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 favored 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.