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 $(N\times N)$-system: \Delta u_i=\lambda\left(\sum_{j=1}^N\sum_{k=1}^N K_{kj}K_{ji}\re^{u_j}\re^{u_k}-\sum_{j=1}^N K_{ji}\re^{u_j}\right)+4\pi\sum_{j=1}^{n_i}\delta_{p_{ij}},\quad i=1,\dots, N; over a doubly periodic domain $\Omega$, with coupling matrix $K$ given by the Cartan matrix of $SU(N+1),$ (see \eqref{k1} 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 \cite{haya1} the authors managed to obtain the existence of one doubly periodic solution via a minimisation procedure, in the spirit of \cite{nota} . Our main contribution in this paper is to show (as in \cite{nota}) 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 $N\ge 6$.Comment: 34 page

On non-topological solutions for planar Liouville Systems of Toda-type

Motivated by the study of non abelian Chern Simons vortices of non topological type in Gauge Field Theory, we analyse the solvability of planar Liouville systems of Toda type in presence of singular sources. We identify necessary and sufficient conditions on the "flux" pair which ensure the radial solvability of the system. Since the given system includes the (integrable) 2 X 2 Toda system as a particular case, thus we recover the existence result available in this case. Our method relies on a blow-up analysis, which even in the radial setting, takes new turns compared with the single equation case

Radial symmetry and symmetry breaking for some interpolation inequalities

We analyze the radial symmetry of extremals for a class of interpolation inequalities known as Caffarelli-Kohn-Nirenberg inequalities, and for a class of weighted logarithmic Hardy inequalities which appear as limiting cases of the first ones. In both classes we show that there exists a continuous surface that splits the set of admissible parameters into a region where extremals are symmetric and a region where symmetry breaking occurs. In previous results, the symmetry breaking region was identified by showing the linear instability of the radial extremals. Here we prove that symmetry can be broken even within the set of parameters where radial extremals correspond to local minima for the variational problem associated with the inequality. For interpolation inequalities, such a symmetry breaking phenomenon is entirely new

Chern--Simons Vortices in the Gudnason Model

We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the ${\cal N}=2$ supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern--Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an-inequality constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.Comment: 39 page

On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities

In this paper we prove some new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities, in any dimension larger or equal than two