Existence theorems for a general 2 x 2 non-Abelian Chern-Simons-Higgs system over a torus
Existence theorems for a general 2 x 2 non-Abelian Chern-Simons-Higgs system over a torus
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环面上一般 2 x 2 非阿贝尔 Chern-Simons-Higgs 系统的存在定理
DOI：
10.1016/j.jde.2017.03.017
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发表时间：
2017
影响因子：
2.4
通讯作者：
Huang Genggeng
中科院分区：
文献类型：
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作者：
Han Xiaosen;Huang Genggeng
本文研究了一般的2× 2非阿贝尔Chern-Simons-Higgs系统，其形式为Δ ui + 1 ε 2（∑ j= 1 2 K j i e u j−∑ j= 1 2∑ k= 1 2 K k j K j i e u j e uk）= 4 π∑ j= 1 N i δ p i j（x），i= 1，2在平坦的2-环面T2上，其中ε> 0，δ p是p处的Dirac测度，N i∈ N（i= 1，2），K是一个非退化的2× 2矩阵，其形式为K=（1+ a− a− B 1+ B），这可能涵盖了当K是Cartan矩阵（秩为2的半单李代数）时物理上有趣的情况。关于T 2上这类系统的存在性结果，文献中通常要求a，B> 0。然而，在几个具有特定规范群的Chern-Simons-Higgs模型中，a，B< 0的系统的可解性问题至今仍是一个悬而未决的问题。我们证明了存在一个常数ε 0> 0，使得当0< ε< ε 0时，该系统在环面上存在一个解，从而部分地解决了这个问题。|一|,|B|是适当的小。此外，当a B≥ 0时，在a，B，N1，N2的适当条件下，该方程组存在山路解.我们的论点是基于微扰方法和山路引理。
In this paper we study a general 2× 2 non-Abelian Chern–Simons–Higgs system of the form Δ u i+ 1 ε 2 (∑ j= 1 2 K j i e u j−∑ j= 1 2∑ k= 1 2 K k j K j i e u j e u k)= 4 π∑ j= 1 N i δ p i j (x), i= 1, 2 over a flat 2-torus T 2, where ε> 0, δ p is the Dirac measure at p, N i∈ N (i= 1, 2), K is a non-degenerate 2× 2 matrix of the form K=(1+ a− a− b 1+ b), which may cover the physically interesting case when K is a Cartan matrix (of a rank 2 semisimple Lie algebra). Concerning the existence results of this type system over T 2, usually in the literature there is a requirement that a, b> 0. However, it is an open problem so far for the solvability about such system with a, b< 0, which naturally appears in several Chern–Simons–Higgs models with some specific gauge groups. We partially solve this problem by showing that there exists a constant ε 0> 0 such that this system admits a solution over the torus if 0< ε< ε 0 provided| a|,| b| are suitably small. Furthermore, if a b≥ 0 in addition, with suitable condition on a, b, N 1, N 2, this system admits a mountain-pass solution. Our argument is based on a perturbation approach and the mountain-pass lemma.