Quartic scalar clouds on fixed Kerr backgrounds

We study stationary nonlinear clouds of a complex scalar field on fixed Kerr and Schwarzschild backgrounds, with an unbounded scalar potential. On Kerr backgrounds we find finite-amplitude Q-clouds satisfying the synchronisation condition ω=mΩH. They occupy two-dimensional regions of the (ΩH,MK) plane and reduce to the corresponding linear clouds as the scalar field amplitude tends to zero. The quartic domain can continue to ΩH=0, where both the synchronised frequency and the Noether charge vanish and the solutions become static nonlinear scalar clouds on a fixed Schwarzschild background. We also compute the existence domain for a normalised, axion-inspired cosine potential.