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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(12),4262-4271
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1012453
R
N
þ˜ap-Kirchhoff•§)•35
ïÄ
444áááuuu
í¢“‰Æ§ô€í¢
ÂvFϵ2021c1113F¶¹^Fϵ2021c1210F¶uÙFϵ2021c1217F
Á‡
©ïÄXe˜ap-Kirchhoffý•§
−

a+b(
Z
R
N
|∇u|
p
)
τ

∆
p
u+V(x)|u|
p−2
u= |u|
q−2
u,x∈R
N
,(1)
š²…f)•35"Ù¥a,b,τ>0,1 <p<N,p<q<p(τ+1)<p
∗
"ÄuC©n§·‚y
•§(1)–•3˜‡š²…)"©̇(J3uØyCq)Sk.5ÚÂñ5"
'…c
p-Kirchhoffý•§§C©n§(PS)S§PohozTevðª
StudyonExistenceofSolutions
forap-KirchhoffEllipticEquation
onR
N
LihuaLiu
YanchengNormalUniversity,Yancheng Jiangsu
Received:Nov.13
th
,2021;accepted:Dec.10
th
,2021;published:Dec.17
th
,2021
©ÙÚ^:4áu.R
N
þ˜ap-Kirchhoff•§)•35ïÄ[J].A^êÆ?Ð,2021,10(12):4262-4271.
DOI:10.12677/aam.2021.1012453
4áu
Abstract
Inthispaper,westudy theexistenceofsolutionsforthefollowingp-Kirchhoffelliptic
equation
−

a+b(
Z
R
N
|∇u|
p
)
τ

∆
p
u+V(x)|u|
p−2
u= |u|
q−2
u,x∈R
N
,(1)
witha,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,PohozTTTevIdentity
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.0
©§·‚•ÄXe˜ap-Kirchhoffý¯K
−

