Uniqueness of bubbling solutions with collapsing singularities

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Journal of Functional Analysis

www.elsevier.com/locate/jfa

Uniqueness of bubbling solutions with collapsing singularities

Youngae Lee^a,∗, Chang-Shou Lin^b

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.

1. Introduction

Weareconcernedwiththefollowingmeanfieldtypeequation:

ΔMu+ρ

h∗e^u

Mh_∗e^udvg −1

= 4π

qi∈Sαi(δqi−1) inM,

Mudvg= 0, (1.1)

where (M,ds) is a compact Riemann surface, ρ > 0, dv_g is the volume form, Δ_M 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 αi∈N,i= 1,· · ·,N. Weassumethath^∗isapositivesmoothfunction.Letu^∗_k beasequenceofblowupsolutions for(1.1),thatis: maxM u^∗_k →+∞as k→+∞.Thenthere isanon-emptyfinite set B (blowupset)suchthat,

ρ h^∗e^u∗k

Mh^∗e^u∗kdvg →

p∈B

β_pδ_p, where β_p∈8πN.

For equation (1.1), we call ρ ^h∗eu∗

Mh^∗e^u∗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]:

Δ_Tu+ 1

ε²e^u(1−e^u) = 4π N j=1

δ_pj 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

uniquenessresultofbubblingsolutionsfor(1.2),thatis,ifuε,1anduε,2aretwosequence of bubblingsolutionsblowingupat thesamepointsundersomenon-degenerate condi- tion, thenu_ε,1=u_2,ε 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:

⎧⎪

⎪⎨

⎪⎪

⎩

ΔMu^∗_t +ρ

h^∗e^u∗t

Mh^∗e^u∗tdvg −1

= 4πd

i=1αi(δ_qi(t)−1) + 4πN

i=d+1αi(δqi−1) inM,

Mu^∗_tdvg= 0,

(1.3)

where limt→0qi(t) = q ∈ {/ qd+1,· · ·,qN}, ∀ i = 1,· · ·,d and qi(t)= qj(t) for i =j ∈ {1,· · ·,d}.Thenthefollowingholds:

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

u^∗_t →w^∗ inC_loc² (M\ {q}) as t→0,andw^∗ satisfies:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

Δ_Mw^∗+ (ρ−8mπ)

h^∗e^w∗

Mh^∗e^w∗ −1

= 4πd

i=1αi−2m

(δq−1) + 4π N i=d+1

αi(δqi−1) in M,

Mw^∗dv_g= 0,

(1.4)

forsome m∈Nwith 1≤m≤[¹₂d

i=1α_i]¹ and ρ>8mπ.

SoTheoremBtellsusthatthemassconcentrationdoesnolongerholdifthecollapsing singularity occurs. Indeed,we havelim_t→0

Mh^∗e^u∗tdv_g <+∞, whichis different from the situation described in Theorem A. We note that Theorem B could be improved providedthatthefollowingnondegeneracyconditionholds:

1 [x] standsfortheintegerpartofx.

Definition1.1.A solutionw^∗of(1.4) issaidnon-degenerate, ifthelinearizedproblem Δ_Mφ+ (ρ−8mπ) h^∗e^w∗

Mh^∗e^w∗dvg

φ−

Mh^∗e^w∗φdvg

Mh^∗e^w∗dvg

= 0,

M

φdv_g = 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, αi∈N for i= 3,· · ·, N. (1.6) Ourmaingoalistoprovethelocaluniquenessofblowupsolutionsof(1.3) withcollapsing singularities.

Theorem1.2. Assume that (1.6)holds andρ∈/8πN.Suppose that u^∗_t,1 andu^∗_t,2 are two blow up solutions of (1.3). Assume that u^∗_t,1,u^∗_t,2 → w^∗ in Cloc(M \ {q}), where w^∗ is a non-degenerate solution of (1.4) with m = 1. Then u^∗_t,1 = u^∗_t,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+e^v= 0 inR², (1.7)

where Δ = 2 i=1

∂²

∂x²_i denotes the standard Laplacian in R². A solution v of (1.7) is completelyclassified[10] suchthat

v(z) =vμ,a(z) = ln 8e^μ

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

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

Lφ:= 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_∂μ^μ,a

(μ,a)=(0,0), Y₁(z) := _1+|z|^z1 2 =−¹₄^∂v_∂a^μ,a₁

(μ,a)=(0,0), Y2(z) := _1+|z|^z2 2 =−¹₄^∂v_∂a^μ,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,thenthereisablowupsolutionu^∗_t of(1.3) suchthatu^∗_t →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’sfunctionfortheLaplaceBeltramioperatorΔ_M onM, thatis

