  We show the counter-intuitive fact that some weighted isoperimetric problems
on the half-space $ \mathbb{R}^N _+ $, for which half-balls centered at the
origin are stable, have no solutions. A particular case is the measure $d\mu =
x_N ^{\alpha } \, dx$, with $\alpha \in (-1,0)$. Some results on stability and
nonexistence for weighted isoperimetric problems on $\mathbb{R}^N $ are also
obtained.
