Existence of non-topological solutions for a skew-symmetric Chern-Simons system

Genggeng Huang; Chang-Shou Lin

paper

Existence of non-topological solutions for a skew-symmetric Chern-Simons system

arXiv:1401.1251

Abstract

We investigate the existence of non-topological solutions $(u_{1}, u_{2})$ satisfying $u_{i} (x) = - 2 β_{i} ln ∣ x ∣ + O (1), as ∣ x ∣ \to + \infty,$ such that $β_{i} > 1$ and $(β_{1} - 1) (β_{2} - 1) > (N_{1} + 1) (N_{2} + 1),$ for a skew-symmetric Chern-Simons system. By the bubbling analysis and the Leray-Schauder degree theory, we get the existence results except for a finite set of curves: $\frac{N _{1}}{β _{1} + N _{1}} + \frac{N _{2}}{β _{2} + N _{2}} = \frac{k - 1}{k}, k = 2, \dots, max (N_{1}, N_{2}) .$ This generalizes a previous work by Choe-Kim-Lin \cite{ChoeKimLin2011}.