  For the abelian self-dual Chern-Simons-Higgs model we address existence
issues of periodic vortex configurations -- the so-called condensates-- of
non-topological type as $k \to 0$, where $k>0$ is the Chern-Simons parameter.
We provide a positive answer to the long-standing problem on the existence of
non-topological condensates with magnetic field concentrated at some of the
vortex points (as a sum of Dirac measures) as $k \to 0$, a question which is of
definite physical interest.
