  In this paper we investigate how the phase diagram of a U(1) symmetric
Higgs-Yukawa system depends on the scalar self coupling $\lambda$. The phase
diagram of similar models with continuous symmetry were extensively studied in
the infinite scalar self coupling $\lambda=\infty$ limit. Recent analytical and
numerical calculations at zero self coupling showed qualitatively different
phase diagram, raising the question of the $\lambda$ dependence of the phase
diagram. Here we use analytical (large $N_f$, perturbative and mean field)
approximations as well as numerical simulations to investigate the system.
