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.


34C11 Growth and boundedness of solutions to ordinary differential equations
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