设为首页 加入收藏 期刊导航 网站地图
  • 首页
  • 期刊
    • 数学与物理
    • 地球与环境
    • 信息通讯
    • 经济与管理
    • 生命科学
    • 工程技术
    • 医药卫生
    • 人文社科
    • 化学与材料
  • 会议
  • 合作
  • 新闻
  • 我们
  • 招聘
  • 千人智库
  • 我要投稿
  • 办刊

期刊菜单

  • ●领域
  • ●编委
  • ●投稿须知
  • ●最新文章
  • ●检索
  • ●投稿

文章导航

  • ●Abstract
  • ●Full-Text PDF
  • ●Full-Text HTML
  • ●Full-Text ePUB
  • ●Linked References
  • ●How to Cite this Article
PureMathematicsnØêÆ,2023,13(3),405-415
PublishedOnlineMarch2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.133044
‘·Üš‚5‘š‚5Ž™•§7Å)
rØ-½5
ëëë|||•••
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2023c131F¶¹^Fϵ2023c31F¶uÙFϵ2023c38F
Á‡
©Ì‡ïÄXe‘k·Ü˜/ªÚò È/ªš‚5 ‘š‚5Å ½™•§7Å)rØ-
½5
i∂
t
ψ+∆ψ+a|ψ|
q
ψ+
1
|x|
α

Z
R
N
|ψ|
p
|x−y|
µ
|y|
α
dy

|ψ|
p−2
ψ= 0,(t,x) ∈[0,T
∗
)×R
N
.
Ù¥N≥3,0 <µ<N,a≥0,2α+µ≤N,0 <q<
4
N
,2−
2α+µ
N
<p<
2N−2α−µ
N−2
,0 <T
∗
≤∞,
¿…ψ(t,x): [0,T
∗
)×R
N
→C´EŠ¼ê.a= 0,
2+2N−2α−µ
N
<p<
2N−2α−µ
N−2
ž,ÏLïá
»OK,y²7Å)rØ-½5"
'…c
š‚5Ž™•§§7Å)§»OK§rØ-½5
StrongInstabilityofStandingWave
SolutionsfortheNonlinearSchr¨odinger
EquationwithMixedNonlinearities
LifangZhao
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:ë|•.‘·Üš‚5‘š‚5Ž™•§7Å)rØ-½5[J].nØêÆ,2023,13(3):405-415.
DOI:10.12677/pm.2023.133044
ë|•
Received:Jan.31
st
,2023;accepted:Mar.1
st
,2023;published:Mar.8
th
,2023
Abstract
Inthispaper,weconsiderthestronginstabil ityofstandingwavesolutionsforthenon-
linearSchr¨odingerequationwithmixedpower-typeandChoquard-typenonlinearities
i∂
t
ψ+∆ψ+a|ψ|
q
ψ+
1
|x|
α

Z
R
N
|ψ|
p
|x−y|
µ
|y|
α
dy

|ψ|
p−2
ψ= 0,(t,x) ∈[0,T
∗
)×R
N
.
WhereN≥3,0<µ<N,a≥0,2α+ µ≤N,0<q<
4
N
,2 −
2α+µ
N
<p<
2N−2α−µ
N−2
,and
ψ(t,x):[0,T
∗
) ×R
N
→Cisthecomplexfunctionwith0<T
∗
≤∞.Whena=0and
2+2N−2α−µ
N
<p<
2N−2α−µ
N−2
,weprovethestronginstabilityofstandingwavesolutionsby
usingblow-upcriterion.
Keywords
NonlinearSchr¨odingerEquation,StandingWaveSolutions,Blow-UpCriterion,Strong
Instability
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó9̇(J
©Ì‡ïÄXe‘k·Ü˜/ªÚòÈ/ªš‚5‘š‚5Ž™•§
(
i∂
t
ψ+∆ψ+a|ψ|
q
ψ+
1
|x|
α
(
R
R
N
|ψ|
p
|x−y|
µ
|y|
α
dy)|ψ|
p−2
ψ= 0,(t,x) ∈[0,T
∗
)×R
N
,
ψ(0) = ψ
0
∈H
1
(R
N
),
(1.1)
Ù¥N≥3,0<µ<N,a≥0,2α+µ≤N,0<q<
4
N
,2−
2α+µ
N
<p<
2N−2α−µ
N−2
,0<T
∗
≤∞,
¿…ψ(t,x) : [0,T
∗
)×R
N
→C´EŠ¼ê.
•§(1.1)äkéõÔnµ.a=0,α=0…p=2ž,T•§¡•Hartree•§,
T•§ƒA…ܯK®²3©z[1–3]¥ïÄ.N=3,a=0,α=0,p=2…µ=2ž,
Pekar3©z[4]¥Ú\T•§^5£ã3êÆÔn¥'u·Ž4zfþfnØ.†dÓž,P.
DOI:10.12677/pm.2023.133044406nØêÆ
ë|•
Lions3©z[5]¥A ^T•§£ã˜‡(3gC^4¥>f,3,«§Ýe,ù«y–Cqu'
uü|©lfNHartee-FocknØ.d,R.Penrose3©z[6]¥JÑù‡•§^5£ãg
ÚåÔŸ.,3ù«œ/eþf~@•´˜«Úåy–.Ïd,T•§Ï~q¡•
Schr¨odinger-Newton•§.•C,ùa•§®²3©z[7–15]¥2•ïÄ.
©Ì‡ïÄ•§(1.1)7Å),=/Xψ(t,x)=e
iωt
u(x))§Ù¥ω∈R´ªÇ,
u∈H
1
(R
n
)´Xeý•§š²…)
−∆u+ωu= a|u|
q
u+

