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

Abstract

For $N\ge 4$, 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 $u\in H_0^1\left(\Omega \right)$ 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 \textrm{ in } Ω\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 $h$.

arXiv admin note: text overlap with arXiv:1702.02202