  We consider the following perturbed critical Dirichlet problem involving the
Hardy-Schr\"odinger operator on a smooth bounded domain $\Omega \subset
\mathbb{R}^N$, $N\geq 3$, with $0 \in \Omega$: $$ \left\{
\begin{array}{ll}-\Delta u-\gamma \frac{u}{|x|^2}-\epsilon
u=|u|^{\frac{4}{N-2}}u &\hbox{in }\Omega u=0 & \hbox{on }\partial \Omega,
\end{array}\right. $$ when $\epsilon>0$ is small and $\gamma< {(N-2)^2\over4}$.
Setting $ \gamma_j=
\frac{(N-2)^2}{4}\left(1-\frac{j(N-2+j)}{N-1}\right)\in(-\infty,0]$ for $j \in
\mathbb{N},$ we show that if $\gamma\leq \frac{(N-2)^2}{4}-1$ and $\gamma \neq
\gamma_j$ for any $j$, then for small $\epsilon$, the above equation has a
positive --non variational-- solution that develops a bubble at the origin. If
moreover $\gamma<\frac{(N-2)^2}{4}-4,$ then for any integer $k \geq 2$, the
equation has for small enough $\epsilon$, a sign-changing solution that
develops into a superposition of $k$ bubbles with alternating sign centered at
the origin. The above result is optimal in the radial case, where the condition
that $\gamma\neq \gamma_j$ is not necessary. Indeed, it is known that, if
$\gamma > \frac{(N-2)^2}{4}-1$ and $\Omega$ is a ball $B$, then there is no
radial positive solution for $\epsilon>0$ small. We complete the picture here
by showing that, if $\gamma\geq \frac{(N-2)^2}{4}-4$, then the above problem
has no radial sign-changing solutions for $\epsilon>0$ small. These results
recover and improve what is known in the non-singular case, i.e., when
$\gamma=0$.
