Some weighted isoperimetric problems on $\mathbb{R}^N _+ $ with stable half balls have no solutions

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.