1.2. PREVIOUS RESULTS 3 and (1.5) Δu1 + 2eu1 eu2 (4e2u1 2e2u2 ) = N1 j=1 δpj Δu2 + 2eu2 2eu1 (4e2u2 4e2u1 2eu1+u2 ) = N2 j=1 δqj in R2, respectively. 1.2. Previous results In the literature, a solution u = (u1,u2) to system (1.2) is called a topological solution if u satisfies ua(z) ln 2 j=1 (K−1)aj as |z| +∞, a = 1, 2, and is called a non-topological solution if u satisfies ua(z) −∞ as |z| +∞, a = 1, 2. (1.6) The existence of topological solutions with arbitrary multiple vortex points was proved by Yang [32] more than fifteen years ago, not only for (1.4) and (1.5), but also for general Cartan matrix including SU(N + 1) case, N 1. However, the existence of non-topological solutions is more difficult to prove. The first result was due to Chae and Imanuvilov [2] for the SU(2) Abelian Chern-Simons equation which is obtained by letting u1(z) = u2(z) = u(z) in the system (1.4) where u satisfies (1.7) Δu + eu(1 eu) = N j=1 δpj in R2. Equation (1.7) is the SU(2) Chern-Simons equation for the Abelian case. This relativistic Chern-Simons model was proposed by Jackiw-Weinberg [12] and Hong- Kim-Pac [11]. For the past more than twenty years, the existence and multiplicity of solutions to (1.7) with different nature (e.g. topological, non-topological, period- ically constrained etc.) have been studied, see [1], [2], [3], [4], [5], [17], [18], [19], [20], [24], [25], [26], [27], [28] and references therein. In [2], Chae and Imanuvilov proved the existence of non-topological solutions for (1.7) for any vortex points (p1, ..., pN ). For the question of existence of non- topological solutions for the system (1.4), an “answer” was given by Wang and Zhang [30] but their proof contains serious gaps. In fact they used a special solution of the Toda system as the approximate solution, but they did not have the full non- degeneracy of the linearized equation of the Toda system and their analysis for the linearized equation is incorrect. Thus, the existence of non-topological solutions has remained a long-standing open problem. Even for radially symmetric solutions (the case when all the vortices coincide), the ODE system of (1.4) or (1.5) is much subtle than equation (1.7). The classification of radial solution is an important issue for future study as long as bubbling solutions are concerned.
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