A Trudinger-Moser inequality on compact Riemannian surface involving Gaussian curvature

0.17 MB·English

by  Yunyan Yang

A Trudinger-Moser inequality on compact Riemannian surface
involving Gaussian curvature
YunyanYang
DepartmentofMathematics,RenminUniversityofChina,Beijing100872,P.R.China
5
1
0
2

n Abstract
a
MotivatedbyarecentworkofX.ChenandM.Zhu(Commun. Math. Stat.,1(2013)369-385),
J
we establish a Trudinger-Moserinequality on compact Riemannian surface without boundary.
9
2 Theproofisbasedonblow-upanalysistogetherwithCarleson-Chang’sresult(Bull. Sci. Math.
110 (1986) 113-127). This inequality is different from the classical one, which is due to L.

] Fontana(Comment. Math. Helv.,68(1993)415-454),sincetheGaussiancurvatureisinvolved.
G
As an application, we improve Chen-Zhu’s result as follows: A modified Liouville energy of
D
conformalRiemannianmetrichasauniformlowerbound,providedthattheEulercharacteristic
. isnonzeroandthevolumeoftheconformalsurfacehasauniformpositivelowerbound.
h
t
a Keywords: Trudinger-Moserinequality,blow-upanalysis,Liouvilleenergy
m 2010MSC: 46E35;58J05
[

1
v
1. Introduction
0
7
Let(Σ,g)beacompactRiemanniansurfacewithoutboundary,K beitsGaussiancurvature
4 g
andχ(Σ)beitsEulercharacteristic.LetW1,2(Σ)bethecompletionofC (Σ)underthenorm
7 ∞
0 1/2
. u = ( u2+u2)dv , (1)
01 k kW1,2(Σ) ZΣ |∇g | g!
5 where is the gradientoperatorand dv is the Riemannian volume element. As a limit case
g g
∇
1 of the Sobolev embeddingtheorem, the Trudinger-Moserinequality [29, 20, 19, 23, 18] plays
:
v an important role in analysis and geometry. In an elegant paper [13], L. Fontana proved that
i D. Adams’ results [1], Trudinger-Moser inequalities for higher order derivatives, still hold on
X
compactRiemannianmanifoldswithoutboundary.Amongthose,thereisthefollowing
r
a
sup eγu2dv <+ , γ 4π. (2)
g
u∈W1,2(Σ), Σudvg=0, Σ|∇gu|2dvg≤1ZΣ ∞ ∀ ≤
R R
Motivated by works of Adimurthi-Druet[2], the author [24, 25, 26] and C. Tintarev [22], we
obtainedin[27]thatforanyα<λ (Σ),thereholds
∗g
sup eγu2dv <+ , γ 4π, (3)
g
u∈W1,2(Σ), Σudvg=0, Σ|∇gu|2dvg−α Σu2dvg≤1ZΣ ∞ ∀ ≤
R R R
Emailaddress:[email protected](YunyanYang)
Preprintsubmittedto*** January30,2015
andextremalfunctionforthisinequalityexists. Hereλ (Σ)isthefirsteigenvalueoftheLaplace-
∗g
Beltramioperatorwithrespecttothemeanvaluezerocondition,namely
λ (Σ)= inf u2dv . (4)
∗g u∈W1,2(Σ),Σudvg=0,Σu2dvg=1ZΣ|∇g | g
R R
Clearlyλ (Σ) > 0 and(3) improves(2). Asaconsequenceof(3), wehaveaweakformofthe
∗g
Trudinger-Moserinequality.Namely,foranyα<λ (Σ),thereexistssomeconstantCdepending
∗g
onlyon(Σ,g)andαsuchthatforallu W1,2(Σ),thereholds
∈
u2dv α (u u)2dv 16πln eudv +16πu C, (5)
g g g g
ZΣ|∇ | − ZΣ − − ZΣ ≥−
whereu= 1 udv . Whenα=0,theinequalitywasobtainedbyDing-Jost-Li-Wang[9].
