非線性橢圓方程及系統中解的唯一性和結構性之探討

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陳志有(Zhi-you Chen) 查詢紙本館藏

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數學系

論文名稱

非線性橢圓方程及系統中解的唯一性和結構性之探討
(Uniqueness and Structure of Solutions for Nonlinear Elliptic Equations and Systems)

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摘要(中)

在本篇文章中，我們將探討橢圓方程和橢圓系統中解的存在性、唯一性及結構性等問題。首先，在橢圓方程中，我們將利用特徵方程來了解某些奇異解的存在性和唯一性，接者再使用 Pohozeaev function 和 Energy function 來了解其他所有解的存在性和結構性。然而在橢圓系統中，我們將透過線性化方程來學解的結構性，進一步結合隱函數定理來證明某些解的唯一性。

摘要(英)

In this article, we consider problems involving the existence, uniqueness, and structure of solutions for (single) elliptic equations and (coupled) elliptic systems. To deal with the elliptic equations, we first employ the characteristic equations to realize the existence and uniqueness of certain singular solutions. Then, specific auxiliary functions, Pohozaev functions and energy functions, will be introduced to conduct the uniqueness for solutions of other types and clarify the complete structure of solutions. On the other hand, for the elliptic systems, we analyze the structure of solutions by means of the corresponding linearized equations. Furthermore, combining the Implicit Function Theorem, the consequences related to the uniqueness of some solutions will be offered as well.

關鍵字(中)

★ 非線性橢圓方程及系統中解的唯一性和結構性之探討

關鍵字(英)

★ Uniqueness and Structure of Solutions for Nonlin

論文目次

Part I. Elliptic Equations
On the Classication of Standing Wave Solutions for the Schrodinger Equation
1. Introduction and Main Results 1
2. Structures and Behaviors of Solutions 7
3. Proofs of the Main Results 12
Uniqueness of Finite Total Curvatures and Structure of Radial Solutions for Nonlinear Elliptic Equations
1. Introduction and Main Results 20
2. Proof of Theorem 1.1 30
3. Zeros and Structure of Entire Solutions 35
4. Proofs of Theorems 1.2-1.5 40
The Structure of Radial Solutions for Elliptic Equations Arising from the Spherical Onsager Vortex
1. Introduction and Main Results 43
2. Preliminaries 52
3. Proofs of Main results 57
On the Blowup and Entire Solutions for a Biharmonic Equation
1. Introduction and Main Results 61
2. Blowup Solutions and Nonexistence Results 65
3. Linearization and Structure of Solution 71
Part II. Elliptic Systems
The Linearization and Uniqueness of Solutions for Elliptic Systems
1. Introduction and Main Results 75
2. Linearization and Uniqueness 77
3. Structure of Solutions 82
The Linearization and Uniqueness of Solutions for Elliptic Systems
1. Introduction and Main Results 86
2. The Non-Degeneracy of Linearized Equations 93
3. Uniqueness of Topological Solution 100
4. Asymptotic Behaviors of All Entire Solutions 104
5. Uniqueness and Solution Structures of the Dirichlet Problem 108
6. The Structures of All Entire solutions 110
7. Reference 121

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指導教授

陳建隆(Jann-long Chern)

審核日期

2008-7-1

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