ΔMG(x, p) + (δp−1) = 0,

M

G(x, p)dσ(x) = 0. (2.1)

Werecallthefollowing assumption:

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

Letu^∗_tbeasequenceofblowupsolutionsof(1.3) andw^∗bethelimitofu^∗_tinTheoremB.

Set

u_t(x) =u^∗_t(x) + 4π 2 i=1

G(x, q_i(t)) + 4π N i=3

α_iG(x, q_i), (2.2)

and

w(x) =w^∗(x) + 4π N i=3

α_iG(x, q_i). (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)}

Mhe^ut−Gtdvg −1 = 0,

Mutdvg= 0, (2.4)

where

G_t(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) inC_loc² (M\{0}) andwsatisfies

Δ_Mw+ (ρ−8π)

h(x)e^w(x)

Mhe^wdvg −1 = 0,

Mwdv_g= 0, w∈C²(M). (2.7)

Weassumethatthelocalisothermal coordinatesystemsatisfies

ds²=e^2ϕ(x)|dx|², ϕ(0) =∇ϕ(0) = 0, (2.8) thatis, e^2ϕΔM = Δ,whereΔ=2

i=1 ∂²

∂x²_i denotesthestandardLaplacianinR².Fixa smallconstantr₀∈(0,¹₂).Itiswell knownthattheconformalfactorϕisasolutionof

−Δϕ=e^2ϕK in Br0(0), (2.9)

whereK(p) istheGaussian curvatureatp∈M.

Letϕ(x) satisfy¯ thefollowing localproblem:

Δ ¯ϕ−e^2ϕρ= 0 in B_r0(0), ϕ(0) =¯ ∇ϕ(0) = 0.¯ (2.10) Wedenote

ψ= 2ϕ+ ¯ϕ. (2.11)

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∈B_r0(0). (2.12) Byusingthelocalcoordinate,wealsoset theregularpartofGreenfunctionG(x,qi(t)) to be

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

2πln|x−q_i(t)|. (2.13) Let

R_t(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

he^ut−Gtdvg−ϕ(x),¯

h₁(x) =ρh(x)e^{ψ(x)−Rt(x)}, h₁(x)>0 inB_r0(0). (2.16) Inordertostudythebehaviourofu¯tneartheorigin,weconsiderthescaledsequence

v_t(y) = ¯u_t(ty) + 6 ln t, x∈B^r0

t (0), (2.17)

whichsatisfies:

⎧⎨

⎩

Δv_t+h_t(y)e^vt(y)= 0 inB^r0 t (0),

Br0

t (0)h_t(y)e^vt(y)dy≤C, (2.18) with

h_t(y) =h₁(ty)|y−e|²|y+e|²=ρh(ty)e^{ψ(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 u_t be a sequence of blow up solutions of (2.4).Then the scaledfunction v_t defined by (2.17) blowsupat0.

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)ρhe^ut−G(2)t dv_g

Mhe^ut−G(2)t dvg

, (2.21)

C_t=1

8h₁(tp_t)|p_t−e|²|p_t+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. Letu_t 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

he^w

⎞

⎠+w(tpt) + 2 lnCt+ 8πR(tpt, tpt)

=O(tlnt), (iii) ρ_t−8π=O(t²lnt), (iv)

Mhe^ut−Gtdvg−_ρ−^ρ_8π

Mhe^wdvg=O(t), (v) |p_t|=O(t).

InordertoproveTheoremD,theauthorsin[23] analyzedthescaledfunctionvtwith thefollowingingredients:Set

It(y) = ln e^λt

(1 +Cte^λt|y−qt|²)², (2.24) whereqtischosensuchthat|qt−pt|1 and

∇yIt(y)

y=pt

=−tρt∇xR(x, tpt)

x=tpt

−t∇xw(x)

x=tpt

.

