We use a recently developed density-functional perturbation theory, which has been applied successfully to predict phase diagrams of systems of attractive particles, to describe the phase diagram of particles interacting via repulsive potentials. We consider potentials composed of a hard-sphere core plus a repulsive term. Specifically, we have investigated square shoulder and repulsive Yukawa terms. We show that, when the range of the interaction is very short, the shoulder potential leads to solid-solid coexistence involving two face-centered cubic structures, in analogy to an attractive square-well potential. Comparison with simulation results shows that the theory is quantitatively correct. If the range of the potentials is sufficiently long, we also find that a body-centered cubic structure can be stabilized. By considering the phase behavior at zero temperature, we argue that several triple points, involving coexistence of fluid and/or solid phases, may occur. A repulsive Yukawa term also shows a region of body-centered cubic stability but, contrary to the square shoulder and attractive Yukawa cases, there is no isostructural solid-solid coexistence. The role of the functional dependence of the interaction potential on particle separation at short distances is discussed and shown to be crucial to generate a solid/solid transition in systems of repulsive particles. Available computer simulation results for this system indicate that the density-functional approximation for the hard-sphere system used in this work, as well as all other currently available approximations, although qualitatively correct, may be overestimating the stability of the body-centered cubic phase.