gauss êß blow-up5gauss!«êß!blow-up5ü pÕ, ç!!ìµoâãb!« ¡ á!©!ƒr2•k.´ç...
TRANSCRIPT
Gauss!«êß!blow-up5ü
pÕ,⇤
ç!!ìµoâãB!«
¡ á
!©!ƒR2•k.´ç⌦˛"òX"Gauss"«êß��uk = Vkeuk3^ásupk(kVkkC1(⌦)+RB euk) < +1e"¬Òú""|^Têß"✏�#K5å$ßXJuk"blow-up:8Sö
òßKÈ?ø"⌦ ⇢⇢ B \Sßuk~%òá~Í!3⌦˛C1¬Ò%GreenºÍGßߘv
êß
��G =
X
p2S
8⇡�p
dûß"Çk
limk!+1
Z
⌦Vke
uk =
X
⌦\S
8⇡
T(Jkm""A¤ø¬"ŸG†á©êß+çp"ò"~^E|$¥!n‹ÿ©
‘ˆ"%á8""
'ÖcµGauss"«êß, blow-up©¤, Lp%O, ✏�#K5
§1 ⁄Û
-f¥Riemann6/(⌃, g)%Rn"&/i\ßPgf¥fp!"›˛ßø(
gf = e2ug (1-1)
K"ÇkGauss!«êß⁄!˛!«êß
��gu = Kfe2u �Kg
�gf = e2uHf ,(1-2)
⇤ƒÍ61
1
Ÿ•Kg⁄Kf©O¥›˛g⁄gfe"Gauss"«ßHg⁄Hf©O¥›˛g⁄gfe"#˛"
«"˛„êß"zòá)Úâ—Riemann6/˛"òá&/ãIXßÖ3dãIX
eGauss"«èKf"œL!$dêßß"Çå±|^Gauss"«K£±9#˛"«H§
Ôƒ&/i\f"5ü"'Xß"Ç'~Iá!ƒ&/i\S"fk"¬Ò߶'3˘
òX"êß•"Gauss"«⁄#˛"«¬Ò%˝kâ("Gauss"«⁄#˛"«"3
˘ê°ßBrezis[1]ßoÒÒ[2]ß)%+[3])ı%Í)[9Ÿ‹äˆQâ—LNı,è
"(J"
3!©•ß"Çê!ƒGauss"«Èu"Kè"3(1-2)•ßäCÜv = 2ußK¥Ñ
��gv = �2�gu = 2Kfe2u � 2Kg = 2Kfe
v � 2Kg
AO/ß,gèIOÓº›˛dx2 +dy2ûß�g = �ßKg = 0ßP2Kf = Vßu¥(1-2)z
è
��v = V ev
§±"ÇêIá!$˛„/™"êß"
!©!ƒR2•k.´ç⌦˛"êßS"
��uk = Vk(x)euk , x 2 ⌦ ⇢⇢ R2
(1-3)
"blow-up5ü""Ç"ÛäÚƒuXeb(µZ
⌦|Vk|euk < ⇤1 È?øk 2 N§·
||Vk � V0||C1(⌦) ! 0, k ! 1
0 < a < |V0| < b < 1, 8x 2 ⌦
(1-4)
øÖ"3¢Íq 2 (1, 2)¶'
rq�2Z
Br(p)|ruk|qdx < ⇤2, 8r 2 R, 8p 2 ⌦ (1-5)
^á(1-4)•1ò™¥⌦˛"'Gauss"«»©ß$1n™¥òá()5"b(ß8"
¥è*{ze°Úá?1"©¤" á(1-5)"òáwÕ`:¥ß"܇30†Cz
e'±ÿC߉N/`ßXJ-vk(x) = uk�x�
�ßKrvk(x) =
1�ruk
�x�
�ßœd"Çk
(�r)q�2Z
B�r(p)|rvk(x)|qdx = (�r)q�2 · 1
�q
Z
B�r(p)|ruk
⇣x�
⌘|qdx
= (�r)q�2 · �2
�q
Z
Br(p)|ruk(t)|qdt = rq�2
Z
Br(p)|ruk(t)|qdt
2
,"Ç!ƒòÑ;&>Riemann6/˛"ÉAêßûß á(1-5)Úg,/*˜v"?
\eòŸ"©¤Écß"Ç"!ƒ6ûÑÿIá^á(1-5)"
c°Æ'J%ß"ÇF"!ƒòáS"ß,Gauss"«Vk1w/¬Ò%,áÆ$
"Gauss"«V0ßÖ"°"°»£$=Reu§k.ûßêß")S""¬Òú""3'+
òÑ"ú"eß)S"{uk}k2Nô7¬Òß ¥œè"ÇêUlêß•'%k�ukkL1k
.ß ÿv±'%uk"C0%O"/"Ç"êßkòáÈ–"5üµå±È%òáv
-,"#Í✏0ß,RB eu < ✏0ûßuk“kC0%O""ÇòÑr˘´5ü°è✏�#K5"
Èu˜v✏�#K5"êßS"ß"Ç å±Ô·ò@*ı"blow-up©¤nÿßö
~òflâ—uk"¬Òú""
òÑ/ßblow-up©¤ù.ná‹©"
1. Blow-up:8⁄blow-up:NC!bubble
blow-up:8“¥˜v
A(p) = limr!0
lim infk!+1
Z
Br(p)euk > 0
"§k:p§§É8‹"Blow-up:ªÄ*êß#K5ß3ßÇNCukÿ¨¬Ò"
/33blow-up:pNCß"Ç UÈ%ò"S"xk ! pßrk ! 0߶'uk(xk +
rkx)~%,á~Í"!kÈ–"¬Ò5üßßǧ¬Ò%"4ÅòÑ*°èbubble"
‰N%"Ç"êßß"Çå±!ƒòá,´çB�(p)˛"êß(1-3)ߟ•B�(p)\S = ;"Gauss"«êß"✏�#K5Úâ"ÇJ¯B�(p)•"òá:"{xk}k2Nßv
uk(xk) = sup
B�(p)uk �! +1, k ! 1xk ! p, k ! 1
ƒkßä‚4åä)nße3,á,+çS/kV0 < 0ßKuk3d+çS"Åå
ä73>.˛+%ßl$3d+çS‹Úÿ¨kblow-up:"03blow-up:pò
(kV0(p) > 0"dûßXJ-
mk = uk(xk), rk = e�mk2
ø!ƒ
vk(x) = uk(xk + rkx)�mk
Ÿ•vk"(¬çè
⌦k = {x : |xk + rkx� p| < �}
KvkÚ˜vêß
��vk = Vk(xk + rkx)evk , x 2 ⌦k
3
|^[1]"#K5(J±9[4]•"òá%á(Jßå$È?ø"R > 0,uk(xk +
rkx)�mk3BRrk(xk)˛1w¬Ò%òáºÍ
w = �2 log
✓1 +
V0(p)
8|x� p|2
◆
ߘv Z
R2V0(p)e
wdx = 8⇡ (1-6)
2. f4Å
d^á(1-5)ßuk~%òá‹2"~Í"ck!Ú3W 1,q(⌦)•f¬Ò"?ò⁄ß