Bydirectcomputation,wehave

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

η_t(y) = v_t(y)−I_t(y)−(G_∗,t(ty)−G_∗,t(tp_t)), (2.26) where

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

η_t(p_t) =v_t(p_t)−I_t(p_t) =O(t²e^−λt), ∇η_t(p_t) = 0. (2.28) Let

Λt,+=

Cte^λt2 , and Λt,−= (Λt,+)⁻¹=e^−λt2

√Ct

(2.29) and η_tbe thescaledfunctionofη_t, thatis

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

TheoremE.[23,Lemma7.1]SupposethattheassumptionsofTheoremDhold.Thenfor any ε∈(0,1),thereexistsaconstant C_ε>0,independentof t>0andz∈B_2R0Λt,+(0) such that

|ηt(z)| ≤Cε(tφt∗+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 →winC_loc(M\{0}), where wisanon-degenerate solutionof(2.7).Wewillusep⁽ⁱ⁾_t ,λ⁽ⁱ⁾_t ,u¯⁽ⁱ⁾_t ,I_t⁽ⁱ⁾, φ⁽ⁱ⁾_t ,C_t⁽ⁱ⁾, q_t⁽ⁱ⁾,v⁽ⁱ⁾_t ,ρ⁽ⁱ⁾_t ,η⁽ⁱ⁾_t ,η_t⁽ⁱ⁾,G⁽ⁱ⁾_∗,t,Λ⁽ⁱ⁾_t,+,Λ⁽ⁱ⁾_t,− todenote pt,λt,u¯t,It,φt,Ct,qt,vt,ρt,ηt,ηt, G_∗,t,Λt,+,Λt,− 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

.

Proof. Recallthatv_t⁽ⁱ⁾(y)=u⁽ⁱ⁾_t (ty)−ln

Mhe^{u(i)t −Gt}dvg−ϕ(ty)¯ + 6ln t satisfies Δv_t⁽ⁱ⁾+ht(y)e^{v(i)t (y)}= 0, (3.1) whereht(y)=ρh(ty)|y−e|²|y+e|²e^{−Rt(ty)+ψ(ty)}.

On∂B2R0(p⁽ⁱ⁾_t ),wehave v_t⁽ⁱ⁾(y) =−ρ⁽ⁱ⁾_t

2π ln|y−p⁽ⁱ⁾_t |+

6−ρ⁽ⁱ⁾_t 2π

lnt+G⁽ⁱ⁾_∗,t(ty)

−ln

M

he^{u(i)t −Gt}dvg+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)e^{v(i)t (y)}

= 2 i=1

(−1)ⁱ⁺¹

∂B2R0(p⁽ⁱ⁾_t )

(ν· ∇v⁽ⁱ⁾_t )(ξ· ∇v_t⁽ⁱ⁾)−1

2(ν·ξ)|∇v⁽ⁱ⁾_t |²

+ 2 i=1

(−1)ⁱ⁺¹

∂B2R0(p⁽ⁱ⁾_t )

(ν·ξ)h_te^v(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· ∇v_t⁽ⁱ⁾)−1

2(ν·ξ)|∇v_t⁽ⁱ⁾|²dy+O(

2 i=1

e^−λ(i)t )

= 2 i=1

(−1)ⁱ

tρ⁽ⁱ⁾_t ξ· ∇xG⁽ⁱ⁾_∗,t(x)

x=tp⁽ⁱ⁾_t +O(tφ⁽ⁱ⁾_t ∗+t²)

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

(LHS) = 2 i=1

(−1)ⁱ⁺¹

B2R0(p⁽ⁱ⁾_t )

ξ·∇h_t(p⁽ⁱ⁾_t ) h_t(p⁽ⁱ⁾_t )

h_t(y)e^{v(i)t (y)}dy

+ 2 i=1

(−1)ⁱ⁺¹

B2R0(p⁽ⁱ⁾_t )

ξ·

∇ht(y)

h_t(y) −∇ht(p⁽ⁱ⁾_t ) ht(p⁽ⁱ⁾_t )

ht(y)e^{v(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)

h_t(y) −∇ht(p⁽ⁱ⁾_t ) ht(p⁽ⁱ⁾_t )

