  We study the three-dimensional (3D) compact U(1) lattice gauge theory coupled
with $N$-flavor Higgs fields by means of the Monte Carlo simulations. This
model is relevant to multi-component superconductors, antiferromagnetic spin
systems in easy plane, inflational cosmology, etc. It is known that there is no
phase transition in the N=1 model. For N=2, we found that the system has a
second-order phase transition line $\tilde{c}_1(c_2)$ in the $c_2$(gauge
coupling)$-c_1$(Higgs coupling) plane, which separates the confinement phase
and the Higgs phase. Numerical results suggest that the phase transition
belongs to the universality class of the 3D XY model as the previous works by
Babaev et al. and Smiseth et al. suggested. For N=3, we found that there exists
a critical line similar to that in the N=2 model, but the critical line is
separated into two parts; one for $c_2 < c_{2{\rm tc}}=2.4\pm 0.1$ with
first-order transitions, and the other for $ c_{2{\rm tc}} < c_2$ with
second-order transitions, indicating the existence of a tricritical point. We
verified that similar phase diagram appears for the N=4 and N=5 systems. We
also studied the case of anistropic Higgs coupling in the N=3 model and found
that there appear two second-order phase transitions or a single second-order
transition and a crossover depending on the values of the anisotropic Higgs
couplings. This result indicates that an "enhancement" of phase transition
occurs when multiple phase transitions coincide at a certain point in the
parameter space.
