Youngae Lee

Existence of mixed type solutions in the Chern–Simons gauge theory of rank two in R2

Abstract We consider the self-dual equations arising from the Chern–Simons gauge theory of rank 2 such as the S U ( 3 ) , S O ( 5 ) , and G 2 Chern–Simons model in R 2 . There are three possible types of solutions in these theories, namely, topological, nontopological, and mixed type solutions. The existence of a mixed type solution for an arbitrary configuration of vortex points has been a long-standing open problem. We construct here a family of mixed type solutions ( u 1 , e , u 2 , e ) , e > 0 for an arbitrary configuration of vortex points under a mild non-degeneracy condition. The main new idea of our construction of an approximate solution is to use different scalings for different components. In the process of the finite dimensional reduction, all estimates for this particular construction become rather delicate.

Bubbling mixed type solutions of the SU(3) models on a torus

We consider a nonlinear elliptic system arising in the study of the SU(3) Chern-Simons model on a two-dimensional flat torus Ω. Solutions of this SU(3) Chern Simons system could be classified as topological, mixed-type, and non-topological solutions. In this paper, we succeed to construct bubbling mixed type solutions. This is the first result for such example in the literature. The analysis for the existence of such solution provides some important insights for us to develop the asymptotic analysis of classifying all mixed-type solution.

Sharp estimates for solutions of mean field equations with collapsing singularity

ABSTRACT The pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26], and Bartolucci-Tarantello [3] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.

Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system

In this paper, we consider a skew-symmetric Chern-Simons system problem with a coupling parameter. Our main goal is that, when the coupling parameter is small, the topological type solutions to this system problem are uniquely determined by the location of their vortex points. This result follows by the bubbling analysis and the non-degeneracy of linearized equations.