼êu

∈W

1,p

)¦I

?Œˆ.?˜Ú,ϕ

q−p

••§(2)

˜‡Ä).

•y²•§(2)Ä)H

‰ê8¥5,·‚‡^keãÚn.

Ún2lim

ε→0

= I

y².dÚn1,•3¼êu

∈D

1,p

)÷v

Q(x)u

dx= 1,

|∇u

dx= I

1wä¼êχ(x)÷v

0 ≤χ(x) ≤1,χ(x) = 1,x∈B

(0),χ(x) = 0,x∈B

(0).

-w

= c

∈W

1,p

),Ù¥:



Q(x)χ



−

,χ

(x) = χ





Q(x)|w

dx= 1,

KdI

½Â

limsup

ε→0

≤limsup

ε→0

(|∇w

+ε

)dx=

|∇w

dx.

5¿n→∞ž,c

→1, χ

→u

1,p

),=w

→u

1,p

).Ïd,-n→∞,

limsup

ε→0

≤

|∇u

dx= I

u

>0•I

ˆ¼ê,KdI

½Â

liminf

ε→0

= liminf

ε→0

(|∇u

+ε

)dx≥liminf

ε→0

|∇u

dx≥I

Ïd,lim

ε→0

= I

.y..2

•§ÄuþãO,·‚y²½n1.

½n1 y².d·K1,•§(2)• 3Ä).ÏLCþO†ØJy²•§(E

)–•3

˜‡Ä).Ïd,(i)¤á.

u

••§(E

)Ä),Kϕ

= ε

q−p

(ε

x)•(2)Ä).Ïd,•3I

ˆ

DOI:10.12677/pm.2023.132016137nØêÆ

œK

¼êu

∈W

1,p

)¦ϕ

= I

q−p

.dÚn2,u

1,p

)¥k..Ïd,3f¿

Âe,ε

→0ž,

1,p

),u

→u

a.e.R

→u

inL

loc

),1 <r<p

∗

|x|<1

dx= 1+

|x|≥1

dx.(6)

dFatouÚn

Q(x)u

dx≥1.

d‰êfeŒëY5,Ún2,ÚI

½ÂŒ

|∇u

dx≤liminf

→0

|∇u

≤liminf

→0

(|∇u

+ε

)dx

=lim

→0

= I

≤

|∇u



Q(x)u



Ïd,I

?Œˆ,=

|∇u

dx= I

>0,

Q(x)u

dx= 1.(7)

?˜Ú§·‚

lim

ε→0

|∇u

dx=

|∇u

dx,lim

ε→0

dx= 0.

?,dfÂñÚÄØª

|ξ

≥|ξ

+p|ξ

p−2

·(ξ

−ξ

)+C(p)|ξ

−ξ

,∀ξ

,ξ

∈R

,p>2

Œ

lim

ε→0

|∇(u

−u

dx= 0.

-ϕ

:=I

q−p

≥0,KdÚn1y²L§ØJyϕ

••§(3)Ä).ŠâÚn2,•

§(2)Ä)ϕ

1,p

)¥Âñ4••§(3)Ä)ϕ

.Ïd,(ii)¤á.

d(6),(7),ÚBr´ezis-LiebÚnØJ

lim

→0

dx=

dx,lim

→0

−u

dx= 0.

DOI:10.12677/pm.2023.132016138nØêÆ

œK

Ïd,é?¿δ>0,

lim

→0

|x|≤

dx=

dx.

5¿

(x) = ε

−

q−p

−

q−p

(ε

−1

x),

Kn→+∞ž,

|x|≥δ

≤

|x|≥δ





|x|≥

Q(x)|u

|x|≥

dx→0.

lim

→0

|x|≤

|∇u

dx=

|∇u

dx,lim

→0

dx= 0,

Kn→∞ž,dÚn2

|x|≥δ

(|∇u

)dx

(|∇u

)dx

= I

−1

|x|≥

(|∇u

+ε

)dx→0.