|x|
−µ
∗

1
|x|
α
|u|
p

1
|x|
α
|u|
p−2
u.(1.2)
ý•§(1.2)ƒéAŠ^•¼†Uþ•¼•:
S
ω
(u) := E(u)+
ω
2
Z
R
N
|u|
2
dx,(1.3)
E(u) :=
1
2
Z
R
N
|∇u|
2
dx−
a
q+2
Z
R
N
|u|
q+2
dx−
1
2p
Z
R
N
Z
R
N
|u|
p
|u|
p
|x|
α
|x−y|
µ
|y|
α
dxdy.
·‚Óž½Âe•¼
K
ω
(u) : = ∂
λ
S
ω
(λu)|
λ=1
= ||∇u||
2
L
2
+ω||u||
2
L
2
−a||u||
q+2
L
q+2
−
Z
R
N
Z
R
N
|u|
p
|u|
p
|x|
α
|x−y|
µ
|y|
α
dxdy,(1.4)
Ú
Q(u) : = ∂
λ
S
ω
(u
λ
)|
λ=1
= ||∇u||
2
L
2
−
aα
1
q+2
||u||
q+2
L
q+2
−
α
2
2p
Z
R
N
Z
R
N
|u|
p
|u|
p
|x|
α
|x−y|
µ
|y|
α
dxdy,(1.5)
Ù¥
u
λ
(x) := λ
N
2
u(λx),α
1
:=
Np
1
2
,α
2
:= Np−2N+2α+µ.(1.6)
e¡,·‚½Â(1.2) š²…)8Ü•
A
ω
:= {u
ω
∈H
1
\{0},S
0
ω
(u
ω
) = 0}.
½½½ÂÂÂ1.1(Ä)[1] )eu
ω
∈A
ω
´S
ω
38ÜA
ω
þ4zUþ),K
G
ω
:= {u
ω
∈A
ω
,S
ω
(u
ω
) ≤S
ω
(v
ω
),∀v
ω
∈A
ω
}.
½½½ÂÂÂ1.2( rØ-½5[1] ).eéu?¿ε>0,•3ψ
0
∈H
1
¦||ψ
0
−u||
H
1
<ε,¿…
±ψ
0
•ÐŠ)ψ(t)3k•žmS»,K7Å)e
iωt
u
ω
(x)rØ-½.
DOI:10.12677/pm.2023.133044407nØêÆ
ë|•
ŠârØ-½5½Â1.2,·‚Äk£˜š‚5Ž™•§²;(Ø.éu²;š‚
5Ž™•§,XJ·‚bЊψ
0
∈Σ := {ψ
0
∈H
1
,xψ
0
∈L
2
},K•-‘p½Æ¤á,=:
1
2
d
dt
Z
R
N
|x|
2
|ψ(t,x)|
2
dx= 2Im
Z
R
N
ψ(t,x)x·∇ψ(t,x)dx.(1.7)
ÏL¦^‘pðªÚ(1.7)ª,Œ•Uþ•¼E(ψ
0
) <0,ù¿›Xš‚5Ž™•3»),Œ„
©z[1].
d,·‚2£˜rØ-½5ƒ'(Ø.7 Å)rØ-½5´BerestyckiÚCazenave
3©z[16] ¥ÄgJÑ.‘, Le Coz 3©z[17] ¥‰Ñ,˜«{´y².¯¢þ, ‡y²7Å
)rØ-½5'…3uïá»OK,̇•{´$^Pohozaev6/N:= {v∈H
1
,Q(v) = 0}
þÄ)C©•x,·‚Œ±¼'…O,=:
Q(ψ(t)) ≤2(S
ω
(ψ
0
)−S
ω
(u
ω
)).
‘,Äu‘pðªÚЊψ
0
Œ
d
2
dt
2
||xψ(t)||
2
L
2
= 8Q(ψ(t)) ≤16(S
ω
(ψ
0
)−S
ω
(u
ω
)) <0,
Ù¥,Q(ψ(t))d(1.5)ª½Â.ddŒ•§(1.1))ψ(t)3k•žmS»,ë•©z[18–20].
•C, ChenÚGuo 3©z[21] ¥ïÄa= 0,α= 0, N= 3,p= 2 œ/e)•35Ú7
Å)rØ-½5.Shi3©z[22]¥XÚ$^8¥; 5Úny²•§(1.1)7Å)3L
2
-g
.,L
2
-.ÚL
2
-‡.œ/e7Å)•35Ú;-½5.Ïd,3L
2
-‡.œ/e,•
§(1.1)7Å)´Ä3k•žmS»,´Ä•3rØ-½7Å)Ò´˜‡ŠïÄ¯K.
©Ì‡ïÄ•§(1.1)Ä7Å)rØ-½5,·‚•Äa=0œ/,ÏLÄ)C
©•x,ïá»OK,ly²7Å)rØ-½5,̇(JXe:
a= 0,u
ω
(x) = ω
1
p−2
eu(ω
1
2
x),Keu÷v•§
∆eu−eu=

|x|
−µ
∗(
1
|x|
α
|eu|
p
)

