  We provide an exact hairy black hole solution to an $n+1$ dimensional complex
scalar field model coupled with gravity. The model is characterized by a
potential with one parameter. Depending on the magnitude of this parameter, the
effective mass square of the scalar field can be both positive and negative. No
matter what case it is, there are temperature ranges in which the hairy black
hole is thermodynamically stable against decaying into its no-hair counterpart.
We use this solution as an AdS/CFT superconductor model and study its
conductivity-frequency relation and find typical features observed in other
AdS/CFT superconductor models.
