## Oscillation and nonoscillation of perturbed half-linear Euler differential equations.(English)Zbl 1164.34012

The authors derive new oscillation and nonoscillation criteria for the perturbed half-linear Euler differential equation $(\Phi(x'))' +\left[\frac{\gamma_p}{t^p} +c(t)\right]\Phi(x)=0,$ where $$\gamma_p:=\left(\frac{p-1}{p}\right)^p$$, $$\Phi(x):=| x| ^{p-2}x$$, and $$p>1$$. These criteria are motivated by the conjectures which were proposed in earlier separate papers by the authors. The methods traditionally use the variational and Riccati techniques.

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations