For the differential equation $y^{(n)} = f(t, y, \ldots, y^{(n-1)})$, where $f: [a,\omega[\times \Delta_{Y_0} \times\Delta_{Y_1} \times\cdots \times \Delta_{Y_{n-1}} \longrightarrow \mathbb{R}$ is a continuous function, $-\infty<a<\omega\le +\infty$, $Y_i$ equals to zero or to $\pm\infty$, $\Delta_{Y_i}$- is some one-sided neighborhood of $Y_i$, $i=0,1,\ldots,n-1$, questions about the existence, asymptotics and about quantity of $P_\omega \left(Y_0, \ldots, Y_{n-1}, \frac{n-i-1}{n-i}\right)$ --- solutions for all $i\in\{1,\ldots,n-1\}$ are investigated under certain restrictions on the function$f$ . Such solutions refer to special cases of class of $P_\omega(Y_0,\ldots,Y_{n-1},\lambda_0)$-solutions where $-\infty \le \lambda_0 \le +\infty$, that was introduced in works of V. M. Evtukhov devoted to the differential equations of Emden-Fowler type of the $n$-th order. Such special cases require their separate consideration because of their specific a priori asymptotic properties. The study of the formulated problems is carried out under the assumption that the differential equation is in some sense asymptotically close to the two-term differential equation with regularly varying nonlinearities.