Existence Of Non-topological Solutions Research Articles

Blow up at infinity in the SU(3) Chern-Simons model, part I

We consider non-topological solutions of a nonlinear elliptic system problem (see (1.4) below) derived from the SU(3) Chern-Simons models in R2. The existence of non-topological solutions even for radial symmetric case has been a long standing open problem. Recently, Choe, Kim, and Lin in [7,8] showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in [7,8] cannot work for an arbitrary configuration of vortex points. In this paper, we develop a new approach by using different scalings for different components of the system to construct a family of non-topological solutions, which blows up at infinity.

On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory

In this paper, by constructing a family of approximation solutions and applying a specific version of the Implicit Function Theorem (please see, e.g. [ 18 ]), we prove the existence of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory.

Self-dual radial non-topological solutions to a competitive Chern–Simons model

We study a non-Abelian Chern–Simons system of rank 2:(Δu1Δu2)+K(eu1eu2)−K(eu100eu2)K(eu1eu2)=(4πN1δ04πN2δ0)in R2, where N1,N2∈N∪{0}, δ0 is the Dirac measure at 0, and K=(aij) is a 2×2 matrix satisfying a11,a22>0, a12,a21<0 and det⁡K>0, including the Cartan matrix B2. The existence of non-topological solutions has remained a long-standing open problem. Here by applying the degree theory, we prove the existence of radial non-topological solutions (u1,u2) satisfying the prescribed asymptotic condition uk(x)=−2αkln⁡|x|+O(1) as |x|→∞ for some αk>1. We also construct bubbling solutions to show that the range of (α1,α2) is optimal in some sense. This generalizes a recent work by Choe, Kim and the second author, where the SU(3) case (i.e. K is the Cartan matrix A2) was investigated.

Uniqueness of Topological Solutions and the Structure of Solutions for the Chern-Simons System with Two Higgs Particles

The existence of topological solutions for the Chern-Simons equation with two Higgs particles has been proved by Lin, Ponce and Yang [16]. However, both the uniqueness problem and the existence of non-topological solutions have been left open. In this paper, we consider the case of one vortex at origin. Among others, we prove the uniqueness of topological solutions and give a complete study of the radial solutions, in particular, the existence of some non-topological solutions.