On the positive solutions of the Matukuma equation

ONTHEPOSITIVESOLUTIONSOFMATUKUMAEQUATIONYiLix1.Introduction.In1930,basedonhisphysicalintuition,T.Matukamapro-posedthefollowingequationasamathematicalmodeltodescribethedynamicsofglobularclusterofstars([M]),u+11+jxj2up=0inR3;(1.1)wherep1andu0isthegravitationalpotentialwithRR3up4(1+jxj2)dxrepre-sentingthetotalmass.Hisaimwastoimproveamodelgivenearlierin1915byA.S.Eddington.(See[NY1,2]foramoredetailedhistoryofthesetwomodels.)SinceMatukumaequation(1.1)isrotationallyinvariant,thestructureofpositiveradialsolutionsu(r;)ofthecorrespondinginitialvalueproblemurr+2rur+11+r2up=0in[0;1);u(0)=0;ur(0)=0;(1.2)wasrststudiedbyMatukuma.Hethenconjecturedthat(i)ifp3,thenu(r;)hasanitezeroforevery0,(ii)ifp=3,thenu(r;)isapositiveentiresolutionwithnitetotalmassforevery0,(iii)ifp3,thenu(r;)isapositiveentiresolutionwithinnitetotalmassforevery0.In1938,Matukumafoundaninterestingexactsolutionu(r;p3)=p3=(1+r2)for(1.2)withp=3whichconrmspartofhisconjecture.SincethenthereseemstobeverylittlemathematicalcontributionintheliteratureonthisequationuntiltherecentworksofW.-M.NiandS.Yotsutani[NY1,2],Y.LiandW.-M.Ni[LN2],andE.S.NoussairandC.A.Swanson[NS].First,itwasobservedin[NY2]and[LN2]thatEddington’smodeldoesnothaveanypositiveentiresolutions(whichperhapsindicatesthatMatukumaequationisindeedabetterphysicalmodel).ConcerningMatukuma’sconjecture,thefollowingresultswereestablishedby[NY2]and[LN1,2]whichshowsthatequation(1.2)isperhapsmoredelicatethanMatukumahadexpected.MathematicsSubjectClassication.Primary35J60.Secdary35B40..ResearchsupportedinpartbytheNationalScienceFoundation..TypesetbyAMS-TEX2TheoremA.Letu=u(r;)bethesolutionof(1.2).(i)If1p5,thenu(r;)hasanitezeroforeverysucientlylarge0.(ii)If1p5,thenthereexistsan0suchthatthesolutionu(r;)ispositivein[0;1)withnitetotalmass.(iii)If1p5,thenu(r;)isapositiveentiresolutionwithinnitetotalmassforeverysucientlysmall0.(iv)Ifp5,thenu(r;)isapositiveentiresolutionwithinnitetotalmassforevery0.Furthermore,forrnear1,(cr1u(r)c1r1ifuisin(ii),(fastdecay),c(logr)1p1u(r)c1(logr)1p1ifuisin(iii)or(iv),(slowdecay),(1.3)wherecissomepositiveconstant.Remark1.1.Thedividingexponentp=5istheso-calledSobolevcriticalpowern+2n2whenn=3.TheoremAgivesanearlycompletedescriptiononthestructureofpositivera-dialsolutionsofequation(1.2).Ontheotherhand,itisaninterestingandnaturalmathematicalquestionthatwhether(1.1)possessesonlypositiveradialentireso-lutions.In[LN2,3,4]wesettlethisproblemconcerningnitetotalmasssolutions.Attemptingtoapplythemethodin[GNN2],oneimmediatelyencountersthefactthatfundamentaltool-(Lemma2.1in[GNN2,p.375]),nolongerholdswhenpiscloseto1.Ourkeynewideaistoobtainpreciseasymptoticexpansionsofsolutionsat1whichturnsouttobesucienttogetthe\movingplaneprocessstartednear1.This\moving-planemethodwasrstdevisedbyA.D.Alexandroin1956andsincethenhasbeenusedbymanymathematicians.(See,e.g.[BN],[CGS],[CL1,2],[GNN1,2],[H],[KKL],[Li],[L2]and[S].)Theresultsin[LN2,3,4]yieldthefollowing.TheoremB.(i)Let1p5.Theneveryboundedpositiveentiresolutionofequation(1.1)withnitetotalmassisradiallysymmetricabouttheoriginandur0inr0.Furthermore,8:u(x)=Cjxjn2+cjxjn2+++cjxjn2+(2k+1)+cjxjn1++++1jxjn1+k+01jxjnnear1;(1.4)3where=(p1)(n2),kistheintegerthatk1(k+1),andC0andcaregenericconstants.(ii)Letp5.Theneveryboundedpositiveentiresolutionof(1.1)hasinnitytotalmass.OneofthekeyingredientsintheproofofTheoremBisadetailedanalysisoftheasymptoticbehaviorofnitetotalmasssolutionsat1whichgetsthemoving-planedevicestartnear1.(See[LN2;Lemma2.3],[LN3;Theorem2.8],and[LN4;page2]),e.g.,oneoftheestimatein[LN2]impliesthateveryboundedpositiveentiresolutionu(x)of(1.1)withnitetotalmassmustbeboundedabovebyc=jxjat1forsomeconstantC0.However,theradialsymmetryofpositivesolutionswithinnitetotalmassof(1.1)isleftopenin[LN2,3,4]duetotheslowdecaypropertyofsuchsolutions(see(1.3)).Themainpurposeofthispaperistosettlethiscasefor1p5.Theorem1.Let1p5.Theneverypositiveentiresolutionuofequation(1.1)isradiallysymmetricabouttheoriginandur0inr0.Nowtounderstandthestructuresofallpositivesolutionsof(1.1)isequivalenttounderstandthestructuresofsuchsolutionsof(1.2),andforwhichwehaveTheorem2.Let1p5andu(r;)bethesolutionof(1.2).Thenthereexistsaunique0,suchthat(i)ifu(x)isapositiveentirenitetotalmasssolutionof(1.1),thenu(x)=u(jxj;)anducanbeexpandedaccordingto(1.4)at1.(ii)ifu(x)isapositiveentiresolutionof(1.1)withinnitetotalmass,thenthereexistsan2(0;)suchthatu(x)=u(jxj;)=u(r)andu(x)=C1(logjxj)1p1pC1(p1)2(n2)log(logr)(logr)pp1+01(logr)pp1!(1.5)at1.4Remark1.2.TheuniquenessofinTheorem2isgivenby[Y].(Seealso[KL]and[KYY]forvariousextensionsofresultsof[Y]),whiletheexpansion(1.5)isderivedby[L1].ThecrucialingredientsoftheproofofTheorem1arethefollowings:rst,theasymptoticbehaviorofpositivesolutionsuof(1.1)isinvestigated;second,furnishedbythestudyoftheirasymptoticbehaviorthemaximumprincipleisobservedtobeapplicableat1totheoperatorL=+K(x)ifK(x)jxj2at1tostartthemoving-planeprocessforsolutionuaslongastheissu