volg(Σ) Σ g
ForanyconfoRrmalmetricg˜ =eug,whereu C2(Σ),theLiouvilleenergyofg˜ reads
∈
g˜
L (g˜)= ln (R dv +R dv ),
g g˜ g˜ g g
ZΣ g
whereR andR aretwicetheGaussiancurvatureK andK respectively.Since
g g˜ g g˜
R =e u(∆ u+R ),
g˜ − g g
wehave
L (g˜)= u2+4K u dv . (6)
g g g g
ZΣ(cid:16)|∇ | (cid:17)
IfΣisatopologicaltwosphereandvol (Σ) = vol (Σ) = 4π,thenitwasprovedbyX.Chenand
g˜ g
M.Zhu[8]thatthereexistssomeconstantCdependingonlyon(Σ,g)suchthat
L (g˜) C. (7)
g
≥−
This is a veryimportantissue in the Calabi flow [6, 8]. Note that (5) can be derivedfrom(3).
One would expect an inequality, which is an analog of (3) and stronger than (7). To state our
results,wefixseveralnotations.Letusfirstdefineafunctionspace
K = u W1,2(Σ): K udv =0 (8)
g g g
( ∈ ZΣ )
andanassociateeigenvalueoftheLaplace-Beltramioperator
λ (Σ)= inf u2dv . (9)
g g g
u∈Kg, Σu2dvg=1ZΣ|∇ |
R
Clearly, K is a closed subspace of W1,2(Σ). While unlike λ (Σ) given as in (4), λ (Σ) is not
g ∗g g
necessarilynonzero. Forexample,λ (Σ) = 0ifK 0. InLemma6below,weshalldescribea
g g
necessaryandsufficientconditionunderwhichλ (Σ≡)>0. Ifα<λ (Σ)andu K ,wewrite
g g g
∈
1/2
u = u2dv α u2dv . (10)
1,α g g g
k k ZΣ|∇ | − ZΣ !
Clearly is an equivalent norm to defined as in (1) on the function space K ,
1,α W1,2(Σ) g
k·k k·k
providedthatα < λ (Σ). The first and the mostimportantresultin thispaper can be stated as
g
follows:
2
Theorem1. Let(Σ,g)beacompactRiemanniansurfacewithoutboundary,K beitsGaussian
g
curvature,andK ,λ (Σ)bedefinedasin(8),(9)respectively. SupposethattheEulercharacter-
g g
isticχ(Σ),0. Thenforanyα<λ (Σ),thereholds
g
sup e4πu2dv <+ , (11)
g
u∈Kg,kuk1,α≤1ZΣ ∞
where isanormdefinedasin(10). Moreover,4πisthebestconstant,inotherwords,if
1,α
k·k
e4πu2 isreplacedbyeγu2 in(11)foranyγ>4π,thentheabovesupremumisinfinity.
An extremely interesting case of Theorem 1 is α = 0. If λ (Σ) > 0, the norm is
g 1,0
well defined on the function space K , and thus (11) is an analog of Fontana’s inequkal·itky (2).
g
Furthermore,thefollowingtheoremrevealstherelationbetweentheTrudinger-Moserinequality
andthetopologyofΣ.
Theorem2. Let(Σ,g)beacompactRiemanniansurfacewithoutboundary,K bedefinedasin
g
(8). ThentheTrudinger-Moserinequality
sup e4πu2dv <+ (12)
g
u∈Kg, Σ|∇gu|2dvg≤1ZΣ ∞
R
holdsifandonlyiftheEulercharacteristicχ(Σ),0.
Letusexplaintherelationbetween(12)and(2)undertheassumptionthatχ(Σ),0.First,we
canseethatbothbestconstantsofthetwoinequalitiesare4π.Second,thesubcriticalinequalities
inbothcasesareequivalent.Precisely,theinequalities
sup eγu2dv <+ , γ<4π,
g
u∈Kg, Σ|∇gu|2dvg≤1ZΣ ∞ ∀
R
holdsifandonlyif(2)holdsforallγ<4π.Third,inthecriticalcase,(12)isindependentof(2).
AnotherinterestingproblemfortheTrudinger-Moserinequalityistheexistenceofextremal
functions. Pioneerworksin thisdirectionwere dueto Carleson-Chang[5], M. Struwe [21], F.
Flucher [12], K. Lin [15], Ding-Jost-Li-Wang [9, 10], and Adimurthi-Struwe[3]. Concerning
theextremalfunctionsfor(11),wehavethefollowing:
Theorem 3. If the Euler characteristic χ(Σ) , 0, then for any γ 4π and α < λ (Σ), where
g
λ (Σ)isdefinedasin(9),thesupremumin(11)canbeattainedby≤somefunctionu K with
g ∗ g
∈
u 1.
∗ 1,α
k k ≤
Onewouldaskwhatwillhappenwhenχ(Σ) = 0. Wetalkaboutthissituationbriefly. From
Lemma6below,weknowthatλ (Σ) = 0. BytheYounginequality2ab ǫa2+ǫ 1b2, ǫ > 0,
g −
≤ ∀
andFontana’sinequality(2),wecanprovethat
sup e4πu2dv <+ , α<0. (13)
g
u∈W1,2(Σ),kuk1,α≤1ZΣ ∞ ∀
For details of the proof of (13), we refer the reader to [11, 28]. Thus (13) is weaker than (2).
Therealsoholds
sup e4πu2dv sup e4πu2dv .
g g
u∈Kg,kuk1,α≤1ZΣ ≤u∈W31,2(Σ),kuk1,α≤1ZΣ
Hence,inthecaseχ(Σ) = 0,(11)isstilltrueandslightlyweakerthan(2). ItwasprovedbyY.
Li [16] that the extremalfunction for (13) exists, while it is open whether or not the extremal
functionfor(11)existsundertheassumptionχ(Σ)=0. Thisissuewillnotbediscussedhere.
FinallyweapplyTheorem1toChen-Zhu’sproblem[8]. Letg˜ = eugbeametricconformal
tog,whereu C2(Σ). Ifχ(Σ),0,wedefineamodifiedLiouvilleenergyofg˜,by
∈
8
L (g˜)= u2+ K u dv . (14)
g ZΣ |∇g | χ(Σ) g ! g
In particular, if Σ is a topological two sphere, then L (g˜) coincides with the Liouville energy
g
L (g˜)definedasin(6). Wedenote
g
C (Σ)= sup e4πu2dv . (15)
g g
u∈Kg, Σ|∇gu|2dvg≤1ZΣ
R
ThefollowingtheoremgeneralizesChen-Zhu’sresult(7).
Theorem 4. Let (Σ,g) be a compactRiemanniansurface without boundary. Supposethat the
Eulercharacteristicχ(Σ) , 0. Foranyconformalmetric g˜ = eugwith u C2(Σ), ifvol (Σ)
g˜
∈ ≥
µvol (Σ)forsomeconstantµ>0,thenthereholds
g
µvol (Σ)
g
L (g˜) 16πln ,
g ≥ C (Σ)
g
whereL (g˜)andC (Σ)aredefinedasin(14)and(15)respectively.
g g
TheproofofTheorem1andTheorem3isbasedonblow-upanalysis.Wefollowthelinesof
[25,27],andtherebyfollowcloselyY.Li[16]andAdimurthi-Druet[2]. Earlierworkshadbeen
donebyDing-Jost-Li-Wang[9,10]andAdimuthi-Struwe[3]. BothTheorem2andTheorem4
areconsequencesofTheorem1. ThefollowinglemmaduetoCarleson-Chang[5]willbeused
inouranalysis.
Lemma 5. Let B be the unitdisc in R2. Assume v is a sequenceof functionsin W1,2(B)
{ ǫ}ǫ>0 0
with v 2dx=1. If v 2dx⇀δ asǫ 0weaklyinsenseofmeasure.Then
B|∇ ǫ| |∇ ǫ| 0 →
R
limsup (e4πv2ǫ 1)dx πe.
ǫ 0 ZB − ≤
→
Anotherkeyingredientinouranalysisisthewell-knownGauss-Bonnetformula(seeforexample
[14],Section3.J.1,p. 176),namely
K dv =2πχ(Σ). (16)
g g
ZΣ
Throughout this paper, we often denote various constants by the same C, also we do not
distinguishsequenceandsubsequence.Theremainingpartofthispaperisorganizedasfollows:
Theorem 1 and Theorem 3 are proved in Section 2; Theorem 2 and Theorem 4 are proved in
Section3andSection4respectively.
