On planar selfdual electroweak vortices (Q1888407)

The authors deal with the analysis of self-dual electroweak vortices that can be reduced to the study of the following elliptic system \[ \begin{aligned} -\Delta u_1&= 4g^2 e^{u_1}+ g^2 e^{u_2}- 4\pi \sum^m_{k=1} n_k\delta(z- z_k),\\ \Delta u_2&= {g^2\over 2\cos^2\theta} (e^{u_2}- \varphi^2_0)+ 2g^2 e^{u_1},\end{aligned}\tag{1} \] where \(\varphi_0\) is a given positive parameter, \(g\) is the \(\text{SU}(2)\)-coupling constant, and \(\theta\in (0,{\pi\over 2})\) is the so-called ``Weinberg angle'', related with the \(U(1)\)-coupling constant \(g_*\) via the relation \(\cos\theta= {g\over (g^2+ g^2_*)^{1/2}}\). The authors mainly interested in planar vortex-type configurations, and consider (1) over \(\mathbb{R}^2\) subject to appropriate decay assumptions at infinity.