Equilibrium core and vortex solutions of Bose-Einstein condensate dark matter around a black hole

We present the construction of stationary solutions of Bose-Einstein condensate dark matter (BECDM) around a point-like gravitational source representing a black hole. The problem is formulated for general axisymmetric configurations, and we focus on two cases: the ground-state core solution and the first nonzero winding number configuration corresponding to a line vortex solution. The stationary equations are solved using an imaginary-time approach, which enables the construction of families of solutions across a wide range of self-interaction and black hole masses. We analyze the impact of these parameters on the density distribution and on the stability properties of the solutions, assessing stability through the turning point criterion based on the enthalpy functional, which allows us to identify stable and unstable branches along each family of solutions. It has been shown in the past that spherical core solutions act as attractors in the collapse of BECDM around black holes in the non-interacting case ($g=0$), supporting their astrophysical relevance. In the present work, the existence of a maximum mass for configurations with attractive self-interaction ($g