Energy Estimates and Harnack Type Inequalities for Prescribing Curvature Type Equations with Boundary Conditions

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Energy Estimates and Harnack Type Inequalities for Prescribing Curvature Type Equations with Boundary Conditions
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Guo, Ying
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Doctorate ( Ph.D.)
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University of Florida
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Mathematics
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ZHANG,LEI
Committee Co-Chair:
HAGER,WILLIAM WARD
Committee Members:
MCCULLOUGH,SCOTT A
CHEN,YUNMEI
ZHOU,LEI

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blowup -- energy -- estimate -- harnack -- inequality -- semilinear -- yamabe
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Abstract:
For a class of semi-linear elliptic equations with critical Sobolev exponents and boundary conditions, we prove pointwise estimates for blowup solutions and energy estimates. A special case of this class of equations is a locally defined prescribing scalar curvature and mean curvature type equation. Furthermore, for a class of prescribing curvature equations on a domain with boundary, we adopt some special assumptions. Green's function was applied elaborately to construct test functions. Based on standard selection process and bubble analysis, we utilize the method of moving spheres to derive a Harnack-type inequality for positive solutions of those equations. As a consequence of this inequality we obtain the energy estimate.
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by Ying Guo.
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Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: ZHANG,LEI.
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Co-adviser: HAGER,WILLIAM WARD.

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ENERGYESTIMATESANDHARNACKTYPEINEQUALITIESFORPRESCRIBINGCURVATURETYPEEQUATIONSWITHBOUNDARYCONDITIONSByYINGGUOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013YingGuo 2

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ACKNOWLEDGMENTS FirstIwouldliketoexpressmysinceregratitudetomyadvisor,Dr.LeiZhang,forhisexcellentacademicguidanceandinvaluableresearchsupport.Thankhisencouragementandeffortstomystudyandlife.Ialsowanttothankmyothercommitteemembers,Dr.YunmeiChen,Dr.ScottMcCullough,Dr.WilliamHagerandDr.LeiZhou,fromwhomIacquiredenormousknowledge,inspirationandhelp.ManythankstoMathewGluckforhisdiscussionandsupport.Lastlybutnotleast,Imustextendmyappreciationtothefaculty,staffandfellowstudentsoftheMathematicsDepartment,whomademyexperienceatUniversityofFloridafruitfulandenjoyable. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 6 CHAPTER 1INTRODUCTION ................................... 7 2ENERGYESTIMATESFORACLASSOFEQUATIONSSEMILINEARELLIPTICEQUATIONS ..................................... 11 2.1OnhalfEuclideanballs ............................ 11 2.2ProofofTheorem 2.1.1 ............................ 14 2.3ProofofTheorem 2.1.2 ............................ 39 3GREEN'SFUNCTIONESTIMATES ........................ 41 4AHARNACK-TYPEINEQUALITYFORAPRESCRIBINGCURVATUREEQUATIONONADOMAINWITHBOUNDARY .................. 43 4.1TheMainTheorem ............................... 44 4.2RescalingandSelection ............................ 46 4.3Blow-upCanOnlyOccurNearB+1\Rn+ .................. 48 4.3.1VanishingofKi(xi) .......................... 49 4.3.2RapidVanishingofKi(xi) ....................... 61 4.3.3CompletionoftheProofofTheorem 4.3.1 .............. 67 4.4ProofofTheorem 4.1.1 ............................ 69 4.4.1VanishingofKi(x0i) .......................... 70 4.4.2ImprovedVanishingRateforKi(x0i) ................ 80 4.4.3CompletionoftheProofofTheorem 4.1.1 .............. 87 4.5EnergyEstimate ................................ 89 REFERENCES ....................................... 91 BIOGRAPHICALSKETCH ................................ 94 5

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyENERGYESTIMATESANDHARNACKTYPEINEQUALITIESFORPRESCRIBINGCURVATURETYPEEQUATIONSWITHBOUNDARYCONDITIONSByYingGuoDecember2013Chair:LeiZhangMajor:MathematicsForaclassofsemi-linearellipticequationswithcriticalSobolevexponentsandboundaryconditions,weprovepointwiseestimatesforblowupsolutionsandenergyestimates.Aspecialcaseofthisclassofequationsisalocallydenedprescribingscalarcurvatureandmeancurvaturetypeequation.Furthermore,foraclassofprescribingcurvatureequationsonadomainwithboundary,weadoptsomespecialassumptions.Green'sfunctionwasappliedelaboratelytoconstructtestfunctions.Basedonstandardselectionprocessandbubbleanalysis,weutilizethemethodofmovingspherestoderiveaHarnack-typeinequalityforpositivesolutionsofthoseequations.Asaconsequenceofthisinequalityweobtaintheenergyestimate. 6

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CHAPTER1INTRODUCTIONThisdissertationismotivatedbythefamousYamabeproblemandsomeexceptionalrelatedresearches.In1960,HidehikoYamabeclaimedthateverysmoothcompactRiemannianmanifoldwithdimensionn3canbedeformedconformallytoasmoothRiemannianstructureofconstantscalarcurvaturein[ 41 ].However,NeilTrudingerdiscoveredanerrorinYamabe'sproofin1968.Asamatteroffact,Yamabepassedawayinhisthirtiesin1960.Ifhewasabletolivelonger,hemightndtheerrorbyhimselfanddemonstratemorebrilliantmathematicaltalent.Trudingertriedtorepairtheproofandsolvedtheproblempartiallybyemployingsomeassumptions[ 40 ].In1976,ThierryAubin[ 2 ]madeprogresssuccessfullybutdidnotresolvetheproblemcompletely.Itwasnotuntil1984thatYamabeproblemwaseventuallyprovedtobeafrmativethankstoRichardSchoen'scontribution[ 37 ].Arelatedtopicisnon-compactYamabeproblem.ZhirenJinannouncedAcounterexampletotheYamabeproblemforcompletenoncompactmanifoldsin1988.Yamabeproblem'sproofrequirestechniquesfromdifferentialgeometry,partialdifferentialequationsandfunctionalanalysis.TheYamabeproblemcanbetranslatedtondingasolutionofasemi-linearellipticequationcalledtheYamabeequation.IthasbeenprovedthattheYamabeequationalwayshasasolution.Ifthemanifoldhasaboundary,youmayaskisitpossibletodeformthemetrictochangethescalarcurvatureandtheboundarymeancurvaturetospecicfunctions(seeCherrier[ 11 ]).Suppose(Mn,g)(n3)isaRiemannianmanifoldwithboundaryM,letbg=u4=(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2)gbeconformaltog,thenthescalarcurvatureRgandboundarymeancurvaturehgofgarerelatedtothescalarcurvatureK(x)andboundarymeancurvaturec(x)ofbgbytheequation8><>:)]TJ /F21 11.955 Tf 9.28 0 Td[(gu+n)]TJ /F8 8.966 Tf 6.96 0 Td[(2 4(n)]TJ /F8 8.966 Tf 6.97 0 Td[(1)Rgu=n)]TJ /F8 8.966 Tf 6.97 0 Td[(2 4(n)]TJ /F8 8.966 Tf 6.96 0 Td[(1)Kun+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,inM,ngu+n)]TJ /F8 8.966 Tf 6.96 0 Td[(2 2hgu=n)]TJ /F8 8.966 Tf 6.97 0 Td[(2 2cun n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,onM, 7

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wheregistheLaplace-Beltramioperatorwithsignconvention)]TJ /F21 11.955 Tf 9.29 0 Td[(g0andngistheunitouternormalvectoronM.IfKandcareconstants,ndingasolutiontotheequationaboveiscalledtheboundaryYamabeproblem.Unlikeitsboundary-freecounterpart,theboundaryYamabeproblemisnotyetcompletelysolved.ImportantprogresshasbeenmadebyEscobar[ 17 18 ],Han-Li[ 23 24 ],Marques[ 35 ],etc.ThereisavastliteratureontheuniformestimateofsolutionstotheboundaryYamabeproblem.Thereadermayreferto[ 14 16 20 23 24 ]andthereferencesthereinformoreinformation.Ourresearchconsistsoftwomainparts,whicharepresentedinChapter2andChapter4.Chapter3pavesthewayforChapter4sincetheestimatesofGreen'sfunctioninChapter3playsanessentialroleconstructtestfunctions.WearesupposedtodiscusstheproblemsonB+Rtomaketheconclusionsuniversal.However,wecanuseB+3orB+1tomaketheproofsmorecleartoread.Byrescaling,theexactvalueofRinB+Rcanbeanypositiverealnumberanddoesnotaffectanything.InChapter2,weconsider8><>:)]TJ /F21 11.955 Tf 9.29 0 Td[(u=g(u),B+3,u xn=h(u),B+3\Rn+,whereu>0isapositivecontinuoussolution,B+3istheupperhalfballcenteredattheoriginwithradius3,gisacontinuousfunctionon(0,)andhislocallyHoldercontinuouson(0,).Ifg(s)=sn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2andh(s)=csn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,theequationaboveisatypicalprescribingcurvatureequation.IfweusedtorepresenttheEuclideanmetric,thenu4 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2disconformaltod.Theequationinthisspecialcasemeansthescalarcurvatureunderthenewmetricis4(n)]TJ /F11 11.955 Tf 11.27 0 Td[(1)=(n)]TJ /F11 11.955 Tf 11.27 0 Td[(2)andtheboundarymeancurvatureunderthenewmetricis)]TJ /F8 8.966 Tf 16.46 4.71 Td[(2 n)]TJ /F8 8.966 Tf 6.96 0 Td[(2c.TheequationwestudiedisverycloselyrelatedtothewellknownYamabeproblemandtheboundaryYamabeproblem.Weputsomespecialassumptions 8

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ongandhtogettheenergyestimateZB+1juj2+u2n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2C,forsomeC>0thatdependsonlyong,handn.Letch:=lims!s)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.41 0 Td[(2h(s).Wefocusonthecasech>0whichdemandsdelicatetechniquestodealwith.Theresultforch0iseasytoprove.TheenergyestimateiscloselyrelatedtotheHarnacktypeinequality(minB+1u)(maxB+2u)C,forsomeC>0.IfwecouldnottakeadvantageofthisHarnacktypeinequality,theenergyestimateseemsimpossible.InChapter3,wepresentGreen'sfunctionestimatestoprovidepreparationforChapter4.InChapter4,weconsider8><>:u+K(x)un+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2=0inB+1Rn+,u>0u xn=c(x)un n)]TJ /F27 6.974 Tf 5.42 0 Td[(2onB+1\Rn+.IfwesetK(x)andc(x)asconstants,theequationaboveisaspecialcaseoftheequationwediscussinChapter2.Asfarasweknow,Harnacktypeinequalitysimilarto(minB+1=3u)(maxB+2=3u)CwasrstdiscoveredforprescribingscalarcurvatureequationswithnoboundarytermbySchoen[ 38 ],Schoen-Zhang[ 39 ]andChen-Lin[ 8 ].In2003,ZhangandLi[ 29 ]provedtheHarnacktypeinequalityfortheequationabovewhenKandcarebothconstants.In2009,L.ZhangprovedtheHarnacktypeinequalityaboveforthecasen=3onlyassumingK>0andctobesmoothfunctions[ 44 ].WeareabletoderivethisHarnacktypeinequalityandtheenergyestimateundernaturalassumptionsonKandcforn4.ItisevidentfromthepreviousworkofZhangandLi[ 30 32 ]thatthisHarnacktypeinequalityisacrucialsteptowardobtainingneestimatesforsolutionsofoftheequationabove.ComparingwiththeresultsofLi-Zhang[ 29 ]and 9

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Zhang[ 44 ],thecaseofn4withnonconstantcoefcientfunctionsismuchharder.ByconstrainingKandcappropriately,weareabletohandlethesenewcomplicationsandderivethedesiredestimates.Harnacktypeinequalitiesandenergyestimatesweresignicantinblowupanalysisforsemilinearellipticequationswithcriticalexponentsinthepasttwodecades.Weobtaintheresultsthroughthestudyoftheasymptoticbehaviorofblowupsolutionsneartheirblowuppoints. 10

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CHAPTER2ENERGYESTIMATESFORACLASSOFEQUATIONSSEMILINEARELLIPTICEQUATIONSWeintendtotreataclassofsemi-linearellipticequationsonhalfEuclideanballs.ThecaseonEuclideanballswithoutboundaryconditionsissignicantlylesscomplex. 2.1OnhalfEuclideanballsInthischapterweconsider 8><>:)]TJ /F21 11.955 Tf 9.29 0 Td[(u=g(u),B+3,u xn=h(u),B+3\Rn+,(2)whereu>0isapositivecontinuoussolution,B+3istheupperhalfballcenteredattheoriginwithradius3,gisacontinuousfunctionon(0,)andhislocallyHoldercontinuouson(0,).ForgandhweassumeGH0:gisacontinuousfunctionon(0,),hisHoldercontinuouson(0,).andGH1:8><>:g(s)s)]TJ /F25 6.974 Tf 8.16 3.54 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2isnon-increasing,lims!g(s)s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(22(0,).s)]TJ /F25 6.974 Tf 12.62 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2h(s)isnon-decreasingandlims!s)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2h(s)<.Let ch:=lims!s)]TJ /F25 6.974 Tf 12.61 3.54 Td[(n n)]TJ /F27 6.974 Tf 5.41 0 Td[(2h(s).(2)Thenifch>0weassumeGH2:sup00: 11

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Theorem2.1.1. Letu>0beasolutionof( 2 )wheregandhsatisfy(GH0)and(GH1).Supposech>0and(GH2)alsohold,then ZB+1juj2+u2n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2C,(2)forsomeC>0thatdependsonlyong,handn.Obviouslyif g(s)=c1sn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,c1>0andh(s)=chsn n)]TJ /F27 6.974 Tf 5.41 0 Td[(2,ch>0,(2)thengandhsatisfytheassumptionsinTheorem 2.1.1 .Theenergyestimate( 2 )forthisspecialcasehasbeenprovedbyLi-Zhang[ 29 ].ItiseasytoseethattheassumptionsofgandhinTheorem 2.1.1 includeamuchlargerclassoffunctions.Forexample,foranynonincreasingfunctionc1(s)satisfyinglims!c1(s)>0andlims!0+c1(s)s4 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2<,g(s)=c1(s)sn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2satisestheassumptionsofg.Similarlyh(s)=c2(s)sn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2foranondecreasingfunctionc2(s)withlims!c2(s)=chandlims!0+jc2(s)js2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2<,satisestherequirementofhinTheorem 2.1.1 .Forthecasech0wehave Theorem2.1.2. Letu>0beasolutionof( 2 )wheregandhsatisfy(GH0)and(GH1).Supposech0andg,hsatisfy(GH3),thentheenergyestimate( 2 )holdsforCdependingonlyong,handn.Ifweallowlims!s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2g(s)=0,thentheenergyestimate( 2 )maynothold.Forexample,letg(s)=1 4(s+1))]TJ /F8 8.966 Tf 6.97 0 Td[(3,thengsatisestheassumptioninTheorem 2.1.2 exceptthatlims!s)]TJ /F25 6.974 Tf 8.17 3.54 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2g(s)=0.Letuj(x)=p x1+j)]TJ /F11 11.955 Tf 10.95 0 Td[(1,itiseasytoverifythatujsatises8><>:)]TJ /F21 11.955 Tf 9.28 0 Td[(uj=g(uj)inB+3,uj xn=0,onB+3\Rn+.Notethath=0inthiscase.Thenclearly( 2 )doesnotholdforuj. 12

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Theenergyestimate( 2 )iscloselyrelatedtothefollowingHarnacktypeinequality: (minB+1u)(maxB+2u)C,(2)whichwasprovedbyLi-Zhang[ 29 ]forthespecialcase( 2 ).Li-Zhang[ 29 ]alsoprovedthe( 2 )for( 2 )using( 2 )intheirargumentinanontrivialway.InthepasttwodecadesHarnacktypeinequalitiessimilarto( 2 )haveplayedanimportantroleinblowupanalysisforsemilinearellipticequationswithcriticalexponents.SomeworksinthisrespectcanbefoundinSchoen[ 38 ],Schoen-Zhang[ 39 ],Chen-Lin[ 8 ],andfurtherextensiveresultscanbefoundin[ 9 21 25 29 30 34 39 43 ]andthereferencestherein.Usuallyforasemilinearequationwithoutboundarycondition,forexample,theconformalscalarcurvatureequationu+K(x)un+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2=0,B3,aHarnackinequalityofthetype)]TJ /F11 11.955 Tf 6.8 -9.69 Td[(maxB1u)]TJ /F11 11.955 Tf 12.28 -9.69 Td[(minB2uCimmediatelyleadstotheenergyestimateZB1juj2+u2n n)]TJ /F27 6.974 Tf 5.41 0 Td[(2CbytheGreen'srepresentationtheoremandintegrationbyparts.However,whentheboundaryconditionasin( 2 )appears,usingtheHarnackinequality( 2 )toderive( 2 )ismuchmoreinvolved.Inordertoderiveenergyestimate( 2 )andpointwiseestimatesforblowupsolutions,LiandZhangprovethefollowingresultsin[ 29 ]:TheoremA(Li-Zhang).Letu>0beasolutionof( 2 )wheregandhsatisfy(GH0),(GH1)and(GH3).Then(max B+1u)(min B+2u)C. 13

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HerewenotethatinTheoremAnosignofchisspecied.Onewouldexpecttheenergyestimate( 2 )tofollowdirectlyfromLi-Zhang'stheorem.Thisisindeedthecaseifch0.Howeverforch>0substantiallymoreestimatesareneededinordertoestablishaprecisepointwiseestimateforblowupsolutions.Asamatteroffact,weneedtoassume(GH2)insteadof(GH3)inordertoobtain( 2 ).Theorganizationoftherestpartofthischapterisasfollows.InsectiontwoweproveTheorem 2.1.1 .InsectionthreeweproveTheorem 2.1.2 usingTheoremAandintegrationbyparts.Insectiontwo,rstweuseaselectionprocesstolocateregionsinwhichthebubblingsolutionslooklikeglobalsolutions.Thenweconsidertheinteractionofthebubblingregions.UsingdelicateblowupanalysisandPohozaevidentityweprovethatbubblingregionsmustbeapositivedistanceapart.Thentheenergyestimate( 2 )follow. 2.2ProofofTheorem 2.1.1 TheproofofTheorem 2.1.1 isbywayofcontradiction.Supposethereisnoenergybound,thenthereexistsasequenceuksuchthat ZB1jukj2+u2n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k!.(2)WeclaimthatmaxB+3=2uk!.Indeed,ifthisisnotthecase,whichmeansthereisauniformboundforukonB+3=2,wejusttakeacut-offfunctionh2Csuchthath1onB+1andh0onB+2nB+3=2andwecanndapositivenumberCsuchthatjhjC.Multiplyingtheequation( 2 )byukh2,usingintegrationbypartsandsimpleCauchy'sinequalityweobtainauniformboundofRB1jukj2,acontradictionto( 2 ). Denition2.2.1. Letfukgbeasequenceofsolutionsof( 2 ).Assumethatxk!x2 B+2andlimk!uk(xk)=.IfthereexistC>0(independentofk)andr>0(independentofk)suchthatuk(x)jx)]TJ /F11 11.955 Tf 12.28 .27 Td[(xjn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2Cforjx)]TJ /F11 11.955 Tf 12.39 .27 Td[(xjr,thenwesaythatxisanisolatedblow-uppointoffukg. 14

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Proposition2.2.1. Letfukgbeasequenceofsolutionsof( 2 )andmaxx2 B+1uk(x)jxjn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2!ask!,thenthereexistsasequenceoflocalmaximumpointsxksuchthatalongasubsequence(stilldenotedasfukg)vk(y):=uk(xk))]TJ /F8 8.966 Tf 6.96 0 Td[(1uk(uk(xk))]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2y+xk)eitherconvergesuniformlyoverallcompactsubsetsofRntoVthatsatises V+AVn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2=0,Rn(2)(whereA=lims!g(s)s)]TJ /F25 6.974 Tf 8.16 3.54 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2),orconvergestoV1denedonfy2Rn;yn>)]TJ /F6 11.955 Tf 9.29 0 Td[(Tg(T:=limk!uk(xk)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2xkn)thatsatises 8><>:V1+AVn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(21=0,Rn\fyn>)]TJ /F6 11.955 Tf 9.28 0 Td[(Tg,V1=yn=chVn n)]TJ /F27 6.974 Tf 5.42 0 Td[(21,yn=)]TJ /F6 11.955 Tf 9.29 0 Td[(T.(2)wherech=lims!s)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2h(s). Remark2.2.1. Allsolutionsof( 2 )aredescribedbytheclassicationtheoremofCaffarelli-Gidas-Spruck[ 6 ].Allsolutionsof( 2 )aredescribedbyLi-Zhu[ 33 ].ProofofProposition 2.2.1 :Letxk2B+1beasequencesuchthatuk(xk)jxkjn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2!ask!.Letdk=jxkjandSk(y)=uk(y)(dk)-138(jy)]TJ /F6 11.955 Tf 10.95 0 Td[(xkj)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2,8y2B+1.SetSk(bxk)=max B(xk,dk)\ft>)]TJ /F14 8.966 Tf 6.97 0 Td[(xkngSk.Then Sk(bxk)Sk(0)=uk(xk)jxkjn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2!.(2) 15

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Letsk=1 2(dk)-137(jxk)]TJ /F18 11.955 Tf 10.96 .51 Td[(bxkj),thenclearly( 2 )canbewrittenas uk(bxk)2n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2sn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2kuk(xk)dkn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2!ask!.(2)Forallx2B+sk(bxk),sinceuk(x)(dk)-137(jx)]TJ /F6 11.955 Tf 10.95 0 Td[(xkj)n)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2uk(bxk)(dk)-137(jxk)]TJ /F18 11.955 Tf 10.95 .5 Td[(bxkj)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2,wehaveuk(x)uk(bxk)dk)-137(jxk)]TJ /F18 11.955 Tf 10.96 .5 Td[(bxkj dk)-137(jx)]TJ /F6 11.955 Tf 10.95 0 Td[(xkjn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2.Usingjx)]TJ /F11 11.955 Tf 12.28 .27 Td[(xkjsk,anddk)-137(jx)]TJ /F6 11.955 Tf 10.95 0 Td[(xkjdk)-137(jxk)]TJ /F11 11.955 Tf 12.28 .27 Td[(xkj)-138(jx)]TJ /F11 11.955 Tf 12.28 .27 Td[(xkjsk,weobtain uk(x)2n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2uk(bxk),forallx2B+sk(bxk).(2)LetMk=uk(bxk)andvk(y)=Mk)]TJ /F8 8.966 Tf 6.96 0 Td[(1uk(Mk)]TJ /F27 6.974 Tf 12.86 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2y+bxk),M)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2ky+xk2B+3.Directcomputationshows vk(y)+(Mkvk(y)))]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2g(Mkvk(y))vk(y)n+2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2=0(2)By( 2 )wehave 0vk(y)2n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 28y2B(0,Mk2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2sk)\fyn)]TJ /F6 11.955 Tf 21.23 0 Td[(Mk2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2bxkng(2)Weconsiderthefollowingtwocases.Caseone:limk!Mk2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2bxkn=.Observing( 2 ),sincebothM2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kskandM2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kxnk!,( 2 )isdenedonjyjlkforsomelk!.Weclaimthatvk!VuniformlyoverallcompactsubsetsofRnandV 16

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satises( 2 )withA=lims!s)]TJ /F25 6.974 Tf 8.17 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2g(s).Indeed,weclaimthatforanyR>1, vk(y)C(R)>0,jyjR.(2)Once( 2 )isestablished,clearlyMkVk!overallBR,thusM)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(Mkvk)=(Mkvk))]TJ /F25 6.974 Tf 8.16 3.54 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2g(Mkvk)vn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k!AVn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2overallcompactsubsetsofRn.ClearlyVsolves( 2 ).Therefore,weonlyneedtoestablish( 2 )forxedR>1.LetR,k:=fy2BR;vk(y)3M)]TJ /F8 8.966 Tf 6.97 0 Td[(1kgandak(y)=M)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(Mkvk)=vk.Thenby(GH1)weseethatinBRnR,kak(y)g(3)v4 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2k4g(3).Fory2R,kweuse(GH2)toobtainak(y)CM)]TJ /F27 6.974 Tf 12.87 3.54 Td[(4 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k,y2R,k.Ineithercaseak(y)isaboundedfunctionandfromvk(y)+ak(y)vk(y)=0inBRandstandardHarnackinequalitywehave1=vk(0)maxBR=2vkC(R)minBR=2vk. 17

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Thus,( 2 )isestablished.ConsequentlyV,asthelimitofvkindeedsolves( 2 ).BytheclassicationtheoremofCaffarelli-Gidas-Spruck[ 6 ],V(y)=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4 m 1+m2jyj2!n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ObviouslyVhasamaximumpointx2Rn.Correspondinglythereexistsasequenceoflocalmaximumpointsofuk,denotedxk,thattendstoxafterscaling.Thus,vkcanbedenedasinthestatementofProposition 2.2.1 .Casetwo:limk!M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kxkn<.InthiscaseweletT=limk!M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kbxkn.Itiseasytoverifythatvksatises8><>:vk(y)+(Mkvk(y)))]TJ /F25 6.974 Tf 8.17 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2g(Mkvk(y))vn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k(y)=0,inM)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2ky+xk2B+3,vk yn=(Mkvk(y)))]TJ /F25 6.974 Tf 12.62 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2h(Mkvk(y))vk(y)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2vk(y),onyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(M2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2kxkn.WeclaimthatforanyR>1,thereexistsC(R)>0suchthat vk(y)C(R)inBR\fyn)]TJ /F6 11.955 Tf 21.24 0 Td[(M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kxkng.(2)Theproofof( 2 )issimilartotheinteriorcase.LetTk=M2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2kxknandpk=(0,,0,)]TJ /F6 11.955 Tf 9.29 0 Td[(Tk)2Rn.OnB(pk,R)\fyn)]TJ /F6 11.955 Tf 21.23 0 Td[(Tkgwewritetheequationforvkas 8>>>><>>>>:vk+akvk=0,inB(pk,R)\fyn>)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg,nvk+bkvk=0,onB(pk,R)\fyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg.(2)whereitiseasytouseassumptionsofgandhtoprovethatjakj+jbkjCforsomeCindependentofkandR.ByaclassicalHarnackinequalitywithboundaryterms(for 18

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example,refertoLemma6.2of[ 42 ]orHan-Li[ 23 ]),wehave1=vk(0)maxB(pk,R=2)\fyn)]TJ /F14 8.966 Tf 13.93 0 Td[(TkgvkC(R)minB(pk,R=2)\fyn)]TJ /F14 8.966 Tf 13.93 0 Td[(Tkgvk.Therefore,vkisboundedbelowbypositiveconstantsoverallcompactsubsets.Thus,thelimitfunctionV1solves( 2 ).ByLi-Zhu'sclassicationtheorem[ 33 ],V1(y)=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4 l 1+l2(jy0)]TJ /F11 11.955 Tf 12.39 .26 Td[(yj2+jyn)]TJ /F11 11.955 Tf 12.38 .26 Td[(ynj2)!n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2,whereyn=chp (n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)n=A=((n)]TJ /F11 11.955 Tf 11.24 0 Td[(2)l),y2Rn)]TJ /F8 8.966 Tf 6.97 0 Td[(1,lisdeterminedbyV1(0)=1.Thus,thelocalmaximumofV1canbeusedtodenevkasinthestatementoftheproposition.Proposition 2.2.1 isestablished.2Proposition 2.2.1 determinestherstpointintheblowupsetk.Theotherpointsinkcanbedeterminedasfollows:ConsiderthemaximumofSk(x)=uk(x)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dist(x,k).IfSk(x)isuniformlybounded,westop.OtherwisethesameselectionprocesswegetanotherblowupprolebyeithertheclassicationtheoremofCaffarelli-Gidas-SpruckorLi-Zhu.Eventuallywehavefqikg2k(i=1,2,..,)thatsatisfy8>>>>><>>>>>: B+rki(qki)\ B+rkj(qkj)=/0,fori6=j,jqki)]TJ /F6 11.955 Tf 10.95 0 Td[(qkjjn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2uk(qkj)!,forj>i,uk(x)2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2dist(x,k)C.andrkiarechosensothatinB+rki(qki),theproleofukiseitherlikeanentirestandardbubbledescribedin( 2 )orapartofthebubbledescribedin( 2 ).Takeanyqk2k,letsk=dist(qk,knfqkg)andletuk(y)=sn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2kuk(qk+sky),ink 19

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wherek:=fy;qk+sky2B+3g.Bytheselectionprocesswehave uk(y)Cjyj)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2,jyj3=4,y2k(2)and uk(0)!.(2)Wefurtherproveinthefollowingpropositionthatukdecayslikeaharmonicfunction: Proposition2.2.2. uk(0)uk(y)jyjn)]TJ /F8 8.966 Tf 6.96 0 Td[(2C,fory2B2=3\k.(2) Remark2.2.2. ThemeaningofProposition 2.2.2 iseachisolatedblowuppointisalsoisolatedsimple. Proof. Directcomputationshowsthatuksatises 8>>>><>>>>:uk(y)+sn+2 2kg(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk)=0,ink,nuk(y)=sn 2kh(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk),onk\fyn=)]TJ /F3 11.955 Tf 1 0 .167 1 342.07 -349.96 Tm[(s)]TJ /F8 8.966 Tf 6.97 0 Td[(1kqkng,(2)LetMk=uk(0).By( 2 )Mk!.Setvk(z)=M)]TJ /F8 8.966 Tf 6.97 0 Td[(1kuk(M)]TJ /F27 6.974 Tf 12.86 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kz),forz2k,wherek:=fz;jzjM2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k,M)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kz2kgNotethatvkisdenedonabiggerset,butfortheproofofProposition 2.2.2 ,weonlyneedtoconsiderthepartink.Directcomputationgives 8>>>><>>>>:vk(z)+l)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(lkvk)=0,z2k,vk zn=l)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kh(lkvk),fzn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg\k,(2) 20

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wherelk=s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kMkandTk=l2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqkn.Weconsiderthefollowingtwocases.Caseone:Tk!.BythesameargumentasintheproofofProposition 2.2.1 ,weknowvk!VinC1,aloc(Rn)whereVsolves( 2 ).Thus,thereexistRk!suchthatkvk)]TJ /F6 11.955 Tf 10.95 0 Td[(VkC1,a(BRk)CR)]TJ /F8 8.966 Tf 6.97 0 Td[(1k.Clearly,( 2 )holdsforjyjM)]TJ /F27 6.974 Tf 12.87 3.54 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kRk,wejustneedtoprove( 2 )forjyj>M)]TJ /F27 6.974 Tf 12.87 3.54 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kRk.SinceV(0)=1isthemaximumpointofV,V(z)=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4(1+jzj2))]TJ /F25 6.974 Tf 8.16 3.54 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2,z2Rn.. Lemma2.2.1. Thereexistsk0>1suchthatforallkk0andr2(Rk,M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k), minBr\kvk2(n(n)]TJ /F11 11.955 Tf 10.95 -.01 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4r2)]TJ /F14 8.966 Tf 6.96 0 Td[(n.(2)ProofofLemma 2.2.1 :Suppose( 2 )doesnothold,thenthereexistrksuchthat vk(z)2(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4r2)]TJ /F14 8.966 Tf 6.97 0 Td[(nk,jzj=rk,z2k.(2)Clearly,rkRk.Letvlk(z)=(l jzj)n)]TJ /F8 8.966 Tf 6.97 0 Td[(2vk(zl),zl=l2z jzj2.Theequationofvlk,bydirectcomputation,is vlk(z)+(l jzj)n+2l)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(lk(jzj l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(2vlk(z))=0,inl(2)wherel:=fz2k;jlj
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Clearly,vlk!VlinC1,aloc(Rn)forxedl>0.Bydirectcomputation,V(z)>Vl(z),forl2(0,1),jzj>lV(z)1,jzj>l.Weshallapplythemethodofmovingspheresforl2(1 2,2).Firstweprovethatforl0=1=2, vk(z)>vl0k(z),z2l.(2)Toprove( 2 ),werstobservethatvk>vl0kinBRnBlforanyxedlargeR.Indeed,vk=vl0konBl0.OnBl0wehavenV>nVl0.Thus,theC1,aconvergenceofvktoVgivesthatvk>vl0knearBl0.Thenbytheuniformconvergencewefurtherknowthat( 2 )holdsonBRnBl0.OnBR,wehave vk(z)((n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4)]TJ /F3 11.955 Tf 1 0 .167 1 250.34 -296.36 Tm[(e)jzj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,jzj=R(2)and vl0k(z)((n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4)]TJ /F11 11.955 Tf 10.95 0 Td[(2e)jzj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,jzjR(2)forsomee>0independentofk.Nextweshallusemaximumprincipletoprovethat vk(z)>((n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4)]TJ /F11 11.955 Tf 10.95 0 Td[(2e)jzj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n>vl0k(z),z2l0nBR.(2)Theproofof( 2 )isbycontradiction.Weshallcomparevkandfk:=((n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4)]TJ /F11 11.955 Tf 10.95 0 Td[(2e)jzj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n.Clearly,vk)]TJ /F6 11.955 Tf 11.6 0 Td[(fkissuperharmonicinl0)]TJ /F6 11.955 Tf 11.6 0 Td[(BRand,by( 2 ),( 2 )and( 2 ),vk)]TJ /F6 11.955 Tf 10.95 0 Td[(fk>0onBRandl0\(Rn+nBR).Ifthereexistsz02l0\fzn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkgand0>vk(z0))]TJ /F6 11.955 Tf 10.95 0 Td[(fk(z0)=minl0nBRvk)]TJ /F6 11.955 Tf 10.95 0 Td[(fk 22

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wewouldhave 0Nkfk(z0)n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2forsomeNk!.Howeverby(GH1)l)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.41 0 Td[(2kh(lkvk(z0))Cvk(z0)n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2.Easytoseeitisimpossibletohavevk(z0)
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Nowweapplythemethodofmovingspherestowl,k.Letlk=inffl2[1 2,(3 2)1=(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2)];vk>vmkinm,8m>lg.By( 2 ),lk>1 2.Weclaimthatlk=(3 2)1=(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2).Supposethisisnotthecasewehavelk<(3 2)1=(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2).Bycontinuitywlk,k0andby( 2 )wlk,k>0ontheoutsideboundary:lkn(Blk[fzn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg).By( 2 ),ifminlkwlk,k=0,theminimumwillhavetoappearonlk.From( 2 )weseethattheminimumdoesnotappearonlkn(Blk[fzn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg).Ifthereexistsx02lk\fzn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg,wehave0Nk(vlkk)n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,inOlk\fzn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg,forsomeNk!.Forvk,(GH2)impliesznvkchvn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k,inOlk\fzn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkgwherech=lims!s)]TJ /F25 6.974 Tf 12.62 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2h(s).Thenitiseasytoseethatwlk>0onfzn=)]TJ /F6 11.955 Tf 9.28 0 Td[(Tkg.ThenHopfLemmaandthecontinuityleadtoacontradictionofthedenitionoflk.Thus,wehaveprovedlk=2.HoweverVlifl>1.Soitisimpossibletohavelimk!lk>1.Thiscontradictionproves( 2 )underCaseone.Lemma 2.2.1 isestablished.FromLemma 2.2.1 wefurtherprovethesphericalHarnackinequalityforvk.Forxedk,consider2Rkr1 2M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kandletvk(z)=rn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2vk(rz). 24

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By( 2 ),vk(z)C.Directcomputationyields8>>>><>>>>:vk(z)+rn+2 2l)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(lkr)]TJ /F25 6.974 Tf 8.16 3.54 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2vk)=0,1 2>>><>>>>:g(1)v4 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kC,iflkr)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2vk(z)1,Cr2l)]TJ /F27 6.974 Tf 12.87 3.53 Td[(4 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k=o(1),iflkr)]TJ /F25 6.974 Tf 8.16 3.54 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2vk(z)1,andjbk(z)j8>>>><>>>>:chv2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kC,iflkr)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2vk(z)1Crl)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k=o(1),iflkr)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2vk(z)1,Hence,akandbkarebothboundedfunctions.Consequently,theequationforvkcanbewrittenas8>>>><>>>>:vk(z)+akvk=0,1 21orTk1.Intherstcasewehavemaxjzj=3=4vk(z)Cminjzj=3=4vk. 25

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Inthesecondcasewehavemaxjzj=1,zn)]TJ /F14 8.966 Tf 13.93 0 Td[(Tk=rvk(z)Cminjzj=1,zn)]TJ /F14 8.966 Tf 13.93 0 Td[(Tk=rvk.Clearly,( 2 )isimplied.Proposition 2.2.2 isestablishedforCaseone.Casetwo:limk!Tk=TRecallthatvksatises( 2 ).BythesameargumentasintheproofofProposition 2.2.1 ,weknowvk!VinC1,aloc(Rn\fyn)]TJ /F6 11.955 Tf 21.23 0 Td[(Tg)whereVsolves( 2 ).Thus,thereexistRk!suchthatkvk)]TJ /F6 11.955 Tf 10.95 0 Td[(VkC1,a(B)]TJ /F25 6.974 Tf 5.42 0 Td[(TRk)CR)]TJ /F8 8.966 Tf 6.97 0 Td[(1k.Clearly,( 2 )holdsforjyjM)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kRk\fyn)]TJ /F6 11.955 Tf 21.53 0 Td[(Tkg,wejustneedtoprove( 2 )forfjyj>M)]TJ /F27 6.974 Tf 12.87 3.54 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kRkg\fyn)]TJ /F6 11.955 Tf 21.23 0 Td[(Tkg.SinceV(0)=1isthemaximumpointofV,V(z)=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4(1+jzj2))]TJ /F25 6.974 Tf 8.16 3.54 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2,z2Rn. Lemma2.2.2. Thereexistsk0>1suchthatforallkk0andr2(Rk,M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k), minBr\kvk2(n(n)]TJ /F11 11.955 Tf 10.95 -.01 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4r2)]TJ /F14 8.966 Tf 6.96 0 Td[(n.(2)ProofofLemma 2.2.2 :Justliketheinteriorcase,supposethereexistrkRksuchthat minBrk\kvk>2(n(n)]TJ /F11 11.955 Tf 10.95 -.01 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4r2)]TJ /F14 8.966 Tf 6.97 0 Td[(nk.(2)Letvk(z)=vk(z)]TJ /F6 11.955 Tf 10.95 0 Td[(Tken)andvlk(z)=(l jzj)n)]TJ /F8 8.966 Tf 6.97 0 Td[(2vk(l2z jzj2).LetDk:=fz;M)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k(z)]TJ /F6 11.955 Tf 10.95 0 Td[(Tken)2k\Brkg 26

