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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(7),4089-4109
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.117437
©êChoquard•§CÒ)•35
ppp777uuu
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Á‡
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©¥§·‚ïÄXe©êChoquard•§
(−∆)
s
u+V(x)u= (|x|
−µ
∗|u|
p
)|u|
p−2
u,x∈R
N
,(P)
Ù¥s∈(0,1)§N≥3§µ∈(0,N)§2 <p<
2N−µ
N−2s
§/∗0“LòÈŽf§(−∆)
s
´©ê.Ê.
dŽf"ÏL(ÜEkelandC©nÚÛ¼ê½n§·‚y²(P) •34UþCÒ)w"d
§·‚„y²wUþî‚ŒuÄUþ§î‚uÄUþü"
'…c
©ê.Ê.dŽf§CÒ)§Choquard•§
ExistenceofSign-ChangingSolutionsfor
aFractionalChoquardEquation
JinhuaGao
DepartmentofMathematics,YunnanNormalUniversity,KunmingYunnan
Received:Jun.1
st
,2021;accepted:Jun.24
th
,2022;published:Jul.1
st
,2022
©ÙÚ^:p7u.©êChoquard•§CÒ)•35[J].A^êÆ?Ð,2022,11(7):4089-4109.
DOI:10.12677/aam.2022.117437
p7u
Abstract
Withanimportantphysicalbackground,thefractionalChoquardequationhasat-
tractedgreatattentionfromthefieldofnonlinearanalysisinrecentyears.Inthis
paper,westudythefollowingfractionalChoquardequation
(−∆)
s
u+V(x)u= (|x|
−µ
∗|u|
p
)|u|
p−2
u,inR
N
,(P)
wheres∈(0,1), N≥3,µ∈(0,N), 2 <p<
2N−µ
N−2s
,“∗”standsfortheconvolutionand (−∆)
s
isthefractionalLaplacianoperator.BycombiningtheEkelandvariationalprinciple
withtheimplicitfunctiontheorem,we provethattheproblem(P)possessesoneleast
energysign-changingsolutionw.Moreover,weshowthattheenergyofwisstrictly
largerthanthegroundstateenergyandlessthantwicethegroundstateenergy.
Keywords
FractionalLaplacian,Sign-ChangingSolutions,ChoquardEquation
Copyright
c
2022byauthor(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.2022.1174374090A^êÆ?Ð
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Ñ(ë„[1],[2]).I‡•Ñ,•§(1.2) )†àžmCzHartree •§
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•Œ±lEinstein-Klein-Gordon ÚEinstein-Dirac XÚíÑ.•õÔnµŒë„[6] 9Ù ¥
ë•©z.
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,
DOI:10.12677/aam.2022.1174374091A^êÆ?Ð
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Ù¥I
µ
: R
N
→RL«Riesz ³,½ÂXe:
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µ
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2
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|x|
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N
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,(1.4)
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s−N
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(1+a(x))(K
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∗|u|
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)|u|
p−2
už,•§(1.4) Ä)•35,Ù¥a(x) 3á?ž”.Shen,Gao
ÚYang [18]ïÄBerestycki-Lions .be©êChoquard•§Ä)•35.
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¤kCÒ¼ê¤f8þ¦å,y²4UþCÒ)•35.
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1
θ
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µ
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+
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g∈L
θ
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r
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N
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R
N
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N
g(x)h(y)
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θ
khk
r
.
51.éuF(u) = |u|
p
,u∈H
s
(R
N
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1
(R
N
),dHardy-Littlewood-Sobolev Øª,
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N
≤p≤
2N−µ
N−2s
ž,
ZZ
R
N
×R
N
F(u(x)F(u(y))
|x−y|
µ
dxdy
k½Â,Ù¥θ=
2N
2N−µ
.
·‚b³¼êV(x)÷v
(V
1
)V(x) ∈C(R
N
,R) …inf V(x) ≥a
1
>0.
(V
2
)é?¿M>0,kmeas{x∈R
N
|V(x) ≤M}<∞.
DOI:10.12677/aam.2022.1174374092A^êÆ?Ð
p7u
Äk,·‚£©êSobolev ˜m.éus∈(0,1),©êSobolev ˜mH
s
(R
N
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H
s
(R
N
) =

u∈L
2
(R
N
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|u(x)−u(y)|
|x−y|
N+2s
2
∈L
2
(R
N
×R
N
)

,
Ù‰ê½Â•
kuk
H
s
= [u]
H
s
+kuk
2
,
Ù¥
[u]
H
s
= (
ZZ
R
N
×R
N
|u(x)−u(y)|
2
|x−y|
N+2s
dxdy)
1
2
L«uGagliardoŒ‰ê.d[21] Œ•,©ê.Ê.dŽf(−∆)
s
Œ½ÂXe:
(−∆)
s
u= −
C
N,s
2
Z
R
N
u(x+y)+u(x−y)−2u(x)
|y|
N+2s
dy(1.5)
…
k(−∆)
s
2
uk
2
2
=
1
2
C
N,s
ZZ
R
N
×R
N
|u(x)−u(y)|
2
|x−y|
N+2s
dxdy,
Ù¥
C
N,s
=

Z
R
N
1−cosξ
1
|ξ|
N+2s
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
−1
,ξ= (ξ
1
,ξ
2
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N
).
•?n•§(1.1) ¥©êŽfšÛÜ5,·‚ò¦^[22] ¥Caffarelli ÚSilvestre uÐ
s−NÚòÿ•{5ïăAòÿ¯K,ù4·‚U^²;ÛܯKC©•{5ïįK
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

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,
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s
(R
N+1
+
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s
(u).3[22] ¥®²y²
(−∆)
s
u= −
1
k
s
lim
y→0
+
y
1−2s
∂w
∂y
(x,y),
Ù¥
k
s
= 2
1−2s
Γ(1−s)
Γ(s)
,
ùpΓ´³ê¼ê.ùp,˜mX
s
(R
N+1
+
)ÚH
s
(R
N
)•À•C
∞
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∞
0
(R
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m,3[23] ¥,‰ê
kwk
X
s
:=