Ïd,(iii)¤á.y..2

3.½n2y²

!ÏL†4••§?1',æ^8¥;5ny²•§(E

)Ä)'uH

‰ê3A½

:?8¥,?¤½n2 y².

Ø”b8¥:x

= 0, ε

•ε.ŠCþO†

ϕ(x) = ε

q−p

u(εx),

Kd•§(E

)

−∆

ϕ+ε

|ϕ|

p−2

ϕ= Q

(x)|ϕ|

q−2

ϕ,x∈R

.(8)

Ù¥:

(x) =











,x∈B

(0),

,x∈B

(

−1,Ù§.

DOI:10.12677/pm.2023.132016139nØêÆ

œK

dÚn1y²L§ØJuyå4¯K

= inf



(|∇u|

+ε

|u|

)dx



u∈W

1,p

(x)|u|

dx= 1



•3ˆ¼êu

∈W

1,p

).-ϕ

= J

q−p

,Kdý•§K5nØÚr•ŒŠnŒ

•ϕ

´•§(8)Ä).

aq/,å4¯K

= inf



|∇u|



u∈D

1,p

(x)|u|

dx= 1



•3ˆ¼êu

∈D

1,p

),…ϕ

= J

q−p

´4••§

−∆

ϕ= K

(x)|ϕ|

q−2

ϕ,x∈R

(9)

Ä),Ù¥:

(x) =







,x∈B

(0),

−1,Ù§.

Äk§·‚y²eãÚn.

Ún3limsup

ε→0

≤J

y².ØJy²J

•3ˆ¼êu

∈D

1,p

),K

(x)u

dx= 1,

|∇u

dx= J

1wä¼êχ(x)÷v

0 ≤χ(x) ≤1,χ(x) = 1,x∈B

(0),χ(x) = 0,x∈B

(0).

(x) = c

∈W

1,p

Ù¥:

(x) = χ







(x)χ

(x)u

(x)dx



−

Q

(x+

) = s

ž,



.Kε>0¿©ž,é∀n<



−r





−r



−|x+



≥2

|x|

= 2.

DOI:10.12677/pm.2023.132016140nØêÆ

œK

džχ

(x) = 0.Ïd,dQ

(x)ÚK

(x)½Â

(x)w



x−



dx=





dx=

(x)w

dx= 1.

¤±§dJ

½Â

limsup

ε→0

≤limsup

ε→0



|∇w

(·−x

/ε)|

+ε

(·−x

/ε)



dx=

|∇w

dx.

w,,n→∞ž,c

→1…χ

→u

1,p

),Kw

(x)→u

1,p

).Ïd,

n→∞ž,limsup

ε→0

≤J

.y..2

£

(x) =











,x∈B

(0),

−1,Ù§,

(x) =











,x∈B

(0),

−1,Ù§,

½Â

(x) =















,x∈B

(0),

−

,Ù§.

β=





q−p

−N





q−p

·‚b

= s

,β<1, ½ös

,β≤1.

53bs

≤s

.dK

ÚK

½Â,x∈B

(0)ž,K

(x) = K

(x) = s

;x/∈B

(0)

ž,K

(x) <−1 = K

(x).Ïd, K

≤K

-u

∈W

1,p

)´J

ˆ¼ê,K

(|∇u

+ε

)dx= J

Q(x)u

dx= 1.

dÚn3Œ, {u

}3D

1,p

)¥k..e¡y²ε>0¿©ž,u

?8¥.

Ún4·‚k

liminf

ε→0





dx>0,lim

ε→0

(0)

dx= 0.

y².œ/1

lim

ε→0

(0)

dx=





dx= 0.

DOI:10.12677/pm.2023.132016141nØêÆ

œK

¯¢þ,ds

>0,s

>0,Ú

(0)

dx+s





dx≥1

•,œ/1Ø¤á.