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bethedomainofvk.ThenDkRn+.Setl:=fz2Dk;jzj>lg.vkandvlkwillbecomparedonlforl2[l0,l1],wherel0=p 1+T)]TJ /F3 11.955 Tf 1 0 .167 1 219.37 -119.54 Tm[(e,l1=p 1+T+eforsomee>0smallandindependentofk.LetVbethelimitofvkinC2loc(Rn+):V(z)=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4(1+jz)]TJ /F6 11.955 Tf 10.95 0 Td[(Tenj2))]TJ /F25 6.974 Tf 8.17 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2.Bydirectcomputation,V>Vl0inRn+nBl0andV1 vk(y)>vl0k(y),y2l0\BR.(2)ForRlargewehave(leta1=(n(n)]TJ /F8 8.966 Tf 6.96 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4)vk(y)(a1)]TJ /F3 11.955 Tf 1 0 .167 1 201.28 -467.21 Tm[(e=5)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nonBR\Rn+andvl0k(y)(a1)]TJ /F11 11.955 Tf 10.95 0 Td[(2e=5)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,jyj>l0.Toprovevk>vl0kinl0nBRwecomparevkwithw=(a1)]TJ /F11 11.955 Tf 10.95 0 Td[(3e=10)jy)]TJ /F6 11.955 Tf 10.95 0 Td[(A1enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n 27

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whereA1=1=(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)cha2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(21.ForRchosensufcientlylargewehavewvl0kinl0nBRandvk>wonBR\l0.Tocomparevkandwoverl0nBR,itiseasytoseethatvk>wonBR\l0andl0n(BR[fzn>0g).Sincevk)]TJ /F6 11.955 Tf 11.05 0 Td[(wissuper-harmonic,theonlythingweneedtoproveisonRn+nBl0 n(vk)]TJ /F6 11.955 Tf 10.95 0 Td[(w)wkonl0nBR.Toobtain( 2 )rstforvkweuse(GH2)tohavenvkcvn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k,zn=0.Ontheotherhand,bythechoiceofA1,weverifyeasilythatnw>cwn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,zn=0.Thus,( 2 )holdsbasedonmeanvaluetheorem.Wehaveprovedthatthemovingsphereprocesscanstartatl=l0:vk>vl0kinl0.Letlbethecriticalmovingsphereposition:l:=minfl2[l0,l1];vmk>vmkinm,8m>l.g. 28

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AsinCaseoneweshallprovethatl=l1,thusgettingacontradictionfromVl1.Forthispurposeweletwl,k=vk)]TJ /F11 11.955 Tf 12.39 .27 Td[(vlk.Toderivetheequationforwl,kwerstrecallfrom( 2 )andthedenitionofvkthat 8>>>><>>>>:vk(z)+l)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(lkvk)=0,z2k,vk zn=l)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kh(lkvk),fzn=0g\k.(2)wherelk=s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kMk.Correspondinglyvlksatises 8>>>><>>>>:vlk+(l jzj)n+2l)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(lk(jzj l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(2vlk(z))=0,inl,vlk zn=(l jzj)nl)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kh(lk(jzj l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(2vlk(z))onl\fzn=0g.(2)LetOlbedenedasbefore.TheninOlwehave,by(GH1)(l jzj)n+2l)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kg(lk(jzj l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(2vlk(z))(vklk))]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2g(lkvk)(vlk)n+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,inOl(l jzj)nl)]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.41 0 Td[(2kh(lk(jzj l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(2vlk(z))(lkvk))]TJ /F25 6.974 Tf 12.61 3.53 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2h(lkvk)(vlk)n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,onOl\fzn=0g.Theinequalitiesaboveyieldwl,k+x1,kwl,k0,inOlnwl,kx2,kwl,k,onOl\fzn=0g.wherex1,k>0andx2,karecontinuousfunctionsobtainedfrommeanvaluetheorem.Itiseasytoseethatthemovingsphereargumentcanbeemployedtoprovethatl=l1,whichleadstoacontradictionfromthelimitingfunctionV.Thus,Lemma 2.2.2 isestablished.2Lemma 2.2.2 guaranteesthatoneachradiusRkr1 2Mktheminimumofvkisalwayscomparabletojzj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n.Re-scalingvkasrn)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2vk(rz)weseethesphericalHarnack 29

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inequalityholdsbythe(GH2)and(GH3).Thus,Proposition 2.2.2 isestablishedinCaseTwoaswell.2 Lemma2.2.3. Letfukgbeasequenceofsolutionsof( 2 )andqk!q2 B+1beasequenceofpointsink.ThenthereexistC>0,r2>0independentofksuchthatuk(qk)uk(x)Cjx)]TJ /F6 11.955 Tf 10.95 0 Td[(qkj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ninjx)]TJ /F6 11.955 Tf 10.95 0 Td[(qkjr2,x2B+3. Proof. Withoutlossofgeneralityweassumethatqksarelocalmaximumpointsofuk.WeconsidertwocasesCaseone:uk(qk)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2qkn!.LetMk=uk(qk)and vk(y)=Mk)]TJ /F8 8.966 Tf 6.96 0 Td[(1uk(Mk)]TJ /F27 6.974 Tf 12.86 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2y+qk),y2k:=fy;M)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2ky+qk2B+3g.(2)TheninthiscasevkconvergesuniformlytoV(y)=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4(1+jyj))]TJ /F25 6.974 Tf 8.17 3.54 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2overallcompactsubsetsofRn.Fore>0smallweletf=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)]TJ /F3 11.955 Tf 1 0 .167 1 190.45 -412.42 Tm[(e)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4(jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n)]TJ /F6 11.955 Tf 10.95 0 Td[(M)]TJ /F8 8.966 Tf 6.97 0 Td[(2k)jyjM2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2konjyjRwhereR>1ischosensothatvk>fonBR.Bydirectcomputation,wehavef yn>Nkfn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,onyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(qknMkforsomeNk!.Itiseasytoseethatvkfonknfyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(M2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2kqkng.Onfyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqkngwehaveyn(vk)]TJ /F3 11.955 Tf 1 0 .167 1 214.78 -571.09 Tm[(f)ch(vk)]TJ /F3 11.955 Tf 1 0 .167 1 281.16 -571.09 Tm[(f).Thus,standardmaximumprincipleimpliesvkfonk.Lemma 2.2.3 isestablishedinthiscase. 30

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NowweconsiderCasetwo:M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqknC.Letvkbedenedasin( 2 ).Inthiscasetheboundaryconditioniswrittenasynvk=(M)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kvk))]TJ /F25 6.974 Tf 12.61 3.54 Td[(n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2h(M)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kvk)vn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k,yn=)]TJ /F6 11.955 Tf 9.29 0 Td[(M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqkn.vkconvergestoV1overallcompactsubsetsofRnfyn)]TJ /F6 11.955 Tf 21.24 0 Td[(Tg,whereT=limk!M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqkn.V1satises( 2 ).ForRlargeande>0small,bothindependentofk,wehavevk(y)(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)]TJ /F3 11.955 Tf 1 0 .167 1 228.03 -262.99 Tm[(e)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,jyj=R.InBR\Rn+wehavetheuniformconvergenceofvktoV1.OurgoalistoprovethatvkisboundedbelowbyO(1)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(noutsideBR.Tothisendletw(y)=(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)]TJ /F11 11.955 Tf 10.95 0 Td[(2e)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4jy)]TJ /F6 11.955 Tf 10.95 0 Td[(A1enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,whereA1=ch(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)1 2)]TJ /F6 11.955 Tf 10.95 0 Td[(T.Thenitiseasytocheckthatw yn>chw(y)n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,onyn=)]TJ /F6 11.955 Tf 9.28 0 Td[(M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqkn.BychoosingRlargerifneededwehavevk(y)>(n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2) A)]TJ /F3 11.955 Tf 1 0 .167 1 181.34 -556.37 Tm[(e)n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 4jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n>w(y),jyj=R,y2Rn+.Thenitiseasytoapplymaximumprincipletoprovevk>winknBR.Lemma 2.2.3 isestablished.2 31

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Letqk12kandqk2beitsnearestoralmostnearestsequenceink:jqk2)]TJ /F6 11.955 Tf 10.95 0 Td[(qk1j=(1+o(1))d(qk1,knfqk1g).Weclaimthat Lemma2.2.4. ThereexistsC>0independentofksuchthat1 Cuk(qk1)uk(qk2)Cuk(qk1). Proof. Letsk=d(qk1,knfqk1g)anduk(y)=sn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2kuk(qk1+sky).Weuseektodenotetheimageofqk2afterscaling(sojekj!1).TheninB1,uk(x)uk(0))]TJ /F8 8.966 Tf 6.97 0 Td[(1jxj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nforjxj1=2.Ononehand,forjxj=1 2wehave,byLemma 2.2.3 appliedtoek,uk(0))]TJ /F8 8.966 Tf 6.97 0 Td[(1(1 2)2)]TJ /F14 8.966 Tf 6.97 0 Td[(nCuk(ek))]TJ /F8 8.966 Tf 6.97 0 Td[(1whichisjustuk(qk1)Cuk(qk2).Ontheotherhand,thesamemovingsphereargumentcanbeappliedtouknearqk2withnodifference.TheHarnacktypeinequalitygivesmaxB(qk2,1=4)\B+3ukminB(qk2,1=2)\B+3ukC.UsingmaxB(qk2,1=4)\B+3ukuk(qk2)andminB(qk2,1=2)\B+3ukminB(qk1,sk)\B+3uk,wehave uk(ek)uk(0))]TJ /F8 8.966 Tf 6.97 0 Td[(1C.(2)Thus,( 2 )givesuk(qk2)Cuk(qk1).Lemma 2.2.4 isestablished.2 Remark2.2.3. Proposition 2.2.2 isnotneededintheproofofLemma 2.2.4 32

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ThefollowinglemmaisconcernedwithPohozaevidentitythatcanbeveriedbydirectcomputation. Lemma2.2.5. Letusolve8>>>><>>>>:u+g(u)=0,inB+s,nu=h(u)on0B+s.ThenZ0B+sh(u)(n)]TJ /F8 8.966 Tf 6.97 0 Td[(1i=1xiiu+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2u)+ZB+s(nG(u))]TJ /F6 11.955 Tf 12.14 8.09 Td[(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2g(u)u) (2)=Z00B+ss(G(u))]TJ /F11 11.955 Tf 12.15 8.09 Td[(1 2juj2+(nu)2)+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2unuwhereG(s)=Rs0g(t)dt. Proposition2.2.3. Thereexistsd>0independentofksuchthatlimk!jqk1)]TJ /F6 11.955 Tf 10.95 0 Td[(qk2jd. Proof. Recallthatsk=(1+o(1))jqk1)]TJ /F6 11.955 Tf 11.1 0 Td[(qk2j.Weprovebywayofcontradiction.Supposesk!0,etMk=uk(0).Weclaimthat Mkuk(y)!ajyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n+b(y)inC2loc(B3=4\knf0g),witha>0,b(0)>0(2)wherek=fy;sky+qk12B+3g.Proofof( 2 ):Asusualweconsiderthefollowingtwocases:Caseone:limk!qk1nM2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k!,andCasetwo:limk!qk1nM2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2k!T
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Recalltheequationforukis( 2 ).MultiplyingMkonbothsidesandlettingk!,itiseasytoseefromtheassumptionsofgandhthatMkuk!hinC2loc(B1nf0g)wherehisaharmonicfunctiondenedinB1nf0g.Thus,h(y)=ajyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n+b(y)forsomeharmonicfunctionb(y)inB1.FromthepointwiseestimateinLemma 2.2.3 weseethata>0.Givenanye>0,wecompareukandwk:=(a)]TJ /F3 11.955 Tf 1 0 .167 1 217.11 -191.27 Tm[(e)(jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n)]TJ /F6 11.955 Tf 10.94 0 Td[(R2)]TJ /F14 8.966 Tf 6.97 0 Td[(nk)onjyjRk.HereRk!islessthanTk.Observethatuk>wkonBRkandjyj=e1fore1sufcientlysmall.Thus,uk>wkbythemaximumprinciple.Letk!wehave,inB1ajyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n+b(y)(a)]TJ /F3 11.955 Tf 1 0 .167 1 238.48 -286.9 Tm[(e)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,B1nBe1.Thenlete!0,whichimpliese1!0wehaveb(y)0inB1.Nextweclaimthatb(0)>0becausebyLemma 2.2.3 andLemma 2.2.4 wehaveajyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n+b(y)a1jy)]TJ /F6 11.955 Tf 10.95 0 Td[(ej2)]TJ /F14 8.966 Tf 6.96 0 Td[(ninB1forsomea1>0,wheree=limk!ek.Thus,b(y)>0whenyisclosetoe,whichleadstob(0)>0.( 2 )isestablishedinCaseone.Casetwo:.AgainwersthaveMkuk!hinC2loc(B)]TJ /F14 8.966 Tf 6.96 0 Td[(T1nf0g)andhisoftheformh(y)=ajyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n+b(y),yn)]TJ /F6 11.955 Tf 21.23 0 Td[(T.Toproveb(y)0wecompare,forxede>0,Mkukwithwk(y)=(a)]TJ /F3 11.955 Tf 1 0 .167 1 189.36 -603.69 Tm[(e)(jy)]TJ /F6 11.955 Tf 10.95 0 Td[(bkenj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n)]TJ /F15 11.955 Tf 10.95 0 Td[((Rk)]TJ /F11 11.955 Tf 10.94 0 Td[(1)2)]TJ /F14 8.966 Tf 6.97 0 Td[(n) 34

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wherebk!0andRk!arechosentosatisfy(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)bkR)]TJ /F8 8.966 Tf 6.97 0 Td[(2k>c0sk,c0=sup0chM)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2ka2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2.ItiseasytoseethatsuchbkandRkcanbefoundeasily.Lethk=Mkuk,and0k=k\fyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tkg.Wedivide0kintotwoparts:E1=fz20k;uk(z)s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2k1g,E2=0knE1.Thenbytheassumptionsonhnhk8><>:c0skhk,x2E2,chM)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2khn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k,x2E1,WiththechoiceofbkandRkitiseasytoverifythatnwkmaxfc0skwk,chM)]TJ /F27 6.974 Tf 12.87 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kwn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kgon0k\BRk.Thus,standardmaximumprinciplecanbeappliedtoprovethathkwkonk\BRk.Lettingk!rstande!0nextwehaveb(y)0inB1\fyn)]TJ /F6 11.955 Tf 22.13 0 Td[(Tg.ThenbyProposition 2.2.4 weseethatb(y)>0whenyisclosetoe,thelimitofek.Thefactnb=0at0impliesb(0)>0.( 2 )isprovedinbothcases.FinallytonishtheproofofProposition 2.2.3 wederiveacontradictionfromeachofthefollowingtwocases:Caseone:limk!M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqk1n>0.InthiscaseweusethefollowingPohozaevidentityonBsfors
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wheregk(s)=sn+2 2kg(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks),G(t)=Zt0g(s)ds,Gk(s)=snkG(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks).Firstweclaimthatfors>0, Gk(s)n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2nsgk(s).(2)Indeed,writingg(t)=c(t)tn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2,weseefrom(GH1)thatc(t)isanon-increasingfunction,thusGk(s)=snkG(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks)=snkZs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks0c(t)tn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dtsnkc(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks)Zs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks0tn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dt=n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2nc(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks)s2n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2=n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2nsgk(s).Replacingsbyukweseethatthelefthandsideof( 2 )isnon-negative.Nextweprovethat limk!M2kZBss(Gk(uk))]TJ /F11 11.955 Tf 12.14 8.09 Td[(1 2jukj2+jnukj2)+n)]TJ /F11 11.955 Tf 10.94 0 Td[(2 2uknuk<0(2)fors>0small.Clearly,after( 2 )isestablishedweobtainacontradictionto( 2 ).Tothisendrstweprovethat M2kGk(uk)=o(1).(2)Indeed,by(GH1)and(GH3)Gk(uk)=snkZs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk0g(t)dt8>>>>><>>>>>:snkRs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk0ctdt,ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kM)]TJ /F8 8.966 Tf 6.96 0 Td[(1k1,snk(R10ctdt+Rs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk1ctn+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dt),ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kM)]TJ /F8 8.966 Tf 6.96 0 Td[(1k>1. 36

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Therefore,Gk(uk)8><>:Cs2kM)]TJ /F8 8.966 Tf 6.96 0 Td[(2k,ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kM)]TJ /F8 8.966 Tf 6.97 0 Td[(1k1,Csnk+CM)]TJ /F27 6.974 Tf 10.87 3.53 Td[(2n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k,ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kM)]TJ /F8 8.966 Tf 6.96 0 Td[(1k>1.Clearly,( 2 )holdsineithercase.Consequently,wewritethelefthandsideof( 2 )asZBs()]TJ /F11 11.955 Tf 10.48 8.09 Td[(1 2sjhj2+sjnhj2+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2hnh)+o(1)whereh(y)=ajyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n+b(y),b(0)>0,a>0.Bydirectcomputation,wehaveZBs()]TJ /F11 11.955 Tf 10.48 8.09 Td[(1 2sjhj2+sjnhj2+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2hnh)=ZBs)]TJ /F15 11.955 Tf 12.14 8.09 Td[((n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)2 ab(0)s1)]TJ /F14 8.966 Tf 6.97 0 Td[(n+O(s2)]TJ /F14 8.966 Tf 6.97 0 Td[(n)dS.Thus,( 2 )isveriedwhens>0issmall.ThesecondcaseweconsiderisCasetwo:limk!M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2kqk1n=0.InthiscaseweusethefollowingPohozaevidentityonB+s:Lethk(s)=sn 2kh(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks),thenwehaveZB+s\Rn+hk(uk)(n)]TJ /F8 8.966 Tf 6.96 0 Td[(1i=1xiiuk+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2uk)+ZB+s(nGk(uk))]TJ /F6 11.955 Tf 12.15 8.09 Td[(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2gk(uk)uk) (2)=ZB+s\Rn+s(Gk(uk))]TJ /F11 11.955 Tf 12.14 8.09 Td[(1 2jukj2+(nuk)2)+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2uknukMultiplyingM2konbothsidesandlettingk!weseebythesameestimateasinCaseonethatthesecondtermonthelefthandsideisnon-negative,therighthandsideisstrictlynegative.Theonlytermweneedtoconsiderislimk!M2kZB+s\Rn+hk(uk)(n)]TJ /F8 8.966 Tf 6.96 0 Td[(1i=1xiiuk+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2uk). 37

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LetH(s)=Rs0h(t)dt,thenfromintegrationbypartswehaveZB+s\Rn+hk(uk)(n)]TJ /F8 8.966 Tf 6.96 0 Td[(1i=1xiiuk+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2uk) (2)=ZBs\Rn+sn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kH(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk)s+ZB+s\Rn+()]TJ /F15 11.955 Tf 9.29 0 Td[((n)]TJ /F11 11.955 Tf 10.95 0 Td[(1)sn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kH(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk)+n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2ukhk(uk))dx0.Forthersttermontherighthandsideof( 2 )weclaim M2ksn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kH(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk)=o(1)onBs.(2)Indeed,by(GH1)and(GH2)jH(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk)j8>>>>><>>>>>:Rs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk0ctdt,ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk1,R10ctdt+Rs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk1ctn n)]TJ /F27 6.974 Tf 5.41 0 Td[(2dt,ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk>1,Usinguk=O(1=Mk)onBswethenhaveM2ksn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kjH(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk)j8><>:O(sk),ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk1,O(sk)+O(M)]TJ /F27 6.974 Tf 12.86 3.53 Td[(2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2k),ifs)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2kuk>1.Thus,( 2 )isveriedandthersttermontherighthandsideof( 2 )iso(1).Therefore,weonlyneedtoestimatethelasttermof( 2 ),whichweclaimisnon-negative.Indeed,fort>0,wewriteh(t)=b(t)tn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2forsomenon-decreasingfunctionb.Thenwehavesn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kH(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks)=sn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kZs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks0h(t)dt=sn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kZs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks0b(t)tn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dtsn)]TJ /F8 8.966 Tf 6.97 0 Td[(1kb(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks)Zs)]TJ /F25 6.974 Tf 6.61 2.75 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks0tn n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dt=n)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2n)]TJ /F11 11.955 Tf 10.95 0 Td[(2b(s)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2ks)s2n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 n)]TJ /F27 6.974 Tf 5.41 0 Td[(2=n)]TJ /F11 11.955 Tf 10.94 0 Td[(2 2n)]TJ /F11 11.955 Tf 10.95 0 Td[(2hk(s)s. 38

