  Ivanov pointed out substantial analytical difficulties associated with
self-gravitating, static, isotropic fluid spheres when pressure explicitly
depends on matter density. Simplification achieved with the introduction of
electric charge were noticed as well. We deal with self-gravitating, charged,
anisotropic fluids and get even more flexibility in solving the
Einstein-Maxwell equations. In order to discuss analytical solutions we extend
Krori and Barua's method to include pressure anisotropy and linear or
non-linear equations of state. The field equations are reduced to a system of
three algebraic equations for the anisotropic pressures as well as matter and
electrostatic energy densities. Attention is paid to compact sources
characterized by positive matter density and positive radial pressure. Arising
solutions satisfy the energy conditions of general relativity. Spheres with
vanishing net charge contain fluid elements with unbounded proper charge
density located at the fluid-vacuum interface. Notably the electric force
acting on these fluid elements is finite, although the acting electric field is
zero. Net charges can be huge ($10^{19}\,C$) and maximum electric field
intensities are very large ($10^{23}-10^{24}\,statvolt/cm$) even in the case of
zero net charge. Inward-directed fluid forces caused by pressure anisotropy may
allow equilibrium configurations with larger net charges and electric field
intensities than those found in studies of charged isotropic fluids. Links of
these results with charged strange quark stars as well as models of dark matter
including massive charged particles are highlighted. The van der Waals equation
of state leading to matter densities constrained by cubic polynomial equations
is briefly considered. The fundamental question of stability is left open.
