A coexistence problem for nonoscillatory solutions to Emden-Fowler type differential equations

We consider a coexistence problem for nonoscillatory solutions to the Emden-Fowler type differential equation (a(t)|x′|^{α}sgn x′)′+b(t)|x|^{β}sgn x=0. (*) For the special case x′′+b(t)|x|^{β}sgn x=0, t≥1, (**), this problem has been posed by Moore and Nehari when 1<β and by Belohorec when 0<β<1. Nonoscillatory solutions to (**) can be classified into three types, according their asymptotic behavior as t→∞, and it is shown that these three types of nonoscillatory solutions cannot simultaneously coexist for (**). When the sublinear case α>β occurs, this result has been recently extended to (*). Here we complete this study, by showing that in any case this triple coexistence for nonoscillatory solutions is impossible also for (*).