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 × N) -system: Δ u i = λ (∑ j = 1 N ∑ k = 1 N K k j K j i e u j e u k − ∑ j = 1 N K j i e u j ) + 4 π ∑ j = 1 n i δ p i j , i = 1 , ... , N ; over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of S U (N + 1) , (see (1.2) below). Here, λ > 0 is the coupling parameter, δ p is the Dirac measure with pole at p and n i ∈ 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. [ABSTRACT FROM AUTHOR]