Renormalization-group methods are applied to the critical behavior of an Ising-like system on an elastic solid of either cubic or isotropic symmetry. Except in the special case where dTcd V=0, the bulk modulus is found to be negative very close to Tc, so that the phase transition at constant pressure must be at least weakly first order. In the isotropic case the solid may be stabilized by pinned boundary conditions, if crystal fracture can be avoided. A "Fisher-renormalized" critical point can then be observed. By contrast, the anisotropic cubic solid will develop a microscopic instability so that Tc cannot be reached, regardless of boundary conditions. Estimates of the size of these effects are given, and contact is made with the Baker-Essam model and a liquid, as limiting cases with a vanishing shear modulus.