Z
R
N+1
+
k
s
|∇w|
2
dxdy

1
2
,
kwk
H
s
(R
N
)
:=

Z
R
N
|2πξ|
2s
|F(u(ξ))|
2
dξ

1
2
=

Z
R
N
|(−∆)
s
2
u|
2
dx

1
2
.
DOI:10.12677/aam.2022.1174374093A^êÆ?Ð
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I‡•Ñ´,i\X
s
(R
N+1
+
)→L
2
∗
s
(R
N
) ´ëY(ë„[23]).Äuþã•{,·‚òïıe
¯K4Uþ(CÒ))•35



−div(y
1−2s
∇w) = 0,x∈R
N+1
+
,
∂w
∂ν
= −V(x)w+(|x|
−µ
∗|w|
p
)|w|
p−2
w,x∈R
N
×{0},
(1.7)
Ù¥
∂w
∂ν
= −
1
k
s
lim
y→0
+
y
1−2s
∂w
∂y
(x,y).
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s
.´„,ew∈X
s
(R
N+1
+
) ´•§(1.7)),Ku(x) =w(x,0) ´(1.1)
).
du ³¼êV(x)Ñy,·‚Ú\Xef˜m
H:= {w∈X
s
(R
N+1
+
) |
Z
R
N
V(x)|w(x,0)|
2
dx<∞}
Ù¥
(w,v) =
Z
R
N+1
+
y
1−2s
∇w∇vdxdy+
Z
R
N
V(x)w(x,0)v(x,0)dx
‰ê½Â•
kwk=

Z
R
N+1
+
y
1−2s
|∇w|
2
dxdy+
Z
R
N
V(x)|w(x,0)|
2
dx

1
2
.
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J(w) =
1
2
Z
R
N+1
+
y
1−2s
|∇w|
2
dxdy+
1
2
Z
R
N
V(x)|w(x,0)|
2
dx
−
1
2p
Z
R
N
(|x|
−µ
∗|w(x,0)|
p
)|w(x,0)|
p
dx.
(1.8)
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1
(H,R)(ë„[8],[9])…éu?
¿w,ϕ∈H,
hJ
0
(w),ϕi=
Z
R
N+1
+
y
1−2s
∇w∇ϕdxdy+
Z
R
N
V(x)w(x,0)ϕ(x,0)dx
−
Z
R
N
(|x|
−µ
∗|w(x,0)|
p
)|w(x,0)|
p−2
w(x,0)ϕ(x,0)dx,
(1.9)
ù`²Uþ•¼J.:´(1.1) f).
ew∈H´(1.1) ˜‡)…w
±
6= 0,…÷v
J(w) = inf{J(v)|v
±
6= 0,J
0
(v) = 0}(1.10)
DOI:10.12677/aam.2022.1174374094A^êÆ?Ð
p7u
Ù¥
w
+
(x,0) = max{w(x,0),0},w
−
(x,0) = min{w(x,0),0}.
K·‚¡w∈H•(1.1) 4UþCÒ).·‚½ÂJƒANehari 6/
N= {w∈H\{0}: hJ
0
(w),wi= 0},(1.11)
…½Â
c:=inf
w∈N
J(w).(1.12)
4·‚£ÁŒ‚5¯K˜²;(J



−∆u= f(x,u)x∈Ω,
u= 0x∈∂Ω.
(1.13)
3[24] ¥,Castro,Cossio ÚNeuberger y² (1.13) •3˜‡CÒ),…§TЕUC˜gÎÒ,
”ù•CÒgêCÒ)•¡•!:).Bartsch,Weth ÚWillem [25]y²TÐCÒ˜g
4Uþ!:)•35.d,3[26] ¥„uyUþOy–.,,ùJ'u(1.13) 
(J•6uXeü«©),=éuu∈H
1
0
(Ω),k
I(u) = I(u
+
)+I(u
−
),(1.14)
hI
0
(u),u
+
i= hI
0
(u
+
),u
+
i, hI
0
(u),u
−
i= hI
0
(u
−
),u
−
i,(1.15)
Ù¥I: H
1
0
(Ω) →R
I(u) =
1
2
Z
Ω
|∇u|
2
dx−
Z
Ω
F(x,u)dx, F(x,u) =
Z
u
0
f(x,τ)dτ
•(1.13) éAUþ•¼.,,d(1.8) ½ÂC©•¼JØ2k†(1.14) Ú(1.15) ƒÓ©).
¯¢þ,
J(w) = J(w
+
)+J(w
−
)−
1
p
Z
R
N
(|x|
−µ
∗|w(x,0)
+
|
p
)|w(x,0)
−
|
p
dx,
hJ
0
(w),w
+
i= hJ
0
(w
+
),w
+
i−
Z
R
N
(|x|
−µ
∗|w(x,0)
−
|
p
)|w(x,0)
+
|
p
dx,
…
hJ
0
(w),w
−
i= hJ
0
(w
−
),w
−
i−
Z
R
N
(|x|
−µ
∗|w(x,0)
+
|
p
)|w(x,0)
−
|
p
dx.
Ïd,3[24](•Œë„[25][26])¥•{qØ·^u·‚¯K(1.1).•(1.1) 4U
DOI:10.12677/aam.2022.1174374095A^êÆ?Ð
p7u
þCÒ),aqu[27],·‚Äk}Á3±eå6/þÏéUþ•¼J•Š
M= {w∈H|w
±
6= 0,hJ
0
(w),w
±
i= 0}.(1.16)
w,,einf
u∈M
J(u) Œˆ,K4´(1.1) ˜‡CÒ).,,3·‚¯K(1.1) ¥,X·‚þ
¡J,•¼JØ2äk5Ÿ(1.14),(1.15),éJ^Ï~•ª5y²M6=∅.Ïd,•y²
8ÜMš˜,·‚/^[7] JÑ˜«†Ye [12]ÚShuai[19] ØÓ•{.·‚y²éu÷v
w
±
6= 0w∈H,•3•˜˜é(d,t) ∈R
+
×R
+
,¦dw
+
+tw
−
∈M,„e©Ún7.•y²
å¯K4´˜‡CÒ),aqu[12] Ú[19],·‚æ^Ekeland C©n[28] ÚÛ¼ê½
n.
·‚1˜‡Ì‡(JLãXe:
½n1.µ∈(0,N),2 <p<
2N−µ
N−2s
.K(1.1) •34UþCÒ)w.
,,dušÛÜ‘(−∆)
s
uÚ|x|
−µ
∗|u|
p
Ñy,·‚Ø•4UþCÒ)w´Ä•UC
˜gÎÒ,Ï•éu÷vsupp(w
1
)∩supp(w
2
) = ∅ü‡¼êw
1
Úw
2
ó,È©
Z
R
N
(−∆)
s
2
w
1
(−∆)
s
2
w
2
dx−
Z
R
N
(|x|
−µ
∗|w
1
|
p
)|w
2
|
p
dx
™7•".
©,˜‡8I´ïá(1.1)4UþCÒ)UþO.¯¢þ,éu•§(1.13),?Û
CÒ)UþÑŒuÄUþü,ù«A53[26]¥Weth ¡•UþO.,,éu¯K
(1.1),·‚(JL²džUþOy–¿Øu),džCÒ)Uþî‚ŒuÄUþ,î‚
uÄUþü.=kXe(J:
½n2.3½n1be,c>0d,)½K)ˆ,…
c<J(w) <2c,
Ù¥w•½n1 ¥‰Ñ4UþCÒ).
©(Xe:312!¥,·‚y²A‡ý5Ún,§‚éy²·‚̇(J–'-
‡.313 !¥,(ÜEkelandC©nÚÛ¼ê½n,·‚y²̇(J.
2.ýÚn
±e´©êSobolev˜mi\(J,ë„[21].
Ún3.t∈[2,2
∗
s
],(V
1
) ¤á.Ki\H→H
s
(R
N
)→L
t
(R
N
) ´ëY.AO/,•3~ê
C
t
>0,¦éu¤ku∈H,
kuk
t
≤C
t
kuk.
d,et∈[1,2
∗
s
),Ki\H
s
(R
N
) →L
t
loc
(R
N
) ´;,Ù¥2
∗
s
=
2N
N−2s
.e(V
1
)−(V
2
) ¤á,K
DOI:10.12677/aam.2022.1174374096A^êÆ?Ð
p7u
éut∈[2,2
∗
s
),H→L
t
(R
N
) ´;.AO/,k
éuz‡2 ≤t<2
∗
s
,H→L
t
(R
N
)´;.(2.1)
e5,·‚ïᕼJ;5(J,=:
Ún4.V(x)÷v^‡(V
1
) Ú(V
2
).Kéu?¿c∈R,J÷v(PS)
c
^‡.
y²:c∈R¿…{w
j
}
j∈N
´H¥˜‡S,¦j→∞ž,
J(w
j
) →c,J
0
(w
j
) →0.
·‚k
c+1+kw
j
k= J(w
j
)−
1
2p
hJ
0
(w
j
),w
j
i
= (
1
2
−
1
2p
)

Z
R
N+1
+
y
1−2s
|∇w
j
|
2
dxdy+
Z
R
N
V(x)|w
j
(x,0)|
2
dx

≥
1
4
kw
j
k
2
.
(2.2)
Ïd,S{w
j
}
j∈N
3H¥k..Šâ(2.1),3f¿Âe,·‚Ø”b
3Hþ,w
j
w
0
,
3L
p
(R
N
)þ,w
j
(x,0) →w
0
(x,0),
w
j
(x,0) →w
0
(x,0) a.e.x∈R
N
.
Ødƒ,Šâ[28],•3g(x,0) ∈L
p
(R
N
)¦
|w
j
(x,0)|≤g(x,0)a.e.x∈R
N
, ∀j∈N.
ŠâV‚››Âñ½n,·‚
Z
R
N
(|x|
−µ
∗|w
j
(x,0)|
p
)|w
j
(x,0)|
p
dx
→
Z
R
N
(|x|
−µ
∗|w
0
(x,0)|
p
)|w
0
(x,0)|
p
dx,
(2.3)
…j→∞ž,
Z
R
N
(|x|
−µ
∗|w
j
(x,0)|
p
)|w
j
(x,0)|
p−2
w
j
(x,0)w
0
(x,0)dx
→
Z
R
N
(|x|
−µ
∗|w
0
(x,0)|
p
)|w
0
(x,0)|
p
dx.
(2.4)
DOI:10.12677/aam.2022.1174374097A^êÆ?Ð
p7u
Ïd,ŠâhJ
0
(w
j
),w
j
i→0 Ú(2.3),j→∞ž,
Z
R
N+1
+
y
1−2s
|∇w
j
|
2
dxdy+
Z
R
N
(V(x)|w
j
(x,0)|
2
)dx→
Z
R
N
(|x|
−µ
∗|w
0
(x,0)|
p
)|w
0
(x,0)|
p
dx.
(2.5)
d,Šâ{w
j
}Ú(2.4)fÂñ5,j→∞ž,
Z
R
N+1
+
y
1−2s
|∇w
0
|
2
+
Z
R
N
V(x)|w
0
(x,0)|
2
dx=
Z
R
N
(|x|
−µ
∗|w
0
(x,0)|
p
)|w
0
(x,0)|
p
dx,
(2.6)
Ïd,j→∞ž,
kw
j
k→kw
0
k,(2.7)
ùL²3H¥{w
j
}
j∈N
rÂñ,ùÒ¤Ún4 y².
w,,ù‡•¼Jäkì´(,=•3ρ,r>0¦infJ(∂B
r
)≥ρ…éu?¿w6=0,
lim
t→∞
J(tw) = −∞.Ïd,·‚k
Ún5.µ∈(0,N),2 <p<
2N−µ
N−2s
.K•§(1.1) k˜‡Ä).
·‚y3y²Ä)ØCÒ(ë„[8],[9]).
Ún6.µ∈(0,N),2 <p<
2N−µ
N−2s
.K•§(1.1) ?ÛÄ)u∈H
s
(R
N
) ØCÒ.
y²:eu=w(x,0) ´(1.1) Ä),Kw∈X
s
(R
N+1
) ´¯K(1.7) ).l,|w|•´˜‡
).Ïd,|u|•´˜‡Ä).§÷v•§
(−∆)
s
|u|+V(x)|u|= (|x|
−µ
∗|u|
p
)|u|
p−1
, x∈R
N
.
¦^u
−
Š•(1.1)¥u¼ê,·‚
0 =
Z
R
N
(−∆)
s
2
u(−∆)
s
2
u
−
dx+
Z
R
N
V(x)uu
−
dx+
Z
R
N
(|x|
−µ
∗|u|
p
)|u|
p−2
uu
−
dx
=
Z
R
N
|(−∆)
s
2
u
−
|
2
dx+
Z
R
N
V(x)|u
−
|
2
dx+
Z
R
N
(|x|
−µ
∗|u|
p
)|u
−
|
p
dx
≥
Z
R
N
|(−∆)
s
2
u
−
|
2
dx+
Z
R
N
V(x)|u
−
|
2
dx≥0.
Ïd,u
−
= 0.l,u≥0.d,eéux
0
∈R
N
,u(x
0
) = 0,K(−∆)
s
u(x
0
) = 0.2Šâ(1.5),
(−∆)
s
u(x
0
) = −
C
N,s
2
Z
R
N
u(x
0
+y)+u(x
0
−y)−2u(x
0
)
|y|
N+2s
dy,
Ïd
Z
R
N
u(x
0
+y)+u(x
0
−y)
|y|
N+2s
dy= 0,
DOI:10.12677/aam.2022.1174374098A^êÆ?Ð
p7u
duu´šK,Œu≡0.ù†u6= 0ƒgñ.u´.Ïd,uØCÒ.
e¡ÚnL²8ÜMš˜.
Ún7.µ∈(0,N),2<p<
2N−µ
N−2s
.Kéu?¿÷vw
±
6=0w∈H,•3•˜˜é
(d,t) = (d
w
,t
w
) ∈R
+
×R
+
,¦d
w
w
+
+t
w
w
−
∈M,=M6= ∅.
y²:·‚æ^[7]Ú[20]¥gŽ,Äk,·‚½Â¼ê
G(d
1
,d
2
) := J(d
1
p
1
w
+
+d
1
p
2
w
−
).
d,éu0 <s
0
,β<NÚ0 <s
0
+β<N,l[20,íØ5.10] Ñ
(|x|
s
0
−N
∗|x|
β−N
)(y) :=
Z
R
N
|z|
s
0
−N
|y−z|
β−N
dz
=
C
N−s
0
−β
C
s
0
C
β
C
s
0
+β
C
N−s
0
C
N−β
|y|
s
0
+β−N
,
(2.8)
Ù¥
C
s
0
:= π
−
s
0
2
Γ(
s
0
2
).
s
0
= β=
N−µ
2
ž,••B,·‚PC(N,µ) :=
C
s
0
+β
C
N−s
0
C
N−β
C
N−s
0
−β
C
s
0
C
β
,Šâ(2.8),
|x|
−µ
= C(N,µ)|x|
−
N+µ
2
∗|x|
−
N+µ
2
.
Ïd
Z
R
N

|x|
−µ
∗(|d
1
p
1
w
+
(x,0)+d
1
p
2
w
−
(x,0)|
p
)

|d
1
p
1
w
+
(x,0)+d
1
p
2
w
−
(x,0)|
p
dx
= C(N,µ)
Z
R
N
|x|
−
N+µ
2
∗

|x|
−
N+µ
2
∗|d
1
p
1
w
+
(x,0)+d
1
p
2
w
−
(x,0)|
p

!
|d
1
p
1
w
+
(x,0)
+d
1
p
2
w
−
(x,0)|
p
dx
= C(N,µ)
Z
R
N
Z
R
N
|x−z|
−
N+µ
2

|z|
−
N+µ
2
∗|d
1
p
1
w
+
(z,0)+d
1
p
2
w
−
(z,0)|
p

dz|d
1
p
1
w
+
(x,0)
+d
1
p
2
w
−
(x,0)|
p
dx
= C(N,µ)
Z
R
N
|z|
−
N+µ
2
∗|d
1
p
1
w
+
(x,0)+d
1
p
2
w
−
(x,0)|
p
Z
R
N
|x−z|
−
N+µ
2
|d
1
p
1
w
+
(z,0)
+d
1
p
2
w
−
(z,0)|
p
dxdz
= C(N,µ)
Z
R
N

|z|
−
N+µ
2
∗|d
1
p
1
w
+
(z,0)+d
1
p
2
w
−
(z,0)|
p

2
dz
= C(N,µ)
Z
R
N

|z|
−
N+µ
2
∗(d
1
|w
+
(z,0)|
p
+d
2
|w
−
(z,0)|
p
)

2
dz.
(2.9)
DOI:10.12677/aam.2022.1174374099A^êÆ?Ð
p7u
Šâ(2.9),·‚k
G(d
1
,d
2
) =
d
2
p
1
2

Z
R
N+1
+
y
1−2s
|∇w
+
|
2
dxdy+
Z
R
V(x)|w
+
(x,0)|
2
dx

+
d
2
p
2
2

Z
R
N+1
+
y
1−2s
|∇w
−
|
2
dxdy+
Z
R
N
V(x)|w
−
(x,0)|
2
dx

−
1
2p
Z
R
N

|x|
−µ
∗(|d
1
p
1
w
+
(x,0)+d
1
p
2
w
−
(x,0)|
p
)

|d
1
p
1
w
+
(x,0)+d
1
p
2
w
−
(x,0)|
p
dx
=
d
2
p
1
2
kw
+
k
2
+
d
2
p
2
2
kw
−
k
2
−
C
N,µ
2p
Z
R
N
|x|
−
N+µ
2
∗

d
1
|w
+
(x,0)|
p
+d
2
|w
−
(x,0)|
p

!
2
dx.
duG(d
1
,d
2
)´ëY¼ê,¿…
G(d
1
,d
2
) =
d
2
p
1
2
kw
+
k
2
+
d
2
p
2
2
kw
−
k
2
−
d
2
1
2p
Z
R
N
(|x|
−µ
∗|w
+
(x,0)|
p
)|w
+
(x,0)|
p
dx
−
d
1
d
2
p
Z
R
N
(|x|
−µ
∗|w
+
(x,0)|
p
)|w
−
(x,0)|
p
dx
−
d
2
2
2p
Z
R
N
(|x|
−µ
∗|w
−
(x,0)|
p
)|w
−
(x,0)|
p
dx
≤
1
2
(d
2
1
+d
2
2
)
1
p
(kw
+
k
2
+kw
−
k
2
)−
1
2p
min{A
1
,A
2
}(d
2
1
+d
2
2
),
Ù¥



