  The functional formulation and one-loop effective action for scalar
self-interacting theory non-linearly coupled with some power of the curvature
are studied. After the explicit one-loop renormalization at weak curvature, we
investigated numerically the phase structure for such unusual phi^4 theory. It
is demonstrated the possibility of curvature-induced phase transitions for
positive values of the curvature power, while for negative values the radiative
symmetry breaking does not take place. The dynamical mechanism for the
explanation of the current smallness of the cosmological constant is presented
for several models from the class of theories under consideration.