3B�(p)É,£çòÑ/ß3?ø"⌦0 ⇢⇢ ⌦ \ S˛§ßdGauss"«êß"✏�#K
5ßå±$0§·
uk ◆ 1, k ! 1
l$dPoincareÿ)™⁄^á(1-5)ßå±y""3™ïu1"~Í"{ck}k2N߶'uk � ck3⌦
0˛1w/¬Ò%òáGreenºÍ"
3. Œf
3blow-up©¤•ß çB� \ BRrk(p)*°èŒfßß ¥blowup©¤•Å
è(J"‹©"òÑ$Ûß"ÇÈJ·òŸŒf"[!"/Œf">.¥òŸ
"ߧ±"ÇIáÊ+ò"E|ßrÈŒf"?ÿ=zèÈŒf>."?ÿ""
Ǣp/œuòáPohozaev./)™ßå±Oé—A(p) = 8⇡ßøÖœLy"
lim�!0
limR!+1
limk!+1
Z
B�(xk)\BRrk(xk)
Vkeukdx = 0
Ú'%
limk!+1
Z
⌦0Vke
ukdx =
X
p2⌦0\S8⇡. (1-7)
˛„˘"(J—kö~Ü*"A¤)3"
1. duêß(1-3)"m‡¥/åu""L1ºÍß0"Çå±ÚŸn)è⌦˛"kÅ
#Borelˇ›ß$=,´U˛©Ÿ"o—/`ßblow-up:8“¥U˛‡8":ߟ
•"záblow-up:“ÈAu˘áˇ›")f"/áblow-up©¤"LߢS˛ç
—ßêk3“U˛‡8”":˛‚¨u)blow-upyñ"
4
2. dGauss-Bonnet(nß3òá&ë•°S2˛AkZ
S2KdA = 2⇡�(S2
) = 4⇡
$3"Ç"ú"eßV = 2KßœdÉA"»©Aè8⇡"œdßÈuS•":pßA(p) =
8⇡Ø¢˛V´ß34Åú/eblow-up:˛“î¥Î2Xòá&ë•°ò*ß$
3blow-up:8É,"°¥#""%$Özáblow-up:7,=Î2Xòá&ë•
°ßÿ¨—yAá•°Î23”òáblow-up:˛"ú/"œd"ÇÚ˘á4Å
°è“bubble”ßß¥È˛„ú/"òá/ñ"èx"
3. (1-7)òÑ*°èU˛/)™ßß"A¤ø¬¥`3/á4ÅLß•U˛7/"ß
§kU˛Å™‡8%blow-up:8˛"
ñdß"Çå±Ú!©"Ãá(Jo(§±e(n"
#n1.1. 8k 2 Nß!uk¥êß
��uk = Vk(x)euk , x 2 ⌦ ⇢⇢ R2
(1-8)
!)",!! Z
⌦|Vk|euk < ⇤1, 8k 2 N
||Vk � V0||C1(⌦) ! 0, k ! 1
0 < a < V0 < b < 1, x 2 ⌦
(1-9)
øÖ!3¢Íq 2 (1, 2)¶"
rq�2Z
Br(p)|ruk|qdx < ⇤2, 8r 2 R, 8p 2 ⌦ (1-10)
KkÖ=k±en´ú/Éòu)µ
1. uk3⌦!?ø;f8˛k.#
2. uk3⌦!?ø;f8˛òó¬Ò#�1#
3. !3kÅ:8S = {a1, a2, · · · , am} ⇢ ⌦߶"
limr!+1
lim infk!+1
Z
Br(ai)Vke
ukdx = 8⇡
5
dûßuk3?ø!⌦0 ⇢⇢ ⌦ \ S˛òó¬Ò#�1ß#Ö!3~Í"ck ! �1ß
¶"uk � ck3W 1,q(⌦)•f¬Ò!3?ø!⌦
0 ⇢⇢ ⌦ \ S˛òó¬Ò#Greenº
ÍGßG˜vêß
��G =
mX
i=1
8⇡�ai
ÖÈ?ø!⌦0 ⇢⇢ ⌦ß"Çk
limk!+1
Z
⌦0Vke
ukdx =
X
p2S\⌦0
8⇡
˘ò(n")©á!"L„/™Å@—y3[1]•"
§2 Gauss!«êß!✏�"K5
§2.1 êß��u = f 2 L1(⌦)!"K5
(⌦èRn˛"m8",êß
��u = f, x 2 ⌦
"m‡f 2 Lp(⌦), 1 < p < 1ûß|^Calderon-Zygmund©)⁄Marcinkiewicz*äß"
Çk';"Lp%O[5]
||u||W 2,p(⌦0) C(||u||Lp(⌦) + ||f ||Lp(⌦)), 8⌦0 ⇢⇢ ⌦
C = C(n, p,⌦,⌦0)
/,f 2 L1(⌦)ûßòÑ'ÿ%˛„%O"È˘´ú/ß[1]•â—*Xe(J
⁄n2.1. -f 2 L1(⌦)ßu 2 W 1,2
0 (⌦)¥êß
��u = f (2-11)
!f)ßKÈ?ø!✏ 2 (0, 4⇡)⁄q 2 (1, 2)ß"Çk
Z
⌦e(4⇡�✏) |u|
kfkL1(⌦) < C(⌦, ✏)
krukLq(⌦) < C(⌦, ✏)kfkL1
(2-12)
Proof. "ÇA^Struwe[6]"ç{"-
ut = max{0,min{u, t}}
6
Kk Z
⌦rutrudx =
Z
⌦fut
§± Z
0ut|ru|2 tkfkL1(⌦)
-u⇤t¥ut"%¸"(|BR| = |⌦|ß|B⇢| = |{u � t}|ßK
u⇤t |B⇢ = t, u⇤t |@BR = 0
du1 p < 1ûß
u 2 W 1,p0 (BR) , u 2 W 1,pÖ3@BR˛Tu = 0£Tè,éf§
0k
inf
u2W 1,20 (BR),u|B⇢=t
Z
BR\B⇢
|ru|2dx = infv2W 1,2(BR),v|@BR
=0,v|@B⇢=t
Z
BR\B⇢
|rv|2dx
/è<§Ÿ$"Ø¢¥ßœLÈZ
BR\B⇢
|rv|2dx
âC©ßå$˘á4,ä7,34v = 0û+%"|^´çBR \B⇢˛˜vA(>ä^
á"N⁄ºÍ"çò5ßå(4,ä3
w = A log r +B
?+%ߟ•A,B"äå^>ä^áçò(("Ü2Oéå$
A =t
log ⇢� logR
B = � t logR
log ⇢� logR
dûk
w =t log r
log ⇢� logR� t logR
log ⇢� logR
rw =t
log ⇢� logR· 1
r2(x1, x2)
0à%"4,äAèZ
BR\B⇢
|rw|2dx =t2
(logR� log ⇢)2
Z ⇡
0
Z R
⇢
1
r2· rdrd✓
= 2⇡(logR� log ⇢)t2
(logR� log ⇢)2= 2⇡
t2
logR� log ⇢
7
§±
2⇡t2
logR� log ⇢= inf
v2W 1,2(BR),v|@BR=0,v|@B⇢=t
Z
BR\B⇢
|rv|2dx
Z
BR\B⇢
|ru⇤|2 Z
0ut|ru|2 tkfkL1(⌦)
$˘)du ����R
⇢
���� � e2⇡t
kfkL1(⌦)
0k
|{u � t}| = ⇡|⇢|2 |⌦|e4⇡ �t
kfkL1(⌦)
dd"Ç'%Xe%O
Z
⌦e(4⇡�✏) u+
kfkL1(⌦) dx =
1X
k=0
Z
kuk+1e(4⇡�✏) u+
kfkL1(⌦) dx
1X
k=0
Z
kue(4⇡�✏) k+1
kfkL1(⌦) dx
1X
k=0
|{u � k}|e(4⇡�✏) k+1
kfkL1(⌦)
|⌦|1X
k=0
e4⇡ �k
kfkL1(⌦) · e
(4⇡�✏) k+1kfk
L1(⌦)
= |⌦|e4⇡
kfkL1(⌦)
1X
k=0
e�✏(k+1)kfk
L1(⌦) < C(⌦, ✏)
(2-13)
aq/ßÈêß
��(�u) = �f
%E˛°"⁄3ßå'%O
Z
⌦e(4⇡�✏) u�
kfkL1(⌦) dx < C(⌦, ✏) (2-14)
œLȺÍex"?Í–m™•àë¶^n4ÿ)™ßå±$0
e(4⇡�✏) u+
kfkL1(⌦) + e
(4⇡�✏) u�kfk
L1(⌦) � e(4⇡�✏) |u|
kfkL1(⌦)
œd(2-13)⁄(2-14)â—
Z
⌦e(4⇡�✏) |u|
kfkL1(⌦) dx
Z
⌦e(4⇡�✏) u+
kfkL1(⌦) dx+
Z
⌦e(4⇡�✏) u�
kfkL1(⌦) dx < C(⌦, ✏) (2-15)
8
˘“y"*(2-12)"1ò™"
è*%O||ru||Lq(⌦)ß"Ç©O%O||ru+||Lq(⌦)⁄||ru�||Lq(⌦)"dYoungÿ)™
��ru+��q = |ru+|q
[(1 + u+)(1 + 2u+)]q2
·⇥(1 + u+)(1 + 2u+)
⇤ q2
1
2/q
|ru+|q
[(1 + u+)(1 + 2u+)]2q
! q2
+
✓1� 1
2/q
◆⇣⇥(1 + u+)(1 + 2u+)
⇤ q2
⌘ 22�q
C
"|ru+|2
(1 + u+)(1 + 2u+)+�(1 + u+)(1 + 2u+)
� 22�q ·
q2
#
= C
"|ru+|2
(1 + u+)(1 + 2u+)+�(1 + u+)(1 + 2u+)
� q2�q
#
(2-16)
"ÇIá©O%O˛™m‡"¸ë3⌦˛"»©"1&ë"»©¥N¥õõ"ß
œè"ÇÆ'y"*È?ø"✏ > 0k
Z
⌦e(4⇡�✏) u+
kfkL1(⌦) dx < C(⌦, ✏)
du✏ > 0ßq 2 (0, 1)ß0µ =(4⇡�✏)(2�q)qkfkL1(⌦)
� 0ßœd
eµu+ � 1 + µu+ +
µ2
2(u+)2
?$å±'%%OZ
⌦
�(1 + u+)(1 + 2u+)
� q2�q dx =
Z
⌦
⇣1 + 3u+ + 2
�u+�2⌘ q
2�qdx
Z
⌦
✓eµu
++
3
µeµu
++
4
µ2eµu
+
◆ q2�q
dx
=
✓1 +
3
µ+
1
µ2
◆ qq�2Z
⌦
⇣eµu
+⌘ q
2�qdx
= C(✏)
Z
⌦e
(4⇡�✏)(2�q)qkfk
L1(⌦)· q2�qu
+
dx
= C(✏)
Z
⌦e(4⇡�✏) u+
kfkL1(⌦) dx
< C(⌦, ✏)
(2-17)
9
èõõ1òëßêá3(2-11)¸>”¶˛log1+2u+
1+u+ ßø5ø%
Z
⌦
|ru+|2
(1 + u+)(1 + 2u+)dx =
Z
⌦
(2 + 2u+ � 1� 2u+)|ru+|2
(1 + u+)(1 + 2u+)dx =
Z
⌦
2|ru+|2
1 + 2u+dx�
Z
⌦
|ru+|2
1 + u+dx
= �Z
@⌦ru+ · log(1 + 2u+) +
Z
⌦ru+ ·r log(1 + 2u+)
+
Z
@⌦ru+ · log(1 + u+)�
Z
⌦ru+ ·r log(1 + u+)
=
Z
⌦(�4u) · log(1 + 2u+)�
Z
⌦(�4u) · log(1 + u+)
=
Z
⌦(�4u) · log 1 + 2u+
1 + u+
=
Z
⌦f · log 1 + 2u+
1 + u+
=
Z
⌦f ·✓log 2� 1
1 + u+
◆
log 2 ·Z
⌦f
log 2 · ||f ||L1(⌦)
(2-18)
È(2-16)™¸‡3⌦˛»©ßø|^(2-18)⁄(2-17)ß“'%
Z
⌦
��ru+��q dx C
Z
⌦
"|ru+|2
(1 + u+)(1 + 2u+)+�(1 + u+)(1 + 2u+)
� q2�q
#dx
< C log 2 · ||f ||L1(⌦) + C · C(⌦, ✏) < C(⌦, ✏)
”nß"Çk Z
⌦|ru�|qdx < C(⌦, ✏) (2-19)
0dn4ÿ)™·$Z
⌦|ru|qdx
Z
⌦|ru+|qdx+
Z
⌦|ru�|qdx < C(⌦, ✏)
˘“y"*(2-12)"1&™"
5P2.1. Ø¢˛˛„!(ÿÈfêß
�(aijui)j = f
觷ߟ•
�|⇠|2 aij⇠i⇠j ⇤|⇠|2
10
⁄n2.1"%Oëâ"Ǹ^Ìÿ"
Ìÿ2.1. -f 2 L1(⌦)ßKÈ?ø!q 2 (1, 2)ßêß(2-11)3W 1,q
0 (⌦)•!)!3Öç
ò"
Proof. çò5d´ç˛N⁄ºÍ"çò5·'ß0êIy""35"+ò"fk 2D(⌦)™ïfµ
kfk � fkL1(⌦) ! 0, k ! 1
-uk˜vêß
��uk = fk, uk|@⌦ = 0
KÈ?ø"✏ 2 (0, 4⇡)ßPoincareÿ)™⁄⁄n2.1â—
kuk � umkW 1,q0 (⌦) C(⌦, ✏)kfk � fmkL1(⌦) ! 0, k ! 1
§±{uk}¥W 1,q0 (⌦)•Cauchy"ß߬Ò%êß(2-11)")u 2 W 1,q
0 (⌦)"
Ìÿ2.2. -f 2 L1(⌦)ßf � 0ßq 2 (1, 2)"!u 2 W 1,q
0 (⌦)˜vêß(2-11)ßKu � 0ßa.e.
