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 nonstandard 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.
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