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 Chem-Simons-Higgs (N x N)-system:Delta u(i) = lambda(Sigma(N)(j=1) Sigma(N)(k=1) K(kj)K(ji)e(uj)e(uk) - Sigma(N)(j=1) k(ji)e(uj)) + 4 pi Sigma(ni)(j=1) delta(pij), i=1, . . . , N;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) is an element of 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 Chem-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. (C) 2019 Elsevier Masson SAS. All rights reserved.