x 2 ⌦"
Proof. -fA= min{f,A}ßK0 fA AßfA 2 L1
(⌦) ⇢ L3(⌦)"|^Lp%O(p =
3)9Sobolevi\W 2,3(⌦) ,! C1
(⌦)ßå'uA 2 C1(⌦)˜vêß
��uA = fA
-A ! +1ßfA5:¸N¬Ò%fßu¥¸N¬Ò(n⁄⁄n2.1¶'
||uA � u||W 1,q0 (⌦) ! 0, A ! 1
/d4åä)n$uA � 0ß8x 2 ⌦ß8A � 0"0u � 0ßa.e. x 2 ⌦"
§2.2 Gauss!«êß!✏�"K5
ÈGauss"«êß��u = V eu$^⁄n2.1ß"Çå±'%Xeö~%á"✏�#K5"
#n2.1. PBèR2•!¸†#""eu˜vêß
��u = V eu, x 2 B (2-20)
Ÿ•
|V | < b < 1, x 2 B (2-21)
11
KÈ?ø!q 2 (1, 2)ß!3✏0 > 0ß&Z
Beudx < ✏0 (2-22)
ûß"Çk
krukW
1, 2q2�q
✓B 1
2
◆ C (✏0, b)�1 + krukLq(B)
�(2-23)
Proof. -v˜vêß
��v = V eu
v|@B = 0
Kd(n^á±9⁄n2.1å$ßÈ?ø0 < ✏1 < 4⇡ß"Çk
Z
Be
4⇡�✏1||V eu||
L1(B)|v|dx < C(✏1)
2(˘á✏1ß+✏0 =4⇡�✏1
b · 2�q2q ßK,
Z
Beudx < ✏0 (2-24)
ûßdHolderÿ)™'
Z
Be
2q2�q |v|dx =
Z
Be
4⇡�✏1b✏0
|v|dx
Z
Be
4⇡�✏1||V eu||
L1(B)|v|dx < C (2-25)
dÿ)™
e|v| � 1 + |v|
±9q 2 (1, 2)û2q
q � 1> q > 1
å'òX"%OZ
B|v|dx < C(✏0, b),
Z
B|v|qdx < C(✏0, b),
Z
B|v|
2q2�q dx < C(✏0, b)
du"Çkêß
��(u� v) = 0
0dN⁄ºÍ"#˛ä5üßÈ?ø"x 2 B 34
u(x)� v(x) =1
|B 14|
Z
B 14(x)
(u� v)dx
12
/dJensenÿ)™ Z
B 14(x)
udx log
Z
B 14(x)
eudx < log ✏0
œd Z
B 14(x)
(u� v)dx Z
B 14(x)
|u� v|dx
Z
B 14(x)
|u|dx+
Z
B 14(x)
|v|dx < C(✏0, b)
u¥c°"#˛ä5üâ—
8x 2 B 34
u(x)� v(x) < C(✏0, b)
§±"Çk Z
B 34
e2q2�qudx =
Z
B 34
e2q2�q ve
2q2�q (u�v)
dx < C(✏0, b)
3B 34˛A^Lp%O
⇣p =
2q2�q
⌘±9Gagliardo-Nirenberg-Sobolevÿ)™ß·'
kr(u� u)kW
1, 2q2�q
✓B 1
2
◆ C
0
B@
0
@Z
B 34
e2q2�qudx
1
A
2�q2q
+
0
@Z
B 34
|u� u|2q2�q dx
1
A
2�q2q
1
CA
C(✏0, b)
1 + ku� uk
W 1,q
✓B 3
4
◆
!
C(✏0, b)
1 + kruk
Lq
✓B 3
4
◆
!
dMorreyÿ)™ß „(nk±eÜ2"Ìÿµ
Ìÿ2.3. 3Ü'n2.1É”!^áeß"Çk
krukC0,↵
✓B 1
2
◆ C(✏0, b)(1 + krukLq(B)) (2-26)
È,¢Í0 < ↵ < 1§·"
5P2.2. 5ø#3'n2.1!y"•"Ç#E5/â—&✏0!/™"Ø¢˛ßè¶'
Ö!(2-25)§·ßdHolderÿ)™å&êá 2q2�q 4⇡�✏1
b✏0§·=å"du✏1 2 (0, 4⇡)ßq 2
(1, 2)ß"Çg,/È✏0!'äkòá%O
0 < ✏0 <2⇡
b
13
§3 êß��uk = Vkeuk!blow-up©¤
!Ÿ"Ç3b((3-28)⁄(3-29)e!ƒêß(1-3)")"blow-up5ü"
§3.1 Blow-up:89Ÿƒ"5ü
§3.1.1 Blow-up:8
e°"Ç!ƒ(1-3)"È?øB•":pß(¬
A(p) = limr!0
lim infk!+1
Z
Br(p)Vke
ukdx
ò"êß��uk = Vkeuk (k 2 N)"blow-up:8S(¬è
S = {p 2 B : A(p) > 0}
Ø¢˛ßA(p)å±*n)è,´“ü˛”"5(/`ßdu{Vkeuk}3L1(⌦)•k.ß0
ß3f⇤ˇ¿e"3òáf"¬Ò%,ákÅ댓ˇ›µ"3blow-up:8•"?ø
:p˛ßkµ({p}) � A(p) > 0ß$=p¥µ")f"
§3.1.2 òáe.#OµÅ{¸!blow-up©¤
ƒkb(Vk3C1(⌦)•"4ÅV0‰k#"e."3˘´ú/ߧ2.2•"✏�#K5
·=â—A(p)"òá#"e.µ
#n3.1. 8k 2 Nß!uk¥êß
��uk = Vk(x)euk , x 2 ⌦ ⇢⇢ R2
(3-27)
!)",!! Z
⌦|Vk|euk < ⇤1, 8k 2 N
||Vk � V0||C1(⌦) ! 0, k ! 1
0 < a < V0 < b < 1, x 2 ⌦
(3-28)
øÖ!3¢Íq 2 (1, 2)¶"
rq�2Z
Br(p)|ruk|qdx < ⇤2, 8r 2 R, 8p 2 ⌦ (3-29)
ep 2 SßKA(p) � a✏0ߟ•✏0X'n2.1§„"
14
Proof. eÿ,ß(A(p) < a✏0ßK"3� > 0߶'Èv-å"k/k
a
Z
B�(p)euk <
Z
B�(p)Vke
uk < a✏0
œd Z
B�(p)euk < ✏0
dûdÌÿ2.3⁄^á3-29å$
krukkC0,↵
✓B �
2(p)
◆ < C(✏0, b)(1+krukLq(B�(p))) < C(✏0, b)(1+C||Vkeuk ||L1(B�(p))) < C(✏0, b)
§±dNewton-Leibniz˙™å$
oscB �
2(p)
uk < C(✏0, b) (3-30)
"ljÛßdûÈv-å"k7k
sup
B �2(p)
uk < C < 1 (3-31)
Ÿ•Cè,#~Í"l$
A(p) = limr!0
lim infk!+1
Z
Br(p)Vke
ukdx lim�!0
lim infk!+1
Z
