Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes

Abstract We construct one-parameter families of static spherically symmetric asymptotically anti-de Sitter black hole solutions $$(\mathcal {M},g_{\epsilon },\phi _{\epsilon })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mi>ϵ</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>ϵ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to the Einstein–Maxwell–(charged) Klein–Gordon equations. Each family bifurcates off a sub-extremal Reissner–Nordström-AdS spacetime $$(\mathcal {M},g_{0},\phi _{0}\equiv 0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>g</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>≡</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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.