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 ## Abstract

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

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

over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of $SU(N + 1)$, (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  the authors managed to obtain the existence of one doubly periodic solution via a minimization procedure, in the spirit of . Our main contribution in this paper is to show (as in ) 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 $N \geq 6$.