Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions

For $n\ge 3$ and $0<ϵ\le 1$, let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded smooth domain and $u_{ϵ}:\mathrm{\Omega }\subset {\mathbb{R}}^n\to {\mathbb{R}}^2$ solve the Ginzburg-Landau equation under the weak anchoring boundary condition: $$\{\begin{array}{lll}-\mathrm{\Delta }{u}_{ϵ}=\frac{1}{{ϵ}^2}(1-\mathrm{\mid }{u}_{ϵ}{\mathrm{\mid }}^2)u_{ϵ}& \mathrm{i}\mathrm{n}\mathrm{\Omega },\frac{\mathrm{\partial }{u}_{ϵ}}{\mathrm{\partial }\nu }+{\lambda }_{ϵ}(u_{ϵ}-g_{ϵ})=0& \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where the anchoring strength parameter ${\lambda }_{ϵ}=K{ϵ}^{-\alpha }$ 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 $u_{ϵ}$ as $ϵ$ goes to zero under the condition that the total modified Ginzburg-Landau energy {satisfies} $F_\epsilon(u_\epsilon,\Omega)\le M|\log\epsilon|$forsome$M>0$.