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研究生:鄭益新
研究生(外文):Yi-Hsin Cheng
論文名稱:橢圓型方程正解的存在性與多解性
論文名稱(外文):Existence, multiplicity of positive solutions for elliptic equations
指導教授:吳宗芳
指導教授(外文):Tsung-Fang Wu
學位類別:博士
校院名稱:國立高雄大學
系所名稱:應用數學系博士班
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:2015
畢業學年度:103
語文別:英文
論文頁數:92
中文關鍵詞:半線性橢圓方程Kirchhoff型式方程山路引理巴萊-斯麥爾理論Nehari 流形
外文關鍵詞:semilinear elliptic equationsKirchhoff type equationsmountain pass theoremPalais-SmaleNehari manifold
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  • 下載下載:17
  • 收藏至我的研究室書目清單書目收藏:0
在這篇論文中,我們探討不定型的半線性橢圓方程以及Kirchhoff型式方程,運用變分法如山路引理,巴萊-斯麥爾理論以及研究Nehari流形的性質來得到正解的存在性與多解性。
In this thesis, we study a class of indefinite semilinear elliptic equations and the Kirchhoff type equations by using the variational methods to obtain the existence and multiplicity of positive solutions. Roughly speaking, we use the mountain pass theorem, the Palais - Smale theory and study properties of the Nehari manifold to investigate the existence and multiplicity.
1 Introduction 1
2 Preliminaries 8
3 Indefinite elliptic problems (I) 11
3.1 Introduction and main result . . . . . . . . . . . . . . . . . . . . 11
3.2 Some preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 The estimate of energy . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Existence of a positive solution . . . . . . . . . . . . . . . . . . 26
3.5 Existence of two positive solutions . . . . . . . . . . . . . . . . 31
3.6 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Indefinite elliptic problems (II) 43
4.1 Introduction and main results . . . . . . . . . . . . . . . . . . . 43
4.2 Variational setting and Preliminaries . . . . . . . . . . . . . . . 45
4.3 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 Concentration for solutions . . . . . . . . . . . . . . . . . . . . 57
5 The Kirchhoff type equations 60
5.1 Introduction and main results . . . . . . . . . . . . . . . . . . . 60
5.2 Some preliminaries result . . . . . . . . . . . . . . . . . . . . . 63
5.3 Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4 Proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 Proof of Theorem 5.8 . . . . . . . . . . . . . . . . . . . . . . . . 76
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