Bubbling Solutions for the SU(3) Chern‐Simons Model on a Torus

We consider the following nonlinear system derived from the SU(3) Chern‐Simons models on a torus Ω:$$(0.1)\,\,\,\left\{ \matrix{ \Delta u_1 - {1 \over {\varepsilon ^2 }}(4e^{2u_1 } - 2e^{2u_2 } - 2e^{u_1 } + e^{u_2 } - e^{u_2 } - e^{u_1 + u_2 } ) \hfill \cr \,\,\,\,\,\,\,\,\, = 4\pi \sum\nolimits_{j = 1}^{N_2 } {\delta p_j^1 } , \hfill \cr \Delta u_2 - {1 \over {\varepsilon ^2 }}(4e^{2u_2 } - 2e^{2u_1 } - 2e^{u_2 } + e^{u_1 } - e^{u_1 + u_2 } ) \hfill \cr \,\,\,\,\,\,\,\,\,\, = 4\pi \sum\nolimits_{j - 1}^{N_2 } {\delta p_j^2 ,} \hfill \cr} \right.$$ where $\delta_p$ denotes the Dirac measure at $p\in\Omega$ . When $\{p_j^1\}_1^{N_1}= \{p_j^2\}_1^{N_2}$ , if we look for a solution with $u_1=u_2=u$ , then (0.1) is reduced to the Chern‐Simons‐Higgs equation:$$(0.2)\,\,\Delta u + {1 \over {\varepsilon ^2 }}e^u (1 - e^u ) = 4\pi \sum\limits_{j = 1}^N {\delta p_j } .$$ The existence of bubbling solutions to (0.1) has been a longstanding problem. In this paper, we prove the existence of such solutions such that $u_1\ne u_2$ even if $\{p_j^1\}_1^{N_1}=\{p_j^2\}_1^{N_2}$ . © 2012 Wiley Periodicals, Inc.