  In the context of an nonuniversal $U(1)'$ extension of the standard model
free from anomalies, we introduce a complex scalar singlet candidate to be dark
matter. In addition, an extra scalar doublet and a heavy scalar singlet are
required to provide masses to all fermions and to break spontaneously the
symmetries. From unitarity and stability of the Higgs potential, we find the
full set of bounds and order relations for the scalar coupling constants. Using
recent data from the CERN-LHC collider, we study the signal strenght of the
diphoton Higgs decay $R_{\gamma \gamma}$, which imposes very stringent bounds
to the scalar couplings and other scalar parameters. We obtain constraints in
different scenarios of the space of parameters, where decays into dark matter
may or may not contribute according to the mass of the scalar dark matter
candidate. By assuming that the lightest scalar boson of the model corresponds
to the observed Higgs boson, we evaluate deviations from the SM of the
trilineal Higgs self-coupling. The conditions from unitarity, stability and
Higgs diphoton decay data allow trilineal deviations in the range $0 \leq
\delta g \lesssim -72\%$.
