1、一個無界區域中的奇攝動非線性橢園型系統第22卷2期1999年6月安徽師范大學(自然科學版)JournalofAnhuiNormalUniversity(NaturalScience)V01.22.No.2Jun.I999文章號;1001-2443(1999)02O100-04ASINGULARLYPERTURBEDNONLINEARELLIPTICSYSTEMlNUNBOUNDEDDOMAINM0Jiaqi(Department0fMathematics.AnhulNormalUnlverslD",241000Wuhu,Anhui,China)Abstract:Thesingular

2、lyperturbedproblemsforthenonlinearellipticsystemsinthehalfspaceRareconsidered.Undersuitableconditions,usingthecomparisontheoremtheexistenceandasymptoticbehaviorOfsolutionfortheboundaryvalueproblemsarestudied.Keywords:ellipticsystem;singularperturbation;comparisontheoremCLCnumber:0l75.29Documentcode:

3、AWeconsiderthefollowingsingularlyperturbedprobleminthehalfspace尼;Iz>0E2L"zf2,1,"2.,N,e),i一1.2,.毋I,'一1).以=0,i一1,2.,Where(1)(Z)工:未矗+c考,4(z)點,vR,2>0,whereeisapositiveparameter,=(I.z,)R.Theauthorstudiedaclassofsingularlyperturbedboundaryvalueproblemsfortheellipticequationsin3

4、"-6.Thispaperinvolvessingularlyper.turbedprobleminthehalfspace.Assumethat1thecoefficientsof工areboundedsmoothfunctionsin愛蘭f0)2】,gandtheirderivativesunitlmthorderareboundedfunctionswithregardtotheirvariablesincorrespondencerangesH3thereexistpositiveconstants,suchthatexp(),i=1,2,N;Wenowconstructth

5、eformalasymptoticsolutionoftheproblem(1)-(2).Thereducedproblemof(1)(2)is,"l,如.,N,0)=0,心,ii,2,N.(*)WealsoassumethatH'thereexistsasolutionUD蘭(ulD,Uzo,UNo)ofsystern(*),andUI¨anditsderiratiresareboundedfunctionsin冠.IetformalexpansionsoftheOutersolutionU;(U1,UU)fortheoriginalproblem(1)(2)be

6、,i=1.2,N.(3)Substituting(3)into(1),developingfine?equatingcoefficientsoflikepowersofrespectively.weobtain收輔日期I10馳兒一26作者前介c羹毫琪(1937一).男.教授正+22卷第2期莫嘉琪:一個無界區域中的奇攝動非線性橢園型系統101.(_r,UU2.,+U.,0)U+F,一,1(2)一0+i1,2,+;,1+2,whereF,aredeterminedfunctionsofU,+r一1,andtheirconstructionsareomitted.Theaboveandbel0w+t

7、hevalueoftermsforthenagetivesubscriptarezero.Fromabovelinearsystem+wecansolveUlJsuccessively.Form(3),weobtaintheoutersolutionU一(UI+U,U)fortheoriginalproblem.Butitmaynotsatisfiestheboundarycondition(2).sothatweneedconstructtheboundarylayertermV=(Vt,y2,+y).Weleadintothevariablesofmultiplescalesinthene

8、ighborhood=0:】r:叢型,P:',(4)whereh(x.'_r.)isafunctiontobedetermined.Forconvenience,westillsubstituteforPbelow.From(4),wehave工一古K.+÷K1+Kz,(5)whereK.一丟,.鑫+摹哦鑫+)善,一丟+善.-in去+莖+疊+驀毒Letthesolutionoforiginalproblem(1)(2)be"一(ul,"2+,un):,E)一U(,e)+V(r,1,',E),i=1,2+,N.Substituting(6)i

9、nto(1)(2),wehaveLV=,Ul+VI,U2+V2,u+y,e)一(_z,u1,u2+,uN,e),i=1,2+,V.=g(.r1,_r一1)一Ui,rP='=0+i一1,2,And1et(6)(7)(8),oi,i=1,2,.(9)J=oSubstituting(9)into(7),(8)andconsidering(5),expandingnonlineartermsine,dequatingthecoefficientsoflikepowersofe,weobtainK0仉o;(r,_r1,.27Ul.+1口,U20+,0+c,0)一(r,_r1,_r,UU拍'

10、;.?,U舯,O),i=1,2,一g(l,'1)一Ur=P:.=0,i=1,2,ArK.一(r,',ul.+lo,U+2o,u+VNOO)V?,1=一K1.ul1一K2(一2】+GiJ,i=1,2,?J=1,2,仉J=一U,t-=P一_r.:0,i=1,2,一1,2,whereG,j,(=1,2,J=1,2,)aresuccessivelydeterminedfunctions,omittedtOO.Let(1O)(11)(12)(13)theirc0nstructi0nsare"一一=爭.d_From(1O)(11),wecanhavegroupofsolution

11、so,=1,2,N,whichpossessboundarylayerbehavior.Andfrom(1Z)(13),wecaoalsoobtain仉_+i一1,2,+Jl,2+successively+whichpossessboundarylayerbehaviortOO.102安徽師范大學(自然科學版)1999Thenwecanconstructthetollowingformalasymptoticexpsnsionsofthesolution=(1.,)/ortheoriginalproblem(1)(2):(u+)e,0<e1.i一1,2,N.(14)Theorem

12、UnderthehypothesesH1H4.Thenthereexistsstleastonesolution=(l,)ofthesingularlyperturbedproblem(1)(2)fortheellipticsystemandtheunitormlyvalidaymptoticexpansions(14)forein冠holds.ProofI七t=Y嘲一.rexp(一t)¨,i=1,2,一+.rexp(一如)+1,i一1,2,wherersrelargeenoughpositiveconstantswhichwillbedecidedbelow,while-+2一U+

13、,i=1.2,NJ=oJoObviously,且.Fromhypotheses,itiseasytoseethatforlargeenoughgi(x",'一1)且,.r=0,i一1,2,.Nowweprovethatz一Cr,_1,u2,.'.?,_,e)0,R,蘆,i=1,2,N,一Cr,一u._2'.?,屆'.?,_.e)0,.r墨.,i=1,2,Intact.thereexistpositiveconstantsM1,suchthatELai一(,u1,_2,e)(15)(16)(17)(18)(19):E土y.一exp(一Sx)g"

14、+1一(,_1,.'一.r一exp(一如).,E】Ly一fl(3r,五1,.,y,.N,e)+(,五l,u2,Y.用,_J.)一(.r,u1,u2,Y山一.r一exp(一)¨,e)一肘+_一fAz,UU'.?,u,o)+jm一:一,UU,u,O)U一Pq+K0.一(r,.rI?-,u1o+口U2.+口,uNo+7Jt0,O)+far,.r1,.r,UU舶.,u,9)+?+,?)rlzexp(一r)一(riMi1+lVl,2)(島一肘)一肘)+1,Selectingr2>M2/forEsufficientlysmall,thenwehaveprovedth

15、einequality(18).Analogously,wecanprovetheinequality(19).Thusfrom(17)(19)andE22,thereestsatleastonesolution=(1,z,.N)anditholdsthat,.r露,0<e1.From(15),(16),weobtain_md-2峨一u,+0().0<1.JDJDTheproofofthetheoremiscompleted.NoteThebriefreportofthispaperispublishedbyJ.Math.Res.Exposition.口K一口K.U

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