Hardy-Sobolev inequality with singularity a curve

Abstract

We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\geq 3$, and $h$ a continuous function on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u\in H^1_0(\Omega)$ to the equation $$ -\Delta u+h u=\rho^{-\sigma}_\Gamma u^{2^*_\sigma-1} \qquad \textrm{ in } \Omega $$ where $2^*_\sigma:=\frac{2(N-\sigma)}{N-2}$, $\sigma\in (0,2)$, and $\rho_\Gamma$ is the distance function to $\Gamma$. For $N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-\Delta+h$ is positive at a point of the curve.