Perturbations of the half-linear Euler–Weber type differential equation ✩

We investigate oscillatory properties of the half-linear second order differential equation r(t)Φ(x ) +c(t)Φ(x) = 0, Φ(x)= |x|p−2x, p >1, viewed as a perturbation of another half-linear differential equation of the same form r(t)Φ(x ) + ˜c(t)Φ(x) = 0. (∗) The obtained oscillation and nonoscillation criteria are formulated in terms of the integral [c(t)− ˜c(t)]× hp(t) dt, where h is a function which is close to the principal solution of (∗), in a certain sense. A typical model of (∗) in applications is the half-linear Euler–Weber differential equation with the critical coefficients Φ(x ) + γp tp + μp tp log2 t Φ(x) = 0, γp := p − 1 p p , μp := 1 2 p −1 p p−1 , we establish oscillation and nonoscillation criteria for perturbations of this equation. Some open problems and perspectives of the further research along this line are also formulated. © 2005 Elsevier Inc. All rights reserved.