a+b(
Z
R
N
|∇u|
p
)
τ

∆
p
u+V(x)|u|
p−2
u= |u|
q−2
u,x∈R
N
,
Ù¥∆
p
u= div(|∇u|
p−2
∇u)´p-LaplacicanŽf.¯K(1)Œ±w¤´dêÆ[Kirchhoff[1]JÑ
˜‡êÆ.í2.'up-Kirchhoff-typeý•§˜k(JŒë„ƒ'©z[2–4].
ÆöG. LiÚH.Ye[5]3R
3
þïÄ(1)p=2,τ= 1žœ/,±9³¼êV(x)÷vXe^
‡:
(V
1
)V(x)3(R
3
,R)ëY…fŒ§(∇V(x),x)∈L
∞
(R
3
)∪L
3
2
(R
3
)§V(x)−(∇V(x),x)≥
0a.ex∈R
3
,Ù¥(·,·)´R
3
þÏ~½ÂSÈ;
DOI:10.12677/aam.2021.10124534263A^êÆ?Ð
4áu
(V
2
)éuA¤kx∈R
3
,V(x)≤liminf
|y|→∞
V(y)= V
∞
<+∞§…3V‚ÿÝf8þkTØ
'Xá;
(V
3
)•3C>0¦C=inf
u∈H
1
(R
3
)\{0}
R
R
3
|∇u|
2
+V(x)|u|
2
R
R
3
|u|
2
>0.
¦‚y3<q<6ž§•§(1)–•3˜‡Ä).3©z[6]¥,N.Ikoma•Ä
2<q≤3ž•§(1)•35.,,éu˜„p-Kirchhoffý•§§•§(1)´Ä••3
aq•35(Jºdup-LaplacicanŽf•3§þã©z2^FËA˜mïÄ•{é
up-Kirchhoffý•§Ø2·^"©¥§É©z[5–7]éu§·‚ò|^ä¼ê!Pohozaev
ðª±9©z[8]¥üN5E|•{(PS)Sk.5O§?|^U?ì´Ún
yÙš²…f)•35.
•Qã·‚̇(J,·‚Äk0˜Sobolev˜mÚ‰ê½Â.••Bå„,·
‚ò
R
Ω
hdsÚ
R
∂Ω
hds©OL«3«•Ω⊂R
3
Ú>.∂ΩþLebegsgueÈ©.-X=W
1,p
(R
N
)
•Ï~Sobolev˜m§Ù‰ê•kuk=(
R
R
N
a|∇u|
p
+ V(x)|u|
p
)
1
p
,1<p<∞.½Âk·k
q
•Ï~
L
q
(R
N
)‰ê.¯¤±•§r∈(p,p
∗
]ž§i\W
1,p
(R
N
) →L
q
(R
N
)´ëY§±9•3˜‡~
êS
r
¦
kuk
r
≤S
r
kuk, ∀u∈W
1,p
(R
N
).(2)
©¥§·‚ò^ͶSobolevØª[9]
S(
Z
R
N
|u|
p∗
)
1
p
∗
≤(
Z
R
N
|∇u|
p
dx)
1
p
,∀u∈C
∞
0
(R
N
),(3)
Ù¥1 <p<N,S´˜‡~ê.e5,·‚b³¼êV(x)÷vXe^‡:
(H
1
)V(x) ∈C
1
(R
N
)§•3~êV
0
,V
1
>0§¦0 <V
0
≤V(x) ≤V
1
,∀x∈R
N
;
(H
2
)1<p<N,N≥3,p<q<p(τ+1)<p
∗
=
Np
N−p
.λ
∗
=
N(q−p)(p
∗
−p(τ+1))
p
∗
(p(τ+1)−q)
§•3λ∈(0,λ
∗
]
ž,
(x·∇V(x))−λV(x) ≤0,∀x∈R
N
.
½n1.1e^‡(H
1
)−(H
2
)÷v§@o•§(1)–•3˜‡š²…f).
51.2(i)p= 2,τ= 1,3 <q<4,N= 3ž§3©z[6]¥,ëêλþ.λ
∗
÷v
N(q−p)(p
∗
−p(τ+1))
p
∗
(p(τ+1)−q)
=
q−2
4−q
>1,
©λ
∗
Œ±w‰[6]3p-Kirchhoffý¯K ¥í2.,3©z[5]¥ ³¼êV(x)÷v»•
^‡x.∇x≤V(x)§©¥(H
2
)»•^‡x.∇x≤λV(x)w•˜„.
(ii)©¥¦+N´y²•§(1)¤ƒA•¼I(u)÷vì´ÚnAÛ^‡,´p<q<
p(τ+1)ž§·‚ØU†y²(PS)Sk.5.•d,·‚òæ^©z[8]¥˜«š†
üN5•{.
DOI:10.12677/aam.2021.10124534264A^êÆ?Ð
4áu
2.ýóŠ
XJéu?Ûv∈X,÷v
(a+bk∇uk
pτ
p
)
Z
R
N
|∇u|
p−2
∇u∇v+
Z
R
N
V(x)|u|
p−2
u=
Z
R
N
|u|
q−2
uv.(4)
K¡u∈X´•§(1)f).
I(u) : X→R´(1)Uþ•¼§½ÂXeµ
I(u) =
1
p
kuk
p
+
b
p(τ+1)
k∇uk
p(τ+1)
−
1
q
Z
R
N
|u|
q
.(5)
N´y²T•¼÷vI∈C
1
(X,R)±9éu?Ûv∈X§ÙGateauxê•
I
0
(u)v= (a+bk∇uk
pτ
p
)
Z
R
N
|∇u|
p−2
∇u∇v+
Z
R
N
V(x)|u|
p−2
v−
Z
R
N
|u|
q−2
uv,(6)
w,,•§(1)f)éAXƒA•¼.:.
Ún2.1([8])X•˜n<â˜m§ Ù‰ê•k.k
X
,±9K⊂R
+
´˜«m.•ÄXe½Â3X
þC
1
¼êq,
I
µ
(u) = A(u)−µB(u),µ∈K,(7)
Ù¥B(u)šK§I
µ
(0) = 0§kuk
X
→∞ž§÷vA(u) →∞½B(u) →∞.
éu?Ûµ∈K,
Γ
µ
= {γ∈(C[0,1],X) : γ(0) = 0,I
µ
(γ(1)) <0}.(8)
XJéu?Ûµ∈K,8ÜΓ
µ
š˜±9
c
µ
=inf
γ∈Γ
µ
max
t∈[0,1]
I
µ
(γ(t)) >0.(9)
@oéuA¤kµ∈K,•3S{u
n
}⊂X¦(i).{u
n
}k.;(ii).I
µ
(u
n
)→c
µ
;(iii).
3X
∗
¥I
0
µ
(u
n
) →0.
©¥§·‚
A(u) =
a
p
kuk
p
+
b
p(τ+1)
k∇uk
p(τ+1)
p
,B(u) =
1
q
Z
R
N
|u|
q
.(10)
·‚ïÄXe•¼q
I
µ
(u) =
a
p
kuk
p
+
b
p(τ+1)
k∇uk
p(τ+1)
−
µ
q
Z
R
N
|u|
q
,(11)
DOI:10.12677/aam.2021.10124534265A^êÆ?Ð
4áu
±9Gateauxê•
I
0
µ
(u)v= (a+bk∇uk
pτ
p
)
Z
R
N
|∇u|
p−2
∇u∇v+
Z
R
N
V(x)|u|
p−2
−µ
Z
R
N
|u|
q−2
uv,
(12)
e5Ún2.2ÚÚn2.3¿›XI
µ
÷vÚn2.1^‡.
Ún2.2b(H
1
)−(H
2
)÷v,Ké¤kµ∈[
1
2
,1],kΓ
µ
6= ∅.
y²:•‡y²•3φ∈W
1,p
(R
N
)¦éu?Ûµ∈[1/2,1]kI
µ
(φ) <0¤á=Œ.Äk,·‚b