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Replacingsbyukintheaboveweseethatthelasttermof( 2 )isnon-negative.Thus,thereisacontradictionin( 2 )inCasetwoaswell.Proposition 2.2.3 isestablished.2WeareinthepositiontonishtheproofofTheorem 2.1.1 .ByProposition 2.2.3 thereisapositivedistancebetweenanytwomembersofk.TheuniformboundofRB+1u2n=n)]TJ /F8 8.966 Tf 6.97 0 Td[(2kfollowsreadilyfromthepointwiseestimatesforblowupsolutionsnearanisolatedblowuppoint.TheuniformboundforRB+1jukj2canbeobtainedbyscalingandstandardellipticestimatesforlinearequations.Thus,wehaveobtainedacontradictionto( 2 ).Theorem 2.1.1 isestablished.2 2.3ProofofTheorem 2.1.2 Ifhisnon-positive,theenergyestimatefollowsfromtheHarnackinequalityinastraightforwardway.Indeed,letG(x,y)beaGreen'sfunctiononB+3suchthatG(x,y)=0ifx2B+3,y2B+3\Rn+andynG(x,y)=0forx2B+3,y2B+3\Rn+.ItiseasytoseethatGcanbeconstructedbyaddingthestandardGreen'sfunctiononB3itsreectionoverRn+.ItisalsoimmediatetoobservethatG(x,y)Cnjx)]TJ /F6 11.955 Tf 10.95 0 Td[(yj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,x2B+3,y2B+2.MultiplyingGonbothsidesof( 2 )andintegratingbyparts,wehaveu(x)+ZB+3\Rn+h(u(y))G(x,y)dSy+ZB+3\Rn+u(y)G(x,y) ndSy=ZB+3g(u(y))G(x,y)dy.Herenrepresentstheouternormalvectorofthedomain.Usingh0andnG0,wehaveu(x)ZB+3g(u(y))G(x,y)dy,x2B+3.Inparticulartakeletu(x0)=minB+2u,thenjx0j=2,thusCmaxB+1uminB+2uZB+1g(u(y))u(y)G(x0,y)dyCZB+3=2g(u)udy.Therefore,wehaveobtainedtheboundonRB+3=2g(u)udy.ToobtaintheboundonRB+1juj2,weuseacut-offfunctionhwhichis1onB+1andis0onB+2nB+3=2andjhj 39

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C.Multiplyinguh2tobothsidesof( 2 )andusingintegrationbypartsandCauchyinequalityweobtainthedesiredboundonRB+1juj2.Theorem 2.1.2 isestablished.2 40

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CHAPTER3GREEN'SFUNCTIONESTIMATESG(y,h)=1 (n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)sn jy)]TJ /F3 11.955 Tf 1 0 .167 1 222.26 -74.22 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n)]TJ /F18 11.955 Tf 10.95 16.86 Td[(jyj l2)]TJ /F14 8.966 Tf 6.97 0 Td[(njyl)]TJ /F3 11.955 Tf 1 0 .167 1 344.36 -74.22 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n!istheGreen'sfunctionwhichhassomegoodproperties.WewouldliketogivesomeestimatesofthisfunctionsothatwecanutilizetheresultdirectlyinChapter4. Lemma3.0.1. LetA=fh2l:jy)]TJ /F3 11.955 Tf 1 0 .167 1 202.59 -188.3 Tm[(hj(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 259.45 -188.3 Tm[(l)=3gB=fh2l:jy)]TJ /F3 11.955 Tf 1 0 .167 1 203.71 -212.21 Tm[(hj(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 260.58 -212.21 Tm[(l)=3andjhj8lgD=fh2l:jhj8lg.ThereexistpositiveconstantsC1andC2dependingonlyonnsuchthatthefollowingestimateshold. 1. Foralll>>><>>>>:Cjy)]TJ /F3 11.955 Tf 1 0 .167 1 208.65 -402.44 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nh2AC(jyj)]TJ /F22 8.966 Tf 1 0 .167 1 206.42 -420.4 Tm[(l)(jhj2)]TJ /F22 8.966 Tf 1 0 .167 1 241.49 -420.4 Tm[(l2) ljy)]TJ /F22 8.966 Tf 1 0 .167 1 224.16 -431.37 Tm[(hjnC(jyj)]TJ /F22 8.966 Tf 1 0 .167 1 302.08 -420.4 Tm[(l)(jhj)]TJ /F22 8.966 Tf 1 0 .167 1 333.16 -420.4 Tm[(l) jy)]TJ /F22 8.966 Tf 1 0 .167 1 312.82 -431.37 Tm[(hjnh2BC(jyj)]TJ /F22 8.966 Tf 1 0 .167 1 206.42 -444.36 Tm[(l)(jhj2)]TJ /F22 8.966 Tf 1 0 .167 1 241.49 -444.36 Tm[(l2) ljy)]TJ /F22 8.966 Tf 1 0 .167 1 224.16 -455.33 Tm[(hjnCjyj)]TJ /F22 8.966 Tf 1 0 .167 1 298.6 -444.4 Tm[(l ljhj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nh2D(3) 2. Foralljyj4l,both G(y,h)C(jhj)]TJ /F3 11.955 Tf 1 0 .167 1 185.17 -501.7 Tm[(l)(jyj2)]TJ /F3 11.955 Tf 1 0 .167 1 233.55 -501.7 Tm[(l2) ljy)]TJ /F3 11.955 Tf 1 0 .167 1 209.7 -518.17 Tm[(hjnCjhj)]TJ /F3 11.955 Tf 1 0 .167 1 305.09 -501.7 Tm[(l ljyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nh2l(3)and G(y,h)Cjy)]TJ /F3 11.955 Tf 1 0 .167 1 232.6 -569.56 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nh2lnl.(3) Proof. Afterpreformingelementarycomputationsinvolvingthemean-valuetheoremweobtain 41

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0snG(y,h)=1 2l2(jyj2)]TJ /F3 11.955 Tf 1 0 .167 1 225.36 -35.86 Tm[(l2)(jhj2)]TJ /F3 11.955 Tf 1 0 .167 1 280.09 -35.86 Tm[(l2)Z10`t(y,h))]TJ /F25 6.974 Tf 8.16 3.53 Td[(n 2dt,where`t(y,h)=tjy)]TJ /F3 11.955 Tf 1 0 .167 1 184.31 -126.02 Tm[(hj2+(1)]TJ /F6 11.955 Tf 10.95 0 Td[(t)jyj l2yl)]TJ /F3 11.955 Tf 1 0 .167 1 317.83 -126.02 Tm[(h2=jyj l2yl)]TJ /F3 11.955 Tf 1 0 .167 1 224.5 -161.22 Tm[(h2)]TJ /F6 11.955 Tf 15.93 8.1 Td[(t l2(jyj2)]TJ /F3 11.955 Tf 1 0 .167 1 305.41 -161.22 Tm[(l2)(jhj2)]TJ /F3 11.955 Tf 1 0 .167 1 360.14 -161.22 Tm[(l2),0t1;(y,h)2llnfy=hg.Foreach(y,h)2llnfy=hg,t7!`t(y,h)isdecreasingandpositive,soforsuch(y,h), 1 2l2(jyj2)]TJ /F3 11.955 Tf 1 0 .167 1 68.96 -304.67 Tm[(l2)(jhj2)]TJ /F3 11.955 Tf 1 0 .167 1 123.69 -304.67 Tm[(l2)jyj l)]TJ /F14 8.966 Tf 6.97 0 Td[(nyl)]TJ /F3 11.955 Tf 1 0 .167 1 219.88 -304.67 Tm[(h)]TJ /F14 8.966 Tf 6.97 0 Td[(nsnG(y,h) (3) 1 2l2(jyj2)]TJ /F3 11.955 Tf 1 0 .167 1 330.84 -336.1 Tm[(l2)(jhj2)]TJ /F3 11.955 Tf 1 0 .167 1 385.57 -336.1 Tm[(l2)jy)]TJ /F3 11.955 Tf 1 0 .167 1 427.33 -336.1 Tm[(hj)]TJ /F14 8.966 Tf 6.97 0 Td[(n.Eachoftheestimatesin( 3 ),( 3 )and( 3 )followimmediatelyfromeither( 3 )orfrom( 4 ).ToshowG(y,h)satises( 3 ),use( 3 )inadditiontothefactthatG(y,h)=G(h,y).Toseethat( 3 ),( 3 ),( 3 )and( 3 )holdfor G:=G(y,h)+G(y,h),observethatsinceG(y,h)0, G(y,h)G(y,h).Thisgivesboth( 3 )and( 3 ).Toshowthat Gsatises( 3 )and( 3 ),observethatG(y,h)satisestheseinequalitieswithyreplacedbyy.Sincejyj=jyjandjy)]TJ /F3 11.955 Tf 1 0 .167 1 203.53 -515.41 Tm[(hjjy)]TJ /F3 11.955 Tf 1 0 .167 1 252.66 -515.41 Tm[(hjfory,h2Rn+,thedesiredinequalitieshold.2 42

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CHAPTER4AHARNACK-TYPEINEQUALITYFORAPRESCRIBINGCURVATUREEQUATIONONADOMAINWITHBOUNDARYInconformalgeometrythewellknownYamabeproblemasksifitisalwayspossibletodeformthemetricofacompactRiemannianmanifoldtomakethescalarcurvatureconstant.TheYamabeproblemcanbetranslatedtondingasolutiontoasemi-linearellipticequationcalledtheYamabeequation.ThroughtheworksofTrudinger[ 40 ],Aubin[ 2 ]andSchoen[ 37 ]itisprovedthattheYamabeequationalwayshasasolution.AcorrespondingquestioniscalledtheYamabecompactnessproblem,whichasksifallsolutionstotheYamabeequationareuniformlyboundedwhenthemanifoldisnotconformallydiffeomorphictothestandardsphere.TheYamabecompactnessproblemwaseventuallyprovedtobeafrmativeifthedimensionofthemanifoldisnogreaterthan24byKhuri-Marques-Schoen[ 28 ],andnegativebyBrendle-Marques[ 5 ]fordimensionsgreaterthan24.Inthischapterwestudythefollowinglocallydenedequation: 8><>:u+K(x)un+2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2=0inB+1Rn+,u>0u xn=c(x)un n)]TJ /F27 6.974 Tf 5.42 0 Td[(2onB+1\Rn+.(4)OurmaingoalinthischapteristoprovethefollowingHarnacktypeinequality: (max B1=3u)(min B2=3u)C(4)forsomeC>0.TheHarnackinequality( 4 )revealsimportantinformationontheinteractionofbubbles.Itimpliesthatallbubbleshavecomparablemagnitudeandstayfarawayfromoneanother.Asaconsequence,anenergyestimateofthefollowingtypeisessentiallyimplied: ZB+1=2juj2+u2n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dxC.(4) 43

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4.1TheMainTheoremSpecically,weassumethroughoutthischapterthatn4andthatKsatises (K1) K2Cn)]TJ /F8 8.966 Tf 6.96 0 Td[(2( B+1),andthereexistsapositiveconstantC0suchthatforallx2 B+1, jK(x)C0jK(x)jn)]TJ /F27 6.974 Tf 5.42 0 Td[(2)]TJ /F25 6.974 Tf 5.42 0 Td[(j n)]TJ /F27 6.974 Tf 5.42 0 Td[(3j=1,,n)]TJ /F11 11.955 Tf 10.95 0 Td[(2.(4) (K2) Thereexistsaconstant>0suchthatboth1 K(x)forallx2 B+1andkK(x)kCn)]TJ /F27 6.974 Tf 5.42 0 Td[(2( B+1). (K3) Kdependsonlyonx1,,xn)]TJ /F8 8.966 Tf 6.97 0 Td[(1.TherearemanyfunctionssatisfyingtheassumptionsonK.OneelementarysuchfunctionisK(x)=1+ n)]TJ /F8 8.966 Tf 6.97 0 Td[(1j=1x2j!a,an)]TJ /F11 11.955 Tf 10.95 0 Td[(2 2.Theatnessassumption (K1) wasusedbyChenandLinin[ 8 ]toderive(amongotherresults)aHarnack-typeinequalityforpositiveclassicalsolutionsofu+K(x)u(n+2)=(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2)=0onB1.OurapproachismotivatedbytheapproachtakenbyChenandLin.However,sincethesituationinthischapterinvolvesB+1insteadofB1,wemustovercomecomplicationsthatwerenotpresentinChenandLin'sboundary-freecase.Themaintheoremofthischapteristhefollowing. Theorem4.1.1. Letubeasolutionof( 4 ).SupposeKsatises (K1) (K2) and (K3) andthatcisconstant.ThereexistconstantsC(n,,C0)>0ande(n,,C0)>0suchthatifc
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ofK.Seeforexample[ 8 ].Forsimplicity,weallowKtoenjoythispropertyonallof B+1.AsacorollarytoTheorem 4.1.1 ,wehavethefollowingenergybound. Corollary4.1.1. Supposeu,K,candeareasinTheorem 4.1.1 .ThereexistsapositiveconstantC(n,,C0)suchthatforallpositivesolutionsuof( 4 ),( 4 )holds.Thisenergyestimateisareectionofthefactthatsocalled`bubbles',thelargelocalmaximumpointsofblow-upsolutionsto( 4 ),muststayfarawayfromeachother.Inviewofthere-scalingu(x)7!R(n)]TJ /F8 8.966 Tf 6.96 0 Td[(2)=2u(Rx),Theorem 4.1.1 impliescorrespondingHarnackinequalitiesonBRingeneral,aslongasthescalarcurvaturefunctionandthemeancurvaturefunctionstillsatisfythesameassumptionsafterscaling.TheproofofTheorem 4.1.1 isbycontradiction.Bythecontradictionassumption,weobtainasequenceofblow-upsolutionsof( 4 ).Aftershowingthatblow-upcanonlyoccurnearB+1\Rn+,weusethemethodofmovingspherestoderiveacontradiction.Thischapterisorganizedasfollows.InSection 4.2 weuseastandardselectionprocessofSchoen[ 37 ]andLi[ 27 ]andtheclassicationtheoremsofCaffarelli-Gidas-Spruck[ 6 ]andLi-Zhu[ 33 ]toobtainaconvenientrescalingoftheblow-upsolutions.InSection 4.3 weshowthatblow-uppointsmustbeclosetoB+1\Rn+,seeProposition 4.3.1 .Thisisachievedthroughthreeapplicationsofthemethodofmovingspheres(MMS).InparticularMMSisrstusedtoshowthatKmustvanishatablow-uppoint,thenMMSisusedagaintoshowthatKmustvanishrapidlyatablow-uppoint,andanalapplicationofMMSisusedtoshowthatblow-upcanonlyoccurnearB+1\Rn+.InSection 4.4 ,weprovetheHarnack-typeinequalityofTheorem 4.1.1 .Asintheproofthatblow-upcanonlyoccurnearB+1\Rn+,theproofofTheorem 4.1.1 isviathreeapplicationofMMS;oncetoshowKvanishesatablow-uppoint,oncetoshowKvanishesrapidlyatablow-uppoint,andnallytocompletetheprooftheTheorem 4.1.1 .InSection 4.5 wegiveanoverviewofhowtoobtaintheenergyestimateinCorollary 4.1.1 fromtheHarnack-typeinequality.SincethederivationofCorollary 4.1.1 from 45

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Theorem 4.1.1 isstandard,onlythemainpointsoftheproofwillbementioned.Theinterestedreadercanconsult,forexample[ 27 ],[ 23 ]and[ 29 ]fordetails.Asnotationalconventions,wewillusethefollowing.Thecriticalexponent(n+2)=(n)]TJ /F11 11.955 Tf 11.23 0 Td[(2)willbedenotedbyn.Wewilluseo(1)todenoteanyquantitythattendstozeroasi!.ThesymbolsC,C1andC2willdenoteconstantsthatdependonlyonnandandwillbedifferentfromlinetoline.Thefunctionsvi,RandURaswellasthedomainsiandl(tobedened)willbeusedinbothSections 4.3 and 4.4 ,butwillhavedifferentdenitionsinthosesections. 4.2RescalingandSelectionSupposetheHarnack-typeinequality( 4 )fails.Foreachi2N,thereisapositivesolutionuiof( 4 )withKreplacedwithKiandcreplacedbycisuchthat 0@max B+1=3ui1A0@min B+2=3ui1A>i.(4)NotethatandC0asgivenintheassumptionsonKareuniformini.Withoutlossofgeneralityweassumelimk!Ki(xi)=n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2).Byastandardselectionprocess,seeforexampleSchoen[ 38 ]andLi[ 27 ]wemaychoosexi2B+1=2\ Rn+suchthat,forsomesi!0,ui(xi)maxB1=3+ui,ui(xi)ui(x)8x2B(xi,si)\Rn+,andui(xi)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2si!.Forsuchxi,( 4 )yields ui(xi)min B+2=3ui>i,(4)whichimpliesui(xi)!.Ifuiarepositivesolutionsof( 4 )andxiarelocalmaximumpointsofuiforwhich( 4 )holds,uiissaidtoblowup,andablow-uppointisthelimitofanyconvergentsubsequenceofxiforwhich( 4 )occurs.Setting 46

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Mi=ui(xi),)]TJ /F14 8.966 Tf 35.5 -2.11 Td[(i=M2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2i,Ti=xin)]TJ /F14 8.966 Tf 6.78 -2.11 Td[(iandEi=B()]TJ /F21 11.955 Tf 9.29 0 Td[()]TJ /F14 8.966 Tf 6.77 -2.11 Td[(ixi,2)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i)\fyn>)]TJ /F6 11.955 Tf 9.29 0 Td[(Tig,(4)andapplyingstandardargumentsusingtheclassicationtheoremsofCaffarelli-Gidas-Spruck[ 6 ]andLi-Zhu[ 33 ]thefunctions vi(y)=1 Miui(xi+)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iy),y2 Ei(4)convergeinC2overnitedomainsinthefollowingtwocases. Case1: IfthereisasubsequencealongwhichTi!,thenafterpassingtoafurthersubsequence,wehavevi!UinC2loc(Rn),where U(y)=1+jyj2)]TJ /F25 6.974 Tf 8.16 3.53 Td[(n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2.(4) Case2: IffTigisboundedthenafterpassingtoasubsequenceweassumethatTiconverges.Inthiscase,afterpassingtoafurthersubsequence,viconvergesinC2overcompactsubsetsofRn\fyn)]TJ /F11 11.955 Tf 22.76 0 Td[(limiTigtoaclassicalsolutionUof 8>>>><>>>>:U+n(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)Un=0y2Rn\fyn>)]TJ /F11 11.955 Tf 10.62 0 Td[(limiTigU yn=limiciUn=(n)]TJ /F8 8.966 Tf 6.96 0 Td[(2)y2fyn=)]TJ /F11 11.955 Tf 10.62 0 Td[(limiTigU(0)=1=maxy2Rn\fyn)]TJ /F8 8.966 Tf 14.93 0 Td[(limiTigU(y).(4)Sincetheselectionprocessandapplicationoftheclassicationtheoremsarestandard,theirapplicationsarenotpresentedhere.Similartechniqueshavebeenusedin[ 8 ],[ 29 ],[ 45 ],[ 44 ],etc.TheproofofTheorem 4.1.1 isnowsplitintotwostepsaccordingto Case1 and Case2 .Intherststepweprove Case1 cannotoccur,whichshowsthatblow-upcannotoccurfarawayfromB+1\Rn+.Inthesecondstep,withtheknowledgethatblow-upcanonlyoccurnearB+1\Rn+,weproveTheorem 4.1.1 47

