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SINGULAR LIMIT SOLUTIONS FOR TWO-DIMENSIONAL ELLIPTIC PROBLEMS WITH EXPONENTIALLY DOMINATED NONLINEARITY

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SINGULAR LIMIT SOLUTIONS FOR TWO-DIMENSIONAL ELLIPTIC PROBLEMS WITH EXPONENTIALLY DOMINATED NONLINEARITYfor,with

    SINGULAR LIMIT SOLUTIONS FOR

    TWO-DIMENSIONAL ELLIPTIC PROBLEMS WITH EXPONENTIALLY

    DOMINATED NONLINEARITY

Chin.Ann.olMath.

    ;22B:3(2001),287-296

    ;SINGULARLIMITSoLUTIoNSFoR

    ;TWo.DIMENSIoNALELLIPTIC

    ;PRoBLEMSWITHEXPoNENTIALIY

    ;DoMINATEDNoNLINEARITY

    ;S.BARAKETYEDONG

    ;Abstract

    ;Theauthorsconsidertheexistenceofsingularlimitsolutionsforafamilyofno

    nlinear

    ;ellipticproblemswithexponentiallydominatednonlinearityandDirichletbo

    undarycondition

    ;andgenerMizetheresultsofI3].

    ;KeywordsAsymptoticbehaviour,Exponentiallydominatednonlinearity,Si

    ngular

    ;perturbation,WeightedHSlderspace

;2000MRSubjectClassification35J65,35B25,35C20

    ;ChineseLibraryClassificationO175.25DocumentCodeA

    ;ArticleID0252?-9599(2001)03?-0287--10

    ;?1.Intr0ducti0n

    ;Thepurposeofthispaperistoconsidertheexistenceofsolutions:QCCRforthe ;followingDirichletproblem:

    ;J,Au=p2/(u)=p2(e”+e’”)inQcC,1:0onOf/,(1.1)

    ;wherey?

    (0,1).Ourmotivationistostudytheexistenceofnon-minimalsolutionswith ;singularlimitastheparameterPtendsto0andextendtheresultsof3,2

    tomoregeneral

    ;functionswhicharejustexponentiallydominated.Theadditionalterme’yieldsthepossi-

    ;bilityofbettersteadystatemodelsforphysicalphenomenahavingexponentialnon——li——nearities

    ;(seeforexample1and61).

    ;Theasymptoticbehaviourofsolutionsoff1.1)iSwellunderstoodthankstothework

    ;ofNagasakiandSuzuki[(fory<1/4)andarecentworkin

    10].TheGreen’sfunction

    ;G(z,zt),definedoverQ×Q,isgiventobetheuniquesolutionof ;fAG(z,z)=87r8:,inQ,

;G(z,z)=0onaQ,

    ;andH(z,Z)=G(z,Z)+4loglZ—ZldenotestheregularpartofGreen’Sfunction.

    ;ManuscriptreceivedMarch15,2000.RevisedSeptember26,2000. ;,Faeult~desSciencesdeTunis,DdpartementdeMathdmatiques,CampusUniversitaire,1060Tunis,

    ;Tunisie.E-mail.Sami.Baraket@fst.rnu.tn

    ;**DdpartementdeMathdmatiques,siteSaint-Martin,Universit6deCergyPontoise,2avenueAdolphe

    ;Chauvin,95302Cergy-Pontoisecedex,France.E-marl:dong.ye~math.U-cergy,fr

    ;

    ;288CHIN.ANN.0FMATHVl01.22Ser.B

    ;Theorem1.1.,.Letbedregularboundeddomaino/C,?(0,1)andP>

    ;beasequenceofsolutionsof(1.1).Assumethat,asPtendsto0,thesequencep ;tosomenontrivialfunctionin(Q).Then,thelimitfunctionsatisfies ;f-Au=87r?,inQ,1+

    ;:0.aQ.

    ;Inaddition,epoint(zl,…,Zk)?Qisacriticalpointofthefunction

    ;:(z?,…,zk)?C_?H(zj,)+?G(,z1).

    ;J:1j?l

    ;0.Letp

    ;contJees

;(1.2)

    ;(1.3)

    ;Inthispaper,wedealwiththeconversequestion:given(Zl,…,Zk)?

