The sterile neutrino is a viable dark matter candidate that can be produced in the early Universe via non-equilibrium processes, and would therefore possess a highly non-thermal spectrum of primordial velocities. In this paper we analyse the process of structure formation with this class of dark matter particles. To this end we construct primordial dark matter power spectra as a function of the lepton asymmetry, $L_6$, that is present in the primordial plasma and leads to resonant sterile neutrino production. We compare these power spectra with those of thermally produced dark matter particles and show that resonantly produced sterile neutrinos are much colder than their thermal relic counterparts. We also demonstrate that the shape of these power spectra is not determined by the free-streaming scale alone. We then use the power spectra as an input for semi-analytic models of galaxy formation in order to predict the number of luminous satellite galaxies in a Milky Way-like halo. By assuming that the mass of the Milky Way halo must be no more than $2times10^{12}M_{odot}$ (the adopted upper bound based on current astronomical observations) we are able to constrain the value of $L_6$ for $M_sle 8$~keV. We also show that the range of $L_6$ that is in best agreement with the 3.5~keV line (if produced by decays of 7~keV sterile neutrino) requires that the Milky Way halo has a mass no smaller than $1.5times10^{12}M_{odot}$. Finally, we compare the power spectra obtained by direct integration of the Boltzmann equations for a non-resonantly produced sterile neutrino with the fitting formula of Viel~et~al. and find that the latter significantly underestimates the power amplitude on scales relevant to satellite galaxies.
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