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

Abstract

<p style='text-indent:20px;'>Our purpose in this paper is to classify the non-topological solutions of equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\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) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \{\delta_{p_i}\}_{i = 1}^k $\end{document}</tex-math></inline-formula> (resp. <inline-formula><tex-math id="M2">\begin{document}$ \{\delta_{q_j}\}_{j = 1}^l $\end{document}</tex-math></inline-formula>) are Dirac masses concentrated at the points <inline-formula><tex-math id="M3">\begin{document}$ \{p_i\}_{i = 1}^k $\end{document}</tex-math></inline-formula>, (resp. <inline-formula><tex-math id="M4">\begin{document}$ \{q_j\}_{j = 1}^l $\end{document}</tex-math></inline-formula>), <inline-formula><tex-math id="M5">\begin{document}$ n_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ m_j $\end{document}</tex-math></inline-formula> are positive integers. Denote <inline-formula><tex-math id="M7">\begin{document}$ N = \sum^k_{i = 1}n_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ M = \sum^l_{j = 1}m_j $\end{document}</tex-math></inline-formula> satisfying that <inline-formula><tex-math id="M9">\begin{document}$ N-M&gt;1 $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>Problem <inline-formula><tex-math id="M10">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> arises from gauged sigma models and we first construct an extremal non-topological solution <inline-formula><tex-math id="M11">\begin{document}$ u $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M12">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> with asymptotic behavior</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ u(x) = -2\ln |x|-2\ln\ln|x|+O(1)\quad{\rm as}\quad |x|\to+\infty $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and with total magnetic flux <inline-formula><tex-math id="M13">\begin{document}$ 4\pi (N-M-1) $\end{document}</tex-math></inline-formula>. And then we do the classification for non-topological solutions of <inline-formula><tex-math id="M14">\begin{document}$ (E) $\end{document}</tex-math></inline-formula> 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.</p>