The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function英文文献资料

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1、ARTICLEINPRESSJ.DifferentialEquations193(2003)481–499TheNeharimanifoldforasemilinearellipticequationwithasign-changingweightfunction1K.J.BrownandYanpingZhangDepartmentofMathematics,Heriot-WattUniversity,Riccarton,EdinburghEH147HQ,UKReceivedSeptember27,200

2、2Abstractn1TheNeharimanifoldfortheequationDuðxÞ¼laðxÞuðxÞþbðxÞjuðxÞjuðxÞforxAOtogetherwithDirichletboundaryconditionsisinvestigated.ExploitingtherelationshipbetweentheNeharimanifoldandfibreringmaps(i.e.,mapsoftheformt-JðtuÞwhereJistheEulerfunctionalassoci

3、atedwiththeequation)wediscusshowtheNeharimanifoldchangesaslchangesandshowhowexistenceandnon-existenceresultsforpositivesolutionsoftheequationarelinkedtopropertiesofthemanifold.r2003ElsevierScience(USA).Allrightsreserved.Keywords:Variationalmethods;Neharima

4、nifold;Indefiniteweightfunctions1.IntroductionConsiderthesemilinearboundaryvalueproblemn1DuðxÞ¼laðxÞuðxÞþbðxÞjuðxÞjuðxÞforxAO;ð1:1ÞuðxÞ¼0forxA@O;ð1:2ÞwhereOisaboundedregionwithsmoothboundaryinRN;l40isarealparameter,1onoNþ2anda;b:O%-Raresmoothfunctionswhic

5、hchangesigninO%:N2Correspondingauthor.Fax:131-451-3249.E-mailaddress:k.j.brown@ma.hw.ac.uk(K.J.Brown).1SupportedbyaJamesWattScholarshipfromHeriot-WattUniversity.0022-0396/03/$-seefrontmatterr2003ElsevierScience(USA).Allrightsreserved.doi:10.1016/S0022-03

6、96(03)00121-9ARTICLEINPRESS482K.J.Brown,Y.Zhang/J.DifferentialEquations193(2003)481–499SimilarproblemshavebeenstudiedbyBindingetal.[3,4]byusingvariationalmethodsandbyAmannandLopez-Gomez[1]byusingglobalbifurcationtheory.Theexistenceandmultiplicityresultscan

7、besummarizedasfollows.Letl1ðaÞdenotethepositiveprincipaleigenvalueoftheproblemDuðxÞ¼laðxÞuðxÞforxAO;uðxÞ¼0forxA@Owithcorrespondingpositiveprincipaleigenfunctionf1:Itisknownthat(i)thereexistsapositivesolutionto(1.1)–(1.2)whenever0olol1ðaÞ;R(ii)ifbfnþ1dxo0;

8、thereexistsd40suchthat(1.1)–(1.2)hasatleasttwopositiveO1solutionswheneverl1ðaÞolol1ðaÞþd:Results(i)and(ii)canbeunderstoodintermsofglobalbifurcationtheoryastheRnþ1signofObf1dxdeterminesthedirectionofbifurcatio