# 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

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## Abstract

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

over a doubly periodic domain Ω, with coupling matrix *K* given by the Cartan matrix of $SU(N + 1)$, (see (1.2) below). Here, $\lambda > 0$ is the coupling parameter, $\delta _{p}$ is the Dirac measure with pole at *p* and $n_{i} \in \mathbb{N}$, 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 $N \geq 3$, 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 $N \geq 6$.

## 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