4
2. ATrudinger-MoserinequalityinvolvingGaussiancurvature
In this section, we prove Theorem 1 and Theorem 3 by using blow-up analysis. We need
severalpreliminaryresultsbeforebeginningtheblow-upprocedure.
Lemma6. λ (Σ)>0ifandonlyifχ(Σ),0.
g
Proof. Letusfirstprovethatλ (Σ) canbeattained. Bydefinitionofλ (Σ) (see(9)before),we
g g
take u K such that u2dv = 1 and u 2dv λ (Σ) as j + . Clearly u is
j ∈ g Σ j g Σ|∇g j| g → g → ∞ j
boundedinW1,2(Σ). ThusR,uptoasubsequenRce,wecanassume
u ⇀u weakly in W1,2(Σ),
j 0
u u strongly in L2(Σ).
j 0
→
Itthenfollowsthat
u2dv =1, u 2dv λ (Σ). (17)
ZΣ 0 g ZΣ|∇g 0| g ≤ g
Sinceu K ,wehave
j g
∈
K u dv = lim K u dv =0. (18)
g 0 g g j g
ZΣ j→+∞ZΣ
By(17)and(18),wehavethatu K attainsλ (Σ).
0 g g
∈
If χ(Σ) , 0, we claim that λ (Σ) > 0. Suppose not. We have λ (Σ) = 0 and thus u C
g g 0
≡
for some constant C. In view of (18), we have by using the Gauss-Bonnet formula (16) that
2πCχ(Σ)=0. HenceC =0. Thiscontradicts u2dv =1. Onthecontrary,ifχ(Σ)=0,thenthe
Σ 0 g
Gauss-Bonnetformulaimpliesthatu c KR foranyc R. Henceλ (Σ)=0. (cid:3)
g g
≡ ∈ ∈
AnimmediateconsequenceofLemma6isthefollowing:
Lemma7. Supposethatχ(Σ),0. Then u K ,wehave u2dv 1 u2dv .
∀ ∈ g Σ g ≤ λg(Σ) Σ|∇g | g
R R
Proposition8. Supposethatχ(Σ),0. Thereholds
sup eγu2dv <+ , γ<4π. (19)
g
u∈Kg, Σ|∇gu|2dvg≤1ZΣ ∞ ∀
R
Proof. Takeanyu K with u2dv 1. Denoteu= 1 udv . Foranyfixedγ< 4π,
∈ g Σ|∇g | g ≤ volg(Σ) Σ g
wecanfindsomeγ ,sayγ =R(γ+4π)/2,andaconstantCdepenRdingonlyonγsuchthat
0 0
γu2 γ (u u)2+Cu2.
0
≤ −
ByLemma7,thereexistssomeconstantCdependingonlyon(Σ,g)suchthat
1
u2 u2dv C.
≤ volg(Σ)ZΣ g ≤
ThenitfollowsfromFontana’sinequality(2)that
eγu2dvg C eγ0(u−u)2dvg C
ZΣ ≤ ZΣ ≤
5
forsomeconstantCdependingonlyon(Σ,g)andγ. Therefore(19)follows. (cid:3)
Weremarkthatifχ(Σ) , 0,thenProposition8indicatesthatFontana’ssubcriticalinequali-
tiesimply(19). Conversely,assuming(19),thenusingthesameargumentasintheaboveproof
wecanget(2)foranyγ<4π.Therefore,(19)isequivalentto(2)withγ<4π.
