Concentrating solutions for the H&amp#233;non equation in R2.pdf

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1、CONCENTRATINGSOLUTIONSFORTHEHENONEQUATIONIN´
ByPIERPAOLOESPOSITO,ANGELAPISTOIAANDJUNCHENGWEIAbstract.Weconsidertheboundaryvalueproblem,,intheunitballwithhomogeneousDirichletboundaryconditionandalargeexponent.Wefindaconditionwhichensurestheexistenc

2、eofapositivesolutionconcentratingoutsidetheoriginatsymmetricpointsasgoesto.Thesametechniquesleadalsotoamoregeneralresultongeneraldomains.Inparticular,wefindthatconcentrationattheoriginisalwayspossible,provided
.1IntroductionandstatementofmainresultsInthispape

3、r,weconsiderthefollowingso-calledHenonequation([16])´in(1.1)inonwhere,isaboundeddomainin
()containingtheorigin,and.Problem(1.1)hasattractedconsiderableattentioninrecentyears.In[18],Niforshowedtheexistenceofaradiallysymm

4、etricsolutionwhenand.When
,numericalcomputationsbyChen,NiandZhou[8]suggestthatforsomeparameters,thegroundstatesolutionsarenonradial.ThiswaspartiallyconfirmedrecentlybySmets,SuandWilliamin[24],inwhichitwasprovedthatforeachif;i

5、f),thereexistssuchthatforthegroundstatesarenonradial.ThefirstauthorissupportedbyM.U.R.S.T.,project“Variationalmethodsandnonlineardifferentialequations”andaPIMSPostdoctoralFellowship.ThesecondauthorissupportedbyM.U.R.S.T.,project“Metodivariazionalietopologici

6、nellostudiodifenomeninonlineari.”ThethirdauthorissupportedbyanEarmarkedGrantfromRGCofHK.249JOURNALD’ANALYSEMATHEMATIQUE,Vol.100(2006)´250P.ESPOSITO,A.PISTOIAANDJ.WEITheyalsoshowedthatforfixed,thegroundstatesolutionmustberadialifiscloseto1.When,theasymptoticbehavi

7、orof(radialornonradial)groundstatesolutionsasisstudiedbyByeonandWangin[3,4],inwhichitisshownthatthegroundstatesolutiondevelopboundaryconcentrations.In,andanotherdirection,when,isfixed,,CaoandPeng[5]showedthatthegroundstatesolutiondevelopsaboundary

8、bubble(hencemustbenonradial).In[19]and[20],multipleboundaryconcentrationshavebeenandconstructedwhen.Inthispa