B �2(p)
VkeCdx beC lim
�!0
⇡�2
4= 0
ÜA(p) > 0gÒ"
èy"‰Û(3-31)߃k5ø%Jensenÿ)™â—Z
B �2(p)
ukdx log
Z
B �2(p)
eukdx < C(✏0) (3-32)
e
sup
B �2(p)
uk ! +1, k ! 1 (3-33)
Kd(3-30)ß"Ç$k
infB �
2(p)
uk ! +1, k ! 1 (3-34)
l$ Z
B �2(p)
ukdx � ⇡�2
4· infB �
2(p)
uk ! +1, k ! 1 (3-35)
Ü(3-32)gÒ" “*§*y""
15
5P3.1. 3˛„y"•ß¢S˛"Çç—&XeØ¢µep 2 SßKÈv&#!�ß7
k
sup
B �2(p)
uk ! 1, k ! 1
#ÿÿV0¥ƒk*!e." “¥“blow-up”i°˛!øg"
5P3.2. *X⁄Û§`ßeV0 6= 0Öp 2 S 6= ;ßK7kV0(p) > 0"Ø¢˛ßeV0(p) < 0ß
duVk1w¬Ò#V0ß7!3p!#'ç⌦1߶"3⌦1˛ÈòÉv&å!k˛kVk <
0"('˘á⌦1ßÚ§k!êß(3-27)Åõ#⌦1˛\±"ƒ"duVk 2 C1(⌦1)ßä‚
IO!˝#êß!*K5%Oß"Çå±"#uk 2 C1(⌦)ß#ÖÈ?ø!kߧ·k
sup
⌦1
uk = sup
@⌦1
uk
*"Ç&+ºÍ"{vk}k2N!4åä:"{xk}k2N3k ! 1û™ïupßœdÈø©
å!kßuk!ÅåäÚ3⌦1!S‹à#"œddr4åä&nßå±"&Èv&å
!k 2 Nß3⌦1˛)kuk = const"*y3"Çb!XV0(p) < 0ß3⌦1˛ÈòÉv&å
!k˛kVk < 0ßœdêß(3-27)3⌦1˛Ú&{&˜v"
5P3.3. 3V0(p) = 0!ú/ß"Çvk1'Ÿ•Ô·!✏�*K5ß"ƒuk!¬Òú#
Úë#çå!(J"œd'Ÿ•!å‹©E|—Ô·3V0(0) > 0!b!e"ÈV0(p) =
0!ú/ß8cÉ'!Ûäèö~,ß'©ÿÉ?ÿ"
Ìÿ3.1. 3'n3.1!^áeßS¥*·:8"
Proof. ˘d(n3.1⁄R⌦ |Vk|euk < ⇤1·=å'"
Ìÿ3.2. 3'n3.1!^áeßS¥kÅ:8"
Proof. ˘¥œè⌦ ⇢⇢ R2"
§3.2 Blow-up:NC!bubble⁄e.!U?
3˛ò!•"ÇÔ·*A(p)"òáe.%OA(p) � a✏0"ä‚1òŸÅ!"5
Pß"ÇåVå±w%˘áe.kıåµ
0 < a✏0 <a
b2⇡ < 2⇡
e°"ÇœL©¤blow-up:NC"bubble5Ú˘áe.J,ñ8⇡"
#n3.2. 3Ü'n3.1É”!^áeß"Çk
A(p) � 8⇡
16
"ÇÚ˘á(n"y"©èo⁄"
⁄n3.1. 3Ü'n3.1É”!^áeßÈS•!?ø:pß!3ò":xk ! p, k ! 1¶"uk(xk) ! 1, k ! 1"
Proof. (p 2 SßdS•:"4·5ßå+� > 0߶'B�(p) \ S = {p}"d5P3.2ß7
ksupB �2(p) uk ! +1"-
uk(xk) = sup
B �2(p)
uk
ø(x0 2 B �2(p)¶'xk ! x0, k ! 1""ljÛ7kx0 = p"eÿ,ßK"3�1 > 0¶
'B�1(x0)\ S = ;"duA(x0) = 0ßå±È%�2 > 0ß v�2 < �1ßÖaqu"Ç3(
n3.1•"‰Û(3-31)ßÈv-å"kk
sup
B �12(x0)
uk < C < 1
AO/ßuk(xk) < CÈv-å"k§·ßÜuk(xk) ! 1, k ! 1gÒ"
e°"Ç-
mk = uk(xk), rk = e�mk2
¥Ñmk ! +1, k ! 1
rk ! 0, k ! 1
2-
vk(x) = uk(xk + rkx)�mk
Ÿ•vk"(¬çè
⌦k = {x : |xk + rkx� p| < �}
durk ! 0, k ! 1ß0⌦k¸N˛,/™uR2"¥$3⌦k˛k
vk 0, 8x 2 ⌦k
vk(0) = sup
⌦k
vk = 0
vk˜vêß
��vk(x) = r2k · (��uk (xk + rkx))
= Vk(xk + rkx)euk(xk+rkx) · e�mk = Vk(xk + rkx)e
vk(x)
17
$=
��vk = Vk(xk + rkx)evk , x 2 ⌦k (3-36)
È?ø2("R > 0ßdu⌦k % R2ß0ÈòÉv-å"kß"Çå±Úêß(3-36)'
‹Åõ%BR˛\±!ƒ"
⁄n3.2.
kvkkL1(BR) C(R) (3-37)
Ÿ•C(R)¥,=ù6uR!*~Í"
Proof. -v0k˜v
��v0k = Vk(xk + rkx)evk , x 2 B2R
v0k��@B2R
= 0
v0k""35dÌÿ2.1'y"3Ü(n3.1É”"^áeßVkevk 2 L1(B2R)ßVkevk � 0ß
0dÌÿ2.2ßv0k � 0ßa.e. x 2 B2R"œdd(2-12)"1ò™å$7k
kv0kkL1(B2R) C(R) (3-38)
5ø%
�(�vk + v0k) = 0, x 2 B2R
œdß%E(n2.1"y""cå‹©ßå$
���vk + v0k�� < C < 1, x 2 BR
ddå$ß(3-38)%.X(3-37)"
l˛°)E"L1k.S"•ƒ+òá1w¬Òf""E‚¥~^"""ÇÚ
Ÿ/\e°"⁄n"
⁄n3.3. !3{vk}!òáf"1w¬Ò#,ºÍw"w˜vêß
��w = V0(p)ew
(3-39)
Ö
w(0) = sup
BR
w = 0 (3-40)
18
Proof. È?ø"¢Í1 < p < 1ß⁄n3.2ç—Vkevk 2 Lp(BR)"dLp%Oßå$S
"{vk}3W 2,p(BR)•k."dup > 1ûW 2,p
(BR)¥gá"ß0"3{vk}"òáf"£ÿîEPè{vk}§3W 2,p
(BR)•f¬Òu,ºÍw"
5ø%êß(3-36)m‡.kvkß0"Çå±È(3-36)¸‡¶à´&62¬!Íß
,!2g¶^Lp%OßÌ${vk}3W 4,p(BR)•k."%E˘òLßß"Ç$0Ø¢
˛È?ø"/Ím > 0ß{vk}3Wm,p(BR)•k."d';"Sobolevi\(n£~
X[7]§5.6.3ßTheorem 6§ß{vk}3C1(BR)•k."AO/ß{vk}3BR˛¥òók.
