在本論文中我們將探討擬線性橢圓方程在Dirichlet 和 Robin 條件下，解的存在性問題；更進一步，我們在Robin 條件下得到解的唯一性。同時，我們對Hardy-Sobolev方程作了徑對稱解的研究，得到在上臨界的條件下最多僅有一個奇異解。

In this thesis, we study the existence of solutions to quasilinear elliptic equations with Dirichlet and Robin conditions, and the uniqueness of solutions under Robin conditions. Also, we investigate the radial symmetric solutions for a Hardy-Sobolev equation and derive the result that there exists at most one positive radial symmetric singular solution for the supercritical case.

Contents
Chapter 1 Introduction………………………………………………1
Chapter 2 Quasilinear Elliptic Equations with Dirichlet Conditions………………………………………………………………5
2.1 The Maximum Principl……………………………………………9
2.2 The  Estimate……………………………………………………13
2.3 The Existence of Strong Solutions…………………………15
Chapter 3 Quasilinear Elliptic Equations with Robin Conditions ……………………………………………………………31
3.1 The Comparison Principle ……………………………………34
3.2   and   Estimates………………………………………………38
3.3 The Existence and Uniqueness of Solutions in   ………41
Chapter 4 Hardy-Sobolev Equations………………………………45
4.1 Asymptotic Behavior s…………………………………………46
4.2 The Uniqueness of Positive Radial Singular Solutions………………………………………………………………56
Appendix A Basic Notations and Fixed Point Theorems………60
A.1 Basic Notations…………………………………………………60
A.2 Fixed Point Theorems……………………………………… …62
Bibliography …………………………………………………………63

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