We consider a smectic liquid crystal film freely suspended between two electrodes and subject to all electric field. For a fixed voltage the system can exhibit qualitatively different types of equilibrium director profile, depending oil the spontaneous polarization and flexoelectric contributions to the bulk free energy. We examine the stability of the calculated director profiles and determine which configurations are energetically favourable. Furthermore, we also observe that, at critical values of the polarization parameters, there exists a bifurcation corresponding to a change in the number of equilibrium profiles. Generally, any configuration where the electric field and spontaneous polarization act in opposing directions is energetically unfavourable. However, we show that these profiles call become metastable and local energy minima when the flexoelectric parameters are relatively large.