A
1
=
R
R
N
(|x|
−µ
∗|w
+
(x,0)|
p
)|w
+
(x,0)|
p
dx,
A
2
=
R
R
N
(|x|
−µ
∗|w
−
(x,0)|
p
)|w
−
(x,0)|
p
dx.
|(d
1
,d
2
)|→+∞ž,G(d
1
,d
2
)→−∞,Gk˜‡Û•ŒŠ:(a,b) ∈R
+
×R
+
.ùp·‚^
Gî‚]5.¯¢þ,P
F(d
1
,d
2
) =
d
2
p
1
2
kw
+
k
2
+
d
2
p
2
2
kw
−
k
2
,
T(d
1
,d
2
) =
Z
R
N
|x|
−
N+µ
2
∗

d
1
|w
+
(x,0)|
p
+d
2
|w
−
(x,0)|
p

!
2
dx,
K
(F
00
d
1
d
2
)
2
−F
00
d
1
d
1
F
00
d
2
d
2
<0.
DOI:10.12677/aam.2022.1174374100A^êÆ?Ð
p7u
Ïd,F´î‚]¼ê.d,-(d
1
,d
2
),(t
1
,t
2
) ∈R
+
×R
+
,λ∈(0,1),K
T(λ(d
1
,d
2
)+(1−λ)(t
1
,t
2
)) = T(λd
1
+(1−λ)t
1
,λd
2
+(1−λ)t
2
)
=
Z
R
N
|x|
−
N+µ
2
∗

(λd
1
+(1−λ)t
1
|w
+
(x,0)|
p
+(λd
2
+(1−λ)t
2
)|w
−
(x,0)|
p

!
2
dx
=
Z
R
N
λ|x|
−
N+µ
2
∗

d
1
|w
+
(x,0)|
p
+d
2
|w
−
(x,0)|
p

+(1−λ)|x|
−
N+µ
2
∗

t
1
|w
+
(x,0)|
p
+t
2
|w
−
(x,0)|
p

!
2
dx
<λ
Z
R
N
|x|
−
N+µ
2
∗

d
1
|w
+
(x,0)|
p
+d
2
|w
−
(x,0)|
p

!
2
dx
+(1−λ)
Z
R
N
|x|
−
N+µ
2
∗

t
1
|w
+
(x,0)|
p
+t
2
|w
−
(x,0)|
p

!
2
dx
= λT(d
1
,d
2
)+(1−λ)T(t
1
,t
2
),
ù`²G´î‚]¼ê,ddÑ(Ø(a,b) ∈R
+
×R
+
´•˜Û4ŒŠ:,¿…∇G(a,b) =
(0,0),ù`²M6= ∅.
d(2.2),e(ؤá.
Ún8.J(w) 3Mþe•k.…r›.
dÚn7,·‚•ÄXeå4¯K
m:= inf{J(u) : w∈M}.(2.10)
Ún9.m<2c.
y²:ŠâÚn5,e´¯K(1.1) ˜‡Ä),=
J
0
(e) = 0,J(e) = c,e>0.
η⊂C
∞
0
(R
N+1
,[0,1])•˜ä¼ê,¦
suppη∈B
1
(0) := {z= (x,y) ∈R
N+1
: |z|≤1},
3B
1
2
(0)þη≡1,3R
N+1
\B
1
(0)þη≡0,¿…|∇η|<1.½Â
w
R
(x,y) := η(
x
R
,y)e(x,0) ≥0,v
R
(x,y) := −η(
x−x
n
R
,y)e(x,0) ≤0,
DOI:10.12677/aam.2022.1174374101A^êÆ?Ð
p7u
Ù¥R>0,x
n
= (0,0,···,0,3R).Ïd,·‚Œ±y²
suppw
R
(x,y)∩suppv
R
(x,y) = ∅,
ùL²w
R
(x,y) +v
R
(x,y)6= 0,¿…w
R
(x,y)6=0,v
R
(x,y)6= 0.Ïd,ŠâÚn7,•3•˜˜é
(d
w
R
,t
v
R
) ∈R
+
×R
+
¦¯u
R
:= d
w
R
w
R
+t
v
R
v
R
∈M…=



d
2
w
R
kw
R
k
2
−d
2p
w
R
A
3
−d
p
w
R
t
p
v
R
B= 0,
t
2
v
R
kv
R
k
2
−t
2p
v
R
A
4
−t
p
v
R,N
d
p
w
R
B= 0.
(2.11)
•{üå„,·‚¦^±eÎÒ:







A
3
=
R
R
N
(|x|
−µ
∗|w
R
|
p
)|w
R
|
p
dx,
B=
R
R
N
(|x|
−µ
∗|v
R
|
p
)|w
R
|
p
dx,
A
4
=
R
R
N
(|x|
−µ
∗|v
R
|
p
)|v
R
|
p
dx.
(2.12)
d,Šâw
R
Úv
R
½Â,
y
1−2s
|∇(w
R
−e)|
2
+V(x)|w
R
−e|
2
≤C

|∇η(
x
R
,y)|
2
e
2
+|η(
x
R
,y)−1|
2
|∇e|
2

∈L
1
(R
N
).
aq/
y
1−2s
|∇(v
R
+e)|
2
+V(x)|v
R
(x,0)+e(x,0)|
2
∈L
1
(R
N
).
Ïd,ŠâV‚››Âñ½n,3H¥,R→∞ž,
w
R
→e,v
R
→−e.(2.13)
Ïd
Z
R
N
(|x|
−µ
∗|w
R
|
p
)|w
R
|
p
dx→
Z
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx,(2.14)
¿…
Z
R
N
(|x|
−µ
∗|v
R
|
p
)|v
R
|
p
dx→
Z
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx.(2.15)
e5,·‚òy²R→∞ž•3(d
0
,t
0
) ∈R
+
×R
+
¦
d
w
R
→d
0
,t
v
R
→t
0
,(d
0
,t
0
) ∈(0,1)×(0,1).(2.16)
¯¢þ,-lim
R→+∞
d
w
R
= +∞.Šâ(2.11) Ú(2.14) ·‚Ñ
0 ≤
t
p
v
R
d
p
w
R
B=
1
d
2p−2
w
R
kw
R
k
2
−A
3
= −
Z
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx+o(1) <0,
DOI:10.12677/aam.2022.1174374102A^êÆ?Ð
p7u
gñ.Ïd,d
w
R
´˜—k..aqu(2.11) Ú(2.15) y²,t
v
R
•´˜—k..Ø”˜„5,
·‚Œ±b•3d
0
,t
0
∈[0,∞),¦R→∞ž,
d
w
R
→d
0
,t
v
R
→t
0
.
ed
0
= 0½t
0
= 0,Šâ(2.11)-(2.15),

