A perturbation method for multiple sign-changing solutions

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1、Calc.Var.(2010)37:8798DOI10.1007/s00526-009-0253-2CalculusofVariationsAperturbationmethodformultiplesign-changingsolutionsN.Hirano·W.ZouReceived:15December2008/Accepted:5May2009/Publishedonline:10June2009©Springer-Verlag2009AbstractWedevelopaperturbationmethodfornon-evenfunctionalswhichproducesp

2、rescribednumberofsign-changingsolutions.TheabstracttheoryisappliedtotheperturbedsubcriticalellipticequationandtheperturbedBrézisNirenbergcriticalexponentproblem.MathematicsSubjectClassification(2000)35J50·58E051IntroductionRecallthefollowingproblem:−
u=

3、u

4、p−2u+f(x,u),u∈H1(),(1.1)0whereisabounde

5、dsmoothdomainofRN(N≥3),2

6、msisoftenreferredtoasperturbationfromsymmetryproblemsandthesymmetryofthecorrespondingfunctionalisbrokencompletely.AlongstandingopenCommunicatedbyA.Malchiodi.N.HiranowassupportedinpartbyYokohamaIndus.Soc.andW.ZouwassupportedbyNSFC(10871109)andYokohamaIndus.Soc.N.HiranoGraduateSchoolofEnvironmenta

7、ndInformationScience,YokohamaNationalUniversity,Tokiwadai,Yokohama,Japane-mail:hirano@math.sci.ynu.ac.jpW.Zou(B)DepartmentofMathematicalSciences,TsinghuaUniversity,100084Beijing,Chinae-mail:wzou@math.tsinghua.edu.cn12388N.Hirano,W.Zouquestionwhicheventodayisnotadequatelysettledis:whetherthesymme

8、tryofthefunc-tionaliscrucialfortheexistenceofinfinitelymanycriticalpoints(cf.Rabinowitz[23,24]andStruwe[29,p.118]).Severalpartialanswershadbeenobtainedinthepast30years.Letussketchthehistory.Thespecialcase−
u=

9、u

10、p−2u+f(x),u∈H1(),(1.2)0wasfirststudiedbyBahriandBerestycki[3]Struwe[28]independently.I

11、nBahri[2],theauthorconsidered(1.2)andprovedthatthereisanopendensesetoffinW−1,2()suchthat(1.2)hasinfinitelymanysolutionsifp<2N/(N−2).InRabinowitz[23,24],theauthorconsideredthegeneralproblem−
u=g(x,u)+f(x,u),u∈H1(),(1.3)0wher