Intheremainingpartofthissection,wealwaysassumeχ(Σ),0andα<λ (Σ).
g
Lemma9. Forany0<ǫ <4π,thereexistssomeu C1(Σ) K with u =1suchthat
ǫ g ǫ 1,α
∈ ∩ k k
e(4π−ǫ)u2ǫdvg = sup e(4π−ǫ)u2dvg. (20)
ZΣ u∈Kg,kuk1,α≤1ZΣ
Proof. Foranyfixed0<ǫ <4π,wechooseu K with u 1suchthatas j + ,
j g j 1,α
∈ k k ≤ → ∞
e(4π−ǫ)u2jdvg sup e(4π−ǫ)u2dvg. (21)
ZΣ →u∈Kg,kuk1,α≤1ZΣ
By Lemma 7, u is bounded in W1,2(Σ). Then we can assume, up to a subsequence, u ⇀ u
j j ǫ
weaklyinW1,2(Σ),u u stronglyinL2(Σ),andu u a.e.inΣ. Assuch,wehave
j ǫ j ǫ
→ →
u 2dv limsup u 2dv .
g ǫ g g j g
ZΣ|∇ | ≤ j + ZΣ|∇ |
→ ∞
Thisimmediatelyleadsto u 1and
ǫ 1,α
k k ≤
u u 2dv 1 u 2 +o (1).
ZΣ|∇g j−∇g ǫ| g ≤ −k ǫk1,α j
Observe
(4π ǫ)u2 (4π ǫ/2)(u u )2+32π2ǫ 1u2.
− j ≤ − j− ǫ − ǫ
ItfollowsfromtheHo¨lderinequalityandProposition8thate(4π−ǫ)u2j isboundedinLq(Σ)forsome
q> 1. Hencee(4π−ǫ)u2j e(4π−ǫ)u2ǫ stronglyinL1(Σ). Thistogetherwith(21)leadsto(20). Since
u K ,wehaveu →K . Itiseasytoseethat u =1.
j g ǫ g ǫ 1,α
∈ ∈ k k
Byastraightforwardcalculation,weknowthatu satisfiestheEuler-Lagrangeequation
ǫ
∆guǫ −αuǫ = λ1ǫuǫe(4π−ǫ)u2ǫ −µǫKg

 µλǫǫ ==R2Σπχ1u(Σ2ǫ)e((cid:16)4λπ1ǫ−ǫR)Σu2ǫudǫveg(4π−ǫ)u2ǫdvg+αRΣuǫdvg(cid:17), (22)
where∆ istheLaplace-Beltramioperator.Ellipticestimateimpliesthatu C1(Σ). (cid:3)
g ǫ
∈
Lemma10. Wehave
lim e(4π−ǫ)u2ǫdvg = sup e(4π−ǫ)u2dvg. (23)
ǫ→0ZΣ u∈Kg,kuk1,α≤1ZΣ
6
Proof. By Lemma9, Σe(4π−ǫ)u2ǫdvg isincreasingwith respectto ǫ > 0. Hence thelimitonthe
left hand side of (23)Rdoesexist, possibly it is infinity. Noting that for any fixed u K with
g
∈
u 1,wehave
1,α
k k ≤
e4πu2dvg =lim e(4π−ǫ)u2dvg lim e(4π−ǫ)u2ǫdvg,
ZΣ ǫ 0ZΣ ≤ǫ 0ZΣ
→ →
andwhence
sup e4πu2dvg lim e(4π−ǫ)u2ǫdvg. (24)
u∈Kg,kuk1,α≤1ZΣ ≤ǫ→0ZΣ
Ontheotherhand,itisobviousthat
lim e(4π−ǫ)u2ǫdvg sup e4πu2dvg. (25)
ǫ→0ZΣ ≤u∈Kg,kuk1,α≤1ZΣ
Combining(24)and(25),weget(23). (cid:3)
We now follow the linesof [25, 27], and therebyfollowclosely Y. Li [16] and Adimurthi-
Druet[2]. Similarblow-upschemehadbeenusedbyDing-Jost-Li-Wang[9,10]andAdimurthi-
Struwe [3]. Denote c = u (x ) = max u . If c is bounded, applying elliptic estimates to
ǫ ǫ ǫ Σ ǫ ǫ
| | | |
(22),wealreadyconcludetheexistenceofextremalfunction.Withoutlossofgenerality,wemay
assumec =u (x ) + andx p Σasǫ 0.
