  Self-interacting dark matter has been proposed as a solution to the
small-scale structure problems, such as the observed flat cores in dwarf and
low surface brightness galaxies. If scattering takes place through light
mediators, the scattering cross section relevant to solve these problems may
fall into the non-perturbative regime leading to a non-trivial velocity
dependence, which allows compatibility with limits stemming from cluster-size
objects. However, these models are strongly constrained by different
observations, in particular from the requirements that the decay of the light
mediator is sufficiently rapid (before Big Bang Nucleosynthesis) and from
direct detection. A natural solution to reconcile both requirements are
inelastic endothermic interactions, such that scatterings in direct detection
experiments are suppressed or even kinematically forbidden if the mass
splitting between the two-states is sufficiently large. Using an exact solution
when numerically solving the Schr\"odinger equation, we study such scenarios
and find regions in the parameter space of dark matter and mediator masses, and
the mass splitting of the states, where the small scale structure problems can
be solved, the dark matter has the correct relic abundance and direct detection
limits can be evaded.
