本論文主要探討三種不同型態之二階非線性橢圓偏微分方程式，其解的存在性、唯一性與結構性之相關定性分析及研究。第一部分中，我們考慮一個與隨機最佳化控制問題(stochastic optimal control problem)相關之方程，其具所謂『梯度約束方程式(gradient constraint equation)』之型態；在其非線性項更加弱化的條件下，我們證得其正解之存在性與唯一性。第二部分中，我們考慮一個座落於雙曲空間(hyperbolic space)中之半線性橢圓偏微分方程；研究其正奇異解在原點的漸近行為並且提供此正奇異解之存在性與唯一性；除此之外，我們透過『Pohozaev恆等式』了解其他不同型態解之特性，藉此進一步獲得某些特定型態解之不存在性。最後，在第三部分中我們考慮所謂的『Hardy-Sobolev方程』；在不同的指數條件下，我們研究其解的存在性、唯一性以及原點或無窮遠之行為。;This dissertation is concerned with studying some second order elliptic partial differential equations. We are devote to establishing some qualitative properties of solutions, including existence, uniqueness and structure of solutions to three specific types of nonlinear elliptic equations. In Part 1, we study a gradient constraint equation which is related to a stochastic optimal control problem. We offer the existence and uniqueness of positive radial solutions with certain behavior under weaker conditions on nonlinearity. In Part 2, we consider a semilinear elliptic equation on the hyperbolic space. The asymptotic behavior, existence and uniqueness of positive singular solutions at the origin are proved. In addition, we discuss the structure of solutions of various types via the Pohozaev identity. Finally, in our last chapter, we deal with the Hardy-Sobolev equations and investigate behaviors, existence and uniqueness of solutions for different exponents.
