Summary: We investigate oscillatory properties of the half-linear second-order differential equation
viewed as a perturbation of another half-linear differential equation of the same form
The obtained oscillation and nonoscillation criteria are formulated in terms of the integral , where 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
and we establish oscillation and nonoscillation criteria for perturbations of this equation. Some open problems and perspectives of the further research along this line are formulated, too.