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 formequation(∗)(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′))′+[γptp+μptplog2t]Φ(x)=0,γp:=(p−1p)p,μp:=12(p−1p)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.