  We investigate the problem of existence and uniqueness of ground states at
fixed mass for two families of focusing nonlinear Schr\"odinger equations on
the line.
  The first family consists of NLS with power nonlinearities concentrated at a
point. For such model, we prove existence and uniqueness of ground states at
every mass when the nonlinearity power is $L^2-$subcritical and at a threshold
value of the mass in the $L^2-$critical regime.
  The second family is obtained by adding a standard power nonlinearity to the
previous setting. In this case, we prove existence and uniqueness at every mass
in the doubly subcritical case, namely when both the powers related to the
pointwise and the standard nonlinearity are subcritical. If only one power is
critical, then existence and uniqueness hold only at masses lower than the
critical mass associated to the critical nonlinearity. Finally, in the doubly
critical case ground states exist only at critical mass, whose value results
from a non--trivial interplay between the two nonlinearities.
