  We numerically investigate free and self-interacting ultralight scalar fields
around black holes in General Relativity. We focus on complex scalar fields
$\Phi$ whose self-interactions are described by the quartic potential $V
\propto \lambda |\Phi|^4$, and ignore the black hole spin in order to
disentangle the effects of self interactions on the boson cloud. Using the
spectral solver Kadath, we compute quasi-equilibrium configurations of the
dominant eigenstates, including their backreaction on the spacetime metric. For
scenarios with $- 10^{-2} \lesssim \lambda \lesssim 10^{-2}$ we find the mass
of the self-interacting scalar cloud to be up to $\sim 70\%$ larger than that
of a free scalar cloud, though the additional backreaction effect on the
spacetime metric is only up to $\sim 1\%$ due to the low-density nature of the
bosonic configurations. In this region of parameter space we observe
approximate quadratic scalings between the mass of the cloud with $\lambda$,
the scalar field amplitude, and the couplings between these two parameters. For
systems with $\lambda$ beyond this range, the eigenfrequencies differ
sufficiently from the known free-test-field values used as inputs in our
numerical setup to make the results, though convergent, physically unreliable.
This bounds the range of $\lambda$ in which the free scalar field solution
remains a good approximation to self-interacting scalar field configurations.
Our work is among the first nonperturbative explorations of self-interacting
bosonic clouds around black holes, yielding detailed new insights into such
systems in the nonlinear regime, while also overcoming technical challenges and
quantifying limitations. Additionally, our results provide useful inputs for
fully dynamical numerical relativity simulations and for future explorations of
spinning black holes and real scalar fields.
