A variational perturbation theory based on the Gibbs-Bogoliubov inequality is used to predict the phase behavior of systems with short-ranged interactions. We art primarily concerned with the disappearance of a stable liquid phase and the occurrence of an isostructural solid-solid transition, and consider two model systems interacting via a hard-sphere attractive Yukawa (HSAY) potential and a so-called m-n potential, a natural extension of the 12-6 Lennard-Jones potential to higher powers. In the variational calculations, a consistent treatment of the fluid and solid phases is aimed at and the hard-sphere system is used as the reference system for both phases. The predicted phase diagrams for the HSAY system with not very short-ranged potential are confirmed to be in good agreement with essentially the same calculations by Hagen and Frenkel [J. Chem. Phys. 101, 4093 (1994)]. The predicted isostructural solid-solid transition for this system, which occurs for a very short-ranged potential, are somewhat different from the Monte Carlo (MC) simulations by Bolhuis et al. [Phys. Rev. E 50, 4880 (1994)]. In particular, the predicted critical range of the potential for the occurrence of this transition is much shorter than the MC result. For the m-n potential system, a stable liquid phase is found to disappear when the attractive potential range becomes comparable to that of C-60. The critical temperature for the m-n potential system, at which the isostructural solid-solid transition terminates, is consistently higher by about 50% than that for the HSAY system, which is interpreted as an effect of the softness of the repulsive core. The solid-solid transition in the high density limit, which could occur for an infinitesimally short-ranged potential, is discussed in some detail for both systems within the framework of the present method.