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Contents lists available atScienceDirect

Journal of Functional Analysis

www.elsevier.com/locate/jfa

Uniqueness of bubbling solutions with collapsing singularities

Youngae Leea,∗, Chang-Shou Linb

a DepartmentofMathematicsEducation,TeachersCollege,KyungpookNational University,Daegu,SouthKorea

b TaidaInstituteforMathematicalSciences,CenterforAdvancedStudyin TheoreticalSciences,NationalTaiwanUniversity,Taipei106,Taiwan

a r t i c l e i n f o a bs t r a c t

Article history:

Received12July2018 Accepted5February2019 Availableonline13February2019 CommunicatedbyH.Brezis

MSC:

35A05 53A30 35J15 58J05

Keywords:

Meanfieldequation Bubblingsolutions Uniqueness

Collapseofsingularities

The seminal work [7] by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in [30] found that the “bubbling implies mass concentration”phenomenamightnotholdifthereisacollapse of singularities. Furthermore, a sharp estimate [23] for the bubblingsolutionshasbeenobtained.Inthispaper,weprove that thereexistsat mostonesequenceofbubblingsolutions ifthecollapsingsingularityoccurs.Themaindifficultycomes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearizedoperator.Itiswell-knownthatthekernelspaceis three dimensional. So the main technical ingredient of the proofistoshow thatthelimitafterre-scalingisorthogonal tothekernelspace.

©2019ElsevierInc.Allrightsreserved.

* Correspondingauthor.

E-mailaddresses:youngaelee@knu.ac.kr(Y. Lee),cslin@math.ntu.edu.tw(C.-S. Lin).

https://doi.org/10.1016/j.jfa.2019.02.002

0022-1236/©2019ElsevierInc. Allrightsreserved.

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1. Introduction

Weareconcernedwiththefollowingmeanfieldtypeequation:

ΔMu+ρ

heu

Mheudvg 1

= 4π

qiSαi(δqi1) inM,

Mudvg= 0, (1.1)

where (M,ds) is a compact Riemann surface, ρ > 0, dvg is the volume form, ΔM is the Laplace–Beltrami operatoron (M,ds), S ⊆M is a finite set of distinct points qi, αqi > 1, and δqi is the Dirac measure at qi. The point qi with Dirac measure is called vortex point or singular source. Throughout this paper, we always assumethat

|M|= 1, h >0 andh ∈C3(M).The equation(1.1) arises invarious different fields.

Inconformal geometry, (1.1) isrelated to the Nirenberg problem of finding prescribed Gaussian curvatureifS =∅,and theexistenceofapositiveconstantcurvaturemetric withconicsingularitiesifS=∅(see[42] andthereferencestherein).Theequation(1.1) is also related to the self-dualequation of therelativistic Chern–Simons–Higgs model.

Fortherecentdevelopmentsrelatedto(1.1),werefertothereadersto[6,4,2,7,15,11–14, 29,33,31,32,35–37,39,42,44,43,5,8,17–21,28,38,41,45] and referencestherein.

In the seminal work [7] by Brezis and Merle, the blow up behavior of solutions for (1.1) hasbeen studiedas follows:

TheoremA.[7,27,4]Given fixedeachvortexpointqi∈S,suppose αiN,i= 1,· · ·,N. Weassumethathisapositivesmoothfunction.Letuk beasequenceofblowupsolutions for(1.1),thatis: maxM uk +∞as k→+∞.Thenthere isanon-emptyfinite set B (blowupset)suchthat,

ρ heuk

Mheukdvg

p∈B

βpδp, where βp8πN.

