JournalsaihpcVol. 36, No. 5pp. 1401–1430

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

  • Xiaosen Han

    Institute of Contemporary Mathematics, School of Mathematics and Statistics, Henan University, Kaifeng 475004, PR China
  • Gabriella Tarantello

    Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy
Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations cover
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In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs (N×N)(N \times N)-system:

Δui=λ(j=1Nk=1NKkjKjieujeukj=1NKjieuj)+4πj=1niδpij,i=1,,N;\mathrm{\Delta }u_{i} = \lambda \left(\sum \limits_{j = 1}^{N}\sum \limits_{k = 1}^{N}K_{kj}K_{ji}\mathrm{e}^{u_{j}}\mathrm{e}^{u_{k}}−\sum \limits_{j = 1}^{N}K_{ji}\mathrm{e}^{u_{j}}\right) + 4\pi \sum \limits_{j = 1}^{n_{i}}\delta _{p_{ij}},\:i = 1,…,N;

over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of SU(N+1)SU(N + 1), (see (1.2) below). Here, λ>0\lambda > 0 is the coupling parameter, δp\delta _{p} is the Dirac measure with pole at p and niNn_{i} \in \mathbb{N}, for i=1,,Ni = 1,…,N. When N=1,2N = 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 N3N \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 3N53 \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 N6N \geq 6.

Cite this article

Xiaosen Han, Gabriella Tarantello, Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1401–1430

DOI 10.1016/J.ANIHPC.2019.01.002