這篇論文主要分為三大部份：第一部份先論述一些如何應用巴萊斯麥爾理論來解決ㄧ個半線性橢圓方程在一些無界定義域正解的存在性問題。並且證明這些方法彼此是等價之外，更近一步證明它們與定義域的指數有一些充分必要的條件及半線性橢圓方程在一些其它無界定義域正解的存在性問題。在這第一部份之中我們另外探討一些定義域的指數 ，巴萊斯麥爾數列與巴萊斯麥爾條件之間的一些關係。
在第二部份，主要利用ㄧ些軸對稱界定義域的對稱破裂，來證明巴萊斯麥爾條件與定義域的指數有一些充分必要的條件，並利用如此的性質，來證明半線性橢圓方程在一些非凸軸對稱界定義域（有可能是無界）三個正解的存在性問題，而且這三個正解其中一個是軸對稱，其它兩個是非軸對稱。

最後，我們證明ㄧ個在無窮帶子的對稱巴萊斯麥爾分解定理，並且應用它去證明半線性橢圓方程在一些無窮帶子的外定義域正解的存在性問題。

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