Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations

Han, Xiaosen; Tarantello, Gabriella

doi:10.1016/j.anihpc.2019.01.002

Han, Xiaosen ¹ ; Tarantello, Gabriella ²

¹ Institute of Contemporary Mathematics, School of Mathematics and Statistics, Henan University, Kaifeng 475004, PR China
² Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1401-1430.

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Résumé

In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs $(N \times N)$ -system:

Δ u_{i} = λ (\sum_{j = 1}^{N} \sum_{k = 1}^{N} K_{k j} K_{j i} e^{u_{j}} e^{u_{k}} - \sum_{j = 1}^{N} K_{j i} e^{u_{j}}) + 4 π \sum_{j = 1}^{n_{i}} δ_{p_{i j}}, i = 1, \dots, 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} \in N

, for

i = 1, \dots, 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 \geq 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 \leq N \leq 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 \geq 6

DOI : 10.1016/j.anihpc.2019.01.002

Mots-clés : Chern–Simons–Higgs equations, Doubly periodic solutions, Mountain-pass solution

Affiliations des auteurs :

Han, Xiaosen ¹ ; Tarantello, Gabriella ²

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     title = {Multiple solutions for the {non-Abelian} {Chern{\textendash}Simons{\textendash}Higgs} vortex equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1401--1430},
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Han, Xiaosen; Tarantello, Gabriella. Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1401-1430. doi : 10.1016/j.anihpc.2019.01.002. http://www.numdam.org/articles/10.1016/j.anihpc.2019.01.002/

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