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 $dmu = x_N ^{alpha } , dx$, with $alpha in (-1,0)$. Some results on stability and nonexistence for weighted isoperimetric problems on $mathbb{R}^N $ are also obtained