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1、PROCEEDINGSOFTHEAMERICANMATHEMATICALSOCIETYVolume136,Number7,July2008,Pages2565–2570S0002-9939(08)09484-7ArticleelectronicallypublishedonMarch14,2008ONDECAYOFSOLUTIONSTONONLINEARSCHRODINGEREQUATIONS¨ALEXANDERPANKOV(CommunicatedbyMichaelWeinstein)Abstract.Wepresentgeneralresults
2、onexponentialdecayoffiniteenergysolutionstostationarynonlinearSchr¨odingerequations.Undercertainnatu-ralassumptionsweshowthatanysuchsolutioniscontinuousandvanishesatinfinity.Thisallowsustointerpretthesolutionasafinitemultiplicityeigen-functionofacertainlinearSchr¨odingeroperatoran
3、d,hence,applywell-knownresultsonthedecayofeigenfunctions.Inthisnoteweconsidertheequation(1)−∆u+V(x)u=f(x,u),x∈Rn,and,underrathergeneralassumptions,deriveexponentialdecayestimatesforitssolutions.Wesupposethat(i)ThepotentialVbelongstoL∞(Rn)andisboundedbelow,i.e.V(x)≥−cloc0forsome
4、c0∈R.Underassumption(i)thelefthandsideofequation(1)definesaself-adjointop-eratorinL2(Rn)denotedbyH.TheoperatorHisboundedbelow.Wesupposethat(ii)Theessentialspectrumσess(H)oftheoperatorHdoesnotcontainthepoint0.Note,however,that0canbeaneigenvalueoffinitemultiplicity.Thenonlinearityo
5、ffissupposedtosatisfythefollowingassumption.(iii)Thefunctionf(x,u)isaCarath´eodoryfunction;i.e.itisLebesguemeasur-ablewithrespecttox∈Rnforallu∈Randcontinuouswithrespecttou∈Rforalmostallx∈Rn.Furthermore,(2)
6、f(x,u)
7、≤c(1+
8、u
9、p−1),x∈Rnu∈R,withc>0and2≤p<2∗,where⎧⎨2n∗ifn≥3,2=n−2⎩∞ifn=
10、1,2,ReceivedbytheeditorsSeptember18,2006,and,inrevisedform,June29,2007.2000MathematicsSubjectClassification.Primary35J60,35B40.Keywordsandphrases.NonlinearSchr¨odingerequation,exponentialdecay.c2008AmericanMathematicalSocietyRevertstopublicdomain28yearsfrompublication2565Licens
11、eorcopyrightrestrictionsmayapplytoredistribution;seehttp://www.ams.org/journal-terms-of-use2566ALEXANDERPANKOVand
12、f(x,u)
13、limesssupx∈Rn=0.u→0
14、u
15、LetEdenotetheformdomainoftheoperatorH,i.e.,thedomainofthecorre-spondingquadraticformor,whichisthesame,thedomainoftheoperatorH1/2.Itiswe
16、ll-knownthatE={u∈H1(Rn):(V(x)+c+1)u(x)∈L2(Rn)}0wherec0
