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

Abstract

We investigate oscillatory properties of the half-linear second order differential equation

$$(r(t)\Phi (x^{\prime }){)}^{\prime }+c(t)\Phi (x)=0,\Phi (x)={|x|}^{p-2}x,p>1,$$

viewed as a perturbation of another half-linear differential equation of the same form

(∗)$$(r(t)\Phi (x^{\prime }){)}^{\prime }+\tilde{c}(t)\Phi (x)=0.$$

The obtained oscillation and nonoscillation criteria are formulated in terms of the integral $\int [c(t)-\tilde{c}(t)]h^p(t)dt$, 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

$$(\Phi (x^{\prime }){)}^{\prime }+[\frac{{\gamma }_p}{t^p}+\frac{{\mu }_p}{t^p{\log }^{2}t}]\Phi (x)=0,{\gamma }_p:=(\frac{p-1}{p}{)}^p,{\mu }_p:=\frac{1}{2}(\frac{p-1}{p}{)}^{p-1},$$

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.