  We study the stability of static as well as of rotating and charged black
holes in (4+1)-dimensional Anti-de Sitter space-time which possess spherical
horizon topology. We observe a non-linear instability related to the
condensation of a charged, tachyonic scalar field and construct "hairy" black
hole solutions of the full system of coupled Einstein, Maxwell and scalar field
equations. We observe that the limiting solution for small horizon radius is
either a hairy soliton solution or a singular solution that is not a regular
extremal solution. Within the context of the gauge/gravity duality the
condensation of the scalar field describes a holographic
conductor/superconductor phase transition on the surface of a sphere.