For equation (1.1), we call ρ heu

Mheu∗dvg the mass distribution of the solution u. Following this terminology, Theorem A states that: When the vortex points are not collapsing,themeanfieldequationpossessesthepropertyoftheso-called“blowupsolu- tionshasthemassconcentrationproperty”.TheversionofTheoremAforthefollowing Chern–Simons–Higgs(CSH)equationwasalsoprovedin[15,Theorem3.1]:

ΔTu+ 1

ε2eu(1−eu) = 4π N j=1

δpj in T, (1.2)

where ε>0 isasmall parameterandT isaflattorus.Theequation (1.2) wasderived from the CSH model to describe vortices in high temperature superconductivity, and has been extensively studiedduring few decades. We referthe readers to [22,40,44,15, 16,33,34,39] and referencestherein. Amongthem,Lin and Yanin[34] proved thelocal

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uniquenessresultofbubblingsolutionsfor(1.2),thatis,ifuε,1anduε,2aretwosequence of bubblingsolutionsblowingupat thesamepointsundersomenon-degenerate condi- tion, thenuε,1=u2 forsmallε>0.Byapplyingtheideain[34],thelocal uniqueness result of bubblingsolutionsof (1.1) wasobtained in[3]. We note thattheworks[34,3]

are concerned with the local uniqueness of bubbling solutions when the vortex points are notcollapsing.

However, there is a big difference when the collapsing singularities are considered.

First,LinandTarantelloin[30] observedanewphenomenasuchthatblow-upsolutions with collapsing singularities might not have the “concentration” property of its mass distribution. The general version was studied in [23]. To describe the results, let us consider thefollowingequation:

⎧⎪

⎪⎨

⎪⎪

ΔMut +ρ

heu∗t

Mheu∗tdvg 1

= 4πd

i=1αi(δqi(t)1) + 4πN

i=d+1αi(δqi1) inM,

Mutdvg= 0,

(1.3)

where limt0qi(t) = q ∈ {/ qd+1,· · ·,qN}, i = 1,· · ·,d and qi(t)= qj(t) for i =j {1,· · ·,d}.Thenthefollowingholds:

TheoremB. [30,23]AssumeαiN, 1≤i≤N.Letut beasequenceofblowupsolutions of (1.3) with ρ ∈/ 8πN. Then ut blows up only at q M. Furthermore, there exists a function w suchthat

ut →w inCloc2 (M\ {q}) as t→0,andw satisfies:

⎧⎪

⎪⎪

⎪⎪

⎪⎩

ΔMw+ (ρ−8)

hew

Mhew∗ 1

= 4πd

i=1αi2m

(δq1) + 4π N i=d+1

αi(δqi1) in M,

Mwdvg= 0,

(1.4)

forsome m∈Nwith 1≤m≤[12d

i=1αi]1 and ρ>8mπ.

SoTheoremBtellsusthatthemassconcentrationdoesnolongerholdifthecollapsing singularity occurs. Indeed,we havelimt→0

Mheutdvg <+, whichis different from the situation described in Theorem A. We note that Theorem B could be improved providedthatthefollowingnondegeneracyconditionholds:

1 [x] standsfortheintegerpartofx.

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Definition1.1.A solutionwof(1.4) issaidnon-degenerate, ifthelinearizedproblem ΔMφ+ (ρ−8) hew

Mhewdvg

φ−

Mhewφdvg

Mhewdvg

= 0,

M

φdvg = 0 (1.5)

onlyadmitsthetrivialsolution.

Usingthetransversalitytheorem,wecanalwayschooseapositivesmoothfunctionh suchthatw is non-degenerate. SeeTheorem 4.1in[24]. Basedonthenon-degeneracy assumption for (1.4), some sharper estimates on the bubbling solutions of (1.3) were obtainedin[23] (seealso section2below).

Forthesimplicity, throughout this paper,we focus onthecase wherethe collapsing vorticesareonlytwo,thatis,

d= 2, α1=α2= 1, αiN for i= 3,· · ·, N. (1.6) Ourmaingoalistoprovethelocaluniquenessofblowupsolutionsof(1.3) withcollapsing singularities.

Theorem1.2. Assume that (1.6)holds andρ∈/8πN.Suppose that ut,1 andut,2 are two blow up solutions of (1.3). Assume that ut,1,ut,2 w in Cloc(M \ {q}), where w is a non-degenerate solution of (1.4) with m = 1. Then ut,1 = ut,2 for sufficiently small t>0.

