Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations
Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations
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DOI：
10.1016/j.anihpc.2019.01.002
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发表时间：
2018-05
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影响因子：
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通讯作者：
Xiaosen Han;G. Tarantello
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文献类型：
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作者：
Xiaosen Han;G. Tarantello
本文研究了一类非交换Chen-Simons-Higgs(N×N)-系统：Δu i=λ(∑j=1 N∑k=1 N K k j K j i e u j e u k−∑j=1 N K j i e u j)+4π∑j=1 n iδp i j，i=1，…，N；在双周期域Ω上，耦合矩阵K由S U(N+1)的Cartan矩阵给出(见下文(1.2))。这里，λ>；0是耦合参数，δp是极点在p且n i∈N的狄拉克测度，对于i=1，…当N=1，2时，已有许多关于这类系统周期可解性的结果，并给出了拓扑型、非拓扑型、混合型和爆破型解的存在性。相反，对于N≥3，仅在最近的[27]中，作者本着[46]的精神，通过最小化过程获得了一个双周期解的存在性。本文的主要贡献是证明了(如文[46])，假设3≤N≤5，给定系统实际上存在第二个“山路”型双周期解。请注意，从物理角度来看，多个解的存在是相关的。事实上，它意味着不同的非阿贝尔Chern-Simons凝聚体共存，共享相同的涡点、能量和通量集(按组分分配)。要克服的主要困难是获得对应的“作用”泛函的所谓帕莱-斯梅尔条件所包含的“紧致性”性质，其有效性对于N≥6仍然是开放的。
In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs (N× N)-system: Δ u i= λ (∑ j= 1 N∑ k= 1 N K k j K j i e u j e u k−∑ j= 1 N K j i e u j)+ 4 π∑ j= 1 n i δ p i j, i= 1,…, N; over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of S U (N+ 1),(see (1.2) below). Here, λ> 0 is the coupling parameter, δ p is the Dirac measure with pole at p and n i∈ N, for i= 1,…, N. When N= 1, 2 many results are now available for the periodic solvability of such system and provide the existence of different classes of solutions known as: topological, non-topological, mixed and blow-up type. On the contrary for N≥ 3, only recently in [27] the authors managed to obtain the existence of one doubly periodic solution via a minimization procedure, in the spirit of [46]. Our main contribution in this paper is to show (as in [46]) that actually the given system admits a second doubly periodic solutions of “Mountain-pass” type, provided that 3≤ N≤ 5. Note that the existence of multiple solutions is relevant from the physical point of view. Indeed, it implies the co-existence of different non-Abelian Chern–Simons condensates sharing the same set (assigned component-wise) of vortex points, energy and fluxes. The main difficulty to overcome is to attain a “compactness” property encompassed by the so-called Palais–Smale condition for the corresponding “action” functional, whose validity remains still open for N≥ 6.