  It is known that a nucleus with charge $Ze$ where $Z>170$ creates
electron-positron pairs from the vacuum. These electrons collapse onto the
nucleus resulting in a net charge $Z_n<Z$ while the positrons are emitted. This
effect is due to the relativistic dispersion law. The same reason leads to the
collapse of electrons to the charged impurity with a large charge number $Z$ in
narrow-band gap semiconductors and Weyl semimetals as well as graphene. In this
paper, a similar effect of electron collapse and charge renormalization is
found for donor clusters in SrTiO$_3$ (STO), but with a very different origin.
At low temperatures, STO has an enormously large dielectric constant. Because
of this, the nonlinear dielectric response becomes dominant when the electric
field is not too small. We show that this leads to the collapse of surrounding
electrons into a charged spherical donor cluster with radius $R$ when its total
charge number $Z$ exceeds a critical value $Z_c\simeq R/a$ where $a$ is the
lattice constant. Using the Thomas-Fermi approach, we find that the net charge
$Z_ne$ grows with $Z$ until $Z$ exceeds another value $Z^*\simeq(R/a)^{9/7}$.
After this point, $Z_n$ remains $\sim Z^*$. We extend our results to the case
of long cylindrical clusters. Our predictions can be tested by creating discs
and stripes of charge on the STO surface.
