Multi-peak solutions for a class of nonlinear

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cBirkh¨auserVerlag,Basel,2002NoDEANonlineardiffer.equ.appl.9(2002)69–911021–9722/02/010069–23$1.50+0.20/0Multi-peaksolutionsforaclassofnonlinearSchr¨odingerequationsAngelaPISTOIADipartimentoMe.Mo.Mat.viaA.Scarpa1600100Romae-mail:pistoia@dmmm.uniroma1.itAbstract.Inthi

2、spaperweconsiderthestudyofpositivesolutionsof2N−ε∆u+λu=f(x,u)onR,whereεisasmallparameter,λ>0andfisanappropriatefunction.Herewefindmulti-peaksolutionsexhibitingconcentrationatanyprescribed“stable”setofzeroesofthefield
NS(P)=[∇xf(P,UP(y))·y]∇UP(y)dy,P∈R,RNwhereUPistheuni

3、queradialsolutionofthelimitequationN−∆UP+λUP=f(P,UP)onR.Conversely,weshowthatthepointsatwhichasequenceofmulti-peaksolu-tionsconcentratemustbezeroesofthefieldS.2000MathematicsSubjectClassification:35J60,35Q55.Keywords:NonlinearSchr¨odingerequation,multi-peaksolutions,Li

4、apunov-Schmidtreduction.0IntroductionInthispaperweconsiderthefollowingnonlinearSchr¨odingerequation:−ε2∆u+V(x)u=g(x,u)onRN,(0.1)70AngelaPistoiaNoDEAwhereεisapositiveparameterandVisstrictlypositivefunction,awayfromzero.Wewillstudytheexistenceofpositivesolutionstoequat

5、ion(0.1)inthelimitcaseasε−→0.Thefirstclassofresultsconcernsthenonlinearityg(s)=

6、s

7、pwith1

8、V.In[21],[22]OhgeneralizedsucharesulttoN≥2andobtainedforsmallεtheexistenceofmultibumpsolutions,whichconcentrateatanygivenfinitecollectionofnondegeneratecriticalpointsofV.In[25]theauthorprovedthatapointatwhichasequenceofpositiveboundstatesconcentratesmustbeacriticalpoi

9、ntofV.In[1]theauthorsshowedtheexistenceofonepeaksolutionsof(0.1)forεsmallconcentratingnearastrictlocalminimumpointorastrictlocalmaximumpointoffiniteorderofV.In[17]theauthorprovedtheexistenceofbothone-bumpandmulti-bumpsolutionswhichconcentrateatanygivenfinitecollectiono

10、fC1stablecriticalpointsofV(hereadegeneracityofanyorderofthederivativeisallowed).Concerningmulti-bumpsolutions,otherresultswereobtai