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1、HÆa¬ÆØ©OntheExistenceofSolutionsforSchr¨odingerequationsAbstractNonlinearproblemsinmathematicsoriginatedfromphysics,chemistry,biology,celestialmechanics,economicandsocialscience.Itismanifestedintheformofavarietyofnonlinearequations.So,themathematicalwordandthescienceworldattachimportancetothe
2、nonlinearfunctionalanalysis.Withthedevelopmentofscienceandtechnology,somenewresultsforthenonlinearfunctionalanalysisanditsapplicationshavebeenobtained.TheexistenceofsolutionsforaclassofSchr¨odingerequationsisoneofthehotspotswhichhasbeendiscussedinrecentyears.Inthispaper,undertheCeramiconditio
3、n,weusethemountainpasstheorytostudytheexistenceofsolutionsforseveralkindsofSchr¨odingerequations.Thethesisisdividedintotheresectionsaccordingtocontents.Chapter1Preference,weintroducethemaincontentsofthispaper.Chapter2InthischapterwemainlystudythenonhomogeneousSchr¨odingerequation2−∆u+V(x)u=f(
4、u)+h(x),x∈R.F(s)f(s)wheref∈C(R)andlim
5、s
6、2=∞,lim
7、s
8、2∗−1=0.ByusingMountain-Pass
9、s
10、→∞
11、s
12、→∞Theorem,weprovetheexistenceofanontrivialsolution.Chapter3ThepurposeofthisworkistodealwiththeexistenceofsolutionforthefollowingquadraticSchr¨odingerequations(−∆u+V(x)u=f(x,u),(1)1Nu∈H(R),whereV∈C(RN,R),H1(RN
13、)istheusualSobolevspace,andf∈C(RN×R,R)whereRuF(x,u)=f(x,t)dt.Moreover,Fissuperquadraticattheoriginandasymptotically0atinfinity.Keywords:MountainPasstheorem;Schr¨odingerequations;Variationalmethod;Asymptoticallyquadratic.iiHÆa¬ÆØ©8¹1ÙXØ...........................................................
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