Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions

For $n\ge 3$ and $0<\epsilon\le 1$, let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $u_\epsilon:\Omega \subset\R^n\to \mathbb R^2$ solve the Ginzburg-Landau equation under the weak anchoring boundary condition: $$\begin{cases} -\Delta u_\epsilon=\frac{1}{\epsilon^2}(1-|u_\epsilon|^2)u_\epsilon &\ {\rm{in}}\ \ \Omega, \frac{\partial u_\epsilon}{\partial\nu}+\lambda_\epsilon(u_\epsilon-g_\epsilon)=0 & \ {\rm{on}}\ \ \partial\Omega, \end{cases} $$ where the anchoring strength parameter $\lambda_\epsilon=K\epsilon^{-\alpha}$ for some $K>0$ and $\alpha\in [0,1)$, and $g_\epsilon\in C^2(\partial\Omega, \mathbb S^1)$. Motivated by the connection with the Landau-De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the {asymptotic behavior} of $u_\epsilon$ as $\epsilon$ goes to zero under the condition that the total modified Ginzburg-Landau energy {satisfies} $F_\epsilon(u_\epsilon,\Omega)\le M|\log\epsilon|$ for some $M>0$.