We continue investigations started in the previous publications by the authors (LNCS, volumes 8136 (2013) and 9570 (2016)). The structure of stationary point sets is established for the family of functions given as linear combinations of an exponent L of Euclidean distances from a variable point to the fixed points in 2D and 3D spaces. We compare the structure of the stationary point sets for several values of the exponent L, focusing ourselves mainly onto the cases of Coulomb potential and Weber facility location problem. We develop the analytical approach to the problem aiming at finding the exact number of stationary points and their location in relation to the parameters involved.