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Some limit-point and limit-circle results for second order Emden-Fowler equations. (English) Zbl 1053.34024

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

${\left(a\left(t\right){y}^{\text{'}}\right)}^{\text{'}}+r\left(t\right){|y|}^{\lambda }\text{sgn}y=0,\phantom{\rule{2.em}{0ex}}\left(\mathrm{E}\right)$

where $\lambda$ is a positive real number, $\lambda \ne 1$, $a\left(t\right)>0,r\left(t\right)>0$, and $a,r\in A{C}_{\text{loc}}^{1}\left({ℝ}^{+}\right)$. It is said that a solution $y\left(t\right)$ of (E) is of nonlinear limit-circle type, if ${\int }_{0}^{\infty }{|y\left(t\right)|}^{\lambda +1}dt<\infty$; 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 ${y}^{\text{'}\text{'}}+{t}^{\delta }{|y|}^{\lambda }\text{sgn}y=0$ (with $\lambda >0,\lambda \ne 1,\delta \ge 0\right)$ and ${y}^{\text{'}\text{'}}+{e}^{t}{|y|}^{\lambda }\text{sgn}y=0$ (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)].

##### MSC:
 34B20 Weyl theory and its generalizations 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34B30 Special ODE (Mathieu, Hill, Bessel, etc.)