Hardy-Sobolev inequality with singularity a curve

Abstract

We consider a bounded domain $\Omega$ of ${\mathbb{R}}^N$, $N\ge 3$, and $h$ a continuous function on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u\in H_0^1(\Omega )$ to the equation $$-\Delta u+hu={\rho }_{\Gamma }^{-\sigma }{u}^{2_{\sigma }^{\ast }-1}\text{ in }\Omega$$ where $2_{\sigma }^{\ast }:=\frac{2(N-\sigma )}{N-2}$, $\sigma \in (0,2)$, and ${\rho }_{\Gamma }$ is the distance function to $\Gamma$. For $N\ge 4$, 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.

27 pages