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.