*(English)*Zbl 1053.34024

The paper deals with qualitative aspects of the solutions of the nonlinear second-order Emden-Fowler equation

where $\lambda $ is a positive real number, $\lambda \ne 1$, $a\left(t\right)>0,r\left(t\right)>0$, and $a,r\in A{C}_{\text{loc}}^{1}\left({\mathbb{R}}^{+}\right)$. It is said that a solution $y\left(t\right)$ of (E) is of nonlinear limit-circle type, if ${\int}_{0}^{\infty}{\left|y\left(t\right)\right|}^{\lambda +1}dt<\infty $; when the improper integral is equal to infinity the solution is said to be of nonlinear limit-point type. Moreover, if all solutions of (E) are of nonlinear limit-circle type then we say that (E) is of nonlinear limit-circle type, and (E) is of nonlinear limit-point type if there exists at least a solution being of nonlinear limit-point type.

Apart from distinguishing the cases ${\int}_{0}^{\infty}du/a\left(u\right)<\infty $ or equal to $\infty $, the authors impose some sufficient conditions on the coefficients $a\left(t\right)$ and $r\left(t\right)$ in order to ensure that (E) has nonlinear limit-circle or limit-point type. The results obtained are very involved for being described here. The authors apply their results to the examples ${y}^{\text{'}\text{'}}+{t}^{\delta}{\left|y\right|}^{\lambda}\text{sgn}y=0$ (with $\lambda >0,\lambda \ne 1,\delta \ge 0)$ and ${y}^{\text{'}\text{'}}+{e}^{t}{\left|y\right|}^{\lambda}\text{sgn}y=0$ (with $\lambda \ge 3$). These examples are used to illustrate how the results obtained in this paper improve similar results contained in [*M. Bartus̆ek*, *Z. Dos̆lá* and *J. R. Graef*, The nonlinear limit-point/limit-circle problem.Boston, MA: Birkhäuser (2004; Zbl 1052.34021)].