Hardy-Sobolev inequality with higher dimensional singularity

For $N\ge 4$, we let $\mathrm{\Omega }$ to be a smooth bounded domain of ${\mathbb{R}}^N$, $\mathrm{\Gamma }$ a smooth closed submanifold of $\mathrm{\Omega }$ of dimension $k$ with $1\le k\le N-2$ and $h$ a continuous function defined on $\mathrm{\Omega }$. We denote by ${\rho }_{\mathrm{\Gamma }}(\cdot ):={\text{dist}}_g(\cdot ,\mathrm{\Gamma })$ the distance function to $\mathrm{\Gamma }$. For $\sigma \in (0,2)$, we study existence of positive solutions $u\in H_0^1\left(\mathrm{\Omega }\right)$ to the nonlinear equation $$-\mathrm{\Delta }u+hu={\rho }_{\mathrm{\Gamma }}^{-\sigma }{u}^{2^{\ast }(\sigma )-1}\text{in }\mathrm{\Omega },$$ where $2^{\ast }(\sigma ):=\frac{2(N-\sigma )}{N-2}$ is the critical Hardy-Sobolev exponent. In particular, we provide existence of solution under the influence of the local geometry of $\text{G}$ and the potential $h$.