  We report the theoretical analysis for tuning the quantum efficiency of solid
state random lasers. Vollhardt-Woelfle theory of photonic transport in
disordered non-conserving and open random media, is coupled to lasing dynamics
and solved positionally dependent. The interplay of non-linearity and
homogeneous non-radiative frequency conversion by means of a Stokes-shift leads
to a reduction of the quantum efficiency of the random laser. At the threshold
a strong decrease of the spot-size in the stationary state is found due to the
increase of non-radiative losses. The coherently emitted photon number per unit
of modal surface is also strongly reduced. This result allows for the
conclusion that Stokes-shifts are not sufficient to explain confined and
extended mode regimes.
