Kirchhoff型问题的解的存在性.pdf

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3、ÄRN¥k.1w«•þKirchho.¯K8Z<a+bjruj2
u=f(x;u),x2,:u=0,x2@.·‚©O•Äš‚5‘f(x;u)3:´‡‚5…3á?´ìC4-‚5œ/,±9f(x;u)3:´ìC‚53á?´‡4-‚5œ/,š²…)•35.311Ù,·‚0'uù‡¯K˜®k(J.12Ù¥,·‚•Äš‚5‘f(x;u)3:´‡‚5…3á?´ìC4-‚5œ/.ÏLOŽéAUþ•¼3":.+,(ÜPereraÚÜ“7äE

4、,^Morsenؘ‡š²

5、…).13Ù¥,·‚•Äš‚5‘f(x;u)3:´ìC‚53á?´‡4-‚5œ/.ùp·‚3á?éf(x;u)‡¦Ambrosetti-Rabinowitz.‡4-‚5^‡.ÏLOŽéAUþ•¼3á.+,(ÜPereraÚÜ“7äE

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7、^ÇmJј‡^‡3·‚¯K¥C«5í213Ù(Ø.ù‡#^‡'13Ù¥Ambrosetti-Rabinowitz.^‡f,•'Ù¦ïÄö¦^üN^‡½Jeanjean^‡f.·‚

8、‰Ñ•35Úõ•5(J´©z¥ƒA(Jí2.'''………ccc:Kirchho.¯K,.+,.:,PalaisõSmale^‡IöÞŒÆa¬ÆØ©AbstractInthisthesis,westudythefollowingKirchhotypeproblem8Z<a+bjruj2
u=f(x;u),x2,:u=0,x2@,whereisaboundedsmoothdomaininRN.Weconsiderthecasesthatthenonlinearityf(x;u)issuperli

9、nearat0andasymptotically4-linearnearinnity,andthenonlinearityf(x;u)isasymptoticallylinearat0but4-superlinearatnearinnity.Inbothcasesweobtainnontrivialsolution.InChapter1,werecallsomeknownresultsinthisdirection.InChapter2,weconsiderthecasethatthenonlineari

10、tyf(x;u)issuper-linearat0andasymptotically4-linearnearinnity.Bycomputingthecriticalgroupsat0oftheenergyfunctionalandusingthetruncatedtechniqueofPereraandZhang,weobtainanontrivialsolutionviaMorsetheory.IniChapter3,weconsiderthecasethatthenonlinearityf(x;u)

11、isasymp-toticallylinearat0but4-superlinearatnearinnity.HereweimposetheAmbrosetti-Rabinowitztype4-superlinearconditiononf(x;u)nearinnity.Again,bycomputingthecriticalgroupsat0oftheenergyfunctionalandu