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

Abstract

For $N\geq 4$, we let $Ω$ be a bounded domain of $\mathbb{R}^N$ and $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ 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 \textrm{ in } Ω\end{equation} where $h$ is a continuous function and $ρ_Γ$ is the distance function to $Γ$. 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 $h$.