Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes

We construct one-parameter families of static spherically symmetric asymptotically anti-de Sitter black hole solutions $(\mathcal{M},g_{\epsilon },{\phi }_{\epsilon })$ to the Einstein-Maxwell-(charged) Klein-Gordon equations. Each family bifurcates off a sub-extremal Reissner-Nordström-AdS spacetime $(\mathcal{M},g_0,{\phi }_0\equiv 0)$. For a co-dimensional one set of black hole parameters, we show that Dirichlet (respectively Neumann) boundary conditions can be imposed for the scalar field. The construction provides a counter-example to a version of the no-hair conjecture in the context of a negative cosmological constant. Our result is based on our companion work [W. Zheng, \emph{Exponentially-growing Mode Instability on the Reissner-Nordström-Anti-de-Sitter black holes}], in which the existence of linear hair and growing mode solutions have been established. In the charged scalar field case, our result provides the first rigorous mathematical construction of the so-called holographic superconductors, which are of particular significance in high-energy physics.