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研究生:吳宗芳
研究生(外文):Tsung-fang Wu
論文名稱:半線性橢圓方程正解的存在性與多樣性
論文名稱(外文):Existence and Multiplicity of Positive Solutions of Semilinear Elliptic Equations
指導教授:王懷權
指導教授(外文):Hwai-chiuan Wang
學位類別:博士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:89
中文關鍵詞:半線性橢圓方程正解巴萊斯麥爾多重解
外文關鍵詞:Semilinear Elliptic EquationsPositive SolutionsPalais-SmaleMultiple Solutions
相關次數:
  • 被引用被引用:0
  • 點閱點閱:166
  • 評分評分:
  • 下載下載:11
  • 收藏至我的研究室書目清單書目收藏:1
這篇論文主要分為三大部份:第一部份先論述一些如何應用巴萊斯麥爾理論來解決ㄧ個半線性橢圓方程在一些無界定義域正解的存在性問題。並且證明這些方法彼此是等價之外,更近一步證明它們與定義域的指數有一些充分必要的條件及半線性橢圓方程在一些其它無界定義域正解的存在性問題。在這第一部份之中我們另外探討一些定義域的指數 ,巴萊斯麥爾數列與巴萊斯麥爾條件之間的一些關係。
在第二部份,主要利用ㄧ些軸對稱界定義域的對稱破裂,來證明巴萊斯麥爾條件與定義域的指數有一些充分必要的條件,並利用如此的性質,來證明半線性橢圓方程在一些非凸軸對稱界定義域(有可能是無界)三個正解的存在性問題,而且這三個正解其中一個是軸對稱,其它兩個是非軸對稱。
最後,我們證明ㄧ個在無窮帶子的對稱巴萊斯麥爾分解定理,並且應用它去證明半線性橢圓方程在一些無窮帶子的外定義域正解的存在性問題。

1 Introduction ....................... 3
2 Preliminary ....................... 8
3 Indexes of Domains ................... 22
3.1 Palais-Smale Values and Index domains ......... 22
3.2 Palais-Smale Conditions ................ 35
3.3 Fundamental Properties of Indexes of Domains ..... 46
3.4 Existence of Positive Solutions ............ 52
3.4.1 In Channel Domains ................. 52
3.4.2 In Manger Domains .................. 54
4 Symmetry Breaking of Axially Symmetric Domains ..... 57
4.1 Basic Definitions and Notations ............ 57
4.2 Symmetric Palais-Smale Conditions ........... 59
4.3 Multiple Positive Solution .............. 63
4.3.1 Finite Strip with Hole ............... 63
4.3.2 Dumbbell Domain ................... 65
5 Palais-Smale Decomposition Theorem ........... 68
5.1 In General Domains .................. 68
5.2 In Axially Symmetric Domains ............. 71
5.3 Applications ..................... 84
5.3.1 The Infinite Strip with Holes ............ 84
5.3.2 The Large Domain in R^{N} ............. 85

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