On super-linear Emden–Fowler type differential equations

We study the second order Emden–Fowler type differential equation ( a ( t ) | x ′ | α sgn x ′ ) ′ + b ( t ) | x | β sgn x = 0 in the super-linear case α < β . Using a Hölder-type inequality, we resolve the open problem on the possible coexistence on three possible types of nononscillatory solutions (subdominant, intermediate, and dominant solutions). Jointly with this, sufficient conditions for the existence of globally positive intermediate solutions are established. Some of our results are new also for the Emden–Fowler equation.