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

Let $N\geq2$ 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

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 $\partial\Omega\backslash\Gamma$.