g
µ
(t) = µ|t|
q−2
t−V(x)|t|
p−2
t,G
µ
(t) =
Z
t
0
g
µ
(x) =
µt
q
q
−
V(x)
p
t
p
(13)
duq>p>1,À·κ
0
>0¦G
1/2
(κ
0
) >0.½Â
l
R
(x) =







κ
0
,if|x|≤R,
κ
0
(R+1−|x|),ifR<|x|≤R+1,
0,if|x|>R+1.
(14)
u´n→∞ž§kl
R
(x) ∈W
1,p
(R
N
),kl
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)ÚG
1/2
(κ
0
)>0§•3¿©ŒR
0
¦
R
R
N
G
1/2
(l
R
0
)≥
1.e5,Pl
R
0
,β
= l
R
0
(x/β).u´k
k∇l
R
0
,β
k
p
p
= β
N−p
kl
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)
dup(τ+1)<p
∗
=
pN
N−p
,u´(N−p)(τ+1) <N.?β→∞ž§kI
µ
(l
R
0
,β
) →−∞§u ´
·Ky.
Ún2.3b(H
1
)−(H
2
)÷v.K•3~êc>0§¦éu¤kµ∈[
1
2
,1]kc
µ
≥c>0.
y²:Šâ(2)Ú(5)§éu¤ku∈W
1,p
(R
N
)Úµ∈[1/2,1]k
I
µ
(u) =
a
p
kuk
p
+
b
p(τ+1)
k∇uk
p(τ+1)
−
µ
q
Z
R
N
|u|
q
,
≥
a
p
kuk
p
−Ckuk
q
,
(17)
DOI:10.12677/aam.2021.10124534266A^êÆ?Ð
4áu
Ï•q>p,¤±•3ρ>0¦I
µ
(u)>0.AO/,éukuk=ρ,kI
µ
(u)≥c>0.½µ∈[
1
2
,1]
Úγ∈Γ
µ
.ŠâΓ
µ
½ÂÚÙëY5,•3t
γ
∈(0,1)¦kγ(t
γ
)k
E
=ρandkγ(1)k>ρ.Ïd,éu
?Ûµ∈[
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Âñ5Úk.5
dÚn2.1-2.3,éua.e.µ∈[1/2,1],•3k.S{u
n
}inX,÷v
I
µ
(u
n
) →c
µ
,(I
µ
(u
n
))
0
→0,sup
n
ku
n
k<T.(19)
duq∈(p,p
∗
)ž§i\X→L
q
(R
N
)´ëY,•d·‚Œ±b
u
n
u, inX;u
n
→u, inL
q
loc
(R
N
);u
n
→u,a.e.in R
N
.(20)
e5,·‚òy²{u
n
}3X¥Âñ.
Ún3.1b(H
1
)−(H
2
)÷v.éu?Ûµ∈[
1
2
,1],XJS{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.
y²:TÚny²Œ±ë„©z[10].
Ún3.2(H
1
)−(H
2
)÷v.{u
n
}•˜(PS)
c
S±9÷v(20),K3X¥§u
n
→u,=,éu¤
kµ∈[1/2,1]§•¼I
µ
÷v(PS)^‡.
y²:d(12),
(I
µ
(u
n
)−I
µ
(u))
0
(u
n
−u) = P
n
+Q
n
+K
n
,(21)
Ù¥
P
n
=

