The authors consider a plasma of point ions in the presence of a non-uniform neutralising background. This background, the source of an external field, may have some of its parameters (density, form of surface profile, etc.) modified, as long as the total charge is maintained. By considering such modifications in the context of the density-functional formalism for the ions, they prove sum rules giving the first and second moments of the ion density rho (z) in terms of other properties (bulk pressure and temperature derivative of surface tension). The Poisson-Boltzmann functional is considered in detail. They show that the first and second moment conditions on rho (z) are verified. They calculate rho (z) exactly for this system, and also perform variational calculations: comparison shows the importance of respecting the asymptotic behavior of rho (z). Variational calculations have been performed, using the density-functional formalism in the square-gradient approximation for systems with plasma parameter Gamma from 1 to 10. For Gamma >3, important oscillations appear in the profile, as shown by recent Monte Carlo calculations. The profiles calculated variationally also show increasing oscillations, but are not in good agreement with the Monte Carlo results. The surface energies are poor even for Gamma =1 showing the inadequacy of the square-gradient expansion for this system.