  The Horava - Lifshitz (HL) theory has recently attracted a lot of interest as
a viable solution to some quantum gravity related problems and the presence of
an effective cosmological constant able to drive the cosmic speed up. We show
here that, in the weak field limit, the HL proposal leads to a modification of
the gravitational potential because of two additive terms (scaling respectively
as $r^2$ and $r^{-4}$) to the Newtonian $1/r$ potential. We then derive a
general expression to compute the rotation curve of an extended system under
the assumption that the mass density only depends on the cylindrical
coordinates $(R, z)$ showing that the HL modification induces a dependence of
the circular velocity on the mass function which is a new feature of the
theory. As a first exploratory analysis, we then try fitting the Milky Way
rotation curve using its visible components only in order to see whether the HL
modified potential can be an alternative to the dark matter framework. This
turns out not to be the case so that we argue that dark matter is still needed,
but the amount of dark matter and the dark halo density profile have to be
revised according to the new HL potential.
