1、关于带第一特征值 1 具 Sobolev 临界指数2的一类椭圆方程 u=1uu22u （x,u ）第 23 卷第 3 期2003 年 6 月黄冈师范学院JournalofHuanggangNormalUniversityVo1.23NO.3Jun.2003Ontheellipticequations一厶一 1+U2 一 2+T(,)withthefirsteigenvalue1,involvingthecritica1Sobolevexponent2RAORuo-feng(Dept.ofSci.,JiujiangCollege,Jiujiang332005,Jiangxi,China)Abst
2、act:Wegavetheexistencetheoremofnontrivialsolutionforaclassofsemilinearellipticequations 一 = 】+Jl一+r(x,),undersomeconditionsonr(x,),andconsideredtheexistencetheoremofnonzerosolutionforaclassofsemilinearellipticequations-Zu=Au-lUl.一+(z),1,whereisthekthdistincteigenvalueoftheeigenvalueproblem-Au=linH3(
3、n)and9,thecriticalSobolevexponent2.一/._.:-9Keywordssemilinearellipticequation;thecriticalSobolevexponent;theDiriehletproblem;theeigenvalue;theconcentraetionCompactnessprinciple;theleastactionprincipleCLCnumber:O175.25Documentcode:AArticleID:1003-8078(2OO3)O3 一 OOO3 一O4关于带第一特征值具 Sobolev 临界指数 2 的一类椭圆方
4、程一=/l1+JJ2+r(,)饶若峰(九江学院理学系,江西九江 332005)摘要:给出了半线性椭圆方程一.+JUJ.+r(x,)的 Dirichlet 问题在对扰动项 r(x,)增加适当条件后非平凡解的存在性定理,以及方程一MAu-IMI-zu:的第 k 个互不相等的特征值 )的非零解的存在性定理.关键词:半线性椭圆方程;Sobolev 临界指数;Diriehlet 问题;特征lIntrOductiOnandmainresults+h(z),( 这里是方程值;集中紧性原理;极小值原理Existenceofnontrivialsolutionforthesemilinearellipticeq
5、uations一“=l+J“J.“+r(x,zf),UH3()(1)hasbeenunknown(wherer(x,“)isinsubcriticalgrowth),foritwasafamousopenproblem.Supposethat/2R( 3)isaboundedsmoothdomain.11()一 3()withrespecttothenorm收稿日期:20O30410.作者简介:饶若峰,男,江西抚州人,硕士,讲师.主要从事非线性椭圆偏微分方程研究 .基金项目:国家自然科学基金资助项目(10071048).!黄冈师范学院第 23 卷II=(fnII.dz)i1.Theweakco
6、nvergentsignisden.tey 一,2=2N一 2lsthekthdistincteigenvalueoftheeigenvalueproblem一一,EHo(/2).(2)Inthispaper1willgivetheexistencetheorem.fn.ntriVia1s.lutionf.rtheDirichletpr.blem(1)?The0rem1ConsideringtheDirichletproblemAu:,+II.-2uf(x,U),EHo(/2),(1)whereDR(N 3)isaclosedandboundedsmoothdomain,ifthefollow
7、ingc.ndii0nhold:(f1)厂(z,)o,v(z,)E/2X(0,+o.);冗一 o,uniformlyaz,(f2)厂(z,):nXRR,isacontinuousfunctioncrslim.乏乏.,un 讧.rm?yf.rauz)thereexistsac.nstant 户 E(1,2.一 1)that 一.,unif0rmlyf0rallz,andF(z,U)f“厂(z,f)dfisJ0anevenfunctionon,thenthereexistsasolution.EHXo(/2)fortheDirichetproblem(1),moreover,COI(u*)orth
8、ereexistssuchapositiveintegerthatf.一(?)+Nwhere)一丢 f.lI 一一 jII+F(x,u)dxisthefunctiona1correspondingtheDirichletproblem(1).Re 眦 rkthatthesoluti.n.fTheorem1isperhaPsazeto-solution;andremarkthatitd0nth0ldthat 厂(z,)O,V(z,)E/2XR.Indeedf(x,)isan.ddfunctionon?Onlyforthe.on 一nience0ftheproofofTheorem1,Isubst
9、itute(1)for(1)?Corollary1Iftheconditi.ns(f1)(f4)hold,F(x,)J. 厂()disaneven 眦 nononthenatleastthereexistsanontrivials.luti.n.EH0x(/2)f.rtheDirichletproblem(1),moreover,c.:(.)orthereexistssuchapositiveintegerthatf.=(*)+备, .S?Remark,wehave,seethepro.f.fTheorem1inE53,V?EH(n),?(z)o,a?e?zn,therewillexistsu
10、chatxO,t11Bp(O);I(tu1)0,astt1?Indeed,bytheproofofTheorem1inE53,wehave:asf+o.()fz(百 1II 一每 II.II2z:1 一 II.II;.f2_.+)一一 o.,(Q)wherePEE(21)whichistheeigenspacec.rresponding1,P1(z)O,Vzn,IlelIl 一1?wekn0wby(Q):thereexistssuchat0O,I(te1) I(t.P1),VtEE0,+oo).Denote(z)一 F(xtoet(x)then(z)isacontinuousfuctionin
11、n,aclosedandboundedsmoothdomain?Therefore,thereex.stssuchanhoER,h()ho,VzE0.I一一豳凰.,1O,一 2S一,JOf,fDOU,StOO,10OahCUSStSXeerehtdna第 3 期饶若峰:关于带第一特征值?5?Corollary2Iftheconditions(f1)(f4)hold,F(x,)一 If(x,t)dtisanevenfunctionon,d0moreover.ifthefollowholds(meas1N),(o2)thenthecondition(01)ofCorollary1holds,ora
