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1、InternationalJournalofBifurcationandChaos,Vol.11,No.3(2001)801{820cWorldScienti cPublishingCompanyNEWTON'SMETHODANDMORSEINDEXFORSEMILINEARELLIPTICPDESJOHNM.NEUBERGERandJAMESW.SWIFTyDepartmentofMathematicsandStatistics,NorthernArizonaUniversityP.O.Box5717,Flagsta ,AZ86011-5717,USAReceivedApril20,
2、2000;RevisedJune1,2000InthispaperweprimarilyconsiderthefamilyofellipticPDEsu+f(u)=0onthesquareregionΩ=(0;1)(0;1)withzeroDirichletboundarycondition.Followingourpreviousanal-ysisandnumericalapproximationswhichreliedonthevariationalcharacterizationofsolutionsascriticalpointsofanction"functional,we
3、considerNewton'smethodonthegradientofthatfunctional.WeuseaGalerkinexpansion,ineigenfunctionsoftheLaplacian,to ndsolutionsofarbitraryMorseindex.Takingf0(0)tobeabifurcationparameter,weanalyzethebifurcationsfromthetrivialsolution,u0,usingsymmetryargumentsandournumericalalgorithm.TheMorseindexofthea
4、pproximatedsolutionsisprovidedandsupportisfoundconcerningseveralexistenceandnodalstructureconjectures.Wediscusstheapplicabilityofthismethodto ndcriticalpointsoffunctionalsingeneral.1.TheSemilinearProblemofcourseorthogonalinboththeSobolevspace1;2H=H0(Ω)andinL2,withinnerproductsTheprimaryequationwe
5、considerisasuperlinearZZellipticzeroDirichletboundaryvalueproblemonapiecewisesmoothboundedregionΩRN.Inthishu;viH=rurvdxandhu;vi2=uvdxΩΩpaper'snumericalinvestigationsweletN=2andΩ=(0;1)(0;1),butthemethodcanbeappliedrespectively(see[Adams,1975;Gilbarg&tootherregions.LetbetheLaplacianoperator.Tru
6、dinger,1983],or[Neuberger,1997a,1997b]).WeseeksolutionstotheboundaryvalueproblemForthePDE(1)onasquareregion,Ω=(0;1)(0;1),itiswellknownthatthe(doubly(u+f(u)=0inΩindexed)eigenvaluesandeigenfunctionsof−are(1)u=0on@Ω:222m;n=(m+n)andRecallthattheeigenvaluesof−withzerom;n=2sin(mx)sin(ny);(2)Dir
7、ichletboundaryconditioninΩsatisfywheremandnrangeoverallpositiveintegers.One0<1<23!1:ofthemaingoalsinthispaperistodemonstratetheimportanceoftheseeigenfunctionstothetheoryofWedesignatethecorrespondingeigenfunctio
