## Half-linear Euler differential equations in the critical case.(English)Zbl 1265.34126

The authors study oscillatory properties of the perturbed half-linear differential equation $\Bigl (r(t)\Phi (x')\Bigr)^{\hskip -2pt{\prime }}+c(t)\Phi (x)=0\,,$ where $$\Phi (x)=| x| ^{p-2}x$$, $$p>1$$, $$r$$, $$c$$ are continuous functions and $$r(t)>0$$. The perturbation is allowed in both the coefficients $$r$$ and $$c$$. The authors extend the results of O.Došlý and S. Fišnarová [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 12, 3756–3766 (2010; Zbl 1207.34041)] and show that the perturbed half-linear Euler equation $\left [\left (1+\frac {\lambda }{\log ^2 t}\right)\Phi (x')\right ]' +\left [\frac {\gamma _p}{t^p}+\frac {\mu }{t^p\log ^2 t}\right ]\Phi (x) =0\,, \qquad \gamma _p:=\left (\frac {p-1}{p}\right)^p,$ considered in the limiting case $$\mu -\lambda \gamma _p=\frac {1}{2}\bigl (\frac {p-1}{p}\bigr)^{p-1}$$, is nonoscillatory.

### MSC:

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

Zbl 1207.34041