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Perturbations of the half-linear Euler — Weber type differential equation. (English) Zbl 1107.34030

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

${\left(r\left(t\right){\Phi }\left({x}^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right){\Phi }\left(x\right)=0,\phantom{\rule{1.em}{0ex}}{\Phi }\left(x\right)={|x|}^{p-2}x,\phantom{\rule{4pt}{0ex}}p>1,$

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

$\left(r\left(t\right){\Phi }\left({x}^{\text{'}}\right)\right)\right]+\stackrel{˜}{c}\left(t\right){\Phi }\left(x\right)=0·\phantom{\rule{2.em}{0ex}}\left(*\right)$

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

${\left({\Phi }\left({x}^{\text{'}}\right)\right)}^{\text{'}}+\left[\frac{\gamma p}{{t}^{p}}+\frac{{\mu }_{p}}{{t}^{p}{log}^{2}t}\right]\varphi \left(x\right)=0,\phantom{\rule{1.em}{0ex}}{\gamma }_{p}:={\left(\frac{p-1}{p}\right)}^{p},\phantom{\rule{1.em}{0ex}}{\mu }_{p}:=\frac{1}{2}{\left(\frac{p-1}{p}\right)}^{p-1},$

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.

##### MSC:
 34C11 Qualitative theory of solutions of ODE: growth, boundedness