EXISTENCE AND UNIQUENESS OF SINGULAR SOLUTION TO

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1、DISCRETEANDCONTINUOUSdoi:10.3934/dcds.2013.33.4613DYNAMICALSYSTEMSVolume33,Number10,October2013pp.4613{4626EXISTENCEANDUNIQUENESSOFSINGULARSOLUTIONTOSTATIONARYSCHRODINGEREQUATIONWITH•SUPERCRITICALNONLINEARITYFouadHadjSelemLaboratoiredeMathematiquesUMR6620-CNRSUniversiteBlaisePas

2、cal,CampusdesCezeaux-B.P.8002663171Aubierecedex,FranceHiroakiKikuchiandJunchengWeiDepartmentofMathematics,TsudaCollege2-1-1Tsuda-machi,Kodaira-shi,Tokyo187-8577,JapanandDepartmentofMathematics,TheChineseUniversityofHongKongShatin,Hong-Kong,China(CommunicatedbyManueldelPino)Abstr

3、act.Inthispaper,westudyasingularsolutiontothefollowingellipticequations:8<
u+jxj2uujujp1u=0;x2Rd;u(x)>0;x2Rd;:u(x)!0asjxj!1;whered3;>0andp>1.InthespiritofMerleandPeletier[9],weshallshowthatincaseofp>(d+2)=(d2),thereisauniquevalue=suchthattheequationwith=hasauniquerad

4、ialsingularsolution.1.Introduction.Inthispaper,weconsiderthefollowingsemilinearellipticequa-tions:8><
u+jxj2uujujp1u=0;x2Rd;(1)u(x)>0;x2Rd;(2)>:u(x)!0asjxj!1;(3)whered3;>0andp>1.Wecaneasilyseethattheequations(1){(3)hasnonontrivialsolutionfor1,where>0bethersteigenvalueo

5、ftheoperator
+jxj2,thatis,=d.11Moreover,incaseofp21,wealsondthatthereisnonontrivialsolutionto(1){(3)byusingthePohozaevidentity,where2=2d=(d2)(see[4]indetail).WeremarkthatfromtheresultofLiandNi[8],anysolutionto(1){(3)becomesradiallysymmetric.Thus,theequations(1){(3)canbered

6、ucedtoanordinarydierentialequations(see(14)below).2010MathematicsSubjectClassication.Primary:35J15,35J61,35J91;Secondary:58F19.Keywordsandphrases.Sobolevsupercritical,singularsolution,harmonicpotential.ThesecondauthorissupportedbytheGrant-inAidforResearchActivityStart-up#2384003

7、7ofJSPS.ThethirdauthorissupportedbyaGeneralResearchGrantentitledTravelingWaveSolutionsandParabolicDeGiorgiConjecture".46134614FOUADHADJSELEM,HIROAKIKIKUCHIANDJUNCHENGWEIHiroseandOhta[5,6]showedthatforeach>1,theequations(1){(3)hasauniquesolutionincaseofp2(1;21).Ontheotherhand,

8、thereisanumericalobservationwhich