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self-adjoint differential equation; oscillation and nonoscillation criteria; variational method; conditional oscillation
Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation \[ (-1)^n(t^{\alpha }y^{(n)})^{(n)}+q(t)y=0 \qquad \mathrm{(*)}\] are established. In these criteria, equation $(*)$ is viewed as a perturbation of the conditionally oscillatory equation \[ (-1)^n(t^{\alpha }y^{(n)})^{(n)}- \frac{\mu _{n,\alpha }}{t^{2n-\alpha }}y=0, \] where $\mu _{n,\alpha }$ is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.
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