Variational reduction for semi-stiff Ginzburg-Landau vortices. (Q2280480)

In this article, the existence of solutions to the Ginzburg-Landau problem \[-\varepsilon^2\Delta u=(1-|u|^2)u\quad\text{ in }\Omega,\] is investigated. Here \(\Omega\subset\mathbb{R}^2\) is a smooth and bounded domain. The above equation is complemented with the boundary conditions \[|u|=1\quad\text{ and }\quad\mathrm{Im}(\overline{u}\partial_\nu u)=0\qquad\text{ on }\partial\Omega.\] The approach relies on a variational reduction method in the spirit of [\textit{M. del Pino} et al., J. Funct. Anal. 239, No. 2, 497--541 (2006; Zbl 1387.35561)]. In particular, it is shown that: \begin{enumerate}\item If \(\Omega\) is simply connected, then a solution with degree one on the boundary always exists;\item If \(\Omega\) is not simply connected then for any \(k\geq 1\), a solution with \(k\) vortices of degree one exists.\end{enumerate}