Z
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx

2
+o(1) =

1
d
2p−2
w
R
kw
R
k
2
−A
3

1
t
2p−2
v
R
kv
R
k
2
−A
4

+o(1) = +∞,
ù´ØŒU.Ïd,·‚(d
0
,t
0
) ∈R
+
×R
+
.Šâ(2.13) Ú(2.14),·‚Œ±



d
2
w
R
kek
2
−d
2p
w
R
R
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx−d
p
w
R
t
p
v
R
R
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx= 0,
t
2
v
R
kek
2
−t
2p
v
R
R
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx−t
p
v
R
d
p
w
R
R
R
N
(|x|
−µ
∗|e|
p
)|e|
p
dx= 0.
(ÜJ
0
(e) = 0,·‚
1
d
2p−2
w
R
−1 =
t
p
v
R
d
p
w
R
,
1
t
2p−2
v
R
−1 =
d
p
w
R
t
p
v
R
.
Ïd,(d
0
,t
0
) ∈(0,1)×(0,1),(2.16) y.
Ïd,·‚d¯u
R
∈MÚ(2.16) Œ,R→+∞ž,
m≤J(¯u
R
) = (
1
2
−
1
2p
)

Z
R
N+1
+
y
1−2s
|∇¯u
R
|
2
dxdy+
Z
R
N
V(x)|¯u
R
(x,0)|
2
dx

= (
1
2
−
1
2p
)(d
2
w
R
kw
R
k
2
+t
2
v
R
kv
R
k
2
)
= (t
2
0
+d
2
0
)(
1
2
−
1
2p
)kek
2
+o(1)
= (t
2
0
+d
2
0
)J(e)+o(1)
= (t
2
0
+d
2
0
)c+o(1)
≤2c.
Úny.
3.CÒ)
!̇8´y²·‚̇(J.
½n1y²
y²:·‚Äky²å4¯K(2.10)4w(´(1.1) ˜‡),=J
0
(w) = 0.
DOI:10.12677/aam.2022.1174374103A^êÆ?Ð
p7u
dÚn8ÚEkelandC©n,·‚Ø”b•34zS{w
n
}⊂M,¦
J(w
n
) ≤m+
1
n
,
J(v) ≥J(w
n
)−
1
n
kw
n
−vk.(3.1)
·‚Œ±y²S{w
±
n
}Ñ3H¥˜—k..Ïd,3f¿Âe,
3H¥,w
±
n
w
±
,
3L
t
(R
N
)¥,éut∈[2,2
∗
s
),w
±
n
(x,0) →w
±
(x,0),
w
±
n
(x,0) →w
±
(x,0) a.e.x∈R
N
.
d,dSobolev ;i\(2.1),·‚
w
+
(x,0) ≥0,w
−
(x,0) ≤0,w
+
(x,0)·w
−
(x,0) = 0 a.e.x∈R
N
.
Ïd,•y²ù‡½n,·‚•Iy²J
0
(w
n
) →0.
e5,éu?¿φ∈C
∞
0
(R
N
)Úz‡n,·‚½ÂT
1
n
,T
2
n
∈C
1
(R
3
,R) Xe:
T
1
n
(σ,k,l) = k(w
n
+σφ+kw
+
n
+lw
−
n
)
+
k
2
−
Z
R
N
(|x|
−µ
∗|(w
n
+σφ+kw
+
n
+lw
−
n
)
+
|
p
)|(w
n
+σφ+kw
+
n
+lw
−
n
)
+
|
p
dx
−
Z
R
N
(|x|
−µ
∗|(w
n
+σφ+kw
+
n
+lw
−
n
)
−
|
p
)|(w
n
+σφ+kw
+
n
+lw
−
n
)
+
|
p
dx,
T
2
n
(σ,k,l) = k(w
n
+σφ+kw
+
n
+lw
−
n
)
−
k
2
−
Z
R
N
(|x|
−µ
∗|(w
n
+σφ+kw
+
n
+lw
−
n
)
−
|
p
)|(w
n
+σφ+kw
+
n
+lw
−
n
)
−
|
p
dx
−
Z
R
N
(|x|
−µ
∗|(w
n
+σφ+kw
+
n
+lw
−
n
)
+
|
p
)|(w
n
+σφ+kw
+
n
+lw
−
n
)
−
|
p
dx,
KT
1
n
(0,0,0) = T
2
n
(0,0,0) = 0.d,Šâc¡ÎÒ, e^w
n
O†A
1
,A
2
¥w,^w
−
n
Úw
+
n
©OO†B¥v
R
Úw
R
,K
∂T
1
n
(σ,k,l)
∂k
|
(0,0,0)
= 2(1−p)A
1
+(2−p)B,
∂T
2
n
(σ,k,l)
∂l
|
(0,0,0)
= 2(1−p)A
1
+(2−p)B
¿…
∂T
1
n
(σ,k,l)
∂l
|
(0,0,0)
=
∂T
2
n
(σ,k,l)
∂k
|
(0,0,0)
= −pB.
d,Šâ[12],[20,½n9.8],
B
2
<A
1
A
2
≤
A
1
+A
2
2
,(3.2)
DOI:10.12677/aam.2022.1174374104A^êÆ?Ð
p7u
-
J(0,0,0) =


