  The cold dark matter (CDM) candidate with weakly interacting massive
particles can successfully explain the observed dark matter relic density in
cosmic scale and the large-scale structure of the Universe. However, a number
of observations at the satellite galaxy scale seem to be inconsistent with CDM
simulation. This is known as the small-scale problem of CDM. In recent years,
it has been demonstrated that self-interacting dark matter (SIDM) with a light
mediator offers a reasonable explanation for the small-scale problem. We adopt
a simple model with SIDM and focus on the effects of Sommerfeld enhancement. In
this model, the dark matter candidate is a leptonic scalar particle with a
light mediator. We have found several regions of the parameter space with
proper masses and coupling strength generating a relic density that is
consistent with the observed CDM relic density. Furthermore, this model
satisfies the constraints of recent direct searches and indirect detection for
dark matter as well as the effective number of neutrinos and the observed
small-scale structure of the Universe. In addition, this model with the favored
parameters can resolve the discrepancies between astrophysical observations and
$N$-body simulations.
