Mixed Type Solutions of the {SU(3)} Models on a Torus

In this paper, we study mixed-type solutions of \({SU(3)}\) Chern–Simons system (see (1.4) below) on a two dimensional flat torus. Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), among other things, Nolasco and Tarantello obtained solutions of (1.4) as minimizers of several functionals closely related to (1.4), and showed that if \({N_1+N_2=1}\), then one of those minimizers turns out to be a mixed-type solution, that is, one component tends to \({\ln\frac{1}{2}}\) pointwise a.e. and the other component converges to a solution of a mean field equation. We call these kinds of solutions mixed-type (I) solutions. In this paper, we prove two main results: (i) the asymptotic analysis of mixed-type (I) solutions with arbitrary configuration of vortex points, and (ii) the existence of mixed-type (I) solutions under a non-degenerate condition. This non-degenerate condition also ensures some uniqueness result. In particular, our results imply that when \({N_1+N_2=1}\), there are only two mixed-type (I) solutions of (1.4).     

On the Entire Radial Solutions of the Chern–Simons SU(3) System

In this paper, we study the entire radial solutions of the self-dual equations arising from the relativistic SU(3) Chern–Simons model proposed by Kao and Lee (Phys Rev D 50:6626–6632, 1994) and Dunne (Phys Lett B 345:452–457, 1995; Nuclear Phys B 433:333–348, 1995). Understanding the structure of entire radial solutions is one of the fundamental issues for the system of nonlinear equations. In this paper, we prove that any entire radial solutions must be one of topological, non-topological and mixed type solutions, and completely classify the asymptotic behaviors at infinity of these solutions. Even for radial solutions, this classification has remained an open problem for many years. As an application of this classification, we prove that the two components u and v have intersection at most finite times.     

Bubbling Solutions for Relativistic Abelian Chern-Simons Model on a Torus

We prove the existence of bubbling solutions for the following Chern-Simons-Higgs equation:

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

The aim of this paper is to complete the program initiated in [51], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov–Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.     

A Nonlinear Elliptic PDE with Two Sobolev–Hardy Critical Exponents

In this paper, we consider the following PDE involving two Sobolev–Hardy critical exponents,

On reducible monodromy representations of some generalized Lamé equation

In this note, we compute the explicit formula of the monodromy data for a generalized Lamé equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemann–Hilbert problem.

     

Non-Topological Multi-Vortex Solutions to the Self-Dual Chern-Simons-Higgs Equation

 In this article, we construct self-dual N-vortex solutions with a large magnetic flux Φ of (2+1)-dimensional relativistic Chern-Simons model, provided that the coupling constant κ is small and the cites of vorticity satisfies

Classification and nondegeneracy of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>+1) Toda system with singular sources

We consider the following Toda system

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where γ _i >?1, δ ₀ is Dirac measure at 0, and the coefficients a _ij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:

$\sum_{j=1}^n a_{ij}\int_{\mathbb{R}^2}e^{u_j} dx = 4\pi(2+\gamma _i+\gamma_{n+1-i}), \quad\forall\;1\leq i \leq n.$

This generalizes the classification result by Jost and Wang for γ _i =0, \(\forall\;1\leq i\leq n\). (ii) We prove that if γ _i +γ _i+1+?+γ _j ?? for all 1≤i≤j≤n, then any solution u _i is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.