Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation.

*(English)*Zbl 1018.58008Summary: In this article, we construct self-dual $N$-vortex solutions with a large magnetic flux ${\Phi}$ of $(2+1)$-dimensional relativistic Chern-Simons model, provided that the coupling constant $\kappa $ is small and the sites of vorticity $\{{p}_{1},...,{p}_{n}\}$ satisfies

$$\sum _{k\ne j}log\left(\right|{p}_{j}-{p}_{k}\left|\right)\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{independent}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}j\xb7\phantom{\rule{2.em}{0ex}}\left(1\right)$$

Our solutions exhibit the bubbling phenomenon at each ${p}_{j}$. Near each vortex ${p}_{j}$, solutions are locally asymptotically symmetric with respect to ${p}_{j}$, and the curvature ${F}_{12}$ tends to a sum of Dirac measures as $\kappa $ tends to zero. By a heuristic argument, it is shown that (1) is also a necessary condition for the existence of multi-vortex solutions which has a locally asymptotically symmetric vortex at ${p}_{j}$, $j=1,2,...,N$.