  We consider the Hardy-H\'enon system $-\Delta u =|x|^a v^p$, $-\Delta v
=|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in
particular with the Liouville property, i.e. the nonexistence of positive
solutions in the whole space ${\mathbb R}^N$. In view of known results, it is a
natural conjecture that this property should be true if and only if
$(N+a)/(p+1)+$ $(N+b)/(q+1)>N-2$. In this paper, we prove the conjecture for
dimension N=3 in the case of bounded solutions and in dimensions $N\le 4$ when
$a,b\le 0$, among other partial nonexistence results. As far as we know, this
is the first optimal Liouville type result for the Hardy-H\'enon system. Next,
as applications, we give results on singularity and decay estimates as well as
a priori bounds of positive solutions.