œ/2.

liminf

ε→0

(0)

dx>0,liminf

ε→0





dx>0.

1,ε





q−p





,ϕ

2,ε

(x) = u





,(10)

liminf

ε→0

(0)

−1





1,ε

dx>0,liminf

ε→0

(0)

2,ε

dx>0.

duϕ

1,ε

Úϕ

2,ε

1,p

)¥k.,K3f¿Âe,ε→0ž,

i,ε

*ϕ

i,0

1,p

),ϕ

i,ε

→ϕ

i,0

loc

),ϕ

i,ε

→ϕ

i,0

a.e.uR

,i= 1,2.

éui= 1,2,dϕ

i,ε

>0,ϕ

i,0

≥0.?˜Ú, ϕ

i,0

6= 0,i= 1,2.ε>0¿©ž,é∀R<

2ε

·‚kB

(

)∩B

(0) = ∅.5¿

−

(x) = K

−

(x),Q

−

(x+

) = K

−

(x),∀x∈B

(0),

(0)

dx+





≥1+

(0)

−

dx+





−

= 1+

(0)

−1





−





1,ε

dx+

(0)

−

2,ε

= 1+

(0)

−1

−

1,ε

dx+

(0)

−

2,ε

= 1+

(0)

−1

−

1,0

dx+

(0)

−

2,0

dx+o(1),

(0)

dx+





dx=

(0)



−1

1,ε

2,ε



(0)



−1

1,0

2,0



dx+o(1).

DOI:10.12677/pm.2023.132016142nØêÆ

œK

-R→∞,·‚k

(0)



−1

1,0

2,0



dx≥1+



−1

−

1,0

−

2,0



dx.

Ïd,

−1

1,0

dx+

2,0

dx≥1.

e¡©n«œ¹?Ø:

−1

1,0

dx≥1,

2,0

dx≤0;(11)

−1

1,0

dx≤0,

2,0

dx≥1;(12)

−1

1,0

dx>0,

2,0

dx>0.(13)

b(11)¤á.dÚn3

≥limsup

ε→0

= limsup

ε→0

(|∇u

+ε

)dx

≥limsup

ε→0

(0)

(|∇u

+ε

)dx+





(|∇u

+ε

)dx

≥

(0)

−1

|∇ϕ

1,0

dx+

(0)

|∇ϕ

2,0

dx.

-R→∞,dβ≤1Ú53

−1

|∇ϕ

1,0

dx≥

−1

|∇ϕ

1,0



−1

1,0



−1

|∇ϕ

1,0



1,0



≥

|∇ϕ

1,0



1,0



≥J

gñ.

DOI:10.12677/pm.2023.132016143nØêÆ

œK

b(12)¤á.aqu(11)?Ø,·‚k

≥limsup

ε→0

(|∇u

+ε

)dx

≥

−1

|∇ϕ

1,0

dx+

|∇ϕ

2,0

|∇ϕ

2,0

dx≥

|∇ϕ

2,0



2,0



≥J

gñ.

b(13)¤á.d53,β≤1,ÚJ

½Â

−1

|∇ϕ

1,0

dx≥

|∇ϕ

1,0



1,0





−1

1,0



≥J



−1

1,0



Ïd,aqu(11)?Ø,·‚k

≥limsup

ε→0

(|∇u

+ε

)dx

≥

−1

|∇ϕ

1,0

dx+

|∇ϕ

2,0

≥J



−1

1,0





2,0



gñ.

Ïd,ÏLüØ(11),(12),Ú(13),œ/2 Ø¤á.