a+bk∇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∇uk
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.2021.10124534267A^êÆ?Ð
4áu
w,
(I
µ
(u
n
)−I
µ
(u))
0
(u
n
−u) →0asn→∞.(22)
dÚn3.1±9H¨olderØª,k
|K
n
|≤µ(ku
n
k
q−1
q
+kuk
q−1
q
)ku
n
−uk
q
→0asn→∞.(23)
½Â‚5•¼g: X→R§
g(ω) =
Z
R
N
|∇u|
p−2
∇u∇ω.
5¿|g(ω)|≤k∇uk
p−1
kωk,Œ±wg3XþëY.du3X¥§u
n
u,
g(u
n
−u) =
Z
R
N
|∇u|
p−2
∇u∇(u
n
−u) →0,asn→∞.(24)
qÏku
n
k3Xþk.,u´|Q
n
|≤Cg(u
n
−u) →0,as n →∞.
nþ?Ø,·‚k|P
n
|→0asn→∞,ù·‚y3X¥§u
n
→u.y..
ŠâÚn3.2,•3S{µ
n
}⊂[
1
2
,1]§µ
n
→1±9n→∞žk§I
T
µ
n
(u
n
) = c
µ
n
,(I
T
µ
n
)
0
(u
n
) =
0§u
n
is´Xe•§
−(a+bk∇u
n
k
p
p
)∆
p
u
n
+V(x)|u
n
|
p−2
u
n
= µku
n
k
q−2
u
n
.(25)
š²…).e5,·‚ò/ÏuPohozavðªy²ku
n
k<T§•d·‚ÄkïáXeðª.
Ún3.3u∈X´Xe•§
−(a+bk∇uk
pτ
p
)∆
p
u+V(x)|u|
p−2
u= µ|u|
q−2
u.(26)
f)§K÷vXeðª
(a+bk∇uk
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)
y²:duu∈X´•§(26)f),ŠâIOKz?Øk[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+bk∇uk
pτ
p
.(28)
u´u∈X•´Xe•§
−∆
p
u= y(x,u).(29)
DOI:10.12677/aam.2021.10124534268A^êÆ?Ð
4áu
).|^Pohozaevð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
§(ؤá.
Ún3.4b(H
1
)−(H
3
)¤á.{µ
n
}⊂[
1
2
,1] ±9{u
n
}⊂X¦µ
n
%1, I
µ
n
= c
µ
n
ÚI
0
µ
n
(u
n
) =
0.u´S{u
n
}3X¥k..
y²:du(I
µ
n
)
0
(u
n
) = 0,u´ŠâÚn3.3,u
n
÷vXeª:
0 = P
µ
(u
n
) = (a+bk∇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)
dI
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

