On a geometric equation with critical nonlinearity

arXiv:math/0106219v4[math.AP]22May2002OnageometricequationwithcriticalnonlinearityontheboundaryVeronicaFelliandMohamedenOuldAhmedouAbstractAtheoremofEscobarassertsthat,onapositivethreedimensionalsmoothcompactRiemannianmanifoldwithboundarywhichisnotconformallyequivalenttothestandardthreedimensionalball,anecessaryandsufficientconditionforaC2func-tionHtobethemeancurvatureofsomeconformalscalarflatmetricisthatHispositivesomewhere.Weshowthat,whentheboundaryisumbilicandthefunctionHispositiveeverywhere,allsuchmetricsstayinacompactsetwithrespecttotheC2normandthetotaldegreeofallsolutionsisequalto−1.MSCclassification:35J60,53C21,58G30.1IntroductionIn[14],Jos´eF.Escobarraisedthefollowingquestion:givenacompactRiemannianmanifoldwithboundary,whenisitconformallyequivalenttoonethathaszeroscalarcurvatureandwhoseboundaryhasaconstantmeancurvature?Thisproblemcanbeseenasa“generalization”tohigherdimensionsofthewellknownRiemannianMappingTheorem.Thelaterstatesthatanopen,simplyconnectedpropersubsetoftheplaneisconformallydiffeomorphictothedisk.Inhigherdimensionsfewregionsareconformallydiffeomorphictotheball.Howeveronecanstillaskwhetheradomainisconformaltoamanifoldthatresemblestheballintwoways:namely,ithaszeroscalarcurvatureanditsboundaryhasconstantmeancurvature.TheaboveproblemisequivalenttofindingasmoothpositivesolutionutothefollowingnonlinearboundaryvalueproblemonaRiemannianmanifoldwithboundary(Mn,g),n≥3:−Δgu+(n−2)4(n−1)Rgu=0,u0,in˚M,∂u∂ν+n−22hgu=cunn−2,on∂M,(P)whereRgisthescalarcurvatureofM,hgisthemeancurvatureof∂M,νistheouternormalvectorwithrespecttog,andcisaconstantwhosesignisuniquelydeterminedbytheconformalstructure.12V.FelliandM.OuldAhmedouForalmostallmanifolds,Escobar[14,16]establishedthat(P)hasasolution.Morerecentlyin[29]thisproblemhasbeenstudiedusingthetoolsofthecriticalpointsatinfinityofA.Bahri[2],seealsoBahri-Coron[4]andBahri-Brezis[3].Goingbeyondtheexistenceresultsoftheabovepaper,weprovedrecentlyin[18]that,when(M,g)islocallyconformallyflatwithumbilicboundarybutnotconformaltothestandardball,allsolutionsof(P)stayinacompactsetwithrespecttotheC2normandthetotaldegreeofallsolutionsisequalto−1.Theheartoftheproofoftheaboveresultissomefineanalysisofpossibleblow-upbehaviourofsolutionsto(P).Morespecificallyweobtainedenergyindependentestimatesofsolutionsto(Lgu=0,u0,in˚M,Bgu=(n−2)uq,on∂M,where11+ε0≤q≤nn−2,Lg=Δg−n−24(n−1)Rg,Bg=∂∂νg+n−22hg.Insteadoflookingforconformalmetricswithzeroscalarcurvatureandconstantmeancurvatureasin(P),onemayalsolookforscalarflatconformalmetricswithboundarymeancurvaturebeingagivenfunctionH;thisproblemisequivalenttofindingasmoothpositivesolutionuto(Lgu=0,u0,in˚M,Bgu=Hunn−2,on∂M.(PH)Suchaproblemwasstudiedin[16]byEscobar,whoprovedthatonapositivethreedimensionalsmoothcompactRiemannianmanifoldwhichisnotconformallyequivalenttothestandardthreedimensionalball,anecessaryandsufficientconditionforaC2functionHtobethemeancurvatureofsomeconformalflatmetricisthatHispositivesomewhere.WerecallthatamanifoldiscalledofpositivetypeifthequadraticpartoftheEulerfunctionalassociatedto(P)ispositivedefinite.Inourworkweassumethattheboundaryisumbilic,thatisthetracelesspartofthesecondfundamentalformvanishesontheboundary.MoreoverweassumethatthefunctionHispositive.Ourfirsttheoremgivesaprioriestimatesofsolutionsof(PH,q)inH1(M)norm.Theorem1.1Let(M,g)beathreedimensionalsmoothcompactRiemannianmanifoldwithumbilicboundary.Thenforallε00kukH1(M)≤C∀u∈[1+εo≤q≤3MH,q,whereCdependsonlyonM,g,ε0,kHkC2(∂M),andthepositivelowerboundofH.Geometricequationwithcriticalnonlinearity3OurnexttheoremstatesthatforanypositiveC2functionH,allsuchmetricsstayboundedwithrespecttotheC2normandthetotalLeray-Schauderdegreeofallthesolutionsof(PH)is−1.Infactweestablishaslightlystrongercompactnessresult.Considerfor1q≤3theproblem(Lgu=0,u0,in˚M,Bgu=Huq,on∂M.(PH,q)WeuseMH,qtodenotethesetofsolutionsofPH,qinC2(M).Wehavethefollowingtheorem.Theorem1.2Let(M,g)beapositivethreedimensionalsmoothcompactRiemannianmanifoldwithumbilicboundarywhichisnotconformallyequivalenttothestandardthreedimensionalball.Then,forany1q≤3andpositivefunctionH∈C2(∂M),thereexistssomeconstantCdependingonlyonM,g,kHkC2,thepositivelowerboundofHandq−1suchthat1C≤u≤CandkukC2(M)≤Cforallsolutionsuof(PH,q).Moreoverthetotaldegreeofallsolutionsof(PH,q)is−1.Consequently,equation(PH,3)hasatleastonesolution.Weremarkthatthehypothesisthat(M,g)isnotconformallyequivalenttothestandardthreedimensionalballisnecessarysince(PH)mayhavenosolutioninthiscaseduetotheKazdan-Warner’sconditionsforsolvability.OntheballsufficientconditionsonHindimensions3and4aregivenin[13]and[17],andperturbativeresultswereobtainedin[9].Finally,letuspointoutthatrecentlyS.Brendle[7,8]obtainedonsurfacessomeresultsrelatedtoours.Heusedcurvatureflowsmethods,inthespiritofM.Struwe[32]andX.X.Chen[10].ThecurvatureflowmethodwasintroducedbyR.Hamilton[20],andusedbyB.Chow[11],R.Ye[33],andBartz-Struwe-Ye[6].Theremainderofthepaperisorganizedasfollows.Insection2weprovidethemainlocalblow-upanalysisgivingfirstsharppointwiseestimatestoasequenceofsolutionsnearisolatedsimpleblow-uppoints,thenwepro