Influence of the curvature in the existence of solutions for a two Hardy-Sobolev critical exponents

For $N\ge 4$, we let $\mathrm{\Omega }$ be a bounded domain of ${\mathbb{R}}^N$ and $\mathrm{\Gamma }$ be a closed curve contained in $\mathrm{\Omega }$. We study existence of positive solutions $u\in H_0^1\left(\mathrm{\Omega }\right)$ to the equation \begin{equation}\label{Atusi} -Δu+hu=λρ^{-s_1}Γu^{2^*{s_1}-1}+ρ^{-s_2}Γu^{2^*{s_2}-1} \qquad $\text{ in }$ Ω\end{equation} where $h$ is a continuous function and ${\rho }_{\mathrm{\Gamma }}$ is the distance function to $\mathrm{\Gamma }$. We prove the existence of a mountain pass solution for this Euler-Lagrange equation depending on the local geometry of the curve and the potential $h$.