Existence theorems for a general 2 x 2 non-Abelian Chern-Simons-Higgs system over a torus

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Han, Xiaosen^1,2  Huang, Genggeng³ 

[1] Henan Univ, Sch Math, Inst Contemporary Math, Kaifeng 475004, Peoples R China

[2] Univ Roma Tor Vergata, Dipartimento Matemat, Via Ric Sci, I-00133 Rome, Italy

[3] Shanghai Jiao Tong Univ, Sch Math Sci, Shanghai 200240, Peoples R China

关键词: Chern-Simons vortices;    Self-dual equations;    Topological solutions;    Doubly periodic solutions;   

DOI  :  10.1016/j.jde.2017.03.017

来源: Elsevier

【 摘 要 】

In this paper we study a general 2 x 2 non-Abelian Chern-Simons-Higgs system of the form Delta u(i) + 1/epsilon(2) (Sigma(2)(j=1) K(ji)e(uj) - Sigma(2)(j=1) Sigma(2)(k=1) K-kj K(ji)e(uj) e(uk)) = 4 pi Sigma(Ni)(j=1) delta(Pij) (x), i = 1, 2 over a flat 2-torus T-2, where epsilon > 0, delta(p) is the Dirac measure at p, N-i epsilon N (i = 1, 2), K is a non-degenerate 2 x 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 T2, 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 epsilon(0) > 0 such that this system admits a solution over the torus if 0 < epsilon < epsilon(0) provided vertical bar a vertical bar, vertical bar b vertical bar are suitably small. Furthermore, if ab >= 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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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