⁄)›ÎY"ßœddArzela-Ascoli(nå$"3{vk}"òáf"£ÿîEPè{vk}§3BR˛òó¬Ò"dr¬Ò⁄f¬Ò”û§·û4Å"çò5ߢáòó¬Ò"
4Å7èw"aq/ßÈ{vk}"à6!Í©O¶^Arzela-Ascoli(nßå$vk¢S˛
¥1w/¬Ò%w""dd"Çå±3êß(3-36)¸‡+4Åß?$'%(3-39)⁄(3-
40)"
Å!"Ç5*§(n3.2"y""
'n3.2!y". d˛„7⁄nß"ÇkZ
BR
V0(p)ewdx = lim
k!+1
Z
BR
Vk(xk + rkx)evkdx = lim
k!+1
Z
BRrk(xk)
e�mkVk(x)eukdx < C
(3-41)
Ÿ•^%mk ! +1±9R⌦k
|Vk|euk < ⇤1"ddåÑß~ÍCÜR&'"3(3-41)¸‡
”û-R ! 1ø|^V0(p) > a > 0ß"Ç'%Z
R2ewdx C < +1 (3-42)
[4]|^+ƒ#°{y"*3^á(3-42)eêß(3-39)kçò)
w = �2 log
✓1 +
V0(p)
8|x� p|2
◆
Öߘv Z
R2ewdx =
8⇡
V0(p)(3-43)
5ø%ßÈ?ø2("r > 0ßdu
rk ! 0, xk ! p, k ! 1
19
0ÈòÉv-å"koUkBRrk(xk) ⇢ Br(p)"œd
A(p) = limR!1
A(p) = limR!1
limr!0
lim infk!1
Z
Br(p)Vke
ukdx
� limR!1
limr!0
lim infk!1
Z
BRrk(xk)
Vkeukdx
� limR!1
limr!0
limk!1
Z
BRrk(xk)
Vkeuk�mkdx
= limR!1
limk!1
Z
BRrk(xk)
Vkevkdx
=
Z
R2V0(p)e
wdx = 8⇡
Ÿ•^%e�mk ! 0, k ! 1±9(3-43)"ñd"Çy"*A(p) � 8⇡"
5P3.4. 3(3-41)•ßœLòá4ÅLßßÈv&å!k"Çå±Ú?¤BRÿ†#⌦k•"
˘pR!?ø5¶""Çå±ÈR'4Åß?#"#(3-42)"
§3.3 uk3⌦\S˛!¬Ò
'L˛ò!Èblow-up:8"?ÿß"Çå±*//èxuk3⌦˛"¬Òú""
,p /2 SûßÈ?ø"✏ > 0ß"Çå±È%� > 0߶',kv-åûßokZ
B�(p)Vke
ukdx < ✏0 (3-44)
”(n3.1•"(3-30)⁄(3-31)ß"Çk
oscB �
2(p)
uk < C < 1, sup
B �2(p)
uk < C
§±ßÈ?ø"⌦0 ⇢⇢ ⌦ \ SßA^;CXå'
osc⌦0
uk < C(⌦,⌦0), sup
⌦0uk < C(⌦,⌦0
)
±e"Ç?ÿ¸´åU5ßøç—uk3⌦ \ S˛"¬Òÿ¨kŸ¶"ú""
§3.3.1 ú/òµ#3x0 2 ⌦ \ S߶$uk(x0) ! �1
dûßd
osc⌦0
uk < C(⌦,⌦0)
å$ß3?ø"⌦0 ⇢⇢ ⌦˛
uk ◆ �1, k ! 1
20
+� > 0߶'B�(x0) \ S = ;"-uk =1
|B�(x0)|RB�(x0)
ukdx""Çk
|uk(x0)� uk| =
�����1
|B�(x0)|
Z
B�(x0)(uk(x0)� uk) dx
�����
1
|B�(x0)|
Z
B�(x0)|uk(x0)� uk| dx
oscB�(x0)
uk < C(⌦,⌦0) < 1
dPoincareÿ)™±9%O(3-29)ß"Çk
kuk � uk(x0)kW 1,q(⌦) C + kuk � ukkW 1,q(⌦) C 0(1 + krukkLq(⌦)) < C 00
œd"Çå±(
uk � uk(x0) + G, k ! 1 in W 1,q(⌦)
È?ø"' 2 D(⌦)ß òá“approximation identity”™";„ß"Çk
limk!+1
Z
⌦rukr'dx = lim
k!+1
Z
⌦Vke
uk'dx =
X
p2SA(p)'(p)
Ÿ•"Ç^%Éc"(ÿµ3?ø⌦0 ⇢⇢ ⌦\S˛ßuk ◆ �1, k ! 1ßœd3SÉ,
˛„»©¬Ò%0"œdß"ÇÈ%"GaquGreenºÍß3©Ÿø¬e˜ve°"
êß
��G =
X
p2SA(p)�p
È?ø2("⌦0 ⇢⇢ ⌦\Sßduosc
⌦0uk < C(⌦,⌦0
)ß0kkuk�uk(x0)kL1(⌦0) < C(⌦,⌦0)ß
œ$euk = euk�uk(x0)+uk(x0)3?ø"Lp•k."aqu⁄n3.3";„ßå±$0ºÍ
"{uk � uk(x0)}k2N¢S˛3?ø⌦0 ⇢⇢ ⌦•1w/¬Ò%G"
§3.3.2 ú/#µ#3:x0 2 ⌦ \ S߶$uk(x0) > �M
3˘´ú/eßÈ?ø"⌦0 ⇢⇢ ⌦ßaquú/ò"?ÿß"Ç$0µ
1. kukkL1(⌦0) < C%
2. uk3W 1,q(⌦)•f¬Ò%òáºÍu0˜vêß
��u0 = V0eu0 +
X
p2SA(p)�p
Öuk3⌦0˛1w¬Ò%u0.
