一个无界区域中的奇摄动非线性椭园型系统

1、一个无界区域中的奇摄动非线性椭园型系统第22卷2期1999年6月安徽师范大学(自然科学版)JournalofAnhuiNormalUniversity(NaturalScience)V01.22.No.2Jun.I999文章号;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)点,vR,2>0,whereeisapositiveparameter,=(I.z,)R.Theauthorstudiedaclassofsingularlyperturbedboundaryvalueproblemsfortheellipticequationsin3

4、"-6.Thispaperinvolvessingularlyper.turbedprobleminthehalfspace.Assumethat1thecoefficientsof工areboundedsmoothfunctionsin爱兰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兰(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收辅日期I10驰儿一26作者前介c羹毫琪(1937一).男.教授正+22卷第2期莫嘉琪:一个无界区域中的奇摄动非线性椭园型系统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:丛型,P:',(4)whereh(x.'_r.)isafunctiontobedetermined.Forconvenience,westillsubstituteforPbelow.From(4),wehave工一古K.+÷K1+Kz,(5)whereK.一丢,.鑫+摹哦鑫+)善,一丢+善.-in去+茎+叠+蓦毒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=争.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安徽师范大学(自然科学版)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,芦,i=1,2,N,一Cr,一u._2'.?,届'.?,_.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)(岛一肘)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

16、K一.口口U+抻U十U一一三夏(卜22卷第2期莫裹琪,一个无界区域中的奇摄动非线性椭园型系统103REFERENCES1NsehAH.IntroductiontopenurbItl啪techniquesEM3.JohnWileySons?NewYoukl19812PsoCv.NonlinearellipticsystemsinunbouoddomalnsEJ3.NoninesrAnalysis,Theory,Mhads8LApplicatio?1994?22I139】1407MoJiaqi,ZhangXisng.$ingulm'lyperturbedbndaryvalueprobLem

17、forhigherorderqsiinearellipticequationsEJJ.ActsMathApplSinica.1997.2Ol5'l8514MoJiaqi.xuYuxing.Thesingularlyperturbedbouodar/valueproteinforhigherordersemilinearellipticequationsEJ3?ActaMathSCi,1997,17"805MoJiaqi.Theaingulsdypaurbednonlinearellipticsystemsinunboundeddomstirm|.ApplMathJUC?199

18、6?l】()l1531586M.Jiaqi.Zh"EHanlin.ThesingularlypartarhdpMemsfornottfocalsemilinesrellipticequatiormJ.ActsMsthApplSinics?1998?21,l71473f00一l一个无界区域中的奇摄动非线性椭园型系统莫嘉琪(安-癖花大革聂覃iIi'l翻100o)017.27(=)J,2摘要;甘论了在半空间Rt,-中的非线性椭固型系统的奇援动迫值l,q题.在适当的紊件下,利用比较定理,对相应边值问题解的存在性和渐近性态作了研究.美羹饲:苎璺型至基奇异援动些整墨墨中田分类号;O175

19、.29文献标识码:A于摇动.兜隧3Ij8cj8cj8cj8c(上接第99页)31Ya丑窖XM,SCUmAM,FuP_F,ets1.Exo-methylenefuuc-tionslizedpolyethylenesring-openzletpo1)aeriza?ductcontrolinorganolanthanlde-c?Iv_edpolymerization/eopolymerlzationJ.Macromoleeules.1994,27I4625462692YssudsH,lhsraESynthesis0|monodisparsepoLyolofinaEJ.KobnshiR

20、onblInshi.1994.51l63764633】oI.n蛔nLK.KiI1l衄CM.BookhsrtM.NewPd(ID-andNi(I1)-hssedtalysts叫polymerization0|ehylenesnd-olefinsJ.JAmChemSoc,1995.117l6414641534】ohp.sonLK.MeckInBS?IookhartM.Coymerizatlonofeth,Iefandpropylenewithfunonalizedvinylmonomersbypalladium(1I)catalystsr'J.】AmChemSoc,1996.118l267Z6835DemingTJtNovakBM?ZillerJW.LivingpolymertlonofhutieaeatbothchainendsvJsabimetallicnickelinitiator.Preparationofhydroxyteleehelicpoly(hutadiene)sodsymmetticpoly'isocyanlde-b-butsdiene-hisocysnlde)elsstomerlctrlblockeopolyrnersEJ.】AmChemSoc.1994.116: