非线性椭圆方程与方程组解的存在性及多重性

《非线性椭圆方程与方程组解的存在性及多重性》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

1、分类号；0175单位代码学号密级Y890】二510005S200306009公开北京工业大学硕士学位论文题目：非线性椭圆方程与方程组解的存在性及多重性英文并罗U：垦嚣l曼至量毡g量△驻旦M卫圣堡!￡圣!g!至Y墼曼型!堡曼至QB题目：盟Q盟L!盥旦△珏旦L量!￡!!g旦!世g旦L量至￡BQ旦L垦垒』量研究生姓名：娄专业：基础数学导师：程曹宗光谱研究方向：非线性泛函分析职称：教授论文报告提交日期：!!塑生i旦学位授予日期授予单位名称和地址：jB室王些盘兰jE堕堕塑塑墨!堑旦!!!曼摘要摘要卜。：二三竺二二㈤r出刊卫：裂：耋㈣tu兰篓(P3)本文

2、共分为四章。第一章介绍了临界点理论一些主要的发展过程以及相关的记号与基本知识；第二章利用喷泉引理证明了四阶椭圆方程(P1)无穷多个弱解的存在性；第三章利用局部环绕理论来讨论半线性四阶椭圆方程(P2)，得到了它有一个非平凡弱解的结论；第四章运用拓扑度理论研究了四阶椭圆方程组(P3)正的弱解存在性的问题．关键词：四阶椭圆问题；多重解；喷泉引理；局部环绕；拓扑度Qn饥概棚0AbstractTheaimofthecontemporarycriticalpointtheoryistransformingtheproblemofsearchings01u

3、tionofequationorequationsystemintotheproblemofinvestigatingthecriticalpointofafunctionJinasuitablespaceE，using80me乱rongmethodsofcriticalpointtheorysuchasmillimaXtheory)Morsetheorytolookforthecritic出pointofthefunctionJ．So100kingforthecriticalpointofthefunctionisthekeyofdeali

4、ngwiththeproblem．Inthisthesis，wewillfirstlycon8iderthefoUowingfourth．ordernonlinearellipticboundaryv出ueprobIem△2让十c△乱=，(茁，钍)溉Q乱=△“=0o礼6n(P1)Whichisobtainedby8imuIatingasecondordereUiptieproblem，Wewmassumec>A1(A1istheprincipaleigeIlvalueof一△in础(Q))，andrelyontheweIl-knownfoun

5、-taintheoremtodiscussthemultipIicityofnontrivial∞lutionfortheabove—mentionedproblem(P1)．second，weresearchthetypeoffourth—orderelIipticboundaryvalueproblem△2u+c△牡+口(z)让=，(。，u)i住Qu：△“=0o扎an(P2)andgaintheexistenceofweaksolution．Last，weconsidertheexis惋nceofanontrivialweaks01uti

6、onforafourth—orderellipticsystemII(马)佗耗0．oo∞∞嘭∞训州圳@@△II=弘u彤，出地州札归‰渤△岔△=Abstr8ctbytopologicaldegreetheoremThisthesisiscomposedoffourchapters．Inchapteronelweintroducethemainde-velopmentofcriticaltheoryandtheconcernedmarksandbasalknowledgeInchaptertwo，wegettheexistenceofmultip

7、lesolutionsofafourth—orderellipticproblem(P1)byusingthefountaintheorem．Inchapterthee，wediscussasemilinear如urth．orderellip—ticproblem(B)withloca王linking，andgetanontrivialweaks。lutionofthisproblemInchapterfour，byapplying竹Letopologicaldegreetheorem，westudytheexistenceofs01utio

8、nforafourth—orderellipticsy8tem(B)Key、vords：Fburth—orderellipticproblems；multiples