The theory of the potential of a (point) impurity ion in a semiconductor involves an expansion of the screening-charge density in terms of the impurity-ion potential. In a recent paper, this problem has been reexamined by taking into consideration the spatial variation of the dielectric constant of the host medium. In this paper, the linearized Poisson equation, with the neglect of a small term, has been solved approximately by making use of an equivalent variational principle. In another recent paper, the spatial variation of the dielectric constant has been ignored and variational principles have been formulated for obtaining approximate solutions to nonlinear Poisson equations of any given order in the impurity-ion potential. The present paper aims at the unification of the above two approaches and presents variational principles for obtaining approximate solutions of nonlinear Poisson equations for the potential of an impurity ion which is located in a medium characterized by a spatially variable dielectric constant.