21
e°"Çy"ßdû7kS = ;ßl$uk3⌦˛S41w¬Ò%u0"
eÿ,ß+p 2 SßdS•:"4·5ßå+� > 0 ¶'B�(p) \ S = {p}"-v˜v
êß��v = V0e
u0
v|@B�(p) = 0
dÌÿ2.2ßv � 0"3B�(p)˛"Çk
��(u0 � v) = A(p)�p
œd
��
✓u0 � v +
A(p)
2⇡log |x� p|
◆= 0, x 2 B�(p) (3-45)
"Ç$0ß,p < 2ûßlog |x� p| 2 W 2,p(B�(p))"y3q 2 (1, 2)ßœdÈêß(3-
45)^Lq%O=$
u0 � v +A(p)
2⇡log |x� p| 2 W 2,q
(B�(p))
$d^á(3-29)⁄%O(2-12)"1&™ß"Çk
kr(u0 � v)kLq(B�(p)) kru0kLq(B�(p))
+ krvkLq(B�(p))< +1
0����r✓u0 � v +
A(p)
2⇡log |x� p|
◆����Lq(B�(p))
kr (u0 � v)kLq(B�(p))+
����A(p)
2⇡
���� kr (log |x� p|)kLq(B�(p))< 1
œdß"Çå±Èêß(3-45)¸‡”û¶ò62¬!Í!2âògLq%Oßl$'%
r✓u0 � v +
A(p)
2⇡log |x� p|
◆2 W 2,q
(B�(p))
0
u0 � v +A(p)
2⇡log |x� p| 2 W 3,q
(B�(p))
ä‚q 2 (1, 2)ß"ÇkSobolevi\W 3,q(B�(p)) ,! C1
(B�(p))ßœd
u0 � v +A(p)
2⇡log |x� p| = O(1), B�(p) 3 x ! p (3-46)
22
˘*"Ç“k Z
B�(p)eu0dx =
Z
B�(p)eu0�vevdx �
Z
B�(p)eu0�v
dx
� C
Z
B�(p)exp
⇣log |x� p|�
A(p)2⇡
⌘dx
= C
Z
B�(p)
1
|x� p|A(p)2⇡
dx
� C
Z
B�(p)
1
|x� p|8⇡2⇡
dx
= C
Z
B�(p)
1
|x� p|4dx = +1
(3-47)
Ÿ•^%(n3.2"(ÿA(p) � 8⇡ß±9ev � 1£œèv � 0§",$ß,òê°ß
dFatou⁄n¥$ Z
B�(p)V0e
u0dx < ⇤1
˘Ü(3-47)gÒ" *ß"Ç“`"*3ú/&e7kS = ;"dûuk3⌦˛S41w
¬Ò%u0"
§4 Blow-up:8!?ò⁄èx
§4.1 òáPohozaev.$%™
Pohozaev./)™¥˝0êß•~^"Û‰"!Ÿ"ÇÚÃá¶^Xe"Pohozaev.
/)™"
⁄n4.1. PBèR2•!¸†#""eu 2 C2(B)¥êß
��u = V eu, x 2 B (4-48)
!)ßKÈ?ø!¢Í⇢ 2 (0, 1)ß3B•§·))™
Z 2⇡
0
✓@u
@r
◆2�����r=⇢
⇢2d✓+4
Z 2⇡
0
@u
@r
�����r=⇢
⇢d✓ =
Z 2⇡
0
✓@u
@✓
◆2�����r=⇢
d✓�2
Z 2⇡
0V eu
�����r=⇢
⇢2d✓+
Z
B⇢
2@V
@rreudx
(4-49)
Proof. dêß(4-48)"ÇkZ
B⇢
�r@u
@r�udx =
Z
B⇢
V eu@u
@rrdx (4-50)
23
òê°ßdGreen˙™å$Z
B⇢
�r@u
@r�udx =
Z
B⇢
r✓r@u
@r
◆·rudx�
Z
@B⇢
r@u
@r(ru · �) dS
=
Z
B⇢
@u
@rrr ·rudx+
Z
B⇢
r@ru
@r·rudx�
Z
@B⇢
r@u
@r(ru · �) dS
=
Z
B⇢
|ru|2 dx+
Z
B⇢
r@ru
@r·rudx�
Z
@B⇢
r@u
@r(ru · �) dS
Ÿ•ß•°"¸†,{ï�å/è
� =
✓x1
r,x2
r
◆= rr
34ãICÜeß/k
r@u
@r=
X @u
@xi· r@x
i
@r=
X @u
@xixi
$œL©‹»©å$
Z
B⇢
r@ru
@r·rudx =
Z 2⇡
0
Z ⇢
0
1
2r2
@|ru|2
@rdrd✓
=
Z 2⇡
0
1
2|ru|2
�����r=⇢
⇢2d✓ �Z 2⇡
0
Z ⇢
0r |ru|2 drd✓
=
Z 2⇡
0
1
2|ru|2
�����r=⇢
⇢2d✓ �Z
B⇢
|ru|2 dx
0k
Z
B⇢
�r@u
@r�udx =
Z
B⇢
|ru|2 dx+
Z 2⇡
0
1
2|ru|2
�����r=⇢
⇢2d✓ �Z
B⇢
|ru|2 dx�Z
@B⇢
r@u
@r
✓X @u
@xixi
r
◆dS
=
Z 2⇡
0
1
2|ru|2
�����r=⇢
⇢2d✓ �Z
@B⇢
r@u
@r· 1r
✓r@u
@r
◆dS
=
Z 2⇡
0
1
2|ru|2
�����r=⇢
⇢2d✓ �Z
@B⇢
r
✓@u
@r
◆2
dS
=
Z 2⇡
0
1
2|ru|2
�����r=⇢
⇢2d✓ �Z 2⇡
0
✓@u
@r
◆2�����r=⇢
⇢2d✓
(4-51)
24
,òê°ß”*/œu©‹»©ß5ø%Z
B⇢
V eu@u
@rrdx =
Z 2⇡
0
Z ⇢
0V@eu
@rr2drd✓
=
Z 2⇡
0V eu
�����r=⇢
⇢2d✓ �Z
B⇢
2V eudx�Z
B⇢
@V
@rreudx
=
Z 2⇡
0V eu
�����r=⇢
⇢2d✓ +
Z
B⇢
2�udx�Z
B⇢
@V
@rreudx
Ÿ•ßd—›(nå±é'Z
B⇢
�udx =
Z
B⇢
r · (ru) dx =
Z
@B⇢
ru · �dS
=
Z
@B⇢
1
r
✓r@u
@r
◆dS =
Z 2⇡
0
@u
@r
�����r=⇢
⇢d✓
ì\cò™=å'%
Z
B⇢
V eu@u
@rrdx =
Z 2⇡
0V eu
�����r=⇢
⇢2d✓ +
Z 2⇡
02@u
@r
�����r=⇢
⇢d✓ �Z
B⇢
@V
@rreudx (4-52)
Å!5ø%
|ru|2�����r=⇢
=
✓@u
@r
◆2�����r=⇢
+1
⇢2
✓@u
@✓
◆2�����r=⇢
(4-53)
Ú(4-51)⁄(4-52)ì\(4-50)ßø|^(4-53)ß"Ç“*§*y""
§4.2 A(p) = 8⇡
#n4.1. 3b!(1-4)⁄(1-5)eßeV0(p) > a > 0ßK"Çk
A(p) = 8⇡
Proof. ÿîòÑ5ßb(S=.ò:p"({xk}k2RX⁄n3.1§„"
È2("� > 0ß3B�(xk)˛A^Pohozaev./)™(4-49)ß"Ç'%
0
@Z 2⇡
0
0
@⇢@uk@r
�����r=⇢
1
A2
d✓ + 4
Z 2⇡
0⇢@uk@r
�����r=⇢
d✓
1
A
⇢=�
=
Z 2⇡
0
✓@uk@✓
◆2�����r=⇢=�
d✓ � 2
Z 2⇡
0⇢2eukVk
�����r=⇢=�
d✓
+O
2�
����@Vk
@r
����L1(B�(xk))
!Z
B�
eukdx
(4-54)
25
Ÿ•ä‚"Ç"b(ßVk3B�(xk)˛1w¬Ò%V0ß0����@Vk
@r
����L1(B�(xk))
< 1
œd
O
2�
����@Vk
@r
����L1(B�(xk))
!= O (�)
dÉc§§3.3.1⁄§§3.3.2"?ÿß"Ç$0eS 6= ;ßK7"3x0 2 ⌦ \S߶'uk(x0) !�1ßk ! 1"dû3?ø"⌦
0 ⇢⇢ ⌦˛§·k