12、tleastthereexistsanontrivialsolutionu.HI(O)fortheDirichletproblem(1),moreover,co=:=I(u.)orthereexistssuchapositiveinteger,lthat1NTheorem2ConsidertheDirichletboundaryvalueproblemAu=2uIuIu+h(z),.,uHI(O),(3)where12CR(3)isaboundedsmoothdomain,q(O,2 一 23.AssumethathL(n),thenproblem(3)hasatleastonesolutio
13、ninH3().Moreover,ifIh(z)dxOholdsatall,thenproblem(3)hasonenonzerosolutioninHi(n).Remark:V(O,+),therewillexistsucha,tkthat(O,.Indeed,fromE4-1wehaveknownthattheeigenvalueproblem(2)hasasequenceofeigenvalues0.2一.Fromtheabovewecanobtainasequenceofeigenvaluesoftheeigenvalueproblme(2),0l2+l.More,thereareal
14、otofpaperstodealwiththeproblem(3)inthecasethat(O,1).Sointhispaperweconsiderproblem(3)inthecasethat,tEl,.2ProofsoftheoremsandcorollarysProofsofTheorem1andCorollary1:TheproofsofTheorem1andCorollary1areperfectlysimilartotheproofofTheorem1inEsJ.NowIgivetheproofofCorollary2.ProofofCorollary2:Itfollowsbyo
15、nedimensionoftheeigenspaceE(21),I(toe1)一一 Le.Idz+F(x,toea)d一 tZo“Jfedz+.?meas,(4)then(01)followsby(O2)and(4),whichcompletestheproofofCorollary2.ProofofTheorem2:Let一1IIII 一 z+南.fII 如一如foruH3().InawaysimilartoTheorem1.4ifi2,wecanprovethatC(H5(),R).ItiswellknownthatuH5()isasolutionofproblem(1)ifandonly
16、ifuisacriticalpointof.SetH一 E(1)0E(2)oOE(2k),wecanknowfromProperty3ofSection3.5in3thateveryE(九 )(=1,2,)isafinitedimensionalsubspaceofH5(),andfromProperty5ofSection3.5in3thatH5()一 E(2.)0E(22)00E(2k)0一 0H.Therefore,foreveryuEHo(g2),thereexistssuchaHthat 一-一 u,andthatH.Ononehand,thefunctioniscoercive,t
17、hatis,()+as l l+.Bythedefinitionofeigenvaluefrom(3.522)ofSection3.5inIs,onehas黄冈师范学院第 23 卷-flld.-fd,-fIv;,Idz+. u.dz,(5)wegetby(5):“)(1 一砉)IIll+(1 一)II UII+lI.dz 一-f.dz.(6)Bytheequivalenceofthenormsonthefinitedimensionalspace,onehas:thereexistssuchaconstantc0,Idxfll“ll,V“H.(7)Itiseasilyunderstoodwhe
18、reThenonehasIq+2u2asII,Mr+2),?“iq+zR.Bytheorthogonalresolutionof(),and(6) (8),itiseasilyobtained9(u)II“II 一 c.IIIIII“IImeasn,(8)(9)wherec.一 min丢,丢(1 一)f0isapositiveconstantrelatedt.thePoincainequality.Hence,thecoercivityofthefunctioniseasilyprovedby(9).Ontheotherhand,thecoercivityofthefunctionisweek
19、lylowersemicontinuum.Indeed,asno.,ifUn 一“in5(n),frompassingtoasubsequencewegetUin;U(z)U(z),l(z)l+l(z)l+,a.e.z力.WegetfromFatouSlemma:liminfIl“(z)ldxIl(z)ldx.J 力 JnTherefore,liminf“)一 liminf1IIII 一“2dz+-rII.dz 一-r.dz)1-m-nfII“II 一 lim-r“:dz+liminflI 升.dzlimahdz1II“II 一.“dz+l“Idz 一.dz 一“)Nowitfollowsfr
20、omtheleastactionprinciple(see,e.g.,Theorem1.1inz3)thathasaminimum.Henceproblem(3)hasonesolution“(n).Moreover,ifIh(x)dx0,itiseasilyprovedJ 力thatthesolutionU(z)0,a.e.z/2,whichcompletestheproofReferences1LiuShuiqiang,TangChunlei.ExistenceandmultiplicityofsolutionsforaclassofsemilinearellipticequationsJ
21、.JMathAnalAppl,2001,257:321331.2MawhinJ,WillemM.CriticalpointtheoryandHamihoniansystemsM.NewYork:SpringerVerlag,1989.3LuWenduan.Variationalmethodsindifferentialequation(inchinese)M-.Chengdu:SichuanUnivPub,1995.4AzoreroJG.AlonsoIP.SomeresultsabouttheexistenceofasecondpositivesolutioninaquasilinearcriticalproMemJ-.IndianUnivMathJ,1994,43:941957.5RaoRuofeng.AClassofresonanceproblemsonboundeddomainsforarbitraryeigenvalue,involvingthecriticalSobolevexponent(inChinese)J.JEastChinaGeologicalInstitute,2003,26(1):95100.ttt IllfltI.flFf
