Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Abstract

Positive solutions of the nonlinear second-order differential equation $(p(t)|x'|^{\alpha - 1} x')' + q(t)|x|^{\beta - 1} x = 0,\alpha > \beta > 0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.