ǫ ǫ ǫ ǫ
→ ∞ → ∈ →
Lemma11. u ⇀0weaklyinW1,2(Σ),u 0stronglyinLq(Σ)forallq 1,and u 2dv ⇀
ǫ ǫ g ǫ g
→ ≥ |∇ |
δ weaklyinsenseofmeasureasǫ 0,whereδ istheusualDiracmeasurecenteredat p.
p p
→
Proof. Since u = 1andu K , itfollowsfromLemma7thatu isboundedinW1,2(Σ).
ǫ 1,α ǫ g ǫ
k k ∈
Precisely,wehave
λ (Σ)+1
u 2dv + u2dv g . (26)
ZΣ|∇g ǫ| g ZΣ ǫ g ≤ λg(Σ) α
−
In view of (26), without loss of generality, we can assume u ⇀ u weakly in W1,2(Σ), and
ǫ 0
u u stronglyinLq(Σ)forallq 1. Itfollowsthat
ǫ 0
→ ≥
u 2dv =1+α u2dv +o (1) (27)
ZΣ|∇g ǫ| g ZΣ 0 g ǫ
and
(u u )2dv =1 u 2dv +α u2dv +o (1). (28)
ZΣ|∇g ǫ − 0 | g −ZΣ|∇g 0| g ZΣ 0 g ǫ
Supposeu . 0. Inviewof(28),Proposition8togetherwiththeHo¨lderinequalityimpliesthat
0
e4πu2ǫ isboundedinLq(Σ)foranyfixedqwith1≤q<1/(1−ku0k21,α).Applyingellipticestimates
to (22), we have that u is uniformly bounded in Σ, which contradicts c + . Therefore
ǫ ǫ
→ ∞
u 0and(27)becomes
0
≡
u 2dv =1+o (1). (29)
g ǫ g ǫ
ZΣ|∇ |
Suppose u 2dv ⇀µinsenseofmeasure. Ifµ,δ ,theninviewof(29)andu 0,wecan
g ǫ g p 0
|∇ | ≡
choosesufficientlysmallr >0andacut-offfunctionφ C1(B (p))suchthat0 φ 1,φ 1
onB (p), φ 4/r 0and ∈ 0 r0 ≤ ≤ ≡
r0/2 |∇g 0|≤ 0
limsup (φu )2dv <1.
g ǫ g
ǫ→0 ZBr0(p)|∇7 |
Let φu = 1 φu dv . Since u 0 stronglyin Lq(Σ) for allq 1, φu 0 as ǫ 0.
ǫ volg(Σ) Σ ǫ g ǫ → ≥ ǫ → →
Using Fontana’siRnequality(2), we concludethate(4π−ǫ)(φuǫ)2 is boundedin Ls(Br0(p))forsome
s > 1. Note that φ 1 on B (p). Applying elliptic estimates to (22), we have that u is
≡ r0/2 ǫ
uniformlyboundedin B (p),whichcontradictsc + again. Therefore u 2dv ⇀ δ ,
andthiscompletestheprro0/o2fofthelemma. ǫ → ∞ |∇g ǫ| g (cid:3)p
Lemma12. λ hasapositivelowerboundandµ isbounded.
ǫ ǫ
Proof. Notethat
ZΣe(4π−ǫ)u2ǫdvg ≤ZΣ(cid:16)1+(4π−ǫ)u2ǫe(4π−ǫ)u2ǫ(cid:17)dvg =volg(Σ)+(4π−ǫ)λǫ.
ThistogetherwithLemma10impliesthatλ hasapositivelowerbound.ByLemma11,wehave
ǫ
thatu isboundedinL2(Σ). Moreover,since
ǫ
1 1
λǫ ZΣ|uǫ|e(4π−ǫ)u2ǫdvg ≤1+ λǫe4πvolg(Σ),
weconcludethatµ isaboundedsequence. (cid:3)
ǫ
Letexp bethe exponentialmapat x andinj (Σ)be theinjectivityradiusof(Σ,g). There
xǫ ǫ g
existsaδ,0 < δ < inj (Σ), suchthatforanyǫ > 0, exp mapstheEuclideandiscB (0) R2
g xǫ δ ⊂
centeredattheoriginwithradiusδontothegeodesicdiscB (x ) Σ. Let
δ ǫ
⊂
rǫ = λǫc−ǫ1e−(2π−ǫ/2)c2ǫ, (30)
p
g (x)=(exp g)(r x), x B (0).
