The Liouville integral for surfaces of constant Gaussian curvature induces a quantized blowup mechanism of solutions to the Gel’fand or Boltzmann–Poisson equation, as is clarified in the previous chapter. The standard argument to approach this phenomenon, however, is the blowup analysis based on the self-similarity of the equation, which is commonly provided in the fundamental equations of mathematical physics. This property is the invariance under transformation involving independent and dependent variables, which causes a lack of compactness of the family of (approximate) solutions. Breaking down of compactness becomes clear by self-similar transformation, to reach a profile of the structure of total set of solutions from the microscopic viewpoint. Meanwhile, a crucial role is taken by the classification of entire solutions, which traces back to Liouville’s theory on harmonic functions. We thus encounter the third of Liouville’s theorems. The first section is a summary. We begin with several fundamental properties of harmonic functions in the second section, and turn to recent results on semilinear elliptic equations with power nonlinearity in the third section, in accordance with the Liouville property concerning entire solutions. We then generalize the result on exponential nonlinearity obtained in the previous chapter in the final section.