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On asymptotic behavior of solutions to Emden-Fowler type higher-order differential equations. (English) Zbl 1374.34096
Summary: For the equation \[ y^{(n)}+|y|^{k}\operatorname{sgn} y=0,\quad k>1,\;n=3,4 \] the existence of oscillatory solutions \[ y=(x^*-x)^{-\alpha}h(\log (x^*-x)),\quad \alpha =\frac{n}{k-1},\;x<x^*, \] is proved, where \(x^*\) is an arbitrary point and \(h\) is a periodic non-constant function on \(\mathbb{R}\). The result on the existence of such solutions with a positive periodic non-constant function \(h\) on \(\mathbb{R}\) is formulated for the equation \[ y^{(n)}=|y|^{k}\operatorname{sgn}y,\quad k>1,\;n=12,13,14. \]

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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