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Variational reduction for semi-stiff Ginzburg-Landau vortices. (English) Zbl 1449.35213
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 [M. del Pino et al., J. Funct. Anal. 239, No. 2, 497–541 (2006; Zbl 1387.35561)]. In particular, it is shown that:
1
If $$\Omega$$ is simply connected, then a solution with degree one on the boundary always exists;
2
If $$\Omega$$ is not simply connected then for any $$k\geq 1$$, a solution with $$k$$ vortices of degree one exists.
##### MSC:
 35J50 Variational methods for elliptic systems 35J66 Nonlinear boundary value problems for nonlinear elliptic equations