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, $α>8/3$, beyond which the equilibrium solutions are unstable.

7 pages, 3 figures, version accepted by Phys Rev D, added a relativistic generalization of the scalar field