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For $N\geq 4$, we let $\Omega$ be a bounded domain of $\mathbb{R}^N$ and$\Gamma$ be a closed curve contained in $\Omega$. We study existence ofpositive solutions $u \in H^1_0\left(\Omega\right)$ to the equation\begin{equation} -\Delta u+hu = \lambda\rho^{-s_1}_\Gammau^{2^*_{s_1}-1}+\rho^{-s_2}_\Gamma u^{2^*_{s_2}-1} \qquad \textrm{ in } \Omega\qquad (0.1)\end{equation} where $h$ is a continuous function and $\rho_\Gamma$is the distance function to $\Gamma$. We prove the existence of a mountain passsolution for this Euler-Lagrange equation depending on the local geometry ofthe curve and the potential $h$.\\ In this paper, we also study existence,symmetry and decay estimates of the positive entire solutions of (0.1) with$\Omega=\mathbb{R}^N$ and $\Gamma$ the real line.

A Nonlinear elliptic PDE involving two Hardy-Sobolev critical exponents in domains with curve singularity