Classification of bifurcation diagrams for semilinear elliptic equations in the critical dimension

Until the discovery of the accelerated expansion of the universe in the late 1990s, the great majority of physicists were convinced that there was not any kind of sound in any kind of vacuum. The latter specification is necessary because nowadays we discuss whether there is the possibility that sound actually does propagate in space which is free of ordinary matter but filled with “dark energy”. The term “vacuum” for such kind of empty space is avoided by most authors as it remains reserved for models of empty space in which the cosmological constant acts as a source of vacuum energy. 

Before this discussion started, evidence for vanishing propagation of sound in a vacuum apparently aroused from an experiment designed by Otto von Guericke in the 17th century although his intention was merely to show that a vacuum actually does exist. This article will show that the interpretation of this experiment as a proof for vanishing sound-propagation in a vacuum has been a fundamental philosophical mistake and furthermore that any theoretical investigations of sound in space which is free of any kind of matter may require a quantum-theoretical approach.