∂T
1
n
(σ,k,l)
∂k
∂T
1
n
(σ,k,l)
∂l
∂T
2
n
(σ,k,l)
∂k
∂T
2
n
(σ,k,l)
∂l


.
d(3.2),k
detJ(0,0,0) = 4(1−p)
2
A
1
A
2
+2(1−p)(2−P)(A
1
+A
2
)B+4(1−p)B
2
≥8(1−p)(2−p)B
2
>0.
ŠâÛ¼ê½n, •3˜‡S{σ
n
}⊂R
+
Úk
n
(σ),l
n
(σ) ∈C
1
(−σ
n
,σ
n
)÷vk
n
(0) = 0,l
n
(0) = 0,
…
T
1
n
(σ,k
n
(σ),l
n
(σ)) = 0, T
2
n
(σ,k
n
(σ),l
n
(σ)) = 0.
Ïd,éu∀σ∈(−σ
n
,σ
n
),
φ
n,σ
:= w
n
+σφ+k
n
(σ)w
+
n
+l
n
(σ)w
−
n
∈M.
d,Šâ(3.1),
J(φ
n,σ
)−J(w
n
) ≥−
1
n
kσφ+k
n
(σ)w
+
n
+l
n
(σ)w
−
n
k.(3.3)
d(3.3),(Ü÷vhJ
0
(w
n
),w
±
n
i= 0VÐmª,
σhJ
0
(w
n
),φi+o(kσφ+k
n
(σ)w
+
n
+l
n
(σ)w
−
n
k) ≥−
1
n
kσφ+k
n
(σ)w
+
n
+l
n
(σ)w
−
n
k
σ
.
(3.4)
du{w
n
}3H¥´˜—k.…detJ(0,0,0) >0, ·‚{k
0
n
(0)}Ú{l
0
n
(0)}Ñ´˜—k..
Ïdσ→0 ž,
o(kσφ+k
n
(σ)w
+
n
+l
n
(σ)w
−
n
k)
σ
→0.
Ïd,Šâ(3.4),σ→0 ž,
|hJ
0
(w
n
),σi|≤
C
n
,
Ù¥C´˜‡†nÃ'~ê.Ïd,·‚J
0
(w
n
)→0,ù`²J
0
(w)=0.d,éu
w
n
∈M,Šâ(2.7),·‚kw
±
n
k→kw
±
k.Ïd,3H¥,n→∞ž,
w
±
n
→w
±
.
d,ŠâHardy-Littlewood-Sobolev ØªÚhJ
0
(w
n
),w
±
n
i= 0,
kw
+
n
k
2
=
Z
R
N
(|x|
−µ
∗|w
+
n
(x,0)|
p
)|w
+
n
(x,0)|
p
dx+
Z
R
N
(|x|
−µ
∗|w
−
n
(x,0)|
p
)|w
+
n
(x,0)|
p
dx
≤C
1
kw
+
n
k
2p
+C
2
kw
−
n
k
p
kw
+
n
k
p
.
(3.5)
DOI:10.12677/aam.2022.1174374105A^êÆ?Ð
p7u
aq/,k
kw
−
n
k
2
=
Z
R
N
(|x|
−µ
∗|w
−
n
(x,0)|
p
)|w
−
n
(x,0)|
p
dx+
Z
R
N
(|x|
−µ
∗|w
+
n
(x,0)|
p
)|w
−
n
(x,0)|
p
dx
≤C
1
kw
−
n
k
2p
+C
2
kw
+
n
k
p
kw
−
n
k
p
.
(3.6)
Ïd,•3˜‡~ê>0,¦éu¤ kn∈N,kw
±
n
k≥.¤±kw
±
k≥>0,ù`²w
±
6= 0¿
…hJ
0
(w),w
±
i= 0.Ïd,w∈M,J(w)= m,=w´J˜‡.:.Ïd,w´¯K(1.1) ˜
‡CÒ).y..
d½n1,·‚•¯K(1.1)k˜‡4UþCÒ)w.·‚y3y²wUþî‚ŒuÄ
Uþ,î‚uÄUþü.
½n2y²
y²:Šâ½n1y²,(1.1)•34UþCÒ)w.
e5,·‚äó
J(w) = m= inf{J(v) |v
±
6= 0,J
0
(v) = 0}.(3.7)
¯¢þ,duJ(w) = m,w´J.:¿…w
±
6= 0,·‚
J(w) = m≥inf{J(v) |v
±
6= 0,J
0
(v) = 0}.
d,Šâ{J(v) |v
±
6= 0,J
0
(v) = 0}⊂M6= ∅,Œ
inf{J(v) |v
±
6= 0,J
0
(v) = 0}≥inf
w∈M
J(w) = m,
ùÒ`²(3.7) ¤á.Ïd,dM⊂N,·‚kJ(w)=m≥c.ŠâÚn6,¯K(1.1)z‡Ä
)ØCÒ,ddÑ(ØJ(w) = m>c.Ïd,ŠâÚn9,·‚kc<m<2c.y..
©Ì‡|^C©•{ïÄ˜a©êChoquard •§CÒ)•35.·‚ïÄ˜a‘
kšÛÜš‚5‘©êChoquard •§,T•§A:´ÓžÑy©êŽfšÛÜ5Ú
š‚5‘šÛÜ5.(ÜEkelandC©nÚÛ¼ê½n,·‚y²T•§•34UþCÒ
)(¤kCÒ)¥äk•$Uþö),…y²ÙUþ0uÄUþ†2 ÄUþƒm.·‚ï
Äòr?Choquard •§(=‘kšÛÜ‘Schr¨odinger •§) ïÄ,•þfåÆ!zÆ!và
Ôn!š‚51Æ+•uÐJønØ|±.
ë•©z
[1]Lieb, E. (1977)ExistenceandUniquenessof theMinimizingSolutionofChoquard’s Nonlinear
Equation.StudiesinAppliedMathematics,57,93-105.
https://doi.org/10.1002/sapm197757293
DOI:10.12677/aam.2022.1174374106A^êÆ?Ð
p7u
[2]Pekar,S.(1954)Untersuchung¨uberdieelektronentheoriederkristalle.AkademieVerlag,
Berlin.
[3]Moroz, I.,Penrose, R. and Tod,P.(1998) Spherically-SymmetricSolutions ofthe Schr¨odinger-
NewtonEquations.ClassicalandQuantumGravity,15,2733-2742.
https://doi.org/10.1088/0264-9381/15/9/019
[4]Bahrami,M., Großardt,A., Donadi,S.andBassi, A.(2014)TheSchr¨odinger-NewtonEquation
andItsFoundations.NewJournalofPhysics,16,7-28.
https://doi.org/10.1088/1367-2630/16/11/115007
[5]Giulini,D.andGroßardt,A.(2012)TheSchr¨odinger-NewtonEquationasaNon-Relativistic
LimitofSelf-GravitatingKlein-Gordon andDirac Fields.ClassicalandQuantumGravity,29,
ArticleID:215010.https://doi.org/10.1088/0264-9381/29/21/215010
[6]Moroz,V.andSchaftingen,J.V.(2017)AGuidetotheChoquardEquation.JournalofFixed
PointTheoryandApplications,19,773-813.https://doi.org/10.1007/s11784-016-0373-1
[7]Alves,C.,N´obrega,A.andYang,M.(2016)Multi-BumpSolutionsforChoquardEquation
withDeepeningPotential Well.CalculusofVariationsandPartialDifferentialEquations,55,
1-28.https://doi.org/10.1007/s00526-016-0984-9
[8]Moroz,V.andSchaftingen,J.V.(2013)GroundstatesofNonlinearChoquardEquations:Ex-
istence,QualitativePropertiesandDecayAsymptotics.JournalofFunctionalAnalysis,265,
153-184.https://doi.org/10.1016/j.jfa.2013.04.007
[9]Moroz,V.andSchaftingen,J.V.(2015)ExistenceofGroundstatesforaClassofNonlinear
ChoquardEquations.TransactionsoftheAmericanMathematicalSociety,367,6557-6579.
https://doi.org/10.1090/S0002-9947-2014-06289-2