œ/3

liminf

ε→0

(0)

dx>0,lim

ε→0





dx= 0.

d(10)ØJϕ

1,ε

÷v

−∆

ϕ+r

−p

|ϕ|

p−2

ϕ= J

−1

x)|ϕ|

q−2

ϕ,x∈R

-ε→0,Kϕ

1,0

∈D

1,P

)÷v

−∆

ϕ= J

(x)|ϕ|

q−2

ϕ,x∈R

DOI:10.12677/pm.2023.132016144nØêÆ

œK

Ù¥µ

|∇ϕ

1,0

dx>0.

dý•§K5nØÚr4Šnϕ

1,0

>0.

5¿3B

(0)þ,Q

−

…éR<



−r



(0)

−1

1,0

dx+o(1) =

(0)

dx+





≥1+

(0)

−

= 1+

(0)

−1

−

1,0

dx+o(1).

-R→∞,K

−1

1,0

dx≥1.

Ïd,dÚn3

≥limsup

ε→0

(|∇u

+ε

)dx

≥

−1

|∇ϕ

1,0

dx≥

−1

|∇ϕ

1,0



−1

1,0



= β

(

−1)

|∇ϕ

1,0



1,0



|∇ϕ

1,0



1,0



≥J

gñ.Ïd,œ/3Ø¤á.

ÏLüØœ/1,2,3,·‚

liminf

ε→0





dx>0,lim

ε→0

(0)

dx= 0.

y..2

e5§·‚y²e¡·K.

·K2lim

ε→0

= J

DOI:10.12677/pm.2023.132016145nØêÆ

œK

y².dÚn4





dx= 1+



(0)∪B





dx−s

(0)

= 1+





dx+o(1).

d(10)

(0)

2,ε

dx= 1+

−

2,ε

dx+o(1).

-ε→0,dÛÜSobolev;i\½nÚFatou Ún

(0)

2,0

dx≥1+

−

2,0

dx,.

Ïd,

2,0

dx≥1.

d‰êfeŒëY5,Ún3,ÚJ

½Â

|∇ϕ

2,0

(

2,0

dx)

≤

|∇ϕ

2,0

dx≤liminf

ε→0

≤limsup

ε→0

≤J

≤

|∇ϕ

2,0

(

2,0

dx)

Ïd,J

3ϕ

2,0

?Œˆ,…lim

ε→0

= J

.y..2

•§·‚¤½n2y².

½n2y².5¿•§(8)•3Ä),KÏLCþO†ØJy²(E

)•3Ä).

Ïd,(i)¤á.

u

>0•(E

)Ä), Kϕ

= ε

q−p

(ε

x)••§(8)Ä).Ïd,•3J

ˆ

¼êu

∈W

1,p

)¦ϕ

= J

q−p

.5¿

2,ε

= J

−

q−p

(·+x

/ε

…d·K2y²

lim

→0

|ϕ

2,ε

dx= 0,

lim

→0

|ϕ

2,ε

dx=

|ϕ

2,0

dx,

lim

→0

|∇ϕ

2,ε

dx=

|∇ϕ

2,0

dx,

Ù¥:ϕ

2,0

•J

ˆ¼ê.Ïd,ε

→0ž, ϕ

2,ε

→ϕ

2,0

1,p

)∩L

).¤±d·

K2,ϕ

→J

−

q−p

2,0

1,p

).(ii)¤á.

DOI:10.12677/pm.2023.132016146nØêÆ

œK

-ε

→0,é?¿δ>0,·‚k

)

(0)

|ϕ

2,ε

|ϕ

2,ε

≤s

(0)

dx→0,

)

(|∇u

+|u

)dx

(|∇u

+|u

)dx

= J

−1

(0)

(|∇ϕ

2,ε

+ε

|ϕ

2,ε

)dx→0.

Ïd,(iii)¤á.y..2

Ä7‘8

I[g,‰ÆÄ7“c‘8(11901531);I[3ÆÄ7”/•ÜŠ‘8(202008330417);úôŽ

g,‰ÆÄ7-:‘8(LZ22A010001).

ë•©z

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thep-Laplacian.NonlinearAnalysis:Theory,MethodsandApplications,54,1-7.

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œK

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