    Qacriticalpoint

    ;ofthefunctiondefinedin(1.3)andgivenuthesolutionof(1.2),doesthereexistafamily

    ;up,solutionsof(1,1),whichconvergestothefunctionasPtendsto07Thiskindof

    ;problemwasconsideredbymanyauthorsinsomespecialcases(seeforinstance[5,9]and

    ;[8).Recently,in[3,BaraketandPacardhaveconstructedafamilypwhichconvergesto

    ;whenPtendsto0,forf(u)=eonageneraldomainQ.Lateron,in[2]thisresult ;wasextendedtothecase,()=e+e.Ywith?(0,7/8),butthemethodseemsnotto

    ;workfor?

    (7/8,1).Here,wewilluseageneralconstructiontosolvetheproblemforall ;3/4<1.Ourmainresultreads:

    ;Theorem1.2.LetQbearegularboundeddomaino/C.Let7?(0,1)and(Zl,…,z)

    ?

    ;Qbeanondegeneratecriticalpointo/thefunction

    definedin(1.3).Then,thereexistsa

    ;oneparameter,nmilyo/solutionspof(1.1),whichconvergesto,solutiono/(

1.2),when

    ;Ptendsto0.

    ;Ourproofisbasedonsomerefinementsofargumentsin[3].Ourpaperisorganized

    ;asfollows.Wewillrecallsomenotationsandresultsof[3]in?2,andweconstructour

    ;approximatesolutionsin?3,whereasharpestimateonapproximatesolutionsisestablished.

    ;Finally,thenonlinearproblemissolvedin?4.Giventhefactthattheproofofourresult

    ;israthertechnical,weshallrestrictourattentiontoonepointblowupsolutio

    ns.Thecase

    ;wheretheremightbemanyblowuppointscanbetreatedcompletelysimilarly,thoughthe

    ;computationsshouldbemoreinvolved.Inthefollowing,weassumethatk=1,y?[3/4,1)

    ;andcdenotesalwaysaconstantindependentofP,evenitsvaluecouldbechangedfromone

    ;linetoanotherone.

    ;?2.KnownResultsin3andRefinements

    ;Forthesakeofcompleteness,werecallsomeusefulnotationsandresultsin3J:Forany

;E,>0and?

    C,defineEtobethesmallestpositivesolutionofP.=8e./(1+g2). ;(clearly,P0(E)whenP-?0).Note

    ;UE,

    ;TZ)=2log(1+E)2log(e+7-2Izl)+2log7-,

    ;(z)=2log(1+E)2log(e+7-2IzlI1+zI)+2log7-+2logI1+3zI. ;Weknowthatue,r(resp.ue,r,)aresolutionsofAu+pe=0onC(resp.c\{z,1+3/3z=

    ;0)).DefineLe,randLe

    asthefollowinglinearizedoperatorsaboute,randUE~T,:

    ;Le,

    ;.r=?p2eu,

    ;,

    ;Le,

    ;.r,=?p2eu.,,.

    ;

    ;No.3S.BARAKET&YE,D.SINGULARLIMITSOLUTIONSFORELLIPTICPROBLEMS289

    ;Tounderstandtheinversionoftheseoperators,weintroducesomeweightedHSlderspaces

    ;asin

    3.Let5={2{)1{beafinitesubsetofQ.Wechooseapositivefunctiond(z),

    ;smoothinQ\suchthatd(z)=llforzsufficientlyclosetoziandset ;[k,a,[a,2a]---(l)+asupd(y(3-2o-(IYd(z),)?【,,zl

    ;Definition2.1.LetQbearegularboundeddomain.C.Forany??andot?

    (0,1),

    ;afinitesetofsingularitiesinQ,thespace,.(Q\)isdefinedtobethecollectionofal

    l

    ;functions?Ck,a(Q\)forwhichthen0

    ;ullk,.,supo.-vlul,a,tr,2口】

    ;~diamn

    ;isbounded.Moreover,define(Q\)=(?(Q\s),=o0non}.

    ;Inallthispaper,wedenotebyBr(z)theballofradiusrcenteredatz,Brwhenthe

    ;centeristheorigin0,and=Br\{0).ThepropertiesofLe,randLe,r,/3aredescrib

    ed

    ;bythefollowingpropositions:

    ;Proposition2.1..1Forall?(1,2)andall

    >0,thereexisttwocontinuouslinear

    ;yoms(?)(resp.,(?))defined#omco

    ;,a[

    ;,B*)into(resp.C)suchthatyorall

    ;,?(B),thesolutionof

    ;Le.

