  This mini-review is dedicated to the 85th birthday of Prof. L. V. Keldysh,
from whom we have learned so much. In this paper we study the potential and
electron density depth profiles in surface accumulation layers in crystals with
a large and nonlinear dielectric response such as SrTiO$_3$ (STO) in the cases
of planar, spherical and cylindrical geometries. The electron gas can be
created by applying an induction $D_0$ to the STO surface. We describe the
lattice dielectric response of STO using the Landau-Ginzburg free energy
expansion and employ the Thomas-Fermi (TF) approximation for the electron gas.
For the planar geometry we arrive at the electron density profile $n(x) \propto
(x+d)^{-12/7}$, where $d \propto D_0^{-7/5} $. We extend our results to
overlapping electron gases in GTO/STO/GTO multi-heterojunctions and electron
gases created by spill-out from NSTO (heavily $n$-type doped STO) layers into
STO. Generalization of our approach to a spherical donor cluster creating a big
TF atom with electrons in STO brings us to the problem of supercharged nuclei.
It is known that for an atom with nuclear charge $Ze$, where $Z > 170$,
electrons collapse onto the nucleus resulting in a net charge $Z_n < Z$. Here,
instead of relativistic physics, the collapse is caused by the nonlinear
dielectric response. Electrons collapse into the charged spherical donor
cluster with radius $R$ when its total charge number $Z$ exceeds the critical
value $Z_c \simeq R/a$, where $a$ is the lattice constant. The net charge
$eZ_n$ grows with $Z$ until $Z$ exceeds $Z^* \simeq (R/a)^{9/7}$. After this
point, the charge number of the compact core $Z_n$ remains $\simeq Z^*$, with
the rest $Z^*$ electrons forming a sparse Thomas-Fermi electron atmosphere
around it. We extend our results to the case of long cylindrical clusters as
well.
