  We present an exactly solvable non-linear elastic model of a volume collapse
transition in an isotropic solid. Integrity of the lattice through the
transition leads to an infinite-range density-density interaction, which drives
classical critical behavior. Nucleation is forbidden within a pressure window
leading to intrinsic hysteresis and an unavoidable discontinuity of the
thermodynamic potential (zeroth order transition). The window shrinks with
increasing temperature ending at a critical point at a temperature related to
the shear modulus. Mixed phases behave non-extensively and show negative
compressibility. We discuss the implications for Ce, SmS, and related systems.