    ;rw

;w=0

    ;B1,

    ;0nOB1

    ;canbeuniquelydecomposedas

    ;(z)=G(,)((.(z)+2??(

    ;(2.1)

    ;whe.(z)=,l(z)=nndz.2,_(z+z’)/2,fora,z?C.Inndditi

    ;thefollowingpropertieshold:

    ;?Assumethat1<<2,thenthelinearoperatorGe, ;riswelldefined#omthespace

    ;(B)intothespace’a(B)andstaysboundedindependentlyofs?(0,1).

    ;?Assumethat-2<<2,thentherestrictionofGe,rtothespaceoffunctionss

    panned

    ;bye’”nif)In>1)iswelldefined.mthespacepO,

    a2\(Izt1*,~intothespacep2,a(B).

    ;?Assumethat>0,thenthelinearfom(?)iswelldefinedin(B)and

    ;boundedindependentlyofs?(0,7_/2).

    ;?Assumethat>1,thenthelinearfom,(?)iswelldefinedin(B)and

    ;boundedindependentlyofs?(0,1).

    ;Proposition2.2-I3JForall?(1,2),all7_>0andall?

    CwithII<1/4,thereexist

    ;so>0andtwocontinuouslinearIoms(?)(他印.H1,

;(?))definedm()

    ;into(resp.csuchthatyoranys?(0,80)and,?(B),thesolutionof ;canbeuniquelydecomposeda8

    ;LE卢叫=,inBI,

    ;=0onOB1

    ;(z)=G}1(,)(z)+Zo,,(,)a,,(z)+2,,Af)?,,.

    ;(2.2)

    ;\,?/

    ;

    ;290CHIN.ANN.oFMATHVo1.22Ser.B ;Inaddition,thereexistsc>0(independentoffandE<go)suchthat

    ;G(,)llz,.,c(1la,,(f)l12,.,+g.21,(,)l+l,,(,)1), ;IH.o,

    ;,,(,)Ic(.IIa,,(f)ll2,.,+IH.o,,(,)f+g2I,,(,)I),

    ;l(,)lc(IIa,,(f)l12,.,+1,(,)l+l,,(,)1).

    ;Moreover

    ;lIG(,)laBIl.,.c(1lO,,G,,(f)IoBll?,.+IIa,,(f)l12,.,+g.21,(,)l+l, ;,(,)I).

    ;Weneedhoweversomerefinementsforprovingourresults.Wedefinethesubs

    paceof

    ;evenfunctionsinc,.(Bt)by

    ;,

;.(B)={,?c’.(B);suchthat/(z)=,(z),Vz?B).(2.3)

    ;Clearly,’a(Bt)isanalgebraforany>0.Furthermore,weCallgetmoreprecise

    ;estimationsofsolutionsof(2.2)whenfiseven,inparticularfortheLoonormliwil~.The

    ;reasonisthatwedonothavetermslikef1(r)e”intheexpansionoff,sotheoperatorsL

    ;orL,areinvertibleon(B)forallin(0,2),insteadof?(1,2)aLsPropositions

    ;2.1and2.2required.Moreprecisely,wehave

    ;Proposition2.3.Assumethat?(0,2),1->0andll1/4aregiven.Thenthere ;existE0>0andcontinuouslinearyomr,

    ;(?),definedmq_2(B1)intoR,such

    ;thatforanyf?(B)andE?

    (0,to),thesolutionof(2.2)canbeuniquelydecomposed

    ;c’sw(z):GE,r,(,)(z)+Ho,

    ;r,(f)OruE,r,(z).Moreover,thereexistsc>0suchthat

    ;lia,(,)Il2.a,6+lo,,(,)lcli/ll0,a’6

    ;SketchofProof.First,weshowthecorrespondingresultforLE, ;r,easilyobtainedby

    ;Proposition2.1.WefollowtheproofofProposition2.2in[3.Instep1,weget

    afunction

    ;?a(B)suchthatL=fandw[oB1isconstantinR.Weprovethentheorthogonal

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