Multiple Solutions for the Non-Abelian Chern--Simons--Higgs Vortex Equations

In this paper we study the existence of multiple solutions for the non-Abelian Chern--Simons--Higgs $(N\times N)$-system: \[ \Delta u_i=\lambda\left(\sum_{j=1}^N\sum_{k=1}^N K_{kj}K_{ji}\re^{u_j}\re^{u_k}-\sum_{j=1}^N K_{ji}\re^{u_j}\right)+4\pi\sum_{j=1}^{n_i}\delta_{p_{ij}},\quad i=1,\dots, N; \] over a doubly periodic domain $\Omega$, with coupling matrix $K$ given by the Cartan matrix of $SU(N+1),$ (see \eqref{k1} 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 \cite{haya1} the authors managed to obtain the existence of one doubly periodic solution via a minimisation procedure, in the spirit of \cite{nota} . Our main contribution in this paper is to show (as in \cite{nota}) 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 $N\ge 6$.