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4.3Blow-upCanOnlyOccurNearB+1\Rn+TheproofofTheorem 4.1.1 reliesondelicateanalysisofthebehaviorofuinearablow-uppoint.Asarststep,weprovethefollowingtheoremwhichsaysthatblow-upcanonlynearB+1\Rn+.Inthistheorem,weonlyrequirectobebounded. Theorem4.3.1. Supposefuigisasequenceofpositivesolutionsof( 4 )thatsatises( 4 )andthatjc(x)jCforallx2B+1\Rn+andsomeC>0.ThereexistsaconstantC1>0independentofisuchthatifxiisalocalmaximizerofuiforwhich( 4 )holds,thenxinui(xi)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2C1,wherexindenotesthenthcoordinateofxi.TheproofofTheorem 4.3.1 isbycontradiction.Specically,MMSwillbeusedthreetimes;rst,inSubsection 4.3.1 toshowthatKi(xi)vanishes,secondinSubsection 4.3.2 toshowthatKi(xi)vanishesrapidly,andnallyinSubsection 4.3.3 tocompletetheproofofTheorem 4.3.1 .Theargumentinthissectionissimilartothatin[ 8 ].HoweverChen-LinusedacomplicatedmovingplanemethodwhichinvolvestwoKelvintransformationsandatranslation.Wemodifytheirapproachbyusingamuchsimplermovingspherestomakethepicturemucheasiertounderstand(see[ 45 ]).LetMi,)]TJ /F14 8.966 Tf 6.77 -2.12 Td[(iandTibeasin( 4 )andconsiderthefunctionsvi(y)=1 Miui(xi+)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iy),y2 B(0,1 8)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i)\fyn)]TJ /F6 11.955 Tf 21.23 0 Td[(Tig(fortheproofofTheorem 4.3.1 ,viisthesameasviin( 4 )andweomitthebarinthenotation).Observethatify2B(0,)]TJ /F14 8.966 Tf 11.59 -2.11 Td[(i=8)\fyn)]TJ /F6 11.955 Tf 21.23 0 Td[(Tig,thenby( 4 )vi(y)=)]TJ /F8 8.966 Tf 6.77 4.93 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(niMiui(xi+)]TJ /F16 8.966 Tf 6.78 4.93 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1iy)C(n)ijyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n.Infact,wemaychooseei!0slowlysuchthat 48

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vi(y)p ijyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,y2B(0,ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i)\fyn)]TJ /F6 11.955 Tf 21.23 0 Td[(Tig.(4)Denei=B(0,ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i)\fyn>)]TJ /F6 11.955 Tf 9.29 0 Td[(Tig,0i=i\fyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tigand00i=in0i.Elementarycomputationsshowthatvisatises 8><>:vi+Hi(y)vni=0y2ivi yn=ci(xi+)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iy)vn=(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2)iy20i,(4)whereHi(y)=Ki(xi+)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1iy).Bythecontradictionhypothesis,thereisasubsequenceofTialongwhichTi!,so Case1 applies.BeforewecanproveTheorem 4.3.1 weneedtoshowthatKi(xi)vanishesrapidly.Thiswillbedoneintwosteps.TherststepshowsthatKi(xi)vanishesandisproveninSubsection 4.3.1 .ThesecondstepshowsthatKi(xi)vanishesrapidlyandisproveninSubsection 4.3.2 .Fornotationalconvenience,insubsections 4.3.1 4.3.3 wewillusejKi(xi)j=di. 4.3.1VanishingofKi(xi) Proposition4.3.1. Thereexistsasubsequencealongwhichdi!0.TheproofofProposition 4.3.1 isbycontradiction.Namely,wesupposethereisd>0suchthatinfidid>0andusethemovingspheremethodtoderiveacontradiction.Byassumption (K3) wemayassumewithnolossofgeneralitythatthereisasubequencealongwhichKi(xi) di!e=(1,0,,0).ForR1xedandtobedetermined,denethetranslations 49

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vR,i(y)=vi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)andUR(y)=U(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)andtheKelvininversionsvlR,i(y)=l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2vR,i(yl)andUlR(y)=l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2UR(yl),wherel>0andyl=l2y=jyj2.Clearly,vR,i,URandtheirKelvininversionsarewell-denedinl=in Bl.Fornotationalconvenience,weset0l=l\fyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(Tig.Settingl=p 1+R2andcomputingdirectly,itiseasytoseethat 8><>:(UR)]TJ /F6 11.955 Tf 10.95 0 Td[(UlR)(y)>0y2lifll.(4)Forl0=Randl1=R+2,wehavel2[l0,l1],soweonlyconsiderlinthisrange.Denewl(y)=vR,i(y))]TJ /F6 11.955 Tf 10.95 0 Td[(vlR,i(y)y2 l.Forconvenience,wesuppressthei-dependenceinthisnotation.Elementarycomputationsshowthatwlsatises 8>>>><>>>>:Liwl(y)=Ql1(y)y2lBiwl(y)=Ql2y20lwl(y)=0y2l\Bl,(4)where Li=+Hi(y)]TJ /F6 11.955 Tf 10.94 0 Td[(Re)x1(y)Bi= yn)]TJ /F6 11.955 Tf 10.95 0 Td[(ci(xi+)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))x2(y)(4)aretheinteriorandboundaryoperatorsrespectively, 50

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x1(y)=nZ10tvR,i(y)+(1)]TJ /F6 11.955 Tf 10.95 0 Td[(t)vlR,i(y)4 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dt(4) x2(y)=n n)]TJ /F11 11.955 Tf 10.95 0 Td[(2Z10tvR,i(y)+(1)]TJ /F6 11.955 Tf 10.95 0 Td[(t)vlR,i(y)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dt(4)areobtainedfromthemeanvaluetheorem, Ql1(y)=(Hi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))(vR,i(y)l)n(4)isanerrortermtobecontrolledbyatestfunctionand Ql2(y)=ci(xi+)]TJ /F16 8.966 Tf 6.78 4.93 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)))]TJ /F6 11.955 Tf 10.95 0 Td[(ci(xi+)]TJ /F16 8.966 Tf 6.78 4.93 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))vlR,i(y)n=(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2))]TJ /F3 11.955 Tf 1 0 .167 1 117.46 -261.8 Tm[(ln)]TJ /F8 8.966 Tf 6.97 0 Td[(2 jyjn+2Ti(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)jyj2vR,i(yl)+2l2DvR,i(yl),yE. (4) Weneedtoconstructatestfunctionhlsuchthatboth hl(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2l(4)and 8><>:Li(wl+hl)(y)0y2lBi(wl+hl)(y)0y20l\ Ol,(4)where Ol=fy2l:(vR,i)]TJ /F6 11.955 Tf 10.95 0 Td[(vlR,i)(y)vlR,i(y)g.(4)Suchatestfunctionisaperturbationofwlthatallowsthemaximumprincipletobeapplied.Forourpurposes,themaximumprincipleonlyneedstoapplyonOlbecausewl>0offofOl.Webeginwithsomehelpfulestimates.Dene 51

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l=fy2l\B2l:y1>2j(y2,,yn)jg. Lemma4.3.1. ThereexistpositiveconstantsC1andC2independentofiandlsuchthatforisufcientlylarge,8><>:Hi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 21.24 0 Td[(C1)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 303.1 -123.4 Tm[(l)y2lHi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)C2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 301.77 -147.31 Tm[(l)y2lnl.Unlessmentionedotherwise,constantsC1,C2areindependentofiandl. Proof. Theproofiselementaryandfollowsfromthedenitionofl,thefactthatK2C1( B+3)andtheassumption0
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soUlR(y)Cjyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n.2CombiningtheresultsofLemmas 4.3.1 and 4.3.2 weobtainl-independentpositiveconstantsa1anda2suchthatboth Ql1(y))]TJ /F6 11.955 Tf 21.23 0 Td[(a1)]TJ /F16 8.966 Tf 6.78 4.93 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 195.97 -187.05 Tm[(l) 1 1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 261.99 -197.19 Tm[(lej2!(n+2)=2y2l(4)and Ql1(y)8><>:a2)]TJ /F16 8.966 Tf 6.78 4.85 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 234.55 -260.27 Tm[(l)y2 la2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 234.55 -284.18 Tm[(l)jyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2lnl.(4)Thefollowinglemmagivesestimatesforthecoefcientfunctionsx1andx2. Lemma4.3.3. ThereexistpositiveconstantsC1andC2suchthatforisufcientlylarge,x1(y)C2jyj)]TJ /F8 8.966 Tf 6.97 0 Td[(4y2(l\Ol)nB4l,x1(y)C1jyj)]TJ /F8 8.966 Tf 6.97 0 Td[(4y2lnl,andx2(y)C2jyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2y20l\ Ol. Proof. Theprooffollowsimmediatelyfromtheexpressionsofx1andx2in( 4 )and( 4 )andLemma 4.3.2 .2ThenextlemmagivesausefulestimateforQl2andisthereasontheproofofTheorem 4.3.1 islessdifcultthantheproofofTheorem 4.1.1 53

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Lemma4.3.4. ThereexistsaconstantC>0suchthatforisufcientlylarge,Ql2(y))]TJ /F6 11.955 Tf 21.23 0 Td[(CTiln)]TJ /F8 8.966 Tf 6.97 0 Td[(2jyj)]TJ /F14 8.966 Tf 6.96 0 Td[(ny20l. Proof. SincekcikLandbyLemma 4.3.2 ,thereisapositiveconstantCsuchthat(ci(xi+)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)))]TJ /F6 11.955 Tf 10.95 0 Td[(ci(xi+)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))(vlR,i(y))2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2Cjyj)]TJ /F8 8.966 Tf 6.96 0 Td[(2CT)]TJ /F8 8.966 Tf 6.97 0 Td[(2i.Ontheotherhand,sincevR,i!URinC2( B2l)andsincejyjTi,ifiissufcientlylarge,jyj2vR,i(yl)+2l2DvR,i(yl),yE1 2jyj2inf BlUR(y))]TJ /F11 11.955 Tf 10.95 0 Td[(4l2kURkC0( Bl)jyj1 4jyj2inf BlUR(y).Lemma 4.3.4 nowfollowsfromthesetwoestimatesandequation( 4 ).2Wenowproceedwiththeconstructionofthetestfunctionhl.LetsndenotetheareaofSn)]TJ /F8 8.966 Tf 6.96 0 Td[(1andletG(y,h)beGreen'sfunctionfor)]TJ /F21 11.955 Tf 9.28 0 Td[(onRnn BlrelativetotheDirichletcondition.Recallthat G(y,h)=1 (n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)sn jy)]TJ /F3 11.955 Tf 1 0 .167 1 219.85 -434.48 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n)]TJ /F18 11.955 Tf 10.95 16.86 Td[(jyj l2)]TJ /F14 8.966 Tf 6.97 0 Td[(njyl)]TJ /F3 11.955 Tf 1 0 .167 1 341.95 -434.48 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n!.(4)EstimatesonGareprovidedinAppendix 3 .Dene hl(y)=ZlG(y,h)Ql1(h)dh.(4)Byconstructionhlsatises8>>>><>>>>:)]TJ /F21 11.955 Tf 9.29 0 Td[(hl(y)=Ql1(y)y2lhl(y)=0y2l\Bl.hl yn(y)=RlG yn(y,h)Ql1(h)dhy20l. 54

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Wehavethefollowingestimatesofhl. Lemma4.3.5. ThereexistsR0sufcientlylargesuchthatifRR0thentherearepositiveconstantsC1andC2suchthathl(y)8><>:)]TJ /F6 11.955 Tf 9.29 0 Td[(C1)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 228.3 -108.59 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nlogly2 l\ B4l)]TJ /F6 11.955 Tf 9.29 0 Td[(C1)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nl)]TJ /F8 8.966 Tf 6.97 0 Td[(1logly2 lnB4landhl(y)8><>:C2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 243.29 -199.81 Tm[(l)l2y2 l\ B4lC2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1ijyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nln+1y2 lnB4l. Proof. Weconsiderseparatelythecasey2 l\ B4landthecasey2 ln B4l.Case1:y2 l\ B4l.SetI1(y)=ZlG(y,h)Ql1(h)dhandI2(y)=ZlnlG(y,h)Ql1(h)dh,sohl(y)=I1(y)+I2(y).BydirectcomputationwehaveZl(jhj)]TJ /F3 11.955 Tf 1 0 .167 1 215.35 -392.08 Tm[(l)2 (1+jh)]TJ /F3 11.955 Tf 1 0 .167 1 201.72 -410.32 Tm[(lej2)(n+2)=2dhClogl,sousing( 4 )theestimateofGreen'sfunctionin( 3 ),theestimateforI1is I1(y))]TJ /F6 11.955 Tf 28.54 0 Td[(C)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1iZlG(y,h)jhj)]TJ /F3 11.955 Tf 1 0 .167 1 297.75 -482.24 Tm[(l (1+jh)]TJ /F3 11.955 Tf 1 0 .167 1 281.63 -500.47 Tm[(lej2)(n+2)=2dh)]TJ /F6 11.955 Tf 28.54 0 Td[(C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 203.55 -525.88 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nZl(jhj)]TJ /F3 11.955 Tf 1 0 .167 1 320.18 -517.78 Tm[(l)2 (1+jh)]TJ /F3 11.955 Tf 1 0 .167 1 306.56 -536.02 Tm[(lej2)(n+2)=2dh)]TJ /F6 11.955 Tf 28.54 0 Td[(C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 203.55 -553.93 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nlogl. (4) ToestimateI2,let 55

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A1=fh2l:jy)]TJ /F3 11.955 Tf 1 0 .167 1 203.9 -29.69 Tm[(hj(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 260.76 -29.69 Tm[(l)=3g,A2=fh2l:jy)]TJ /F3 11.955 Tf 1 0 .167 1 203.9 -53.59 Tm[(hj(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 260.76 -53.59 Tm[(l)=3andjhj8lg,A3=fh2l:jhj8lg,(4)anduse( 4 )towriteI2(y)3k=1Ik2(y),whereIk2(y)=)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1iZAknlG(y,h)(jhj)]TJ /F3 11.955 Tf 1 0 .167 1 241.12 -150.61 Tm[(l)jhj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(ndh,k=1,2,3.UsingLemma 3.0.1 andperformingroutineintegralestimatesusingjhj)]TJ /F3 11.955 Tf 1 0 .167 1 386.98 -180.99 Tm[(lCjy)]TJ /F3 11.955 Tf 1 0 .167 1 443.7 -180.99 Tm[(hjforI22(y)weobtainIk2(y)C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 229.29 -252.72 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nk=1,2,3.CombiningthiswiththeestimateforI1(y)givenin( 4 )andusingRlweseethatifRissufcientlylargethenhl(y))]TJ /F6 11.955 Tf 21.23 0 Td[(C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 213.09 -354.82 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nlogly2 l\ B4l.Toestimatehl(y)fory2 l\ B4l,observethattheonlynegativetermaboveisI1(y),soweonlyneedtoestimatejI1(y)j.Using( 4 )and( 3 ),wehavejI1(y)jC)]TJ /F16 8.966 Tf 6.77 4.93 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iZlG(y,h)(jhj)]TJ /F3 11.955 Tf 1 0 .167 1 213.57 -468.88 Tm[(l)dhC)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i lZA1jy)]TJ /F3 11.955 Tf 1 0 .167 1 182.16 -505.42 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ndh+ZA2(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 292.71 -497.33 Tm[(l)(jhj2)]TJ /F3 11.955 Tf 1 0 .167 1 342.46 -497.33 Tm[(l2) ljy)]TJ /F3 11.955 Tf 1 0 .167 1 318.6 -513.8 Tm[(hjn(jhj)]TJ /F3 11.955 Tf 1 0 .167 1 393.4 -505.42 Tm[(l)dh!C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 151.95 -535.28 Tm[(l)l2,wherewehaveusedjhj)]TJ /F3 11.955 Tf 1 0 .167 1 140.04 -571.14 Tm[(lCjy)]TJ /F3 11.955 Tf 1 0 .167 1 196.63 -571.14 Tm[(hjforh2A2.ThiscompletestheproofofLemma 4.3.5 inthecasey2 l\ B4l. 56

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Case2:y2 lnB4l.LetI1andI2beasinCase1sothath1=I1+I2.Using( 3 )and( 4 )wehave I1(y))]TJ /F6 11.955 Tf 28.54 0 Td[(C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iZljhj)]TJ /F3 11.955 Tf 1 0 .167 1 219.23 -63.63 Tm[(l ljyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(njhj)]TJ /F3 11.955 Tf 1 0 .167 1 326.2 -63.63 Tm[(l (1+jh)]TJ /F3 11.955 Tf 1 0 .167 1 310.09 -81.87 Tm[(lej2)(n+2)=2dh)]TJ /F6 11.955 Tf 28.54 0 Td[(C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nl)]TJ /F8 8.966 Tf 6.97 0 Td[(1logl. (4) ToestimateI2set D1=fh2l:jhj2jyjgD3=fh2l:jy)]TJ /F3 11.955 Tf 1 0 .167 1 196.54 -215.55 Tm[(hj
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Lemma 4.3.5 isestablished.2Wehavethefollowingestimatefortheboundaryderivativeofhl. Lemma4.3.6. Thetestfunctionhlsatiseshl yn(y)=(1)jyj)]TJ /F14 8.966 Tf 6.96 0 Td[(n,y20l. Proof. BydirectioncomputationwehavesnG yn(y,h)y20l=hn)]TJ /F6 11.955 Tf 10.95 0 Td[(yn jy)]TJ /F3 11.955 Tf 1 0 .167 1 189.1 -193.67 Tm[(hjn)]TJ /F18 11.955 Tf 10.95 16.86 Td[(l jyjnyl)]TJ /F3 11.955 Tf 1 0 .167 1 290.17 -185.29 Tm[(h)]TJ /F14 8.966 Tf 6.97 0 Td[(n Tijhj l2+hn!.Partitionlasin( 4 ).Thenuse( 4 )andperformstandardintegralestimatesusingyn=)]TJ /F6 11.955 Tf 9.29 0 Td[(TiandlB(0,ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i)toobtainZDkG yn(y,h)Ql1(h)dh=(1)jyj)]TJ /F14 8.966 Tf 6.97 0 Td[(n,k=1,,4.2Byconstructionofhlandsincehl0inlwehaveLi(wl+hl)(y)0fory2l.Moreover,byLemmas 4.3.3 4.3.6 weobtainBi(wl+hl)(y)0fory20l\ Ol,sohlsatises( 4 ).Thenextstepistoshowthatthemovingsphereprocesscanstart. Lemma4.3.7. Ifiissufcientlylarge,then wl0(y)+hl0(y)>0,y2l0.(4) Proof. IfR1Risxedandlarge,thenbytheconvergenceofwl0toUR)]TJ /F6 11.955 Tf 11.34 0 Td[(Ul0R,thepropertiesofUR)]TJ /F6 11.955 Tf 10.95 0 Td[(Ul0RandLemma 4.3.5 wehave(wl0+hl0)(y)>0y2l0\ BR1.Weonlyneedtoshow( 4 )fory2l0nBR1.Bydirectcomputationitiseasytoseethatthereexistse0(l0)>0suchthat 58