1
|x|
α
|eu|
p−2
eu.(1.8)
AÏ,ÏLOŽ·‚Œ•
E(u
ω
)
s
c
||u
ω
||
2(1−s
c
)
L
2
= E(eu)
s
c
||eu||
2(1−s
c
)
L
2
,(1.9)
||∇u
ω
||
s
c
L
2
||u
ω
||
1−s
c
L
2
= ||∇eu||
s
c
L
2
||eu||
1−s
c
L
2
,(1.10)
Ù¥,
s
c
=
N
2
−
2+N−2α−µ
2(p−1)
.
½½½nnn1.1N≥3,
2+2N−2α−µ
N
<p<
2N−2α−µ
N−2
.ψ∈C([0,T
∗
),H
1
)´•§(1.1)3a=0
DOI:10.12677/pm.2023.133044408nØêÆ
ë|•
ž),eE(ψ
0
) >0…
(
E(ψ
0
)
s
c
||ψ
0
||
2(1−s
c
)
L
2
<E(u)
s
c
||u||
2(1−s
c
)
L
2
,
||∇ψ
0
||
s
c
L
2
||ψ
0
||
1−s
c
L
2
>||∇u||
s
c
L
2
||u||
1−s
c
L
2
,
(1.11)
Ù¥,u´(1.8)Ä),K•§(1.1)7Å)ψ(t)3k•žmS».
½½½nnn1.2-a=0,N≥3,0<µ<N,α≥0,2α+µ≤N,
2+2N−2α−µ
N
<p<
2N−2α−µ
N−2
,…
u
ω
´(1.2)Ä),KÄ7Å)ψ(t,x) = e
iωt
u
ω
(x)´rØ-½.
©|„(Xe:31!¥,·‚‰Ñ˜ý•£.31n!¥,·‚òy²½
n1.1.31o!¥,·‚òy²½n1.2.
2.ý•£
!·‚̇£˜®•(J.
ÚÚÚnnn2.1(ÛÜ·½5[2])N≥3,0<µ<N,α≥0,2α+ µ≤N,0<q<
4
N−2
,
2 −
2α+µ
N
<p<
2N−2α−µ
N−2
.eψ
0
∈H
1
(R
N
),K•3T=T(kψ
0
k
H
1
)¦•§(1.1)•3•˜)
ψ∈C([0,T),H
1
).2-[0,T
∗
)´)ψ(t)4Œ•3«m,eT
∗
<∞,Klim
t→T
∗
kψ(t)k
H
1
= ∞.,
,ψ(t)÷vŸþ†UþÅð,=éu?¿0 ≤t<T
∗
,k
kψ(t)k
2
L
2
= kψ
0
k
2
L
2
, E(ψ(t)) = E(ψ
0
).
ÚÚÚnnn2.2( Br´ezis-LiebÚn[23] )0<p<∞.XJ{f
n
}´˜‡L
p
(R
N
) ˜mþ k .S
,…÷v{f
n
}3L
p
(R
N
)˜mþA??Âñuf,Kk
lim
n→+∞
(||f
n
||
p
L
p
−||f
n
−f||
p
L
p
−||f||
p
L
p
) = 0.
ƒq,k
lim
n→+∞
Z
R
N

|x|
−µ
∗

1
|x|
α
|f
n
|
p

1
|x|
α
|f
n
|
p−2
f
n
=
Z
R
N

|x|
−µ
∗

|f
n
−f|
p
|x|
α

|f
n
−f|
p−2
|x|
α
|f
n
−f|+
Z
R
N

|x|
−µ
∗

1
|x|
α
|f|
p

1
|x|
α
|f|
p−2
f.
ÚÚÚnnn2.3(Hardy-Littlewood-SobolevØª[24])N≥3,p>1,r>1,0<µ<N,
α≥0,2α+µ≤N,u∈L
p
(R
N
),v∈L
r
(R
N
),K,•3˜‡~êC(α,µ,N,p,r)÷v




Z
R
N
Z
R
N
u(x)v(y)
|x|
α
|x−y|
µ
|y|
α
dxdy




≤C(α,µ,N,p,r)kuk
L
p
(R
N
)
kvk
L
r
(R
N
)
,
Ù¥
1
p
+
1
r
+
2α+µ
N
= 2.
ÚÚÚnnn2.4( Gagliardo-NirenbergØª[22] )N≥3,0<µ<N,α≥0,2α+ µ≤N,
DOI:10.12677/pm.2023.133044409nØêÆ
ë|•
2−
2α+µ
N
<p<
2N−2α−µ
N−2
,K
Z
R
N
Z
R
N
|u|
p
|u|
p
|x|
α
|x−y|
µ
|y|
α
dxdy≤C
α,µ,p
||u||
2p−Np+2N−2α−µ
L
2
||∇u||
Np−2N+2α+µ
L
2
,
Ù¥,•Z~ê•
C
α,µ,p
=
2p
2p−Np+2N−2α−µ

