  We study potential and electron density depth profiles in accumulation,
inversion and depletion layers in crystals with large and nonlinear dielectric
response such as $\mathrm{SrTiO_3}$. We describe the lattice dielectric
response using the Landau-Ginzburg free energy expansion. In accumulation and
inversion layers we arrive at new nonlinear dependencies of the width $d$ of
the electron gas on applied electric field $D_0$. Particularly important is the
predicted electron density profile of accumulation layers (including the
$\mathrm{LaAlO_3/SrTiO_3}$ interface) $n(x) \propto (x+d)^{-12/7}$, where $d
\propto D_0^{-7/5}$ . We compare this profile with available data and find
satifactory agreement. For a depletion layer we find an unconventional
nonlinear dependence of capacitance on voltage. We also evaluate the role of
spatial dispersion in the dielectric response by adding a gradient term to the
Landau-Ginzburg free energy.
