摘要

本論文我們分成兩個部分:(一)自對偶陳-西蒙斯 CP (1)模型 (二)一些非線性橢圓系統。
我們考慮來自陳-西蒙斯平面的物質場理論與陳-西蒙斯規範場在CP (1)形式下的交互作用的非線性方程。我們在單一渦流點的情況下建立了非拓樸解之通量的銳利區域,也得到了所有形態的解的分類之證明。更進一步,我們也給了在參考文獻 [21]的定理1.3完全的結果與證明。
在另一方面,我們在 p ≥ n+2 的條件下證明了 (0.1)非線性橢圓系統
n−2
⎧

1Φq
⎨ ΔΦ1 − Φ1 + θ1Φp 2 =0

ΔΦ2 − Φ2 + θ2Φ1qΦp
⎩ 2 =0 ,
where n> 2, θ1,θ2 > 0, p> 1, and q> 0,不具有兩者皆奇異的正的全域徑解以及在原點為中心,半徑 R的有界定義域上也不具有兩者皆奇異的正的徑解。如果我們將 (0.1)的徑解延拓到負值,我們證得了全域徑解的存在性,而且更進一步,我們得到在每一個有限區間徑解的行為都分別被兩個二次多項式上下界住的結果。事實上,我們證明了其他更一般形式的非線性橢圓系統的徑的全域解的存在性。更進一步,我們得到 (0.1)非線性橢圓系統正的徑解的行為的一些特徵。

Abstract

We separate this thesis into two parts: (I) The Self-Dual Chern-Simons CP (1) Model (II) Some Nonlinear Elliptic Systems.
We consider the nonlinear equation arising from the Chern-Simons theory of planar matter fields interacting with the Chern-Simons gauge field in a CP (1) invariant fashion. Then we establish the sharp region of flux for non-topological solutions and prove the clas¬sification of solutions of all types in the case of one vortex point. Moreover, we also give the complete result of Theorem 1.3 in [21].
On the other hand, we prove that the systems of nonlinear elliptic equations 

 ∆Φ1 − Φ1 + 81Φp 1Φ2 q =0
(0.1)

∆Φ2 − Φ2 + 82Φ1qΦp
 2 =0 where n> 2, 81,82 > 0, p> 1, and q> 0, do not possess any both singular positive radial solutions on the entire domain Rn \{0} and on any bounded domain Ω \{0}, where Ω is an open ball with radius R and the center at the origin if p ≥ n+2 Also, we prove the
n−2. existence of radial entire solutions of (0.1) if we extend the values of radial solutions of (0.1) to negative, and furthermore we obtain that the behaviors of the radial solutions of (0.1) on each finite in¬terval [0,R] are bounded by two quadratic polynomials, respectively. Actually, we prove the existence of radial entire solutions for other more general nonlinear elliptic systems. Furthermore, we also obtain some characteristics of the behaviors of positive radial solutions of (0.1).

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