Existence of steady stable solutions for the Ginzburg-Landau equation in a domain with nontrivial topology

Abstract:

Let $N\geq 2$ and $\Omega$ $\subset \mathbb {R}^{N}$ be a bounded domain with boundary $\partial \Omega$. Let $\Gamma \subset \partial \Omega$ be closed. Our purpose in this paper is to consider the existence of stable solutions $u\in H^{1}(\Omega ,\mathcal {\mathbb {C}})$ of the Ginzburg-Landau equation \[ \left \{ \begin {array} [c]{rlll} -\Delta u(x) & = & \lambda (w_{0}^{2}(x)-\left \vert u\right \vert ^{2})u & \qquad \qquad \text {in }\Omega ,\\ u & = & g & \qquad \qquad \text {on }\partial \Omega \backslash \Gamma ,\\ \frac {\partial u}{\partial \nu } & = & 0 & \qquad \qquad \text {on }\Gamma \end {array} \right . \] where $\lambda >0,$ $w_{0}\in C^{2}(\overline {\Omega },\mathbb {\mathbb {R}^{+}})$ and $g\in C^{2}(\partial \Omega \backslash \Gamma )$ such that $\left \vert g(x)\right \vert =w_{0}(x)$ on $\partial \Omega \backslash \Gamma$.