Weremark thatthe studyof blow upsolutions of(1.3) with collapsing singularities is alsoimportant to compute thetopological degree forthe Toda systemas noticed in [24,26], where the degree counting of the whole system is reduced to computing the degreeofthecorrespondingshadowsystem(see[24,Theorem1.4]).Thusitisinevitable to encounter with the phenomena of collapsing singularities when we want to find a prioribound for solutions ofan associated shadow system. Forthe details,we referto thereadersto[24,26].

InordertoproveTheorem1.2, weneedtoanalyzetheasymptoticbehaviourofζt=

u(1)t,∗u(2)t,∗

u(1)t,∗u(2)t,∗L∞(M).Afterasuitablescalingonsmalldomainofq,ζtconvergestoanentire solutionofthelinearizedproblemassociated totheLiouvilleequation:

Δv+ev= 0 inR2, (1.7)

where Δ = 2 i=1

2

∂x2i denotes the standard Laplacian in R2. A solution v of (1.7) is completelyclassified[10] suchthat

v(z) =vμ,a(z) = ln 8eμ

(1 +eμ|z+a|2)2, μ∈R, a= (a1, a2)R2. (1.8)

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The freedom in thechoice of μand ais due to theinvariance of equation (1.7) under dilationsand translations.ThelinearizedoperatorLrelativetov0,0 isdefinedby,

:= Δφ+ 8

(1 +|z|2)2φ inR2. (1.9) In[1,Proposition1],ithasbeen provedthatanykernel ofLisalinearcombinationof Y0,Y1,Y2,where

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

Y0(z) := 1−|z|1+|z|22 =1 +1+2|z|2 = ∂v∂μμ,a

(μ,a)=(0,0), Y1(z) := 1+|z|z1 2 =14∂v∂aμ,a1

(μ,a)=(0,0), Y2(z) := 1+|z|z2 2 =14∂v∂aμ,a2

(μ,a)=(0,0).

(1.10)

TheorthogonalitytoY1,Y2 canbeobtainedbyapplyingasuitablePohozaev-typeiden- titiesasin[34].However,duetothenon-concentrationofmass,wemeetanewdifficulty to show the orthogonality with Y0. In order to overcome this obstacle, we apply the Green’s representation formula with some delicate analysis. This idea comes from the recent work[25]. In[25], itwasprovedthatif w isanon-degenerate solutionof(1.4), andtheassumptions(1.6) andρ∈/8πNhold,thenthereisablowupsolutionut of(1.3) suchthatut →w inCloc(M\ {q}).

Thispaperisorganizedasfollows.Insection2,wereviewsomeknownsharpestimates forblowupsolutionsof(1.3).Insection3,weanalyzethelimitbehaviorofζtinM\ {q} and a small neighborhood of q respectively. Finally, we prove Theorem 1.2 by using suitablePohozaev-typeidentitiesandGreen’srepresentationformula.

2. Preliminary

LetG(x,p) denotetheGreen’sfunctionfortheLaplaceBeltramioperatorΔM onM, thatis

ΔMG(x, p) + (δp1) = 0,

M

G(x, p)(x) = 0. (2.1)

Werecallthefollowing assumption:

d= 2, α1=α2= 1, αiN for i= 3,· · ·, N.

Letutbeasequenceofblowupsolutionsof(1.3) andwbethelimitofutinTheoremB.

Set

ut(x) =ut(x) + 4π 2 i=1

G(x, qi(t)) + 4π N i=3

αiG(x, qi), (2.2)

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and

w(x) =w(x) + 4π N i=3

αiG(x, qi). (2.3) Wemaychooseasuitablecoordinatecenteredatqand

q= 0, q1(t) =te, q2(t) =−te, where e is a fixed unit vector in S1. Wecanrewriteequation(1.3) as follows

ΔMut+ρ

h(x)e ut(x)−Gt(x)

Mheut−Gtdvg 1 = 0,

Mutdvg= 0, (2.4)

where

Gt(x) := 4πG(x, te) + 4πG(x,−te), and (2.5) h(x) :=h(x) exp(4π

N i=3

αiG(x, qi))0, h∈C3(M), h(0)>0. (2.6) FromTheoremB,wehavethatut(x)→w(x)+8πG(x,0) inCloc2 (M\{0}) andwsatisfies

