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

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 $\Omega$, 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 $i=1,\dots ,N$. 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 $N\ge 3$, 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 $3\le N\le 5$. 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 $N\ge 6$.