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In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs -system:
over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of , (see (1.2) below). Here, is the coupling parameter, is the Dirac measure with pole at p and , for . When 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 , only recently in  the authors managed to obtain the existence of one doubly periodic solution via a minimization procedure, in the spirit of . Our main contribution in this paper is to show (as in ) that actually the given system admits a second doubly periodic solutions of “Mountain-pass” type, provided that . 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 .
Cite this article
Xiaosen Han, Gabriella Tarantello, Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 5, pp. 1401–1430DOI 10.1016/J.ANIHPC.2019.01.002