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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 \[ \bigl(r(t)\Phi(x')\bigr)'+c(t)\Phi(x)=0,\quad \Phi (x)= |x|^{p-2}x,\;p>1, \] viewed as a perturbation of another half-linear differential equation of the same form \[ \bigl(r(t)\Phi(x')\bigr)] + \widetilde c(t)\Phi(x)= 0.\tag{*} \] The obtained oscillation and nonoscillation criteria are formulated in terms of the integral \(\int[c(t)-\widetilde c(t)] \times h^p(t)dt\), 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 \[ \bigl(\Phi(x')\bigr)'+\left[\frac {\gamma p}{t^p}+\frac{\mu_p}{t^p \log^2t}\right]\varphi(x)=0,\quad\gamma_p:=\left( \frac{p-1}{p} \right)^p,\quad\mu_p:=\frac 12\left(\frac{p-1}{p}\right)^{p-1}, \] 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 Growth and boundedness of solutions to ordinary differential equations
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