A Nonlinear elliptic PDE with curve singularity on the boundary

Let $\mathrm{\Omega }$ be a bounded domain of ${\mathbb{R}}^{N+1}$ ($N\ge 3$) with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ and $\mathrm{\Sigma }$ be a closed submanifold contained on $\mathrm{\partial }\mathrm{\Omega }$ and containing $0$. We are interesting in the existence of positive $H^1(\mathrm{\Omega })$-solution of the following Hardy-Sobolev trace type equation \begin{equation*} \begin{cases} -Δu+u=0 \qquad & $\text{ in }\mathrm{\Omega }$\\ \displaystyle\frac{\partial u}{\partial ν}= ρ_Σ^{-s} u^{q_s-1} \qquad & $\text{ on }\mathrm{\partial }\mathrm{\Omega }$, \end{cases} \end{equation*} where $\nu$ is the unit outer normal of $\mathrm{\partial }\mathrm{\Omega }$, ${\rho }_{\mathrm{\Sigma }}:\mathrm{\partial }\mathrm{\Omega }\to \mathbb{R}$ is the distance function in $\mathrm{\partial }\mathrm{\Omega }$ to the curve $\mathrm{\Sigma }$: $${\rho }_{\mathrm{\Sigma }}(x):=\underset{y\in \mathrm{\Sigma }}{\inf }{d}_{\tilde{g}}(x,y)$$ and for $0\le s<1$, $q_s:=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev exponent. The existence of solution may depend on the local geometry of the boundary $\mathrm{\partial }\mathrm{\Omega }$ and $\mathrm{\Sigma }$ at $0$ or in the shapes of the domain $\mathrm{\Omega }$ and its boundary $\mathrm{\partial }\mathrm{\Omega }$.