  We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$
continuous functions on $\Omega$. Let $\Gamma$ be a closed curve contained in
$\Omega$. We study existence of positive solutions $u \in
H^1_0\left(\Omega\right)$ to the perturbed Hardy-Sobolev equation: $$ -\Delta
u+h u+bu^{1+\delta}=\rho^{-\sigma}_\Gamma u^{2^*_\sigma-1} \qquad \textrm{ in }
\Omega, $$ where $2^*_\sigma:=\frac{2(N-\sigma)}{N-2}$ is the critical
Hardy-Sobolev exponent, $\sigma\in [0,2)$, $0< \delta<\frac{4}{N-2}$ and
$\rho_\Gamma$ is the distance function to $\Gamma$. We show that the existence
of minimizers does not depend on the local geometry of $\Gamma$ nor on the
potential $h$. For $N=3$, the existence of ground-state solution may depends on
the trace of the regular part of the Green function of $-\Delta+h$ and or on
$b$. This is due to the perturbative term of order ${1+\delta}$.