2p−Np+2N−2α−µ
Np−2N+2α+µ

Np−2N+2α+µ
2
kQ
p
k
2−2p
L
2
,
Ù¥,Q
p
´Xeý•§Ä)
−∆Q
p
+Q
p
=
1
|x|
α

|x|
−µ
∗(
1
|x|
α
|Q
p
|
p
)

|Q
p
|
p−2
Q
p
.
AÏ,3L
2
-.œ/e,=:p=
2+2N−2α−µ
N
ž,•Z~ê•C
α,µ,p
= pkQk
2−2p
L
2
.
d,±ePohoˇzaevðª¤á:
||∇Q
p
||
2
L
2
=
2p−Np+2N−2α−µ
Np−2N+2α+µ
||Q
p
||
2
L
2
=
Np−2N+2α+µ
2p
Z
R
N
Z
R
N
|Q
p
|
p
|Q
p
|
p
|x|
α
|x−y|
µ
|y|
α
dxdy.
3.»OK
!·‚y²½n1.1.
Äk,·‚bψ
0
∈H
1
,ψ∈C([0,T
∗
]),H
1
)´•§(1.1)),…•3δ>0¦
sup
t∈[0,T
∗
)
K(ψ(t)) ≤−δ<0,(3.1)
K,7Å)ψ(t)3k•žmS»,=:T
∗
<+∞.
3L
2
-‡.œ/e,=:s
c
>0,·‚•ÄE(ψ
0
) ≥0œ/,d(1.11)ªŒ•





E(ψ
0
)
s
c
||ψ
0
||
2σ
L
2
<E(u)
s
c
||u||
2σ
L
2
,
||∇ψ
0
||
L
2
||ψ
0
||
σ
L
2
>||∇u||
L
2
||u||
σ
L
2
,
(3.2)
Ù¥,
σ:=
1−s
c
s
c
=
2p−α
2
α
2
−2
.
¢Sþ,dÚn2.4Œ•
C
α,µ,p
=
R
R
N
R
R
N
|u|
p
|u|
p
|x|
α
|x−y|
µ
|y|
α
dxdy
||∇u||
α
2
L
2
||u||
2p−α
2
L
2
.(3.3)
DOI:10.12677/pm.2023.133044410nØêÆ
ë|•
2(ÜþãPohoˇzaevðª,Œ
C
α,µ,p
=
2p
α
2
1
(||∇u||
L
2
||u||
σ
L
2
)
α
2
−2
.(3.4)
ÏLOŽk
E(u)||u||
2σ
L
2
=
α
2
−2
2α
2
(||∇u||
L
2
||u||
σ
L
2
)
2
.(3.5)
e¡òE(ψ(t))ü>Óž¦±||ψ(t)||
2σ
L
2
,(ÜÚn2.4,·‚k
E(ψ(t))||ψ(t)||
2σ
L
2
=
1
2
||∇ψ(t)||
2
L
2
||ψ(t)||
2σ
L
2
−
1
2p
Z
R
N
Z
R
N
|ψ(t)|
p
|ψ(t)|
p
|x|
α
|x−y|
µ
|y|
α
dxdy||ψ(t)||
2σ
L
2
≥
1
2
(||∇ψ(t)||
L
2
||ψ(t)||
σ
L
2
)
2
−
C
α,µ,p
2p
(||ψ(t)||
σ
L
2
)
Np−2N+2α+µ
= f(||∇ψ(t)||
L
2
||ψ(t)||
σ
L
2
).
Ù¥,
f(x) :=
1
2
x
2
−
C
α,µ,p
2p
x
α
2
.
d(3.4)ªŒ•,¼êf3«m(0,x
0
)þ4~,3«m(x
0
,∞)þ4O,Ù¥
x
0
=