ǫ ∗xǫ ǫ ∀ ∈ δrǫ−1
Foranyfixedβ,0<β<4π,weeestimate
rǫ2eβc2ǫ =λǫc−ǫ2e−(4π−ǫ−β)c2ǫ ≤c−ǫ2ZΣu2ǫeβu2ǫdvg →0, (31)
herewehaveusedProposition8andLemma11. Inparticular,r 0. Thisleadsto
ǫ
→
g ξ in C2 (R2), (32)
ǫ → loc
where ξ denotesthe Euclideanmetreic. Define two sequences of blow-upfunctionson the Eu-
clideandiscB (0)by
δrǫ−1
ψ (x)=c 1u (exp (r x)), ϕ (x)=c (u (exp (r x)) c ).
ǫ −ǫ ǫ xǫ ǫ ǫ ǫ ǫ xǫ ǫ − ǫ
It is first discovered by Adimurthi and M. Struwe [3] that the above function sequences are
suitableforthiskindofproblems.Bytheequation(22),wehaveonB (0),
δrǫ−1
−∆gǫψǫ =αrǫ2ψǫ +c−ǫ2ψǫe(4π−ǫ)(1+ψǫ)ϕǫ −rǫ2c−ǫ1µǫKg, (33)
−∆egǫϕǫ =αrǫ2c2ǫψǫ +ψǫe(4π−ǫ)(1+ψǫ)ϕǫ −rǫ2cǫµǫKge, (34)
e e
8
whereK (x)=K (exp (r x)).Itiseasytoseethat∆ ψ 0inL (R2), ψ 1andψ (0)=1.
Applyingg ellipticgestimxǫateǫsto (33) andnoting(32),gwǫ eǫh→aveψ ∞locψinC| 1ǫ|(≤R2), wheǫreψisa
e e ǫ → loc
distributionalsolutionto
∆ ψ=0 in R2, ψ 1, ψ(0)=1.
ξ
− | |≤
ThentheLiouvilletheoremimpliesthatψ 1onR2.
Inviewof(31),∆ ϕ isboundedinB ≡foranyfixedR>0.Notealsothatϕ (x) 0=ϕ (0)
forall x B (0). Agǫppǫlyingelliptic estiRmatesto (34), we haveϕ ϕ inC1ǫ (R≤2), wherǫeϕ
∈ δrǫ−1 e ǫ → loc
satisfies
∆ ϕ=e8πϕ in R2, ϕ(0)=0=supϕ. (35)
ξ
− R2
Moreover,wehave
e8πϕ(x)dx limsup e(4π−ǫ)(u2ǫ(expxǫ(rǫx))−c2ǫ)dx
ZBR(0) ≤ ǫ→0 ZBR(0)
= limsup e(4π−ǫ)(u2ǫ(expxǫ(y))−c2ǫ)rǫ−2dy
ǫ→0 ZBRrǫ(0)
1
= limǫ→s0upλǫ ZBRrǫ(xǫ)u2ǫe(4π−ǫ)u2ǫdvg
1 (36)
≤
byusing(30),changeofvariables,andψ 1inC1 (R2). Inviewof(35)and(36),aresultof
ǫ → loc
ChenandLi[7]impliesthat
1
ϕ(x)= log(1+πx2), x R2. (37)
−4π | | ∀ ∈
Asaconsequence
e8πϕdx=1. (38)
ZR2
Inconclusion,weobtainthefollowing:
Proposition13. ψ 1inC1 (R2)andϕ ϕinC1 (R2),whereϕsatisfies(37)and(38).
ǫ → loc ǫ → loc
Proposition 13 providesthe convergencebehavior of u near the blow-uppoint p. For the
ǫ
convergencebehaviorofu awayfrom p,wehavethefollowing:
ǫ
Proposition14. c u ⇀GweaklyinW1,q(Σ)forall1<q<2,andc u GinC1 (Σ p )
ǫ ǫ ǫ ǫ → loc \{ } ∩
L2(Σ),whereGisaGreenfunctionsatisfying
∆ G αG =δ 1+α ΣGdvgK in Σ
 g − p− 2πRχ(Σ) g (39)
 ΣGKgdvg =0.
