Author Description

For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system has the following form: ∗{@l@ l@Δu1 + (∑i=12K1xieui − ∑i = 12∑j = 12euiK1ieujKij) = 4π∑j = 1N1δpjΔu2. in R2, * where K is the Cartan matrix of rank 2 2. There are three Cartan matrix of rank 2: A2, B2 and G2. A long-standing open problem for this equation is the question of the existence of non-topological solutions. In this paper, we consider the A2 and B2 case. We prove the existence of non-topological solutions under the condition that either N2∑N1j=1pj = N1∑N2j=1qj or N2∑N1j=1pj ≠ N1∑N2j=1qj and N1,N2 > 1, |N1−N 2|≠1 N1,N2>1,|N1−N2|≠1. We solve this problem by a perturbation from the corresponding A2 and B2 Toda system with one singular source.