2p
C
α,µ,p
α
2

1
α
2
−2
= ||∇u||
L
2
||u||
σ
L
2
.
aq,d(3.4)ªÚ(3.5)ªŒ•,
f(||∇u||
L
2
||u||
σ
L
2
) = E(u)||u||
2σ
L
2
.
Ïd,(ÜÚn2.1Ú(1.11)ª,éu?¿t∈[0,T
∗
],·‚k
f(||∇ψ(t)||
L
2
||ψ(t)||
σ
L
2
) ≤E(ψ(t))||ψ(t)||
2σ
L
2
= E(ψ
0
)||ψ
0
||
2σ
L
2
<E(u)||u||
2σ
L
2
= f(||∇u||
L
2
||u||
σ
L
2
),
ƒq,ŠâëY5Ú(1.11)ª,éu?¿t∈[0,T
∗
),·‚k
||∇ψ(t)||
L
2
||ψ(t)||
σ
L
2
>||∇u||
L
2
||u||
σ
L
2
.(3.6)
qduE(ψ
0
)||ψ
0
||
2σ
L
2
<E(u)||u||
2σ
L
2
,…•3η>0v,¦
E(ψ
0
)||ψ
0
||
2σ
L
2
≤(1−η)E(u)||u||
2σ
L
2
,
DOI:10.12677/pm.2023.133044411nØêÆ
ë|•
Ïd,·‚Œ
K(ψ(t))||ψ(t)||
2σ
L
2
= α
2
E(ψ(t))||ψ(t)||
2σ
L
2
−
α
2
−2
2
||∇ψ(t)||
2
L
2
||ψ(t)||
2σ
L
2
= α
2
E(ψ
0
)||ψ
0
||
2σ
L
2
−
α
2
−2
2
(||∇ψ(t)||
L
2
||ψ(t)||
σ
L
2
)
2
≤α
2
(1−η)E(u)||u||
2σ
L
2
−
α
2
−2
2
(||∇u||
L
2
||u||
σ
L
2
)
2
= −ηα
2
E(u)||u||
2σ
L
2
,(3.7)
ùL²δ= ηα
2
E(u)||u||
2σ
L
2
ž(3.1)ª¤á,•§(1.1))ψ(t)3k•žmS».y..
4.rØ-½5
!·‚y²½n1.2.
u
ω
∈G
ω
,K
S
ω
(u
λ
ω
) =
λ
2
2
||∇u
ω
||
2
L
2
+
ω
2
||u
ω
||
2
L
2
−
λ
α
2
2p
Z
R
N
Z
R
N
|u
ω
|
p
|u
ω
|
p
|x|
α
|x−y|
µ
|y|
α
dxdy,
∂
λ
S
ω
(u
λ
ω
) = λ||∇u
ω
||
2
L
2
−
α
2
2p
λ
α
2
−1
Z
R
N
Z
R
N
|u
ω
|
p
|u
ω
|
p
|x|
α
|x−y|
µ
|y|
α
dxdy=
Q(u
λ
ω
)
λ
.
dþã(JŒ•,∂
λ
S
ω
(u
λ
ω
) = 0kš")
2p||∇u
ω
||
2
L
2
α
2
R
R
N
R
R
N
|u
ω
|
p
|u
ω
|
p
|x|
α
|x−y|
µ
|y|
α
dxdy
!
1
α
2
−2
= 1.
AO/,dPohoˇzaevðªŒ•,Q(u
ω
) = 0žþãª¤á,Ïd,·‚k



