  We examine the effects of strain and cation substitution on the
superconducting phase of polar semiconductors near a ferroelectric quantum
phase transition with a model that combines a strong coupling theory of
superconductors with a standard microscopic framework for displacive polar
modes coupled to strain degrees of freedom. Our calculations reveal that the
superconducting transition temperature $T_c$ is enhanced by proximity to the
ferroelectric instability from the disordered side, while it is generally
suppressed in the ordered phase due to its increase in dielectric stiffness and
a reduction of critical fluctuations from dipolar induced anisotropies. The
condensation of the pairing phonon excitations generates a kink in $T_c$ at a
charge density that is generally lower than that of the quantum critical point
(QCP) and where both superconducting and ferroeletric orders set-in. We apply
our model to SrTiO$_3$ and find that the anti-adiabatic limit places the kink
nearly at its QCP. As the QCP is pushed to higher charge densities with either
tuning parameter, we find that the dome narrows and sharpens. Our model is in
qualitatively and fair quantitatively agreement with the recent observation of
overlapping ferroelectric-like and superconducting instabilities in n-doped
Sr$_{1-x}$Ca$_x$TiO$_3$ and strain tunning of $T_c$ in n-doped SrTiO$_3$. We
compare our results to previous models invoking order-disorder lattice dynamics
to describe the pairing excitations.