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Ul0R(y)(1)]TJ /F11 11.955 Tf 10.95 0 Td[(5e0)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n,jyjR1.(4)Moreover,bychoosingR1largerifnecessary,wemaysimultaneouslyachieve UR(y)1)]TJ /F3 11.955 Tf 1 0 .167 1 208.21 -105.97 Tm[(e0 2jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n,jyj=R1.(4)Asanimmediateconsequenceof( 4 )andtheconvergenceofvR,itoURwehavevl0R,i(y)(1)]TJ /F11 11.955 Tf 10.95 0 Td[(4e0)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(ny2l0nBR1.Sincehl0(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(ninl0Lemma 4.3.7 willbeestablishedonceweshowvR,i(y)>(1)]TJ /F3 11.955 Tf 1 0 .167 1 199.17 -270.46 Tm[(e0)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(ny2l0nBR1.Thiswillbeachievedviathemaximumprinciple.BytheconvergenceofvR,itoUR,inequality( 4 )and( 4 ),ifiissufcientlylargethefunctionfi(y)=vR,i(y))]TJ /F15 11.955 Tf 10.95 0 Td[((1)]TJ /F3 11.955 Tf 1 0 .167 1 264.93 -372.57 Tm[(e0)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nissuperharmonicinl0nBR1andpositiveonBR1.Moreover,by( 4 ),fi(y)Cp ijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n,y2l0\fjyj=ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(ig.Bythemaximumprinciple,iffiattainsanonpositiveminimumvalueon l0nBR1,thisvaluemustbeachievedon0l0.Weshowthatthiscannothappen.Accordingly,supposeyi20l0satises miny2 l0nBR1fi(y)=fi(yi)0.(4)Sinceyiisaminimizer,fi yn(yi)0.Ontheotherhand,using( 4 ),( 4 )andtheassumptionTi!,ifiissufcientlylargethen 59

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fi yn(yi)=ci(xi+)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))vR,i(yi)n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2)]TJ /F6 11.955 Tf 10.95 0 Td[(C(e0)Tijyij)]TJ /F14 8.966 Tf 6.97 0 Td[(nsupikcikC0( B+3))]TJ /F6 11.955 Tf 10.95 0 Td[(CTijyij)]TJ /F14 8.966 Tf 6.97 0 Td[(n<0,acontradiction.2WithLemma 4.3.7 provenwecannallyproveProposition 4.3.1 ProofofProposition 4.3.1 ByLemma 4.3.7 ,l=supfl2[l0,l1]:(wm+hm)(y)0inmforalll0mlgiswelldened.Wewillshowthatl=l1>lwhich,togetherwith( 4 )andtheestimatehl(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nforl0ll1contradictstheconvergenceofvR,itoUR.Supposel>>><>>>>:Li(wl+hl)(y)0y2lBi(wl+hl)(y)0y20l\ Ol(wl+hl)(y)=0y2l\Bl.By( 4 )andtheestimatehl(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,wehave(wl+hl)(y)>0y2l\fjyj=ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(ig.Thestrongmaximumprinciplenowensuresthat(wl+hl)(y)>0fory2l.ByHopf'slemma, 60

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n(wl+hl)(y)>0y2Bl,wherenistheouterunitnormalvectoronBl(pointingintol).Exploitingthecontinuityofl7!wl+hloncemoreweobtainacontradictiontothemaximalityofl.Proposition 4.3.1 isestablished.2 4.3.2RapidVanishingofKi(xi)InthissectionweshowthatKi(xi)vanishesquickly. Proposition4.3.2. ThereexistsaconstantC>0suchthatforalli,di)]TJ /F14 8.966 Tf 6.77 4.94 Td[(n)]TJ /F8 8.966 Tf 6.96 0 Td[(3iC.AsintheproofofProposition 4.3.1 ,theproofofProposition 4.3.2 isbycontradiction.Foreaseofnotation,set`i=d1 n)]TJ /F27 6.974 Tf 5.42 0 Td[(3i)]TJ /F14 8.966 Tf 6.78 -2.12 Td[(iandpasstoasubsequenceforwhich`i!.AsintheproofofProposition 4.3.1 weassumethatd)]TJ /F8 8.966 Tf 6.97 0 Td[(1iKi(xi)!eandconsiderthefunctionsvR,iandURaswellastheirKelvininversionsvlR,iandUlR.Withwlasbefore,theequalitiesin( 4 )arestillsatisedandweseektoconstructatestfunctionhlsuchthat( 4 )and( 4 )hold.Beforeconstructinghl,webeginwithsomeusefulestimates.Thefollowingestimateisanalogoustotheestimategiveninlemma 4.3.1 Lemma4.3.8. ThereexistpositiveconstantsC1andC2suchthatforisufcientlylarge,8><>:Hi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 21.24 0 Td[(C1)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 289.74 -564.66 Tm[(l)y2lHi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)C2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 288.41 -588.57 Tm[(l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjjy2l. Proof. TheprooffollowsroutinelyfromtheassumptionsonKandTaylor'stheorem.2 61

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ByLemmas 4.3.8 and 4.3.2 weobtainpositivel-independentconstantsa1anda2suchthatQl1(y))]TJ /F6 11.955 Tf 21.23 0 Td[(a1)]TJ /F16 8.966 Tf 6.78 4.93 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 213.03 -83.68 Tm[(l)(1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 271.7 -83.68 Tm[(lej2))]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 2y2landQl1(y)a2)]TJ /F16 8.966 Tf 6.78 4.93 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 219.68 -161.88 Tm[(l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.96 0 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(ny2l.Wearenowreadytoconstructthetestfunctionhl.Inthiscase,theconstructionofhlismoredelicatethaninSubsection 4.3.1 .Indeed,hlasdenedin( 4 )isnotbeguaranteedtobenonpositive.Thiscreatesextratermsintheinteriorequationforwl+hlthatmustbecontrolled.ToovercomethisweuseQl1toconstructafunctionbQlanddenehlbyintegratingGreen'sfunctionagainstbQl.TheadvantageofthisdenitionisthatbQlwillcontrolbothQl1andtheextratermscreatedbythepossibilityofhlbeingpositive.ToconstructbQl,rstdeneCl=fy2l\B(0,3l=2):y1>4j(y2,,yn)jgandletflbeanysmoothfunctionsatisfyingbothfl(y)=8><>:)]TJ /F14 8.966 Tf 10.48 5.29 Td[(a1 2(1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 224.82 -484.07 Tm[(lej2))]TJ /F9 8.966 Tf 6.96 0 Td[((n+2)=2y2Cl2a2n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.96 0 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(ny2lnland)]TJ /F11 11.955 Tf 10.48 8.1 Td[(3 4a1(1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 126.2 -580.28 Tm[(lej2))]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 2fl(y)3a2n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.96 0 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(ny2 lnCl.Set 62

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bQl(y)=)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 236.43 -35.86 Tm[(l)fl(y)y2l(4)andobservethatbQlenjoystheestimates bQl(y)8><>:)]TJ /F14 8.966 Tf 10.48 5.29 Td[(a1 2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 187.04 -121.04 Tm[(l)(1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 245.71 -121.04 Tm[(lej2))]TJ /F9 8.966 Tf 6.96 0 Td[((n+2)=2y2 Cl3a2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 183.68 -144.95 Tm[(l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2 lnCl(4)and bQl(y)8><>:C)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 213.21 -217.74 Tm[(l)y2l\B4lC)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 213.21 -241.65 Tm[(l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2lnB4l.(4)Moreover,wehave (Ql1)]TJ /F18 11.955 Tf 12.76 3.16 Td[(bQl)(y)8><>:)]TJ /F14 8.966 Tf 10.48 5.29 Td[(a1 4)]TJ /F16 8.966 Tf 6.77 4.85 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 204.07 -320.92 Tm[(l)(1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 262.74 -320.92 Tm[(lej2))]TJ /F9 8.966 Tf 6.96 0 Td[((n+2)=2y2 l)]TJ /F6 11.955 Tf 9.29 0 Td[(a2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 204.03 -344.83 Tm[(l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2 lnl.(4)Denehl(y)=ZlG(y,h)bQl(h)dh.Byconstruction,hlsatises 8>>>><>>>>:)]TJ /F21 11.955 Tf 9.28 0 Td[(hl(y)=bQl(y)y2lhl(y)=0y2Blhl yn(y)=RlG yn(y,h)bQl(h)dhy20l.(4)Thenextlemmaprovidesusefulestimatesforhl. 63

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Lemma4.3.9. IfRandiaresufcientlylarge,thentherearepositiveconstantsC1andC2suchthatbothhl(y)8><>:)]TJ /F6 11.955 Tf 9.29 0 Td[(C1)]TJ /F16 8.966 Tf 6.78 4.85 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 194.12 -68.34 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nlogly2 B4lC2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1y2 lnB4land hl(y)8><>:C2)]TJ /F16 8.966 Tf 6.78 4.85 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idil2(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 188.86 -177 Tm[(l)y2 l\ B4lC2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nl1+n+`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1y2 lnB4l=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n. (4) Proof. Writehl(y)=I1(y)+I2(y),with I1(y)=ZClG(y,h)bQl(h)dhandI2(y)=ZlnClG(y,h)bQl(h)dh.(4)Weconsiderseparatelythecasey2 l\ B4landthecasey2 lnB4l.Case1:y2 l\ B4l.Inthiscase,usingtheestimatesforbQlin( 4 )andtheestimatesforGin( 3 )estimatingsimilarlyto( 4 )weobtain I1(y))]TJ /F6 11.955 Tf 28.54 0 Td[(C)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 268.21 -518.39 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nlogl. (4) ToestimateI2(y),letA1,A2andA3beasin( 4 )andwriteI2(y)=3k=1Ik2(y),whereIk2(y)=ZAknClG(y,h)bQl(h)dh 64

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Performingroutineintegralestimatesusing`i!andLemma 3.0.1 yieldsIk2(y)C)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 237.93 -114.06 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nk=1,2,3.CombiningthiswiththeestimateforI1(y)givenin( 4 )andchoosingRsufcientlylargeweobtainhl(y))]TJ /F6 11.955 Tf 21.23 0 Td[(C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 218.01 -216.17 Tm[(l)l)]TJ /F8 8.966 Tf 6.97 0 Td[(1logly2 l\ B4l.Toshow( 4 )fory2 l\ B4lweonlyneedtoestimatejI1(y)j.Using( 4 )andtheestimatesforG(y,h)inLemma 3.0.1 ,wehave jI1(y)jZClG(y,h)bQl(h)dhC)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi lZA1jy)]TJ /F3 11.955 Tf 1 0 .167 1 187.49 -366.77 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ndh+ZA2(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 298.04 -358.67 Tm[(l)(jhj2)]TJ /F3 11.955 Tf 1 0 .167 1 347.79 -358.67 Tm[(l2) ljy)]TJ /F3 11.955 Tf 1 0 .167 1 323.94 -375.14 Tm[(hjn(jhj)]TJ /F3 11.955 Tf 1 0 .167 1 398.73 -366.77 Tm[(l)dh!C)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 157.29 -396.63 Tm[(l)l2. (4) Case2:y2 lnB4l.By( 4 )and( 3 )wehaveI1(y))]TJ /F6 11.955 Tf 28.54 0 Td[(C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nl)]TJ /F8 8.966 Tf 6.97 0 Td[(1logl.ToestimateI2(y),letD1,D2,D3andD4beasin( 4 )andlet 65

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Ik2(y)=RDknClG(y,h)bQl(h)dhsothatI2(y)=4k=1Ik2(y).Foreachk=1,,4weuseboth( 4 )and( 3 )toestimateIk2(y).Fork=1wehaveI12(y)C)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idiZD1G(y,h)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijhjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1)]TJ /F14 8.966 Tf 6.97 0 Td[(ndhC)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n l)]TJ /F8 8.966 Tf 6.97 0 Td[(1+`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1!.Fork=2,3,4,theintegralsIk2areminor.AfterperformingroutineintegralestimateswehaveIk2(y)C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nn)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1.k=2,3,4.CombiningtheestimatesforIk2(y),k=1,,4,weget jI2(y)jC)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n l)]TJ /F8 8.966 Tf 6.97 0 Td[(1+`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj+n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1!=(1))]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n.(4)CombiningtheestimatesforI1andI2weobtainapositiveconstantCsuchthatforRsufcientlylarge hl(y)C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n `)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1!.(4)Noticeinparticularthathl(y)neednotbenegative.Toshow( 4 ),by( 4 ),weonlyneedtoestimatejI1(y)j.By( 4 )andsinceG(y,h)Cjy)]TJ /F3 11.955 Tf 1 0 .167 1 87.1 -506.06 Tm[(hj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ninClwehavejI1(y)jZClG(y,h)bQl(h)dhC)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nl1+n.Lemma 4.3.9 isestablished.2 66

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Lemma4.3.10. Thetestfunctionsatisestheestimatehl yn(y)=(1)jyj)]TJ /F14 8.966 Tf 6.96 0 Td[(ny20l. Proof. Use( 4 ),di=(1)andjyjei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(itoobtainbQl(y)=(1))]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1ijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(1)]TJ /F14 8.966 Tf 6.97 0 Td[(nfory2l.NowproceedasintheproofofLemma 4.3.6 .2 ProofofProposition 4.3.2 By( 4 ),( 4 )andLemmas 4.3.3 and 4.3.9 ,wehaveafterincreasingRifnecessaryandforilargeLi(wl+hl)(y)=(Ql1)]TJ /F18 11.955 Tf 12.76 3.16 Td[(bQl)(y)+Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)x1(y)hl(y)0y2l\Ol.Moreover,bylemmas 4.3.3 4.3.4 4.3.9 ,and 4.3.10 weobtainBi(wl+hl)0y20l.ArguingsimilarlytotheproofofLemma 4.3.7 weseethatthemovingsphereprocesscanstartatl=l0,thenarguingsimilarlytotheproofofProposition 4.3.1 weobtainacontradictionto`i!.Proposition 4.3.2 isestablished.2 4.3.3CompletionoftheProofofTheorem 4.3.1 Witharapidvanishingratefordiinhand,wearereadytoproveTheorem 4.3.1 ProofofTheorem 4.3.1 Inthisproofweconsiderthefunctionsvi,U(notshiftedbyRe)aswellastheirKelvininversionsvli(y)=l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2vi(yl)andUl(y)=l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2U(yl)fory2l=in Bl.Inthiscase,withl=1directcomputationyields8><>:(U)]TJ /F6 11.955 Tf 10.95 0 Td[(Ul)(y)>0y2Rnn Blifll, 67

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andweconsiderlbetweenl0=1=2andl1=2.Setwl(y)=vi(y))]TJ /F6 11.955 Tf 10.95 0 Td[(vli(y)y2l.Thenwlsatisesequations( 4 )-( 4 )withR=0.Westillneedtoconstructatestfunctionhlsuchthat( 4 )and( 4 )hold.Becauseoftherapidvanishingrateofdi,theconstructionwillbesimple.ByanapplicationofTaylor'sTheorem,assumption (K1) andProposition 4.3.2 ,wehave Hi(yl))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)C)]TJ /F8 8.966 Tf 6.78 4.94 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(nijyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2y2l.(4)Sincel2,usingtheconvergenceofvlitoUlandthepropertiesofUl,wehave vli(y)Cjyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2l.(4)Usingthisand( 4 )intheexpressionofQl1wehave Ql1(y)C)]TJ /F8 8.966 Tf 6.77 4.93 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(nijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(4y2l.(4)Moreover,asinLemma 4.3.4 withR=0and1=2l2,wehave Ql2(y))]TJ /F6 11.955 Tf 21.23 0 Td[(CTijyj)]TJ /F14 8.966 Tf 6.97 0 Td[(ny20l.(4)Sethl(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F8 8.966 Tf 6.77 4.94 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(ni(l)]TJ /F8 8.966 Tf 6.97 0 Td[(1)]TJ 10.95 .05 Td[(jyj)]TJ /F8 8.966 Tf 6.97 0 Td[(1)0,y2 l,wherea>0istobedetermined.BydirectcomputationandsincelB(0,ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i),hlisseentosatisfy 68

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8>>>>>>><>>>>>>>:hl(y))]TJ /F6 11.955 Tf 21.23 0 Td[(a)]TJ /F8 8.966 Tf 6.77 4.86 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(nijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(3y2lhl(y)=0y2Blhl(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2lhl yn(y)=aTi)]TJ /F8 8.966 Tf 6.78 4.86 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(nijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(3=(1)jyj)]TJ /F14 8.966 Tf 6.97 0 Td[(ny20l.(4)Combining( 4 )and( 4 ),afterchoosingasufcientlylargeweobtainLi(wl+hl)(y)0y2l.Moreover,by( 4 ),Lemma 4.3.3 and( 4 )wehaveBi(wl+hl)(y)0,y20l\Ol,whereinthiscaseOlisasin( 4 )withR=0.ArguingsimilarlytotheproofofLemma 4.3.7 showsthatthemovingsphereprocesscanstartatl0=1=2,thenarguingasintheproofofProposition 4.3.1 yieldsacontradiction.Theorem 4.3.1 isestablished.2 4.4ProofofTheorem 4.1.1 InthissectionweprovetheHarnack-typeinequality.TheproofissimilartotheproofofTheorem 4.3.1 inthatthreeapplicationofMMSwillbeapplied;rstinSubsection 4.4.1 toshowthatKvanishesatablow-uppoint,secondinSubsection 4.4.2 toshowthatKvanishedrapidlyatablow-uppoint,andnallyinSubsection 4.4.3 tocompletetheproofofTheorem 4.1.1 .TheessentialdifferencebetweentheproofofTheorem 4.1.1 andtheproofofTheorem 4.3.1 isthatintheproofofTheorem 4.1.1 ,duetoTheorem 4.3.1 ,thecomplicationspresentedbytheboundaryequationsarenotminor.ThismakestheconstructionofthetestfunctionsmuchmoredelicateintheproofofTheorem 4.1.1 thanintheproofofTheorem 4.3.1 .TominimizethecomplicationscausedbythepresenceofB+1\Rn+,weassumethroughoutSection 4.4 thatcisconstant.Considerthefunctions 69

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vi(y)=1 Mivi(xi+)]TJ /F16 8.966 Tf 6.77 4.93 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Tien))=1 Miui(x0i+)]TJ /F16 8.966 Tf 6.78 4.93 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1iy),wherex0i=(x1,,xn)]TJ /F8 8.966 Tf 6.97 0 Td[(1,0)istheprojectionofxiontoRn)]TJ /F8 8.966 Tf 6.97 0 Td[(1.Bythetheequationsforvi,standardelliptictheory,theselectionprocessandbytheclassicationtheoremofLiandZhu[ 33 ],thereisasubsequencealongwhichbothTiconvergesandviconvergesinC2loc( Rn+)toaclassicalsolutionUof( 4 ).Lettingc0=limici,theclassicationtheoremofLiandZhu[ 33 ]gives U(y)= g g2+jy)]TJ /F6 11.955 Tf 10.95 0 Td[(t0enj2!n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2,(4)whereg=8><>:1+c20 (n)]TJ /F8 8.966 Tf 6.96 0 Td[(2)2)]TJ /F8 8.966 Tf 6.97 0 Td[(1ifc001ifc0>0andt0=gc0 n)]TJ /F11 11.955 Tf 10.95 0 Td[(2.Moreover,limiTi=8><>:0ifc00c0=(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)ifc0>0.WebeginbyderivingapreliminaryvanishingrateforKi(x0i).Forconvenience,throughoutSection 4.4 weusethenotationKi(x0i)=di. 4.4.1VanishingofKi(x0i) Proposition4.4.1. Thereexistsasubsequencealongwhichdi!0.TheproofissimilartotheproofofProposition 4.3.1 ,themajordifferencebeingthatinthiscase,atestfunctionmustbeconstructedtocontroltermsintheboundaryequation.Supposethepropositionwerefalseandletd>0satisfyinfidid>0.Byassumption (K3) weassumewithnolossofgeneralitythatd)]TJ /F8 8.966 Tf 6.97 0 Td[(1iKi(x0i)!e.ForR1xedandtobedetermined,weconsiderthefunctions 70