∂
λ
S
ω
(u
λ
ω
) >0,λ∈(0,1),
∂
λ
S
ω
(u
λ
ω
) <0,λ∈(1,∞).
dþã(JŒ•,éu?¿λ>0…λ6= 1,·‚kS
ω
(u
λ
ω
) <S
ω
(u
ω
).qdu||u
λ
ω
||
L
2
= ||u
ω
||
L
2
,
Kéu?¿λ>1,k
E(u
λ
ω
) <E(u
ω
).(4.1)
e¡,éu?¿λ
n
>1,·‚-Њψ
0,n
(x)=u
λ
n
ω
(x)=λ
N
2
n
u
ω
(λ
n
x),…lim
n→∞
λ
n
=1.ŠâÚ
n2.2,·‚Œ•,3H
1
(R
N
)˜m¥,ψ
0,n
→u
ω
.Ïd,d(4.1)ª·‚Œ±íÑ
E(ψ
0,n
) <E(u
ω
),
||∇ψ
0,n
||
L
2
= λ
n
||∇u
ω
||
L
2
>||∇u
ω
||
L
2
.
DOI:10.12677/pm.2023.133044412nØêÆ
ë|•
(Ü(1.9)ªÚ(1.10)ª,·‚Œ±
E(ψ
0,n
)
s
c
||ψ
0,n
||
2(1−s
c
)
L
2
<||∇u
ω
||
s
c
L
2
||u
ω
||
2(1−s
c
)
L
2
= E(u)
s
c
||u||
2(1−s
c
)
L
2
,
||∇ψ
0,n
||
s
c
L
2
||ψ
0,n
||
1−s
c
L
2
>||∇u
ω
||
s
c
L
2
||u
ω
||
1−s
c
L
2
= ||∇u||
s
c
L
2
||u||
1−s
c
L
2
,
Ù¥u´(1.8)Ä),…s
c
=
N
2
−
2+N−2α−µ
2(p−1)
.
Šâ½n1.1Œ•,±ψ
0,n
•ÐŠ)ψ
n
3k•žmS».Ïd,7Å)´rØ-½.y..
ë•©z
[1]Cazenave,T.(2003)SemilinearSchr¨odingerEquations(CourantLectureNotesinMathe-
matics,Vol.10).NewYorkUniversity,CourantInstituteofMathematicalSciences/American
MathematicalSociety,NewYork/Providence,RI.
[2]Feng,B.andYuan,X.(2015)OntheCauchyProblemfortheSchr¨odinger-HartreeEquation.
EvolutionEquationsandControlTheory,4,431-445.https://doi.org/10.3934/eect.2015.4.431
[3]Feng,B.andZhang,H.(2018)StabilityofStandingWavesfortheFractionalSchr¨odinger-
HartreeEquation.JournalofMathematicalAnalysisandApplications,460,352-364.
https://doi.org/10.1016/j.jmaa.2017.11.060
[4]Pekar,S.I.(1954)Untersuchung¨uberdieElektronentheoriederKristalle.AkademieVerlag,
Berlin.
[5]Lieb,E.H.(1977)ExistenceandUniquenessoftheMinimizingSolutionofChoquard’sNon-
linearEquation.StudiesinAppliedMathematics,57,93-105.
https://doi.org/10.1002/sapm197757293
[6]Penrose,R.(1996)OnGravity’sRoleinQuantumStateReduction.GeneralRelativityand
Gravitation,28,581-600.https://doi.org/10.1007/BF02105068
[7]Chen,Z.,Shen,Z.andYang,M.(2017)InstabilityofStandingWavesforaGeneralized
ChoquardEquationwithPotential.JournalofMathematicalPhysics,58,ArticleID:011504.
https://doi.org/10.1063/1.4974251
[8]Du, L., Gao, F. and Yang, M. (2022) Existence andQualitative Analysis for NonlinearWeight-
edChoquardEquations.https://arxiv.org/abs/1810.11759
[9]Feng,B.(2018) OntheBlow-UpSolutionsfortheFractionalNonlinearSchr¨odingerEquation
withCombinedPower-TypeNonlinearities.CommunicationsonPureandAppliedAnalysis,
17,1785-1804.https://doi.org/10.3934/cpaa.2018085
[10]Feng,B.,Liu,J.,Niu,H.andZhang,B.(2020)StrongInstabilityofStandingWavesfora
Fourth-OrderNonlinearSchr¨odingerEquationwiththeMixedDispersions.Nonlinear Analysis,
196,ArticleID:111791.https://doi.org/10.1016/j.na.2020.111791
DOI:10.12677/pm.2023.133044413nØêÆ
ë|•
[11]Feng,B.andWang,Q.(2021)StrongInstabilityofStandingWavesfortheNonlinear
Schr¨odingerEquationinTrappedDipolarQuantumGases.JournalofDynamicsDifferential
Equations,33,1989-2008.https://doi.org/10.1007/s10884-020-09881-0