ΔMw+ (ρ−8π)

h(x)ew(x)

Mhewdvg 1 = 0,

Mwdvg= 0, w∈C2(M). (2.7)

Weassumethatthelocalisothermal coordinatesystemsatisfies

ds2=e2ϕ(x)|dx|2, ϕ(0) =∇ϕ(0) = 0, (2.8) thatis, e2ϕΔM = Δ,whereΔ=2

i=1 2

∂x2i denotesthestandardLaplacianinR2.Fixa smallconstantr0(0,12).Itiswell knownthattheconformalfactorϕisasolutionof

Δϕ=e2ϕK in Br0(0), (2.9)

whereK(p) istheGaussian curvatureatp∈M.

Letϕ(x) satisfy¯ thefollowing localproblem:

Δ ¯ϕ−e2ϕρ= 0 in Br0(0), ϕ(0) =¯ ∇ϕ(0) = 0.¯ (2.10) Wedenote

ψ= 2ϕ+ ¯ϕ. (2.11)

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Inviewof(2.8) and(2.10),wenote that

xψ(x) =∇ψ(0) +O(|x|) =O(|x|), xϕ(x) =¯ O(|x|),

xϕ(x) =O(|x|) for x∈Br0(0). (2.12) Byusingthelocalcoordinate,wealsoset theregularpartofGreenfunctionG(x,qi(t)) to be

R(x, qi(t)) =G(x, qi(t)) + 1

2πln|x−qi(t)|. (2.13) Let

Rt(x) : = 4πR(x, te) + 4πR(x,−te). (2.14) Therefore wecanformulatethelocalversionof(2.4) around0 asfollows:

Δ¯ut+h1(x)|x−te|2|x+te|2e¯ut(x)= 0 inBr0(0), (2.15) where

¯

ut(x) =ut(x)ln

M

heut−Gtdvg−ϕ(x),¯

h1(x) =ρh(x)eψ(x)Rt(x), h1(x)>0 inBr0(0). (2.16) Inordertostudythebehaviourofu¯tneartheorigin,weconsiderthescaledsequence

vt(y) = ¯ut(ty) + 6 ln t, x∈Br0

t (0), (2.17)

whichsatisfies:

⎧⎨

Δvt+ht(y)evt(y)= 0 inBr0 t (0),

Br0

t (0)ht(y)evt(y)dy≤C, (2.18) with

ht(y) =h1(ty)|y−e|2|y+e|2=ρh(ty)eψ(ty)Rt(ty)|y−e|2|y+e|2. (2.19) In[23], thefollowingresultwasobtained.

TheoremC. [23,Theorem 1.2,Section5]Assumethat(1.6)holdsandρ∈/8πN.Suppose that ut be a sequence of blow up solutions of (2.4).Then the scaledfunction vt defined by (2.17) blowsupat0.

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Now we are going to give refined estimates than those providedin Theorem B and TheoremCunderthenon-degeneracyassumptionfor(2.7).Tostateourresult,wefixa constantR0>2,anddefine thefollowingnotations:

λt= max

Br0(0)vt=vt(pt), (2.20) ρt=

BtR0(tpt)ρheut−G(2)t dvg

Mheut−G(2)t dvg

, (2.21)

Ct=1

8h1(tpt)|pt−e|2|pt+e|2, (2.22) φt(x) =ut(x)−w(x)−ρtG(x, tpt). (2.23) Letφt=φtC1(M\B2R0t(tpt)).Thenwehavethefollowingresult.

TheoremD.[23,Theorem 1.4, Section5]Assume that (1.6)holds andρ∈/8πN. Letut bethesequenceof blow upsolutions of (2.4) andw+ 8πG(x,0)be itslimit inM\ {0}. If wisanon-degenerate solutionof (2.7),then

(i) φt=O(tlnt), (ii)

λt+ 2 lnt−ln

ρ ρ−8π

M

hew

⎠+w(tpt) + 2 lnCt+ 8πR(tpt, tpt)

=O(tlnt), (iii) ρt8π=O(t2lnt), (iv)

MheutGtdvgρρ8π

Mhewdvg=O(t), (v) |pt|=O(t).