Moreover,GcanbedecompoRsedas
1
G = f(r)logr+A +ψ , (40)
−2π p α
whererdenotesthegeodesicdistancefrom p, f(r)isanonnegativesmoothdecreasingfunction,
which is equal to 1 in B (p), and to zero for r inj (Σ), A is a constant real number,
injg(Σ)/2 ≥ g p
ψ C1(Σ)withψ (p)=0.
α α
∈
9
Proof. With a slight modification of proofs of ([25], Lemmas 4.5-4.9), we obtain c u ⇀ G
ǫ ǫ
weakly in W1,q(Σ) for all 1 < q < 2, and c u G in C1 (Σ p ) L2(Σ), where G is a
ǫ ǫ → loc \ { } ∩
distributional solution to (39). It is known ([4], Section 4.10, p. 106) that there exists some
functionh L (Σ)suchthat
∞
∈
1
∆ f(r)logr =δ +h
g −2π ! p
inthedistributionalsense. Hence
∆ G+ 1 f(r)logr =αG h 1+α ΣGdvgK (41)
g 2π ! − − 2πRχ(Σ) g
inthedistributionalsense. SinceG Ls(Σ)forany s 1bytheSobolevembeddingtheorems,
∈ ≥
the terms on the right hand side of (41) belong to Ls(Σ) for all s 1. By elliptic estimates,
G+ 1 f(r)logr C1(Σ),whichimplies(40). ≥ (cid:3)
2π ∈
In the following, we shall derive an upper boundof the integrals Σe(4π−ǫ)u2ǫdvg. There are
two waysto obtainthe upperbound: One isto use the capacityestimRate whichis dueto Y. Li
[16]; The other is to employ Carleson-Chang’s estimate (Lemma 5), which was first used by
Li-Liu-Yang[17]. Hereweprefertothesecondway. Inviewof(40),wehave
∂G
G2dv = α G2dv G ds
ZΣ\Bδ(p)|∇g | g ZΣ\Bδ(p) g−Z∂Bδ(p) ∂ν g
1+α Gdv
Σ g K Gdv
− 2πχR(Σ) ZΣ\Bδ(p) g g
1 1
= log +A +α G 2+o (1).
2π δ p k k2 δ
Henceweobtain
1 1 1
u 2dv = log +A +α G 2+o (1)+o (1) . (42)
ZΣ\Bδ(p)|∇g ǫ| g c2ǫ 2π δ p k k2 δ ǫ !
Lets =sup u andu =(u s )+. Thenu W1,2(B (p)).By(42)andthefactthat
ǫ ∂Bδ(p) ǫ ǫ ǫ − ǫ ǫ ∈ 0 δ
e e
u 2dv =1 u 2dv +α u2dv ,
ZBδ(p)|∇g ǫ| g −ZΣ\Bδ(p)|∇g ǫ| g ZΣ ǫ g
wehave
1 1 1
u 2dv τ =1 log +A +o (1)+o (1) .
Z |∇g ǫ| g ≤ ǫ − c2 2π δ p δ ǫ !
Bδ(p) ǫ
Now we choose an isotheermal coordinate system (U,φ; x1,x2 ) near p such that B (p) U,
2δ
φ(p) = 0, and the metric g = eh(dx12 +dx22) forsome{functio}nh C1(φ(U))with h(0)⊂= 0.
∈
Clearly, for any δ > 0, there exists some c(δ) > 0 with c(δ) 0 as δ 0 such that dv
g
(1+c(δ))dxandφ(B (p)) B (0) R2. Notingthatu =→0outsideB→(p),wehave ≤
δ δ(1+c(δ)) ǫ δ
⊂ ⊂
e
(u φ 1)2dx= u 2dv = u dv τ .
ǫ − g ǫ g g ǫ g ǫ
ZBδ(1+c(δ))(0)|∇ ◦ | Zφ−1(Bδ(1+c(δ))(0))|∇ | ZBδ(p)|∇ | ≤
10
e e e