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1、UniquenessofPositiveRadialSolutionsofAu+f(u)=0inR"KEVlNIV[CLEOD&JAMESSERRIN1.IntroductionInthispaperwediscusstheuniquenessofsmoothsolutionsu(r)ofthegroundstateproblemn--1u"+u'+f(u)=O,r>0r(GS)u'(O)=O,u(r)-+0asr-+oou(r)>0forr~O.Heren>Iisaconstantandf(u)isanassi
2、gnedfunctiondeterminingtheparti-cularformoftheproblem;byasmoothsolution,wemeanafunctionofclassCl[0,eo)AC2(0,oo).Wemakethefollowingstandingassumptionsonthefunctionf(u):(i)fEC1[0,~);f(0)=0,f'(0)=--m<:0,(ii)thereexistsaconstant0~:>0suchthatf(u)<0foruE(0,00,f(u)>
3、0foruE(0roo),(iii)f'(~)>0.Theaboveproblemarisesnaturallyinthestudyofpositiveclassicalsolutions(groundstates)oftheproblemAuq-f(u)=OinRn(I)u(x)~OasIxl-+~~Indeed,itwasprovedbyGIDAS,Nt&NmENBERG[11]thatanypositivesolutionof(I)mustberadialwithrespecttosomeoriginofc
4、o-ordinatesXo(andmono-tonicallydecreasingasonemovesoutwardfromXo),atleastiffisassumedtobeofclassC1+aonsomeinterval[0,y),y>0.Thus,solutionsof(I)canbetreatedassolutionsof(GS),withrdenotingtheradialvariableandnthedimension.116K.McLEOD&J.SERRINThephysicallyimport
5、antcasesare,ofcourse,n=1,2,3,4;whenn----1,both(GS)and(I)areeasilysolvedbyquadrature,andhenceforsimplicitywehaveomitteddiscussionofthiscase.Thefactthatsolutionsof(I)mustberadialdoesnotitselfguaranteeuniqueness,for,apartfromtranslations,thequestionofwhetherorno
6、tmorethanoneradialfunctioncansatisfy(GS)isclearlyunan-sweredby[11].OurmethodsalsoapplytopositiveradialsolutionsoftheDirichletproblemAu-bf(u)-=OinB(0,R)u=0on0B(0,R)andtheexteriorNeumannproblemAu+f(u)=0inIxl~>R0uOn0,Ix[Ru-+0asIx[-->oo,seesection8.Inpractice,the
7、equationAtt-}-f(u)=0arisesinthestudyofphasetransi-tionsofvanderWaalsfluids[4,19,23],inpopulationgenetics[7,9]andinnuclearphysics.Inthelattercase,TAKAHASm[22]obtainedapairofcoupledpartialdifferentialequationsthatSYNGE[21]laterreducedto(I)withf(u)=--u+u2.Simila
8、rly,FINKELSTEINetal.[8]obtained(I)withf(u)=--u-t-ua.Insection3weconsiderthecasef(u)-~--u+upindetail,asanillustrationofourconclusions.Quitegeneralconditionsonfthatensuretheexistenceofposit
