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Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation. (English) Zbl 1018.58008
Summary: 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 $$\{p_1,\ldots,p_n\}$$ satisfies $\sum_{k\neq j} \log (|p_j-p_k|)\text{ is independent of }j.\tag{1}$ 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 $$p_j$$, $$j=1,2,\ldots,N$$.

##### MSC:
 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences 81T13 Yang-Mills and other gauge theories in quantum field theory
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