Asymptotic formulae for solutions of half-linear differential equations

We establish asymptotic formulae for regularly varying solutions of the half-linear differential equation (r(t)|y′|α−1sgny′)′=p(t)|y|α−1sgny,where r, p are positive continuous functions on [a, ∞) and α ∈ (1, ∞). The results can be understood in several ways: Some open problems posed in the literature are solved. Results for linear differential equations are generalized; some of the observations are new even in the linear case. A refinement on information about behavior of solutions in standard asymptotic classes is provided. A precise description of regularly varying solutions which are known to exist is given. Regular variation of all positive solutions is proved.