×h_t(y)e^{λ(i)t +η(i)t +G(i)∗,t(ty)−G(i)∗,t(tp(i)t )}

(1 +C_t⁽ⁱ⁾e^λ(i)t |y−q_t⁽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)

h_t(y) −∇ht(p⁽ⁱ⁾_t ) ht(p⁽ⁱ⁾_t )

ht(y)e^{λ(i)t +η(i)t +G(i)∗,t(ty)−G(i)∗,t(tp(i)t )} (1 +C_t⁽ⁱ⁾e^λ(i)t |y−q_t⁽ⁱ⁾|²)² dy

=

B2R0(p⁽ⁱ⁾_t )

ξ·

∇

∇ht(p⁽ⁱ⁾_t ) h_t(p⁽ⁱ⁾_t )

·(y−p⁽ⁱ⁾_t ) +O(|y−p⁽ⁱ⁾_t |²)

×ht(y)e^{λ(i)t +η(i)t +G(i)∗,t(ty)−G(i)∗,t(tp(i)t )} (1 +C_t⁽ⁱ⁾e^λ(i)t |y−q_t⁽ⁱ⁾|²)² dy

=

B2R0 Λ(i) t,+

(0)

ξ·

∇

∇h_t(p⁽ⁱ⁾_t ) h_t(p⁽ⁱ⁾_t )

·(Λ⁽ⁱ⁾_t,−)z+O(t²|z|²)

×ht((Λ⁽ⁱ⁾_t,−)z+p⁽ⁱ⁾_t )e^{η(i)t +G(i)∗,t(t(Λ(i)t,−)z+tp(i)t )−G(i)∗,t(tp(i)t )} C_t⁽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 )

·(Λ⁽ⁱ⁾_t,−)z+O(t²|z|²)

×ht(p⁽ⁱ⁾_t )(1 +O(t|z|) +O(|η⁽ⁱ⁾_t |) +O(t²)) C_t⁽ⁱ⁾(1 +|z|²)² dz

=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) ofh_t,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,and∇²(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\B_2R0t(tp⁽¹⁾_t ).

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

η_t⁽¹⁾(y)−η_t⁽²⁾(y)

=

v⁽¹⁾_t (y)−I_t⁽¹⁾(y)−(G⁽¹⁾_∗,t(ty)−G⁽¹⁾_∗,t(tp⁽¹⁾_t ))

−

v_t⁽²⁾(y)−I_t⁽²⁾(y)−(G⁽²⁾_∗,t(ty)−G⁽²⁾_∗,t(tp⁽²⁾_t ))

=u⁽¹⁾_t (ty)−ln

M

he^{u(1)t −Gt}dvg−λ⁽¹⁾_t + 2 ln(1 +C_t⁽¹⁾e^λ(1)t |y−q_t⁽¹⁾|²)

−

u⁽²⁾_t (ty)−ln

M

he^{u(2)t −Gt}dvg−λ⁽²⁾_t + 2 ln(1 +C_t⁽²⁾e^λ(2)t |y−q_t⁽²⁾|²)

+O(t). (3.12)

ByTheoremD,wehave

M

he^{u(1)t −Gt}dvg−

M

he^{u(2)t −Gt}dvg=O(t), (3.13) λ⁽ⁱ⁾_t + 2 lnt+ 2 lnC_t⁽ⁱ⁾+ 8πR(tp⁽ⁱ⁾_t , tp⁽ⁱ⁾_t )

−ln ρ

ρ−8π

M

he^w

+w(tp⁽ⁱ⁾_t ) =O(tlnt), (3.14)

and

C_t⁽¹⁾−C_t⁽²⁾= ρh(tp⁽¹⁾_t )|p⁽¹⁾_t −e|²|p⁽¹⁾_t +e|²e^{−Rt(tp(1)t )+ψ(tp(1)t )} 8

−ρh(tp⁽²⁾_t )|p⁽²⁾_t −e|²|p⁽²⁾_t +e|²e^{−Rt(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∈B_2R0(p⁽¹⁾_t ),wewanttoestimate

2 ln(1 +C_t⁽¹⁾e^λ(1)t |y−q⁽¹⁾_t |²)−2 ln(1 +C_t⁽²⁾e^λ(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=q_t⁽¹⁾+ Λ⁽¹⁾_t,−z.Then wehavefory∈B2R0(p⁽¹⁾_t ),