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

Abstract

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 of positive solutions $u \in H^1_0\left(\Omega\right)$ to the equation \begin{equation}\label{Atusi} -\Delta u+hu=\lambda\rho^{-s_1}_\Gamma u^{2^*_{s_1}-1}+\rho^{-s_2}_\Gamma u^{2^*_{s_2}-1} \qquad \textrm{ in } \Omega \end{equation} where $h$ is a continuous function and $\rho_\Gamma$ is the distance function to $\Gamma$. 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 $h$.