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×N)-system:Δui=λ(∑j=1N∑k=1NKkjKjieujeuk−∑j=1NKjieuj)+4π∑j=1niδpij,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, λ>0 is the coupling parameter, δp is the Dirac measure with pole at p and ni∈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≥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≤N≤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≥6.