uk ◆ �1, k ! 1
œdß%E§§3.3.1•"y"ß"Ç$0"3ò"¢Í{ck}k2Nß v
ck ! �1, k ! 1
¶'uk � ck1w/¬Ò%,áGreenºÍG˜v
��G = A(p)�p
œd"Çk
��
✓uk � ck +
A(p)
2⇡log |x� p|
◆! 0, k ! 1 (4-55)
duck¥~Íßœd"Çå±|^^á(3-29)⁄
,p < 2û log |x� p| 2 W 2,p(B�(p))
˘òØ¢Èêß
��
✓G+
A(p)
2⇡log |x� p|
◆= 0, x 2 B�(p)
±9ÈŸ¸‡”û¶ò62¬!Í!'%"êßâLq%O"aqu§§3.3.2•¶^Sobolevi
\(n";„£Ø¢˛˘ái\3§§3.3.1"ú/e$U§·§ß"Ç'%
G+A(p)
2⇡log |x� p| 2 W 3,q
(B�(p)) ,! C1(B�(p))
œdß"Çå±'%'(3-46)ç‰N"–m™
uk � ck ! G = �A(p)
2⇡log |x� p|+⇥(p) +
X
i=1,2
ai�xi � pi
�+O(|x� p|2), k ! 1, x ! p
26
Ÿ•ck ! �1ß⇥(p), ai(i = 1, 2)˛è~Í"ñdßòê°"Çk
0
@Z 2⇡
0
0
@⇢@uk@r
�����r=⇢
1
A2
d✓ + 4
Z 2⇡
0⇢@uk@r
�����r=⇢
d✓
1
A
⇢=�
=
✓�A(p)
2⇡
◆2
�4·A(p)
2⇡+O(�), � ! 0
,òê°ßƒk/œu4ãICÜ"Là™ß"Çk
Z 2⇡
0
����@uk@✓
����2�����r=⇢=�
d✓ =
Z 2⇡
0(�a1� sin ✓ + a2� cos ✓)
2d✓ +O(�)
= O(�), � ! 0
Ÿgd^áR⌦ |Vk|euk < ⇤1ß"Ç$0
Z 2⇡
0⇢2eukVk
�����r=⇢=�
d✓ = �2Z 2⇡
0eukVk
�����r=⇢=�
d✓ = O(�), � ! 0
d,duV0 > a > 0ß0^áR⌦ |Vk|euk < ⇤1%.XÈv-å"kk
Z
⌦euk < C < 1
l$
O
2�
����@Vk
@r
����L1(B�(xk))
!Z
B�
eukdx = O (�)
Z
B�
eukdx = O (�) , � ! 0
œdßÚ˛°©¤"àëì\(4-54)ß"Ç'%
✓�A(p)
2⇡
◆2
� 4 · A(p)
2⇡= O (�) , � ! 0
-� ! 0ß"Ç“'%A(p) = 8⇡"
k*˘á(nß"Ç“å±ÈS 6= ;ú/eºÍ"{uk}"¬Òkçç¶"èx"
Ìÿ4.1. 3'n4.1!b!eßÈ?ø!⌦0 ⇢⇢ ⌦ß"Çk
limk!+1
Z
⌦0Vke
ukdx =
X
p2⌦0\S8⇡.
Proof. "Ç$0ßœèS 6= ;ßdûêUu)§§3.3.1•"ú/"(p 2 Sß+�0v-,
¶'B�0(p) \ S = {p}"d§§3.3.1•"?ÿß"Çå±+xk ! pßrk ! 0߶'
uk(xk + rkx)� uk(xk) ! w = �2 log
✓1 +
V0(p)
8|x� p|2
◆, k ! 1
27
Ÿ•ºÍuk(xk + rkx)"(¬ç⌦k¸N˛,/™uR2"”ûß"ÇÑ$0"3ck !�1߶'È?ø"� < �0ß{uk � ck}k2N3B�0(p)\B�(p)˛1w¬Ò%GreenºÍGßG3
©Ÿø¬e˜vêß
��G = 8⇡�p, x 2 B�0(p)
3(n4.1"y"•ß"ÇÆ'w%ßå±(
G = �4 log |x� p|+⇥(p) + ai�xi � pi
�+O(|x� p|2)
Ϗk
limR!+1
limk!+1
Z
BRrk(xk)
Vkeukdx = lim
R!+1
Z
BR
V0(p)ewdx = 8⇡
⁄
lim�!0
limk!+1
Z
B�0(xk)\B�(xk)
Vkeukdx = lim
�!0lim
k!+1eckZ
B�0(xk)\B�(xk)
Vkeuk�ckdx = 0
øÖ,k ! 1ûkB�0(xk) ! B�0(p)ß"ÇêIáy"
lim�!0
limR!+1
limk!+1
Z
B�(xk)\BRrk(xk)
Vkeukdx = 0 (4-56)
Ø¢˛ßduZ
B�(x0)\BRrk(xk)
Vkeukdx = �
Z
B�(xk)\BRrk(xk)
�ukdx = �Z
@B�(xk)
@uk@r
dS+
Z
@BRrk(xk)
@uk@r
dS
œ$,k ! +1ûß"Çk
�Z
@B�(xk)
@uk@r
dS �! �Z
@B�(p)
@G
@rdS =
Z 2⇡
0
Z
r=�
4
r· rdrd✓ = 8⇡ +O(�), � ! 0
”û
Z
@BRrk(xk)
@uk@r
dS �!Z
@BR
@w
@rdS =
Z 2⇡
0
�4V0(p)8 R2
1 +V0(p)8 R2
d✓ = �8⇡V0(p)8 R2
1 +V0(p)8 R2
0
lim�!0
limR!+1
limk!+1
Z
B�(xk)\BRrk(xk)
Vkeukdx = 8⇡ � 8⇡ = 0
˘“*§*y""
28
Î"©z
[1] H. Brezis and F. Merle. Uniform estimates and blow-up behavior for solutions of ��u = v(x)eu
in two dimensions. Comm. Partial Di↵erential Equations, 16(8):1223–1253.
[2] Y. Li and I. Shafrir. Blow-up analysis for solutions of ��u = veu in dimension two. Indiana
Univ. Math. J., 43(4):1255–1270, 1994.
[3] C.C. Chen and C.S. Lin. Topological degree for a mean field equation on riemann surfaces.
(English summary) Comm. Pure Appl. Math, 56(12):1667–1727, 2003.
[4] W. Chen and C. Li. Classification of solutions of some nonlinear elliptic equations. Duke Math.
J., 63(3):615–622, 1991.
[5] David Gilbarg and Neil S. Trudinger. Elliptic Partial Di↵erential Equations of Second Order.
Springer-Verlag, Berlin Heidelberg, 2001.
[6] M. Struwe. Positive solutions of critical semilinear elliptic equations on non-contractible planar
domains. J. Eur. Math. Soc., 2(4):329–388, 2000.
[7] Lawrence C. Evans. Partial Di↵erential Equations. American Mathematical Society, Provi-
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29