Solutions of half-linear differential equations in the classes Gamma and Pi

We study asymptotic behavior of (all) positive solutions of the non-oscillatory half-linear differential equation of the form (r(t)|y'|^ {alpha-1} sgn y')'=p(t)|y|^{alpha-1}sgn y, where alpha>1 and r,p are positive continuous functions, with the help of the Karamata theory of regularly varying functions and the de Haan theory. We show that increasing resp. decreasing solutions belong to the de Haan class Gamma resp. Gamma- under suitable assumptions. Further we study behavior of slowly varying solutions for which asymptotic formulas are established. Some of our results are new even in the linear case alpha=2.