Sign-Changing Solutions for Critical Equations with Hardy Potential

We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr"odinger operator on a smooth bounded domain $Omega subset mathbbR^N$, $Ngeq 3$, with $0 in Omega$: $$ left eginarrayll-Delta u-gamma racu|x|^2-epsilon u=|u|^rac4N-2u &hboxin Omega u=0 & hboxon partial Omega, endarray ight. $$ when $epsilon>0$ is small and $gamma< (N-2)^2over4$. Setting $ gamma_j= rac(N-2)^24left(1-racj(N-2+j)N-1 ight)in(-infty,0]$ for $j in mathbbN,$ we show that if $gammaleq rac(N-2)^24-1$ and $gamma eq gamma_j$ 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 $gamma rac(N-2)^24-1$ 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 $gammageq rac(N-2)^24-4$, 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 $gamma=0$.