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Classification of non-topological solutions of an elliptic equation arising from self-dual gauged Sigma model

  • * Corresponding author

    * Corresponding author
This work is supported by NNSF of China, No: 12071189 and 12001252, by the Jiangxi Provincial Natural Science Foundation, 20202BAB201005 and No: 20202ACBL201001, by the Science and Technology Research Project of Jiangxi Provincial Department of Education, No: 200307 and 200325
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  • Our purpose in this paper is to classify the non-topological solutions of equations

    $ -\Delta u +\frac{4e^u}{1+e^u} = 4\pi\sum\limits_{i = 1}^k n_i\delta_{p_i}-4\pi\sum^l\limits_{j = 1}m_j\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2,\;\;\;\;\;\;(E) $

    where $ \{\delta_{p_i}\}_{i = 1}^k $ (resp. $ \{\delta_{q_j}\}_{j = 1}^l $) are Dirac masses concentrated at the points $ \{p_i\}_{i = 1}^k $, (resp. $ \{q_j\}_{j = 1}^l $), $ n_i $ and $ m_j $ are positive integers. Denote $ N = \sum^k_{i = 1}n_i $ and $ M = \sum^l_{j = 1}m_j $ satisfying that $ N-M>1 $.

    Problem $ (E) $ arises from gauged sigma models and we first construct an extremal non-topological solution $ u $ of $ (E) $ with asymptotic behavior

    $ u(x) = -2\ln |x|-2\ln\ln|x|+O(1)\quad{\rm as}\quad |x|\to+\infty $

    and with total magnetic flux $ 4\pi (N-M-1) $. And then we do the classification for non-topological solutions of $ (E) $ with finite magnetic flux. This solves a challenging long standing problem. We believe that our approach is novel and applies to other types of equations.

    Mathematics Subject Classification: Primary: 35R06, 35A01; Secondary: 81T13.


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