  We consider two-dimensional (2D) "artificial atoms" confined by an axially
symmetric potential $V(\rho)$. Such configurations arise in circular quantum
dots and other systems effectively restricted to a 2D layer. Using the
semiclassical method, we present the first fully self-consistent and analytic
solution yielding equations describing the density distribution, energy, and
other quantities for any form of $V(\rho)$ and an arbitrary number of confined
particles. An essential and nontrivial aspect of the problem is that the 2D
density of states must be properly combined with 3D electrostatics. The
solution turns out to have a universal form, with scaling parameters
$\rho^2/R^2$ and $R/a_B^*$ ($R$ is the dot radius and $a_B^*$ is the effective
Bohr radius).