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vR,i(y)=vi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)=1 Miui(x0i+)]TJ /F14 8.966 Tf 6.78 -2.12 Td[(i(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))(4)whicharewell-denedin B+(0,)]TJ /F14 8.966 Tf 11.59 -2.11 Td[(i=4).Similarlyto( 4 ),wemaychooseei!0slowlysothat vR,i(y)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(np i,y2B(0,ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i)\ Rn+,(4)sointhiscaseweset i=B+(0,ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i),00i=i\Rn+and00i=in0i.(4)ElementarycomputationsshowthatvR,isatises 8><>:vR,i(y)+Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)vR,i(y)n=0y2ivR,i yn(y)=civR,i(y)n=(n)]TJ /F8 8.966 Tf 6.96 0 Td[(2)y20i,(4)whereHi(y)=Ki(x0i+)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iy).Moreover,vR,iconvergesinC2overcompactsubsetsof Rn+to UR(y)=U(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)= g g2+jy)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)]TJ /F6 11.955 Tf 10.95 0 Td[(t0enj2!n)]TJ /F27 6.974 Tf 5.42 0 Td[(2 2.(4)Forl>0letyl=l2y=jyj2andconsidertheKelvininversions UlR(y)=l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2UR(yl),andvlR,i(y)=l jyjvR,i(yl).(4)whicharewell-denedintheclosureofl=inBl.Lettingl=(g2+t20+R2)1=2,elementarycomputationsshowthat 8><>:(UR)]TJ /F6 11.955 Tf 10.95 0 Td[(UlR)(y)>0y2lifll.(4) 71

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Setl0=Randl1=R+2.Sincel0>>><>>>>:Liwl=Qly2lBiwl=0y20lwl(y)=0y2l\Bl,(4)whereLi=+Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)x1(y)Bi= yn)]TJ /F6 11.955 Tf 10.95 0 Td[(cix2(y)aretheinteriorandboundaryoperatorsrespectively, x1(y)=nZ10tvR,i(y)+(1)]TJ /F6 11.955 Tf 10.95 0 Td[(t)vlR,i(y)4 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dt(4) x2(y)=n n)]TJ /F11 11.955 Tf 10.95 0 Td[(2Z10tvR,i(y)+(1)]TJ /F6 11.955 Tf 10.95 0 Td[(t)vlR,i(y)2 n)]TJ /F27 6.974 Tf 5.42 0 Td[(2dt(4)areobtainedfromthemean-valuetheoremandQl(y)=Hi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)(vR,i(y))nisanerrortermthatwillbecontrolledwithtestfunctions.Specically,wewillconstructatestfunctionhl(y)suchthatboth hl(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n(4)and 72

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8>>>><>>>>:Li(wl+hl)(y)0y2l\OlBi(wl+hl)(y)0y20l\ Ol(wl+hl)(y)=0y2l\Bl,(4)whereOl=fy2l:(vR,i)]TJ /F6 11.955 Tf 10.95 0 Td[(vlR,i)(y)vlR,i(y)g.Thiswillallowthemaximumprincipletobeapplied.NotethatthemaximumprincipleonlyneedstoholdonOl.Thisisbecauseof( 4 );ifiissufcientlylarge,then(wl+hl)>0inlnOl.Beforeweconstructhlwerecordsomeestimatesthatwillbeusefulwhenderivingpropertiesofhlafteritisconstructed.Wedenethespecialsubsetofll=fy2l\ B2l:y1>2j(y2,,yn)jg.BytheassumptionsonKandtheconvergenceofvR,itoURwehave8><>:Hi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 21.24 0 Td[(C1)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 315.54 -388.89 Tm[(l)y2lHi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)C2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 314.21 -412.79 Tm[(l)y2l.Moreover,similarlytoLemma 4.3.2 ,therearepositiveconstantsC1,C2suchthatforlargei,bothC1jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nvlR,i(y)C2jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(ny2lnl,andC1l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2 1 1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 193.47 -599 Tm[(lej2!(n)]TJ /F8 8.966 Tf 6.97 0 Td[(2)=2vlR,i(y)2y2l.Therefore,therearepositivel-independentconstantsa1,a2suchthat 73

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Ql(y))]TJ /F6 11.955 Tf 21.23 0 Td[(a1)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 194.23 -37.13 Tm[(l) 1 1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 260.24 -47.27 Tm[(lej2!(n+2)=2y2l.(4)and Ql(y)8><>:a2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 218.64 -110.35 Tm[(l)jyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2ln( Bl[l)a2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 218.64 -134.26 Tm[(l)y2 l.(4)Finally,x1andx2stillsatisfytheconclusionsofLemma 4.3.3 .Nowweconstructthetestfunctionhlwhichwillbethesumoftwofunctionsh1andh2.Thersttestfunctionh1issimilartothetestfunctionconstructedin( 4 ).Thesecondtestfunctionh2willcontrolthebadtermson0lintroducedbyh1.LetG(y,h)beGreen'sfunctionfor)]TJ /F21 11.955 Tf 9.29 0 Td[(onRnn BlrelativetotheDirichletcondition.TheexpressionforG(y,h)isgivenin( 4 ).Lety=(y1,,yn)]TJ /F8 8.966 Tf 6.96 0 Td[(1,)]TJ /F6 11.955 Tf 9.29 0 Td[(yn)denotethereectionofyacrossRn+andset G(y,h)=G(y,h)+G(y,h).Deneh1(y)=Zl G(y,h)Ql(h)dh.Clearly,h1satisesthefollowing 8>>>><>>>>:)]TJ /F21 11.955 Tf 9.28 0 Td[(h1(y)=Ql(y)y2lh1(y)=0y2l\Blh1 yn0y20l.(4) Lemma4.4.1. ThereexistsR0sufcientlylargesuchthatifRR0thentherearepositiveconstantsC1andC2suchthat 74

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h1(y)8><>:)]TJ /F6 11.955 Tf 9.29 0 Td[(C1)]TJ /F16 8.966 Tf 6.78 4.85 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 227.38 -30.88 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nlogly2 l\ B4l)]TJ /F6 11.955 Tf 9.29 0 Td[(C1)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nl)]TJ /F8 8.966 Tf 6.97 0 Td[(1logly2 lnB4l(4)and jh1(y)j8><>:C2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 242.36 -122.11 Tm[(l)l2y2 l\ B4lC2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1ijyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nln+1y2 lnB4l.(4)TheproofofLemma 4.4.1 issimilartotheproofofLemma 4.3.5 andisomitted.Nowwedenethesecondtestfunctionh2.Letg:[l,)![0,)beasmoothpositivefunctionsatisfying g(r)=8>>>><>>>>:l)]TJ /F14 8.966 Tf 7.57 -4.98 Td[(r ln)]TJ /F14 8.966 Tf 12.14 4.71 Td[(n)]TJ /F8 8.966 Tf 6.96 0 Td[(1 2r2 l+l 2(n)]TJ /F11 11.955 Tf 10.95 0 Td[(3)lr3lsmoothpositiveconnection3lr4ll)]TJ /F8 8.966 Tf 6.97 0 Td[(1)]TJ /F6 11.955 Tf 10.95 0 Td[(r)]TJ /F8 8.966 Tf 6.96 0 Td[(14lr,(4)where`smoothpositiveconnection'meansthereisaconstantM(l)>0suchthatbothkgkC2([3l,4l])Mand g(r)1 M3lr4l.(4)Elementarycomputationsshowthat8><>:g(l)=0,g(r) rl=12g(r) r2l>0,l1forl
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Moreover,wehaveboth 2g(r) r2)]TJ /F6 11.955 Tf 12.14 8.09 Td[(n)]TJ /F11 11.955 Tf 10.95 0 Td[(1 rg(r) r=8><>:(n)]TJ /F11 11.955 Tf 10.95 0 Td[(1)(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)l)]TJ /F8 8.966 Tf 6.97 0 Td[(1l0xedbuttobedetermined(willbechosensufcientlysmallanddependingonn,,landM)deneh2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iynjyj)]TJ /F14 8.966 Tf 6.97 0 Td[(ng(jyj)y2 l.Clearly,h2<0inl,h20onl\(Bl[Rn+)and jh2(y)j8>>>><>>>>:a)]TJ /F16 8.966 Tf 6.78 4.85 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1il1)]TJ /F14 8.966 Tf 6.97 0 Td[(n(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 246.7 -337.09 Tm[(l)y2 l\ B3laM)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1il1)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2( l\ B4l)nB3la)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj1)]TJ /F14 8.966 Tf 6.96 0 Td[(nl)]TJ /F8 8.966 Tf 6.97 0 Td[(1y2 lnB4l (4) =(1)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n.Performingelementarycomputationsandusingthepropertiesofggivenin( 4 )and( 4 )weobtain h2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iynjyj)]TJ /F14 8.966 Tf 6.97 0 Td[(n 2g(jyj2) jyj)]TJ /F6 11.955 Tf 12.14 8.1 Td[(n)]TJ /F11 11.955 Tf 10.95 0 Td[(1 jyjg(jyj) jyj!8>>>><>>>>:0y2l\B3la)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iMl1)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2(l\B4l)nB3la)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2lnB4l, (4) 76

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whereadenotesaconstantoftheformC(n)a.Also,using( 4 ),( 4 )and( 4 )weobtain h2 yn(y)y20l=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj)]TJ /F14 8.966 Tf 6.96 0 Td[(ng(jyj)8>>>><>>>>:)]TJ /F11 11.955 Tf 10.34 .27 Td[(a)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1il)]TJ /F14 8.966 Tf 6.97 0 Td[(n(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 266.93 -130.39 Tm[(l)y20l\ B3l)]TJ /F11 11.955 Tf 10.34 .27 Td[(aM)]TJ /F8 8.966 Tf 6.97 0 Td[(1)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1il)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2(0l\ Bl)nB3l)]TJ /F11 11.955 Tf 10.34 .27 Td[(a)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1il)]TJ /F8 8.966 Tf 6.97 0 Td[(1jyj)]TJ /F14 8.966 Tf 6.97 0 Td[(ny20lnB4l. (4) Lethl=h1+h2.Sinceeachofh1andh2arenon-positive,using( 4 )weobtain 8>>>><>>>>:Li(wl+hl)(y)Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)x1(y)h1(y)+h2y2lBi(wl+hl)(y)cix2(y)jh1(y)j+h2 yn(y)y20l(wl+hl)(y)=0y2l\Bl.(4)Moreover,sinceHi(y)]TJ /F6 11.955 Tf 11.12 0 Td[(Re))]TJ /F8 8.966 Tf 6.96 0 Td[(1,usingLemma 4.3.3 ,equation( 4 )and( 4 )weseethata=a(M,l)maybechosensufcientlysmalltoachieveLi(wl+hl)(y)0y2l.Nowconsidertheboundaryinequalityin( 4 ).Ifci0thenBi(wl+hl)0on0lholdstriviallyash2=yn0.Weonlyneedtoconsiderthecaseci>0.By( 4 )and( 4 )thereisaconstantC(M,l)>0suchthatjh1(y)j8>>>><>>>>:C)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1i(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 232.96 -544.75 Tm[(l)y20l\ B3lC)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1iy2(0l\ Bl)nB3lC)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(ny20lnBl.Combiningthiswithlemma 4.3.3 and( 4 )weseethatthereise(n,,l,M,a)>0suchthatifc0
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Bi(wl+hl)(y)0y20l\ Ol.Thenextlemmaensuresthatthemovingsphereprocesscanstart. Lemma4.4.2. Thereexistse>0sufcientlysmallandi02Nsuchthatifc00y2l0. Proof. IfR1Risanyxedlargeconstant,theforisufcientlylarge,wl0+hl0>0inl0\BR1.ThisisbecauseofthepropertiesofUR)]TJ /F6 11.955 Tf 11.2 0 Td[(Ul0R,theconvergenceofwl0toUR)]TJ /F6 11.955 Tf 11.01 0 Td[(Ul0Randtheestimatehl0=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n.Weonlyneedtoshowpositivityofwl0+hl0onl0nBR1.Byperformingelementaryestimatesitiseasytoseethatthereexistse0(g,t0,l0)>0suchthatUl0R(y)(1)]TJ /F11 11.955 Tf 10.95 0 Td[(5e0)gn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2jy)]TJ /F6 11.955 Tf 10.94 0 Td[(enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,jyjR1.ByincreasingR1ifnecessary,wemaysimultaneouslyachieve UR(y)1)]TJ /F3 11.955 Tf 1 0 .167 1 184.86 -423.75 Tm[(e0 2gn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2jy)]TJ /F6 11.955 Tf 10.95 0 Td[(enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n,jyj=R1.(4)AsanimmediateconsequenceoftheseinequalitiesandtheconvergenceofvR,itoUR,ifiissufcientlylargewehave vl0R,i(y)(1)]TJ /F11 11.955 Tf 10.95 0 Td[(4e0)gn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2jy)]TJ /F6 11.955 Tf 10.94 0 Td[(enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2l0nBR1.(4)Nowsuppose c0<(n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)(2g))]TJ /F8 8.966 Tf 6.97 0 Td[(1(1)]TJ /F3 11.955 Tf 1 0 .167 1 262.83 -617.63 Tm[(e0))]TJ /F8 8.966 Tf 6.97 0 Td[(2=(n)]TJ /F8 8.966 Tf 6.96 0 Td[(2).(4) 78

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Weshowthatifiissufcientlylarge,then vl0R,i(y)>(1)]TJ /F3 11.955 Tf 1 0 .167 1 176.49 -59.77 Tm[(e0)gn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2jy)]TJ /F6 11.955 Tf 10.95 0 Td[(enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2l0nBR1.(4)By( 4 )andtheconvergenceofvR,itoUR,ifiissufcientlylarge,thenvR,i(y)>(1)]TJ /F3 11.955 Tf 1 0 .167 1 171.8 -137.97 Tm[(e0)gn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2jy)]TJ /F6 11.955 Tf 10.95 0 Td[(enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2l0\BR1.Therefore,fi(y)=vR,i(y))]TJ /F15 11.955 Tf 10.95 0 Td[((1)]TJ /F3 11.955 Tf 1 0 .167 1 242.25 -216.17 Tm[(e0)gn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2jy)]TJ /F6 11.955 Tf 10.95 0 Td[(enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nissuperharmonicinl0nBR1andpositiveon l0\BR1.Moreover,by( 4 ),ifiissufcientlylarge,fi(y)C(n,)p ijy)]TJ /F6 11.955 Tf 10.95 0 Td[(enj2)]TJ /F14 8.966 Tf 6.97 0 Td[(ny2l0\fjyj=ei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(ig.Bythemaximumprinciple,iffiachievesanonpositiveminimumon l0nBR1,itmustoccuron0l0nBR1.However,thisisimpossible.Indeed,supposeyi20l0nBR1satises min l0nBR1fi(y)=fi(yi)0.(4)Sinceyin=0wehave fi yn(yi)=civi(yi)n n)]TJ /F27 6.974 Tf 5.42 0 Td[(2)]TJ /F15 11.955 Tf 10.95 0 Td[((n)]TJ /F11 11.955 Tf 10.95 0 Td[(2)(1)]TJ /F3 11.955 Tf 1 0 .167 1 276.08 -524.08 Tm[(e0)gn)]TJ /F27 6.974 Tf 5.41 0 Td[(2 2jyi)]TJ /F6 11.955 Tf 10.95 0 Td[(enj)]TJ /F14 8.966 Tf 6.97 0 Td[(n.(4)Ifc00eitherci<0orbothci0andci=(1).Ifci<0thenf yn(yi)<0.If0ci=(1)thenby( 4 )and( 4 ),ifiissufcientlylargethenfi yn(yi)<0.Finally,ifc0>0theusing( 4 )oncemorealongwiththesmallnessassumption( 4 )weobtain 79

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f yn(yi)<0forisufcientlylarge.Inanycase,fi yn(yi)<0soyiisnotaminimizerforfi.2 ProofofProposition 4.4.1 WithLemma 4.4.2 proven,themovingsphereprocesscanstartatl=l0.Sincehlsatises( 4 )and( 4 ),wecanshowthatforl=supfl2[l0,l1]:wm(y)+hm(y)0inmforalll0ml1g,wehavel=l1.ThiscontradictstheconvergenceofvR,itoUR.2 4.4.2ImprovedVanishingRateforKi(x0i)Inthissubsectionwederiveafastvanishingratefordi. Proposition4.4.2. ThereexistsaconstantC>0suchthat)]TJ /F14 8.966 Tf 6.77 4.94 Td[(n)]TJ /F8 8.966 Tf 6.96 0 Td[(3idiC.TheproofofProposition 4.4.2 issimilarinspirittotheproofofProposition 4.3.2 .ThedifferenceisthattheproofofProposition 4.4.2 requiresasecondtestfunctiontocontrolanunfavorableboundarytermintroducedbythersttestfunction. Proof. AsintheproofofProposition 4.3.2 ,theproofofProposition 4.4.2 isbycontradictionandwepasstoasubsequenceforwhichboth`i!andd)]TJ /F8 8.966 Tf 6.97 0 Td[(1iKi(x0i)!e.ForR1xedandtobedetermined,letvR,ibeasin( 4 )andleti,0iand00ibeasin( 4 ).AsintheproofofProposition 4.4.1 ,vR,isatisesboth( 4 )and( 4 )andconvergestoUR(y)inC2overcompactsubsetsof Rn+,whereURisgivenby( 4 ).LettingUlRandvlR,idenotetheKelvininversionsofURandvR,iasin( 4 ),westillhave( 4 ).Weonlyconsiderlbetweenl0=Randl1=R+2.Lettingwlbeasin( 4 ),westillhave( 4 ),soweneedtoconstructhlthatsatisesboth( 4 )and( 4 ).Westartwithsomehelpfulestimates. 80

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Lemma4.4.3. ThereexistpositiveconstantsC1andC2suchthatforisufcientlylarge,bothHi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 21.24 0 Td[(C1)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 308.52 -83.68 Tm[(l)y2l.andHi(yl)]TJ /F6 11.955 Tf 10.95 0 Td[(Re))]TJ /F6 11.955 Tf 10.95 0 Td[(Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)8><>:C2)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 275.97 -168.85 Tm[(l)y2lC2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 275.97 -192.76 Tm[(l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjjy2lnl.TheproofofLemma 4.4.3 issimilartotheproofofLemma 4.3.8 andisomitted.ByLemma 4.4.3 andLemma 4.3.2 ,weobtainpositivel-independentconstantsa1anda2suchthatbothQl(y))]TJ /F6 11.955 Tf 21.23 0 Td[(a1)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 210.62 -318.45 Tm[(l)(1+jy)]TJ /F3 11.955 Tf 1 0 .167 1 269.29 -318.45 Tm[(lej2))]TJ /F25 6.974 Tf 8.16 3.53 Td[(n+2 2,y2landQl(y)a2)]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 217.27 -372.74 Tm[(l)n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=0`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.96 0 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(n,y2l.LetbQlbeasin( 4 ).Theestimatesin( 4 ),( 4 )and( 4 )arestillsatised.Deneh1(y)=Zl G(y,h)bQl(h)dh,y2l.Thenh1satises8>>>><>>>>:)]TJ /F21 11.955 Tf 9.29 0 Td[(h1(y)=bQl(y)y2lh1(y)=0y2l\Blh1 yn(y)=0y20l.AsintheproofofLemma 4.3.9 ,westillhavepositiveconstantsC1andC2suchthatboth 81

