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

Han, Xiaosen; Tarantello, Gabriella

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 5, pp. 1401-1430.

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

$$\mathrm{\Delta }{u}_i=\lambda (\sum _{j=1}^N\sum _{k=1}^N{K}_{kj}{K}_{ji}{\mathrm{e}}^{u_j}{\mathrm{e}}^{u_k}-\sum _{j=1}^N{K}_{ji}{\mathrm{e}}^{u_j})+4\pi \sum _{j=1}^{n_i}{\delta }_{p_{ij}},i=1,\dots ,N;$$

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 $i=1,\dots ,N$. 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 $N\ge 6$.

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