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We show the counter-intuitive fact that some weighted isoperimetric problemson the half-space $ \mathbb{R}^N _+ $, for which half-balls centered at theorigin 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 andnonexistence for weighted isoperimetric problems on $\mathbb{R}^N $ are alsoobtained.

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