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 (M,gε,φ ε) ( M , g ϵ , ϕ ϵ ) to the Einstein–Maxwell–(charged) Klein–Gordon equations. Each family bifurcates off a sub-extremal Reissner–Nordström-AdS spacetime (M,g₀,φ ₀≡ 0) ( M , g 0 , ϕ 0 ≡ 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 (Zheng in Commun. Math. Phys. 406:260, 2024), 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.