On oscillation of solutions to quasi-linear Emden–Fowler type higher-order differential equations

Existence and behavior of oscillatory solutions to nonlinear equations with regular and singular power nonlinearity are investigated. In particular, the existence of oscillatory solutions is proved for the equation
\begin{gather*} y^{(n)}+P(x,y,y',\ldots,y^{(n-1)})|y|^k\ {\rm sign}\,y=0,\\ n\ge 2,\,\,\,k\in \mathbb {R},\,\,\, k>1,\,\,\, P\neq0,\,\,\,\,P\in C(\mathbb{R}^{n+1}). \end{gather*}
A criterion is formulated for oscillation of all solutions to the quasilinear even-order differential equation
\begin{gather*} y^{(n)}+\sum_{i=0}^{n-1}a_{j}(x)\;y^{(i)}+p(x)\;|y|^{k} {\rm sign} y=0,\\ p\in C(\mathbb{R}),\,\,a_j\in C(\mathbb{R}),\,\,\,j=0,\dots,{n-1},\,\,\, k>1,\,\, n=2m,\,\, m\in\mathbb{N}, \end{gather*}
which generalizes the well-known Atkinson's and Kiguradze's criteria.
The existence of quasi-periodic solutions is proved both for regular ($k>1$) and singular ($0<k<1$) nonlinear equations
$$ y^{(n)}+p_0\,|y|^{k} {\rm sign} y=0, \quad n>2,\quad k\in \mathbb {R},\quad k>0,\,\,\,k\neq1, \quad p_0\in \mathbb {R}, $$
with $(-1)^{n}p_0>0.$ A result on the existence of periodic oscillatory solutions is formulated for this equation with $n=4,\,\,k>0,\,\,k\neq1,\,\,p_0<0.$