  We analyze the ground state phase diagram of attractive lattice bosons, which
are stabilized by a three-body onsite hardcore constraint. A salient feature of
this model is an Ising type transition from a conventional atomic superfluid to
a dimer superfluid with vanishing atomic condensate. The study builds on an
exact mapping of the constrained model to a theory of coupled bosons with
polynomial interactions, proposed in a related paper [11]. In this framework,
we focus by analytical means on aspects of the phase diagram which are
intimately connected to interactions, and are thus not accessible in a mean
field plus spin wave approach. First, we determine shifts in the mean field
phase border, which are most pronounced in the low density regime. Second, the
investigation of the strong coupling limit reveals the existence of a new
collective mode, which emerges as a consequence of enhanced symmetries in this
regime. Third, we show that the Ising type phase transition, driven first order
via the competition of long wavelength modes at generic fillings, terminates
into a true Ising quantum critical point in the vicinity of half filling.
