Using a variational method the binding energy has been calculated for a shallow donor impurity and the donor-related photoionization cross-section in 1D and 0D GaAs low-dimensional systems. The dependence on the binding energy and the photoionization cross-section for a hydrogenic donor impurity in the finite potential model are discussed and the results are presented as a function of the radius, polarization of the photon, applied hydrostatic pressure, and photon energy. The calculations for the pressure effects are performed both in the direct and indirect GaAs gap regime. Calculations are presented for an on-axis (on-center) impurity in the wire (in the dot).