InordertoproveTheoremD,theauthorsin[23] analyzedthescaledfunctionvtwith thefollowingingredients:Set

It(y) = ln eλt

(1 +Cteλt|y−qt|2)2, (2.24) whereqtischosensuchthat|qt−pt|1 and

yIt(y)

y=pt

=−tρtxR(x, tpt)

x=tpt

−t∇xw(x)

x=tpt

.

Bydirectcomputation,wehave

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|qt−pt|=O(te−λt). (2.25) Fory∈B2r0(pt),weset

ηt(y) = vt(y)−It(y)(G∗,t(ty)−G∗,t(tpt)), (2.26) where

G,t(x) =ρtR(x, tpt) +w(x). (2.27) It iseasytoseethat

ηt(pt) =vt(pt)−It(pt) =O(t2eλt), ∇ηt(pt) = 0. (2.28) Let

Λt,+=

Cteλt2 , and Λt,= (Λt,+)1=eλt2

√Ct

(2.29) and ηtbe thescaledfunctionofηt, thatis

ηt(z) =ηt((Λt,−)z+pt) for |z| ≤2R0Λt,+. Forηt(z),wehavethefollowingestimate

TheoremE.[23,Lemma7.1]SupposethattheassumptionsofTheoremDhold.Thenfor any ε∈(0,1),thereexistsaconstant Cε>0,independentof t>0andz∈B2R0Λt,+(0) such that

t(z)| ≤Cε(t+t2)(1 +|z|)ε. 3. Uniquenessoftheblow upsolutionswithmassconcentration

ToproveTheorem1.2isequivalenttoprovethelocaluniquenessofblowupsolutions of theequation(2.4).Toshowit,we arguebycontradiction andsupposethat(2.4) has twodifferentblowupsolutionsu(1)t andu(2)t ,whichsatisfyu(1)t ,u(2)t →winCloc(M\{0}), where wisanon-degenerate solutionof(2.7).Wewillusep(i)t ,λ(i)t ,u¯(i)t ,It(i), φ(i)t ,Ct(i), qt(i),v(i)t ,ρ(i)t ,η(i)t ,ηt(i),G(i)∗,t(i)t,+(i)t,− todenote pt,λt,u¯t,It,φt,Ct,qt,vt,ρt,ηt,ηt, G,tt,+t, insection2correspondingto u(i)t , i= 1,2,respectively.

From TheoremD,wehave|p(i)t |=O(t) fori= 1,2.Inthefollowing lemma,weshall improve theestimation for|p(1)t −p(2)t |.

Lemma 3.1.|p(1)t −p(2)t |=O t2lnt

.

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Proof. Recallthatvt(i)(y)=u(i)t (ty)ln

Mheu(i)t Gtdvg−ϕ(ty)¯ + 6ln t satisfies Δvt(i)+ht(y)ev(i)t (y)= 0, (3.1) whereht(y)=ρh(ty)|y−e|2|y+e|2eRt(ty)+ψ(ty).

On∂B2R0(p(i)t ),wehave vt(i)(y) =−ρ(i)t

2π ln|y−p(i)t |+

6−ρ(i)t 2π

lnt+G(i),t(ty)

ln

M

heu(i)t Gtdvg+φ(i)t (ty)−ϕ(ty),¯ (3.2)

whereG(i),t(x)=ρ(i)t R(x,tp(i)t )+w(x).

