# 臺灣博碩士論文加值系統

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 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 12 Preliminaries 83 Indefinite elliptic problems (I) 113.1 Introduction and main result . . . . . . . . . . . . . . . . . . . . 113.2 Some preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 The estimate of energy . . . . . . . . . . . . . . . . . . . . . . . 163.4 Existence of a positive solution . . . . . . . . . . . . . . . . . . 263.5 Existence of two positive solutions . . . . . . . . . . . . . . . . 313.6 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 384 Indefinite elliptic problems (II) 434.1 Introduction and main results . . . . . . . . . . . . . . . . . . . 434.2 Variational setting and Preliminaries . . . . . . . . . . . . . . . 454.3 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . 524.4 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . 544.5 Concentration for solutions . . . . . . . . . . . . . . . . . . . . 575 The Kirchhoff type equations 605.1 Introduction and main results . . . . . . . . . . . . . . . . . . . 605.2 Some preliminaries result . . . . . . . . . . . . . . . . . . . . . 635.3 Proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . . . 695.4 Proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . . . 745.5 Proof of Theorem 5.8 . . . . . . . . . . . . . . . . . . . . . . . . 76References 78
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 1 半線性橢圓方程正解的存在性與多樣性

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