  We study the Yukawa model with one scalar and one axial scalar fields,
coupled to $N$ copies of Dirac fermions, in curved spacetime background. The
theory possesses a reach set of coupling constants, including the scalar terms
with odd powers of scalar fields in the potential, and constants of non-minimal
coupling of the scalar fields to gravity. Using the heat-kernel technique and
dimensional regularization, we derive the one-loop divergences, describe the
renormalization of the theory under consideration and calculate the full set of
beta- and gamma-functions for all coupling constants and fields. As a next
step, we construct the renormalized one-loop effective potential of the scalar
fields up to the terms linear in scalar curvature. This calculation includes
only the contributions from quantum scalar fields, and is performed using
covariant cut-off regularization and local momentum representation. Some
difficulties of the renormalization group approach to the effective potential
in the case under consideration are discussed.