Forany unit vectorξ R2, we apply thePohozaev identity to (3.1) by multiplying ξ· ∇v(i)t ,andobtain

2 i=1

(1)i+1

B2R0(p(i)t )

(ξ· ∇ht)ev(i)t (y)

= 2 i=1

(1)i+1

∂B2R0(p(i)t )

(ν· ∇v(i)t )(ξ· ∇vt(i))1

2(ν·ξ)|∇v(i)t |2

+ 2 i=1

(1)i+1

∂B2R0(p(i)t )

(ν·ξ)htev(i)t , (3.3)

whereν denotestheunitnormalof∂B2R0(p(i)t ).From(2.12),wehave

|∇yϕ(ty|=t|∇tyϕ(ty|=O(t2|y|) for |y| ≤ r0

t . (3.4)

Fortherighthandside of(3.3),wecanuse(3.2),TheoremD,andTheorem Etoget (RHS) of (3.3)

= 2 i=1

(1)i+1

∂B2R0(p(i)t )

(ν· ∇v(i)t )(ξ· ∇vt(i))1

2(ν·ξ)|∇vt(i)|2dy+O(

2 i=1

eλ(i)t )

= 2 i=1

(1)i

(i)t ξ· ∇xG(i)∗,t(x)

x=tp(i)t +O((i)t +t2)

=O(t2lnt). (3.5) Forthelefthandsideof (3.3),byusing TheoremD, wegetthat

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(LHS) = 2 i=1

(1)i+1

B2R0(p(i)t )

ξ·∇ht(p(i)t ) ht(p(i)t )

ht(y)ev(i)t (y)dy

+ 2 i=1

(1)i+1

B2R0(p(i)t )

ξ·

∇ht(y)

ht(y) −∇ht(p(i)t ) ht(p(i)t )

ht(y)ev(i)t (y)dy

= 2 i=1

(1)i+1ρ(1)t

ξ·∇ht(p(i)t ) ht(p(i)t )

+ 2 i=1

(1)i+1

B2R0(p(i)t )

ξ·

∇ht(y)

ht(y) −∇ht(p(i)t ) ht(p(i)t )

×ht(y)eλ(i)t +η(i)t +G(i)∗,t(ty)−G(i)∗,t(tp(i)t )

(1 +Ct(i)eλ(i)t |y−qt(i)|2)2 dy+O(t2lnt).

(3.6)

Bythechangeofvariablez= Λ(i)t,+(y−p(i)t ) and

B2R0 Λ(i) t,+

(0) zk

(1+|z|2)2dz= 0 fork= 1,2, we seethat

B2R0(p(i)t )

ξ·

∇ht(y)

ht(y) −∇ht(p(i)t ) ht(p(i)t )

ht(y)eλ(i)t +η(i)t +G(i)∗,t(ty)G(i)∗,t(tp(i)t ) (1 +Ct(i)eλ(i)t |y−qt(i)|2)2 dy

=

B2R0(p(i)t )

ξ·

∇ht(p(i)t ) ht(p(i)t )

·(y−p(i)t ) +O(|y−p(i)t |2)

×ht(y)eλ(i)t +η(i)t +G(i)∗,t(ty)G(i)∗,t(tp(i)t ) (1 +Ct(i)eλ(i)t |y−qt(i)|2)2 dy

=

B2R0 Λ(i) t,+

(0)

ξ·

∇ht(p(i)t ) ht(p(i)t )

·(i)t,)z+O(t2|z|2)

×ht((Λ(i)t,−)z+p(i)t )eη(i)t +G(i)∗,t(t(i)t,−)z+tp(i)t )−G(i)∗,t(tp(i)t ) Ct(i)(1 +|z+ Λ(i)t,+(p(i)t −q(i)t )|2)2 dz

=

B2R0 Λ(i) t,+

(0)

ξ·

∇ht(p(i)t ) ht(p(i)t )

·(i)t,−)z+O(t2|z|2)

×ht(p(i)t )(1 +O(t|z|) +O((i)t |) +O(t2)) Ct(i)(1 +|z|2)2 dz

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=O(t2lnt) fori= 1,2, (3.7) hereweused TheoremEinthelastline.

From(3.5)–(3.7),wehave

∇ht(p(1)t )

ht(p(1)t ) −∇ht(p(2)t )

ht(p(2)t ) =O(t2lnt). (3.8) Byusing theexpression (2.19) ofht,wesee that

∇ht(p(1)t )

ht(p(1)t ) −∇ht(p(2)t ) ht(p(2)t )

=(ln|y−e|2|y+e|2)|y=p(1)t − ∇(ln|y−e|2|y+e|2)|y=p(2)t +O(t|p(1)t −p(2)t |).