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h1(y)8><>:)]TJ /F6 11.955 Tf 9.29 0 Td[(C1)]TJ /F16 8.966 Tf 6.77 4.85 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 174.44 -42.54 Tm[(l)l)]TJ /F14 8.966 Tf 6.97 0 Td[(nlogly2 l\ B4lC2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1y2 lnB4l(4)and jh1(y)j8><>:C2)]TJ /F16 8.966 Tf 6.77 4.85 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idil2(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 187.27 -127.49 Tm[(l)y2 l\ B4lC2)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idijyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nl1+n+`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1y2 lnB4l=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n. (4) Fortheconstructionofh2,thesecondpartofthetestfunction,weconsiderseparatelythecasec0<0andthecasec00.Case1:c0<0.Inthiscaseforilargewehaveci<0.Letgi:[l,)![0,)begivenbygi(r)=8>>>><>>>>:l1)]TJ /F14 8.966 Tf 6.97 0 Td[(nrn)]TJ /F14 8.966 Tf 12.15 4.71 Td[(n)]TJ /F8 8.966 Tf 6.97 0 Td[(1 2lr2+l 2(n)]TJ /F11 11.955 Tf 10.95 0 Td[(3)lr3lsmoothpositiveconnection3lr4llogr l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`1)]TJ /F14 8.966 Tf 6.97 0 Td[(jirj)]TJ /F8 8.966 Tf 6.97 0 Td[(14lr,where`smoothpositiveconnection'meansthereisapositiveconstantM(n,,l)suchthatbothgi(r)1 Mfor3lr4landkgikC2([3l,4l])M.Byelementaryestimateswehave2gi(r) r2)]TJ /F3 11.955 Tf 1 0 .167 1 145.64 -499.65 Tm[(gi(r) r8>>>><>>>>:0lr3l)]TJ /F6 11.955 Tf 9.29 0 Td[(M3lr4l)]TJ /F6 11.955 Tf 9.29 0 Td[(Cn)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=1`1)]TJ /F14 8.966 Tf 6.97 0 Td[(jirj)]TJ /F8 8.966 Tf 6.97 0 Td[(34lr.Seth2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.93 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi`)]TJ /F8 8.966 Tf 6.97 0 Td[(1iynjyj)]TJ /F14 8.966 Tf 6.96 0 Td[(ngi(jyj)0jyjl, 82

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whereaisapositiveconstantwhichistobedetermined.Bydirectcomputationandusingthepropertiesofgiwehaveboth h2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.93 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi`)]TJ /F8 8.966 Tf 6.97 0 Td[(1iynjyj)]TJ /F14 8.966 Tf 6.96 0 Td[(n2gi(jyj) jyj)]TJ /F6 11.955 Tf 12.14 8.1 Td[(n)]TJ /F11 11.955 Tf 10.94 0 Td[(1 jyjgi(jyj) jyj8>>>><>>>>:0ljyj3laM)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi`)]TJ /F8 8.966 Tf 6.97 0 Td[(1il1)]TJ /F14 8.966 Tf 6.97 0 Td[(n3ljyj4la)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(nn)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=1`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj4ljyj, (4) andh2 yn(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ijyj)]TJ /F14 8.966 Tf 6.97 0 Td[(ngi(jyj)8><>:0y20l\ B4l)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj)]TJ /F14 8.966 Tf 6.97 0 Td[(n`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1y20lnB4l,whereadenotesaconstantoftheformC(n)a.Moreover,byelementaryestimateswehaveh2(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(n.Sethl=h1+h2.Bytheestimatesofh1andh2wehavehl(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(ninl.Itremainstoshowthathlsatises( 4 ).Clearly,wl+hlvanishesonl\Bl,soweonlyneedtoshowthedifferentialinequalitiesin( 4 ).Sinceh20andsinceeachofh2andh1 ynvanishon0l,wehave8><>:Li(wl+hl)(y)(Ql)]TJ /F18 11.955 Tf 12.76 3.16 Td[(bQl)(y)+Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)x1(y)h1(y)+h2(y)y2lBi(wl+hl)(y)=jcijx2(y)h1(y)+h2 yn(y)y20l.Fory2 l\ B3l,eachofh1andh2arenonpositivesowehavebothLi(wl+hl)(y)0fory2 l\ B3landBi(wl+hl)(y)0fory20l\ B3l.Fory2( l\ B4l)nB3l, 83

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h10.Inaddition,usingboththeestimatesofQl)]TJ /F18 11.955 Tf 13.13 3.16 Td[(bQlin( 4 )and( 4 ),since`)]TJ /F8 8.966 Tf 6.97 0 Td[(1i=(1),foranychoiceofawehaveLi(wl+hl)(y)C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idil)]TJ /F8 8.966 Tf 6.97 0 Td[(1)]TJ /F14 8.966 Tf 6.97 0 Td[(n(aM`)]TJ /F8 8.966 Tf 6.97 0 Td[(1il2)]TJ /F6 11.955 Tf 10.95 0 Td[(a2)0y2(l\ B4l)nB3lprovidediissufcientlylarge.Moreover,sinceeachofh1andh2 ynarenonpositiveforjyj4lwehaveBi(wl+hl)(y)0fory20l\ B4l.Finally,ifjyj4lwemustaccountforthepossibilitythath10.Byconstructionofh2andtheestimatesofx2andh1giveninLemma 4.3.3 and( 4 )respectively,afterchoosinga(n,)sufcientlylarge,wehaveBi(wl+hl)(y)C(jcij)]TJ /F11 11.955 Tf 11.34 .27 Td[(a))]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj)]TJ /F14 8.966 Tf 6.97 0 Td[(n `)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.97 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1!0y2(0l\ Ol)nB4l.FortheinteriorinequalitywehaveLi(wl+hl)(y))]TJ /F16 8.966 Tf 6.78 4.94 Td[()]TJ /F8 8.966 Tf 6.96 0 Td[(1idijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(n )]TJ /F11 11.955 Tf 10.34 .27 Td[(a2jyj+C1`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+ajyj)]TJ /F11 11.955 Tf 12 .27 Td[(a2jyj2+C2n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1(1+ajyj)]TJ /F11 11.955 Tf 12 .27 Td[(a2jyj2)!y2(l\Ol)nB4l.Therefore,bychoosingR=R(a,a2)largerifnecessary,wehaveLi(wl+hl)(y)0fory2(l\Ol)nB4l.Estimates( 4 )and( 4 )aresatisedinthecasec0<0.Case2:c00.Inthiscase,eitherci>0or0)]TJ /F6 11.955 Tf 21.23 0 Td[(ci=(1).Forthiscasewesetgi(r)=8>>>><>>>>:l1)]TJ /F14 8.966 Tf 6.97 0 Td[(nrn)]TJ /F14 8.966 Tf 12.15 4.71 Td[(n)]TJ /F8 8.966 Tf 6.97 0 Td[(1 2lr2+l 2(n)]TJ /F11 11.955 Tf 10.95 0 Td[(3)lr3lsmoothpositiveconnection3lr4ll1+n+`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(14lr, 84

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where`smoothpositiveconnection'meansthereisani-independentconstantM(l)>0suchthatbothgi(r)M)]TJ /F8 8.966 Tf 6.96 0 Td[(1for3lr4landkgikC2([3l,4l])M.Sincegi(l)=0,gi(3l)=Cland2gi(r) r2>0forlr3l,thereisaconstantC>0suchthatgi(r)C(r)]TJ /F3 11.955 Tf 1 0 .167 1 32.45 -83.68 Tm[(l)forlr3l.Moreover,bydirectcomputationandelementaryestimateswehave2gi(r) r2>)]TJ /F6 11.955 Tf 10.48 8.09 Td[(n)]TJ /F11 11.955 Tf 10.95 0 Td[(1 rgi(r) r8>>>><>>>>:0lr3l)]TJ /F6 11.955 Tf 9.29 0 Td[(M3lr4l)]TJ /F6 11.955 Tf 9.29 0 Td[(Cn)]TJ /F8 8.966 Tf 6.97 0 Td[(3j=1`)]TJ /F14 8.966 Tf 6.97 0 Td[(jirj)]TJ /F8 8.966 Tf 6.97 0 Td[(34lr.Nowseth2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idiynjyj)]TJ /F14 8.966 Tf 6.97 0 Td[(ngi(jyj)0jyjl.Bydirectcomputationandelementaryestimateswehavebothh2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.93 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idiynjyj)]TJ /F14 8.966 Tf 6.97 0 Td[(n 2gi(jyj) jyj2)]TJ /F6 11.955 Tf 12.15 8.1 Td[(n)]TJ /F11 11.955 Tf 10.95 0 Td[(1 jyjgi(jyj)jyj!8>>>><>>>>:0ljyj3laM)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idil1)]TJ /F14 8.966 Tf 6.97 0 Td[(n3ljyj4la)]TJ /F16 8.966 Tf 6.78 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(nn)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=1`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj4ljyjand h2 yn(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj)]TJ /F14 8.966 Tf 6.97 0 Td[(ngi(jyj) (4) 8>>>><>>>>:)]TJ /F11 11.955 Tf 10.34 .27 Td[(a)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idil)]TJ /F14 8.966 Tf 6.97 0 Td[(n(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 176.52 -561.23 Tm[(l)y2`l\ B3l)]TJ /F11 11.955 Tf 10.34 .27 Td[(a)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idil)]TJ /F14 8.966 Tf 6.97 0 Td[(nM)]TJ /F8 8.966 Tf 6.97 0 Td[(1y2(0l\ B4l)nB3l)]TJ /F11 11.955 Tf 10.34 .27 Td[(a)]TJ /F16 8.966 Tf 6.77 4.86 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idijyj)]TJ /F14 8.966 Tf 6.97 0 Td[(nl1+n+`)]TJ /F8 8.966 Tf 6.97 0 Td[(1ilogjyj l+n)]TJ /F8 8.966 Tf 6.96 0 Td[(3j=2`)]TJ /F14 8.966 Tf 6.96 0 Td[(jijyjj)]TJ /F8 8.966 Tf 6.97 0 Td[(1y20lnB4l,whereadenotesaconstantoftheformCa. 85

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Sethl=h1+h2.Thenhl(y)=(1)jyj2)]TJ /F14 8.966 Tf 6.97 0 Td[(nandhl=0onl\Bl.Weneedtoshowthatthedifferentialinequalitiesin( 4 )holdsoweconsider8><>:Li(wl+hl)(y)(Ql)]TJ /F18 11.955 Tf 12.76 3.16 Td[(bQl)(y)+Hi(y)]TJ /F6 11.955 Tf 10.95 0 Td[(Re)x1(y)h1(y)+h2y2lBi(wl+hl)(y)jcijx2(y)jh1(y)j+h2 yn(y)y20l.Fory2 l\ B3lbothofh1andh2arenonpositive,soLi(wl+hl)(y)0onthisset.Moreover,inviewof( 4 )and( 4 ),onceaischosenwemaychoosee>0dependingonn,,l,asuchthatBi(wl+hl)(y)C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 215.43 -242.53 Tm[(l)(jcij)]TJ /F3 11.955 Tf 1 0 .167 1 260.98 -242.53 Tm[(l)]TJ /F14 8.966 Tf 6.97 0 Td[(na)0y20l\ B3lwheneverci
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Bi(wl+hl)(y)C)]TJ /F16 8.966 Tf 6.77 4.94 Td[()]TJ /F8 8.966 Tf 6.97 0 Td[(1idi(jcij)]TJ /F11 11.955 Tf 12 .27 Td[(a)jyj)]TJ /F14 8.966 Tf 6.96 0 Td[(ngi(jyj)0y2(0l\Ol)nB4l.Wehaveshownthathlsatises( 4 )whenc00.ArguingasintheproofofLemma 4.4.2 showsthatthemovingsphereprocesscanstartatl=l0.ThenarguingasintheproofofProposition 4.4.1 weobtainacontradictiontotheconvergenceofvR,itoUR.Proposition 4.4.2 isestablished.2 4.4.3CompletionoftheProofofTheorem 4.1.1 Witharapidvanishingratefordiinhand,analapplicationofthemethodofmovingsphereswillprovetheorem 4.1.1 .Therapidvanishingrateofdimakestheconstructionofthetestfunctionsimple. ProofofTheorem 4.1.1 weconsidervi,UandtheirKelvininversionsvli(y)=l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2vi(yl)andUl(y)=l jyjn)]TJ /F8 8.966 Tf 6.97 0 Td[(2U(yl)fory2l=in Bl.Inthiscase,withl=1directcomputationyields8><>:(U)]TJ /F6 11.955 Tf 10.95 0 Td[(Ul)(y)>0y2Rnn Blifll,andweconsiderlbetweenl0=1=2andl1=2.Setwl(y)=vi(y))]TJ /F6 11.955 Tf 10.95 0 Td[(vli(y)y2l.Thenwlsatisesequations( 4 )-( 4 )withR=0.Westillneedtoconstructatestfunctionhlsuchthat( 4 )and( 4 )hold.Notethat( 4 )stillholds.ByProposition 4.4.2 and( 4 )wehave Ql(y)C)]TJ /F8 8.966 Tf 6.77 4.94 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(nijyj)]TJ /F8 8.966 Tf 6.97 0 Td[(4y2l.(4) 87

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Leth1(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a1)]TJ /F8 8.966 Tf 6.77 4.93 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(niln+2(l)]TJ /F8 8.966 Tf 6.97 0 Td[(1)]TJ 10.95 .05 Td[(jyj)]TJ /F8 8.966 Tf 6.97 0 Td[(1)jyjl,wherea1isapositiveconstantwhichistobedetermined.Routinecomputationsshowthath1satises 8>>>><>>>>:h1)]TJ /F6 11.955 Tf 21.24 0 Td[(a1)]TJ /F8 8.966 Tf 6.77 4.86 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(niln+2jyj)]TJ /F8 8.966 Tf 6.96 0 Td[(3y2lh1 yn=0y2l\Rn+h1(y)=0y2l\Bl.(4)Moreover, jh1(y)j8><>:a1)]TJ /F8 8.966 Tf 6.78 4.86 Td[(2)]TJ /F14 8.966 Tf 6.97 0 Td[(niln(jyj)]TJ /F3 11.955 Tf 1 0 .167 1 253.05 -245.74 Tm[(l)l>>><>>>>:r)]TJ /F3 11.955 Tf 1 0 .167 1 182.22 -353.53 Tm[(l+n)]TJ /F8 8.966 Tf 6.97 0 Td[(1 2l(r)]TJ /F3 11.955 Tf 1 0 .167 1 244.45 -353.53 Tm[(l)2l0suchthatbothg(r)1 M3lr4landkgkC2([l,))M.Deneh2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a2)]TJ /F8 8.966 Tf 6.77 4.94 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(niynjyj)]TJ /F8 8.966 Tf 6.97 0 Td[(2g(jyj)y2l, 88

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wherea2isapositiveconstanttobedetermined.Routinecomputationsyield 8>>>>>>><>>>>>>>:h2(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a2)]TJ /F8 8.966 Tf 6.77 4.86 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(niynjyj)]TJ /F8 8.966 Tf 6.97 0 Td[(22g(jyj) jyj2+n)]TJ /F8 8.966 Tf 6.97 0 Td[(3 jyjg(jyj) jyj)]TJ /F8 8.966 Tf 12.14 5.95 Td[(2(n)]TJ /F8 8.966 Tf 6.96 0 Td[(2) jyj2g(jyj) jyjy2RnnBlh2 yn(y)=)]TJ /F6 11.955 Tf 9.29 0 Td[(a2)]TJ /F8 8.966 Tf 6.77 4.86 Td[(2)]TJ /F14 8.966 Tf 6.96 0 Td[(niy)]TJ /F8 8.966 Tf 6.96 0 Td[(2g(y)y2Rn+nBlh2(y)=0y2Bl[Rn+jh2(y)j=(1)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(nljyjei)]TJ /F14 8.966 Tf 6.77 -2.11 Td[(i(4)Moreover,jh2(y)jC2ui(xi))]TJ /F8 8.966 Tf 6.97 0 Td[(2jyj)]TJ /F8 8.966 Tf 6.96 0 Td[(1g(jyj)C2Mui(xi))]TJ /F8 8.966 Tf 6.96 0 Td[(2jyj)]TJ /F8 8.966 Tf 6.96 0 Td[(1=(1)jyj2)]TJ /F14 8.966 Tf 6.96 0 Td[(n.Sethl(y)=h1(y)+h2(y).Sinceeachofh1andh2arenonpositiveinl,using( 4 ),( 4 )and( 4 ),weseethata1maybechosensufcientlylargeanddependingona2suchthatLi(wl+hl)(y)0inl.Now,ifci0,thenBi(wl+hl)0in0l\ B4lholdstrivially.Ifci>0,thenusingtheestimatesforjh1jandjh2jalongwithlemma 4.3.3 and( 4 ),weseethatthereis0
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uiswell-approximatedbyastandardbubble,themajorityofwhoseenergyisinthisneighborhood.ThekeyinformationrevealedbytheHarnack-typeinequalityisthatthedistancebetweenthelocalmaximizersofuisnottoosmall.Duetothelocalnatureoftheequationsconsideredinthischapter,theapproachincontrollingthisdistancebetweenmaximizersofuisslightlydifferentthantheapproachusedin[ 27 ]sowementionitnow.Forthelocalequations,itisnotpossibletondtwolocalmaximizersofuthataremutuallyclosesttoeachother.Eachlocalmaximizercertainlyhasasecondmaximizerwhichisclosesttoit,buttheremaybeathirdlocalmaximizerwhosedistancetothesecondlocalmaximizerissmallerthanthedistancefromtherstlocalmaximizertothesecondlocalmaximizer.Toovercomethisdifculty,rescaletheequationsothatthedistancefromtherstlocalmaximizertothenearestlocalmaximizerisone.TheHarnack-typeinequalityforcesthevaluesofuatthesetwolocalmaximumpointstobecomparable.Thecomparabilityofthesetwomaximumvaluesensuresthatnotwobubblescantendtothesameblow-uppoint.Indeed,iftwobubblestendtothesameblow-uppoint,thenaharmonicfunctionwithpositivesecond-ordertermcanbeconstructed.ThisfunctionwillgiveacontradictioninthePohozaevidentity.Withthedistancebetweenlocalmaximizersofucontrolled,onecanusestandardelliptictheorytoshowthatnearalargelocalmaximum,ubehaveslikearapidlydecayingharmonicfunction.ThisbehavioryieldstheenergyestimateinCorollary 4.1.1 90

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BIOGRAPHICALSKETCH YingGuowasborninChina.Shereceivedherbachelor'sdegreeinpureandappliedmathematicsfromNankaiUniversityin2004.There,shecontinuedhereducationandearnedhermaster'sdegreeinappliedmathematicsin2007.From2007to2013,YingstudiedingraduateschoolattheUniversityofFlorida,whichconferredamaster'sdegreeinindustrialandsystemsengineeringandaDoctorofPhilosophydegreeinmathematics. 94