Abstract We construct one-parameter families of static spherically symmetric asymptotically anti-de Sitter black hole solutions (M, g, ) (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, 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.
Weihao Zheng (2026) studied this question.