ku
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

ku
n
k
q
q
≥a

1
p
−
1
p(τ+1)

k∇u
n
k
p
p
+(a+bk∇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

ku
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´{üOŽ
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
3L
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.2021.10124534269A^êÆ?Ð
4áu
ùq¿›Xu
n
3L
p
¥k..y..
4.š²…)•35
½n1.1y²dÚn3.4,·‚b½ku
n
k≤T,¤±k
I(u
n
) = I
µ
n
(u
n
)+(µ
n
−1)ku
n
k
q
q
.(36)
쥵
n
→1,Œy{u
n
}3Iþ÷v(PS)^‡.¯¢þ,{u
n
}k.5¿›X{I
µ
n
}k..,,
I
0
(u
n
)v= I
0
µ
n
(u
n
)v+(µ
n
−1)
Z
R
N
|u
n
|
q−2
u
n
v,v∈X.(37)
u´I
0
(u
n
) →0±9{u
n
}´˜k.(PS)S.ŠâÚn2.3,{u
n
}k˜Âñf.b½u
n
→˜u
0
.
u´I
0
(˜u
0
)=0.ŠâÚn2.2,I(˜u
0
)=lim
n→∞
I(u
n
)=lim
n→∞
I
µ
n
(u
n
)≥c>0§¤±˜u
0
´¯
K(1)š²…).ù·‚Ò¤T½ny².
Ä7‘8
ô€Žpg‰¡þ‘8(20KJD110001)"
ë•©z
[1]Kirchhoff,G.(1883)Vorlesungen¨uberMechanik.Teubner,Leipzig.
[2]Corrˆea,F.J.S.A.andFigueiredo,G.M.(2006)OnaEllipticEquationofp-KirchhoffTypevia
VariationalMethods.BulletinoftheAustralianMathematicalSociety,74,263-277.
https://doi.org/10.1017/S000497270003570X
[3]Corrˆea,F.J.S.A.andFigueiredo,G.M.(2009)Onap-KirchhoffEquationviaKrasnoselskii’s
Genus.AppliedMathematicsLetters,22,819-822.https://doi.org/10.1016/j.aml.2008.06.042
[4]Liu, D.and Zhao, P.(2012) Multiple Nontrivial Solutions toa p-Kirchhoff Equation.Nonlinear
Analysis,75,5032-5038.https://doi.org/10.1016/j.na.2012.04.018
[5]Li,G.andYe,H.(2014)ExistenceofPositiveGroundStateSolutionsfortheNonlinear
KirchhoffTypeEquationsinR
3
.JournalofDifferentialEquations,257,566-600.
https://doi.org/10.1016/j.jde.2014.04.011
[6]Ikoma,N.(2015)Existence ofGroundStateSolutions totheNonlinearKirchhoffTypeEqua-
tionswithPotentials.DiscreteandContinuousDynamicalSystems,35,943-966.
https://doi.org/10.3934/dcds.2015.35.943
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[7]Li,Y.H.,Li,F.Y.andShi,J.P.(2012)ExistenceofaPositiveSolutiontoKirchhoffType
ProblemswithoutCompactnessConditions.Journalof DifferentialEquations, 253, 2285-2294.
https://doi.org/10.1016/j.jde.2012.05.017
[8]Jeanjean,L.(1999)OntheExistenceofBoundedPalais-SmaleSequenceandApplicationsto
aLandesman-Lazer-TypeSetonR
N
.ProceedingsoftheRoyalSocietyofEdinburghSection
A,129,787-809.https://doi.org/10.1017/S0308210500013147
[9]Willem,M.(1996)MinimaxTheorem.Birkh¨auser,Inc.,Boston,MA.
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N
.AppliedMathematicsLetters,52,176-182.https://doi.org/10.1016/j.aml.2015.09.007
[11]Kuzin, I. andPohozaev, S. (1997) Entire Solutions of SemilinearElliptic Equations. In:Brezis,
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Birkh¨auser,Boston,MA.
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