We consider three-dimensional lattice SU($N_c$) gauge theories with multiflavor ($N_f>1$) scalar fields in the adjoint representation. We investigate their phase diagram, identify the different Higgs phases with their gauge-symmetry pattern, and determine the nature of the transition lines. In particular, we study the role played by the quartic scalar potential and by the gauge-group representation in determining the Higgs phases and the global and gauge symmetry-breaking patterns characterizing the different transitions. The general arguments are confirmed by numerical analyses of Monte Carlo results for two representative models that are expected to have qualitatively different phase diagrams and Higgs phases. We consider the model with $N_c = 3$, $N_f=2$ and with $N_c=2$, $N_f= 4$. This second case is interesting phenomenologically to describe some features of cuprate superconductors.
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