在第一部份，我們首先考慮具備哈帝位勢及臨界非線性項的橢圓 方程，並且在位勢項做了相當一般性的假設，探討其奇異解的節 構。一般而言，我們可以證明存在唯一一個特殊奇異解及無窮多個 奇異解在此特殊奇界解附近震盪。我們也學習這些圍繞在此唯一奇 異解的特殊解的極限行為。我們的結果可以應用在不同的問題上 面，例如純量場方程、細胞重製模型以及卡法芮利-科恩-尼倫柏格不 等式。另外，我們也個別討論了三個橢圓方程，並依據每個方程的 特徵考慮其在超臨界的情況下解在無窮遠處的行為或分類所有解節 構。在第二部份，我們證明了來自於乘積阿貝爾規範場論的橢圓系 統其非拓樸解的存在性。;For the first part, we consider the structure of singular solutions for elliptic equations with the Hardy potential and critical nonlinearity under quite general conditions on the potential terms. In general, it is shown that there exists a unique special singular solution, and other infinitely many singular solutions are oscillatory around the special singular solution. We also study the asymptotic behavior of the solutions around the singular point. Our results can be applied to various problems such as the scalar field equation, a self-replication model and the Cafarelli-Kohn- Nirenberg inequality. In particular, we discuss the three elliptic equations separately and to consider the asymptotic behavior of the solutions at infinity under supercritical case or classify all the solutions structure according to the characteristic of each equation. For the second part, we prove the existence of Non-Topological solutions for the elliptic system arising from a product Abelian Gauge Field theory.