[10]Gao,F.andYang,M.(2017)OnNonlocalChoquardEquationswithHardy-Littlewood-
SobolevCriticalExponents.JournalofMathematicalAnalysisandApplications,448,1006-
1041.https://doi.org/10.1016/j.jmaa.2016.11.015
[11]Clapp,M.andSalazar,D.(2013)PositiveandSignChangingSolutionstoaNonlinear
ChoquardEquation.JournalofMathematicalAnalysisandApplications,407,1-15.
https://doi.org/10.1016/j.jmaa.2013.04.081
[12]Ye,H.(2015)TheExistenceofLeastEnergyNodalSolutionsforSomeClassofKirchhoff
EquationsandChoquardEquationsinR
N
.Journal ofMathematical Analysis and Applications,
431,935-954.https://doi.org/10.1016/j.jmaa.2015.06.012
[13]Ghimenti,M. andSchaftingen, J.V.(2016)NodalSolutionsforthe Choquard Equation.Jour-
nalofFunctionalAnalysis,271,107-135.https://doi.org/10.1016/j.jfa.2016.04.019
[14]Ghimenti,M.,Moroz,V.andSchaftingen,J.V.(2017)LeastActionNodalSolutionsforthe
QuadraticChoquardEquation.ProceedingsoftheAmericanMathematicalSociety,145,737-
747.https://doi.org/10.1090/proc/13247
DOI:10.12677/aam.2022.1174374107A^êÆ?Ð
p7u
[15]d’Avenia,P.,Siciliano,G.andSquassina,M.(2015)ExistenceResultsforaDoublyNonlocal
Equation.SaoPauloJournalofMathematicalSciences,9,311-324.
https://doi.org/10.1007/s40863-015-0023-3
[16]d’Avenia, P., Siciliano, G.and Squassina,M.(2015)On FractionalChoquard Equations.Math-
ematicalModelsandMethodsinAppliedSciences,25,1447-1476.
https://doi.org/10.1142/S0218202515500384
[17]Chen,Y.and Liu,C. (2016)GroundStateSolutionsfor Non-AutonomousFractionalChoquard
Equations.Nonlinearity,29,1827-1842.https://doi.org/10.1088/0951-7715/29/6/1827
[18]Shen,Z.,Gao,F.andYang,M.(2016)GroundstatesforNonlinearFractionalChoquardE-
quationswithGeneralNonlinearities.MathematicalModelsandMethodsinAppliedSciences,
39,4082-4098.https://doi.org/10.1002/mma.3849
[19]Wei,S.(2015)Sign-ChangingSolutionsforaClassofKirchhoff-TypeProbleminBounded
Domains.JournalofDifferentialEquations,259,1256-1274.
https://doi.org/10.1016/j.jde.2015.02.040
[20]Lieb,E.andLoss,M.(1997)Analysis,SecondEdition.In:GraduateStudiesinMathematics,
Vol.14,AmericanMathematicalSociety,Providence,RI.
[21]Nezza, E.,Palatucci, G. and Valdinoci,E.(2012) Hitchhiker’sGuide tothe Fractional Sobolev
Spaces.BulletindesSciencesMath´ematiques,136,521-573.
https://doi.org/10.1016/j.bulsci.2011.12.004
[22]Caffarelli,L.andSilvestre,L.(2006)AnExtensionProblemRelatedtotheFractionalLapla-
cian.CommunicationsinPartialDifferentialEquations,32,1245-1260.
https://doi.org/10.1080/03605300600987306
[23]Br¨andle, C.,Colorado,E.,Pablo,A.andS´anchez, U.(2010)AConcave-ConvexEllipticProb-
lemInvolving theFractionalLaplacian.ProceedingsoftheRoyalSocietyofEdinburghSection
A:Mathematics,143,39-71.https://doi.org/10.1017/S0308210511000175
[24]Castro,A.,Cossio,J.andNeuberger,J.(1997)ASign-ChangingSolutionforaSuperlinear
DirichletProblem.TheRockyMountainJournalofMathematics,27,1041-1053.
https://doi.org/10.1216/rmjm/1181071858
[25]Bartsch,T.,Weth,T.andWillem,M.(2005)PartialSymmetryofLeastEnergyNodalSolu-
tionstoSomeVariationalProblems.Journald’AnalyseMath´ematique,96,1-18.
https://doi.org/10.1007/BF02787822
[26]Weth,T.(2006) EnergyBoundsfor Entire NodalSolutionsof AutonomousSuperlinearEqua-
tions.CalculusofVariationsandPartialDifferentialEquations,27,421-437.
https://doi.org/10.1007/s00526-006-0015-3
[27]Bartsch,T. and Weth, T. (2005) Three Nodal Solutions of Singularly Perturbed Elliptic Equa-
tionsonDomainswithoutTopology.Annalesdel’InstitutHenriPoincar´eC,NonLin´eaire,
22,259-281.https://doi.org/10.1016/j.anihpc.2004.07.005
DOI:10.12677/aam.2022.1174374108A^êÆ?Ð
p7u
[28]Willem,M.(1996)MinimaxTheorems.SpringerScience+BusinessMedia,Berlin.
https://doi.org/10.1007/978-1-4612-4146-1
DOI:10.12677/aam.2022.1174374109A^êÆ?Ð

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