Asymptotic classification of solutions to the second-order Emden–Fowler type differential equation with negative potential

Consider the second-order differential equation of Emden–Fowler type with negative potential $y'' - p\left(x, \, y,\, y'\right) |y|^k \, \mathrm{sgn} \, y = 0$. The function $p\left(x, \, y, \, y'\right)$ is assumed positive, continuous, and Lipschitz continuous in $y$, $y'.$ In the case of singular nonlinearity ($0<k<1$) the solutions to above equation can behave in a special way not only near the boundaries of their domains but also near internal points of the domains. This is why a notion of maximally uniquely extended solutions is introduced. Asymptotic classification of non-extensible solutions to above equation in case of regular nonlinearity ($k>1$) and classification of maximally uniquely extended solutions to above equation in case of singular nonlinearity ($0<k<1$) are obtained.