渐近线性Schrdinger方程的解.pdf

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u+V(x)u=g(x;u)x2RN(s1)u2H1(RN)e•§(s1)š‚5‘g÷ve¡b:(g):•3šK¼êb(x)2L2(RN)Úb(x)2L2(RN)L1(RN)…÷v:121Ng2C(RR),jg(x;s)jb1(x)jsj+b2(x)G(x;s)(g):g(x;s)s0;lim=1éx2RNA??¤á,js

4、j!1jsjg(x;s)lim=0éx2RN˜—¤á.jsj!1jsjK•§(s1))•35.1nÙ¥,·‚ïá2ÂQ:½n,¿A^u¹±Ï³Schr•odinger•§
u+V(x)u=u+g(x)f(u)(s2)u2H1(RN)e•§(s2)÷vXe^‡:(V):V(x)2C(RN),´ZN-±Ï,…=2(
+V)(g):g2L2(RN)L1(RN),g(x)0,x2RNf(u)(f):f(x)2C(R),÷vlimjuj!1u=0,f(u)u0,u2RIöÞŒÆa¬ÆØ©K–•3˜‡).''

5、'………ccc:C©{;.:;ý•§;ìC‚5;Schr•odinger•§;Q:½n;ì´Ún;2‚7½n;(PS)cSIIöÞŒÆa¬ÆØ©AbstractInthisthesis,byusingvariationalmethodswestudytheexistenceofsolu-tiontoasymptoticallylinearSch•odingerequations.Inchapterone,theintroduction,wereviewsomebackgroundoftheprob-lemsstudied

6、inthethesis.Inchaptertwo,weconsidertheexistenceofsolutionsofaclassofresonantasymptoticallylinearSch•odingerequationsoftheformN
u+V(x)u=g(x;u)x2R(s1)1Nu2H(R)wherethenonlinearitygsatisesthefollowingconditions:(g):g2C(RNR)andjg(x;s)jb(x)jsj+b(x)withb(x)2L2(RN)1122an

7、db(x)2L2(RN)L1(RN)1G(x;s)(g):g(x;s)s0;lim=1a.e.x2RN,jsj!1jsjg(x;s)lim=0uniformlyinx2RN.jsj!1jsjWeobtaintheexistenceofatlestonesolutionsforequation(s1).Inchapterthree,Weestablishageneralizedsaddlepointtheorem,whichisaninnitedimensionalgeneralizationoftheclassica

8、lsaddlepointtheoremofRabinowitz.ThenweillustrateitsapplicationtoanasymptoticallylinearSchrodingerequationwithperiodicpotentialofthe