Hardy-Sobolev inequality with singularity a curve

We consider a bounded domain $\mathrm{\Omega }$ of ${\mathbb{R}}^N$, $N\ge 3$, and $h$ a continuous function on $\mathrm{\Omega }$. Let $\mathrm{\Gamma }$ be a closed curve contained in $\mathrm{\Omega }$. We study existence of positive solutions $u\in H_0^1(\mathrm{\Omega })$ to the equation $$-\mathrm{\Delta }u+hu={\rho }_{\mathrm{\Gamma }}^{-\sigma }{u}^{2_{\sigma }^{\ast }-1}\text{ in }\mathrm{\Omega }$$ where $2_{\sigma }^{\ast }:=\frac{2(N-\sigma )}{N-2}$, $\sigma\in (0,2)$,and$\rho_\Gamma$isthedistancefunctionto$\Gamma$\mathrm{.}For$N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-\mathrm{\Delta }+h$ is positive at a point of the curve.