4 1. INTRODUCTION 1.3. Main results In this paper, we give an affirmative answer to the existence of non-topological solutions for the system with Cartan matrix A2 and B2. Our main theorem can be stated as follows. Theorem 1.1. Let {pj}j=1, N1 {qj}j=1 N2 R2. If either (a) N2 N1 j=1 pj = N1 N2 j=1 qj or (b) N2 N1 j=1 pj = N1 N2 j=1 qj and N1,N2 1, |N1 N2| = 1, then there exists a non-topological solution (u1,u2) of problem ( 1.4) and ( 1.5) respectively. Non-topological solutions play very important role in the bubbling analysis of solutions to (1.2). Therefore, our result is only the first step towards understanding the solution structure of non-topological solution of (1.2). For further study on non-topological solutions for the Abelian case, we refer to [3] and [5]. We will prove Theorem 1.1 in three cases which we describe below: Assumption (i): N2 N1 j=1 pj = N1 N2 j=1 qj , N1 = N2 (1.8) Assumption (ii): N2 N1 j=1 pj = N1 N2 j=1 qj , N1 = N2 , N1,N2 1 (1.9) or N2 N1 j=1 pj = N1 N2 j=1 qj , |N1 N2| = 1 , N1,N2 1 Assumption (iii):N2 N1 j=1 pj = N1 N2 j=1 qj, N1 = N2, N1 = 1 or N2 = 1. (1.10) If we can prove the existence of non-topological solutions under the above three assumptions separately, then it is easy to see that Theorem 1.1 is proved. So in the following, we will prove the theorem under the three assumptions respectively. 1.4. Sketch of the proof for the case A2 In the following, we will outline the sketch of our proof for the A2 case, the proof for B2 is similar. First, let us recall the proof of the existence of non-topological solutions of Chae and Imanuvilov [2] for single SU(2) equation (1.7). Getting rid of the Dirac measure, (1.7) is equivalent to (1.11) Δu + Πj=1|z N pj|2eu = Πj=1|z N pj|4e2u in R2. After a suitable scaling transformation, the equation (1.11) becomes (1.12) Δ ˜ U + Πj=1|z N εpj|2e ˜ U = ε2Πj=1|z N εpj|4e2 ˜ U in R2
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