[12]Feng, B., Chen,R. and Liu, J. (2021)Blow-Up Criteriaand Instability of NormalizedStanding
WavesfortheFractionalSchr¨odinger-Choquard Equation.Advancesin Nonlinear Analysis, 10,
311-330.https://doi.org/10.1515/anona-2020-0127
[13]Feng,B.andZhu,S.(2021)StabilityandInstabilityofStandingWavesfortheFractional
NonlinearSchr¨odingerEquations.JournalofDifferentialEquations,292,287-324.
https://doi.org/10.1016/j.jde.2021.05.007
[14]Wang,Y.andFeng,B.(2019)SharpThresholdsofBlow-UpandGlobalExistenceforthe
Schr¨odinger EquationwithCombined Power-Type and Choquard-Type Nonlinearities. Bound-
aryValueProblems,2019,ArticleNo.195.https://doi.org/10.1186/s13661-019-01310-6
[15]Zhang,J.andZhu,S.(2017)StabilityofStandingWavesfortheNonlinearFractional
Schr¨odingerEquation.JournalofDynamicsandDifferentialEquations,29,1017-1030.
https://doi.org/10.1007/s10884-015-9477-3
[16]Berestycki,H.andCazenave,T.(1981)Instabilit´edes´etatsstationairesdansles´euationsde
Schr¨odingeretdeKlein-Gordonnonlin´eires.ComptesRendusdel’Acad´emiedesSciences—
SeriesI,293,489-492.https://zbmath.org/0492.35010
[17]LeCoz,S.(2008)AnoteonBerestycki-Cazenave’sClassicalInstabilityResultforNonlinear
Schr¨odingerEquations.AdvancedNonlinearStudies,8,455-463.
https://doi.org/10.1515/ans-2008-0302
[18]Feng,B.,Chen,R.andWang,Q.(2020)InstabilityofStandingWavesfortheNonlinear
Schr¨odinger-PoissonEquationintheL
2
-CriticalCase.JournalofDynamicsandDifferential
Equations,32,1357-1370.https://doi.org/10.1007/s10884-019-09779-6
[19]Weinstein,M.I.(1983)NonlinearSchr¨odingerEquationsandSharpInterpolationEstimates.
CommunicationsinMathematicalPhysics,87,567-576.https://doi.org/10.1007/BF01208265
[20]Ohta,M. (1995) Blow-Up SolutionsandStrongInstability of StandingWavesfortheGeneral-
izedDavey-StewartsonSysteminR
2
.Annales de l’Institut Henri Poincar´e-Physique Th´eorique,
63,111-117.https://eudml.org/doc/76684
[21]Chen,J.andGuo,B.(2007)StrongInstabilityofStandingWavesforaNonlocalSchr¨odinger
Equation.PhysicaD:NonlinearPhenomena,227,142-148.
https://doi.org/10.1016/j.physd.2007.01.004
[22]Shi, C.(2022)Existence ofStableStandingWavesfor theNonlinearSchr¨odingerEquationwith
MixedPower-TypeandChoquard-TypeNonlinearities.AIMSMathematics,7,3802-3825.
https://doi.org/10.3934/math.2022211
DOI:10.12677/pm.2023.133044414nØêÆ
ë|•
[23]Br´ezis,H. andLieb,E.H.(1983)ARelationbetweenPointwiseConvergenceofFunctionsand
ConvergenceofFunctionals.ProceedingsoftheAmericanMathematicalSociety,88,486-493.
https://doi.org/10.2307/2044999
[24]Lieb,E.H.(1983) SharpConstants intheHardy-Littlewood-SobolevandRelated Inequalities.
AnnalsofMathematics,118,349-374.https://doi.org/10.2307/2007032
DOI:10.12677/pm.2023.133044415nØêÆ

版权所有：汉斯出版社 (Hans Publishers) Copyright © 2023 Hans Publishers Inc. All rights reserved.