  Axions and axion-like particles are a leading model for the dark matter in
the Universe; therefore, dark matter halos may be boson stars in the process of
collapsing. We examine a class of static boson stars with a non-minimal
coupling to gravity. We modify the gravitational density of the boson field to
be proportional to an arbitrary power of the modulus of the field, introducing
a non-standard coupling. We find a class of solutions very similar to Newtonian
polytropic stars that we denote "quantum polytropes." These quantum polytropes
are supported by a non-local quantum pressure and follow an equation very
similar to the Lane-Emden equation for classical polytropes. Furthermore, we
derive a simple condition on the exponent of the non-linear gravitational
coupling, $\alpha>8/3$, beyond which the equilibrium solutions are unstable.