(3.9)

Notethat|p(1)t −p(2)t |=O(t) fromTheoremD,and2(ln|y−e|2|y+e|2)|y=0isinvertible.

So(3.8) and(3.9) yieldthat|p(1)t −p(2)t |=O(t2lnt),andthuswecompletetheproofof Lemma3.1. 2

Nowwearegoingtoestimate u(1)t −u(2)t L(M). Lemma3.2.

u(1)t −u(2)t L(M)=O(tlnt).

Proof. We note that M\B2R0t(tp(2)t ) M \B2R1t(tp(1)t ) for some R1 > 0. For x M\B2R0t(tp(1)t ),weseefromTheorem Dthat

u(1)t (x)−u(2)t (x)

= (w(x) +ρ(1)t G(x, tp(1)t ) +φ(1)t (x))(w(x) +ρ(2)t G(x, tp(2)t ) +φ(2)t (x)) (3.10)

=ρ(1)t G(x, tp(1)t )−ρ(2)t G(x, tp(2)t ) +O(tlnt).

TogetherwithTheoremD andTheoremE,wehaveforsomeθ∈(0,1), u(1)t (x)−u(2)t (x) =−ρ(1)t

2π(ln|x−tp(1)t | −ln|x−tp(2)t |) +O(tlnt)

= O(t|p(1)t −p(2)t |)

θ|x−tp(1)t |+ (1−θ)|x−tp(2)t |+O(tlnt) (3.11)

=O(|p(1)t −p(2)t |) +O(tlnt) =O(tlnt) for x∈M\B2R0t(tp(1)t ).

WenotethatB2R0t(tp(2)t )⊆B2R2t(tp(1)t ) forsomeR2>0.Fory∈B2R0(p(1)t ),wesee that

(13)

ηt(1)(y)−ηt(2)(y)

=

v(1)t (y)−It(1)(y)(G(1),t(ty)−G(1),t(tp(1)t ))

vt(2)(y)−It(2)(y)(G(2),t(ty)−G(2),t(tp(2)t ))

=u(1)t (ty)ln

M

heu(1)t −Gtdvg−λ(1)t + 2 ln(1 +Ct(1)eλ(1)t |y−qt(1)|2)

u(2)t (ty)ln

M

heu(2)t −Gtdvg−λ(2)t + 2 ln(1 +Ct(2)eλ(2)t |y−qt(2)|2)

+O(t). (3.12)

ByTheoremD,wehave

M

heu(1)t −Gtdvg

M

heu(2)t −Gtdvg=O(t), (3.13) λ(i)t + 2 lnt+ 2 lnCt(i)+ 8πR(tp(i)t , tp(i)t )

ln ρ

ρ−8π

M

hew

+w(tp(i)t ) =O(tlnt), (3.14)

and

Ct(1)−Ct(2)= ρh(tp(1)t )|p(1)t −e|2|p(1)t +e|2eRt(tp(1)t )+ψ(tp(1)t ) 8

−ρh(tp(2)t )|p(2)t −e|2|p(2)t +e|2eRt(tp(2)t )+ψ(tp(2)t ) 8

=O(|p(1)t −p(2)t |) =O(t2lnt),

(3.15)

whichimply

λ(1)t −λ(2)t =O(tlnt). (3.16) Fory∈B2R0(p(1)t ),wewanttoestimate

2 ln(1 +Ct(1)eλ(1)t |y−q(1)t |2)2 ln(1 +Ct(2)eλ(2)t |y−q(2)t |2).

Inviewof(2.25) andLemma3.1,wehave

|q(1)t −q(2)t | ≤ 2 i=1

|p(i)t −q(i)t |+|p(1)t −p(2)t |=O(t2lnt). (3.17)

Lety=qt(1)+ Λ(1)t,−z.Then wehavefory∈B2R0(p(1)t ),

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