  We consider the slightly subcritical elliptic problem with Hardy term $$
\left\{ \begin{aligned} -\Delta u-\mu\frac{u}{|x|^2} &=
|u|^{2^{\ast}-2-\epsilon}u &&\quad \text{in } \Omega\subset\mathbb{R}^N, \\\ u
&= 0&&\quad \text{on } \partial \Omega, \end{aligned} \right. $$ where
$0\in\Omega$ and $\Omega$ is invariant under the subgroup $SO(2)\times\{\pm
E_{N-2}\}\subset O(N)$; here $E_n$ denots the $n\times n$ identity matrix. If
$\mu=\mu_0\epsilon^\alpha$ with $\mu_0>0$ fixed and $\alpha>\frac{N-4}{N-2}$
the existence of nodal solutions that blow up, as $\epsilon\to0^+$, positively
at the origin and negatively at a different point in a general bounded domain
has been proved in \cite{BarGuo-ANS}. Solutions with more than two blow-up
points have not been found so far. In the present paper we obtain the existence
of nodal solutions with a positive blow-up point at the origin and $k=2$ or
$k=3$ negative blow-up points placed symmetrically in
$\Omega\cap(\mathbb{R}^2\times\{0\})$ around the origin provided a certain
function $f_k:\mathbb{R}^+\times\mathbb{R}^+\times I\to\mathbb{R}$ has stable
critical points; here $I=\{t>0:(t,0,\dots,0)\in\Omega\}$. If
$\Omega=B(0,1)\subset\mathbb{R}^N$ is the unit ball centered at the origin we
obtain two solutions for $k=2$ and $N\ge7$, or $k=3$ and $N$ large. The result
is optimal in the sense that for $\Omega=B(0,1)$ there cannot exist solutions
with a positive blow-up point at the origin and four negative blow-up points
placed on the vertices of a square centered at the origin. Surprisingly there
do exist solutions on $\Omega=B(0,1)$ with a positive blow-up point at the
origin and four blow-up points on the vertices of a square with alternating
positive and negative signs. The results of our paper show that the structure
of the set of blow-up solutions of the above problem offers fascinating
features and is not well understood.
