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Zbl 1053.34024
Bartušek, Miroslav; Graef, John R.
Some limit-point and limit-circle results for second order Emden-Fowler equations.
(English)
[J] Appl. Anal. 83, No. 5, 461-476 (2004). ISSN 0003-6811; ISSN 1563-504X/e

The paper deals with qualitative aspects of the solutions of the nonlinear second-order Emden-Fowler equation $$(a(t)y')'+r(t)\vert y\vert ^{\lambda}\text{sgn}y=0,\tag E$$ where $\lambda$ is a positive real number, $\lambda\neq1$, $a(t)>0, r(t)>0$, and $a,r \in AC_{\text{loc}}^{1}(\Bbb{R}^{+})$. It is said that a solution $y(t)$ of (E) is of nonlinear limit-circle type, if $\int_{0}^{\infty}\vert y(t)\vert ^{\lambda+1}dt<\infty$; 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. \par Apart from distinguishing the cases $\int_{0}^{\infty}du/a(u)<\infty$ or equal to $\infty$, the authors impose some sufficient conditions on the coefficients $a(t)$ and $r(t)$ 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 $y''+t^{\delta}\vert y\vert ^{\lambda}\text{sgn}y=0$ (with $\lambda>0, \lambda\neq1, \delta\ge0)$ and $y''+e^{t}\vert y\vert ^{\lambda}\text{sgn}y=0$ (with $\lambda\ge3$). These examples are used to illustrate how the results obtained in this paper improve similar results contained in [{\it M. Bartusek}, {\it Z. Doslá} and {\it J. R. Graef}, The nonlinear limit-point/limit-circle problem.Boston, MA: Birkhäuser (2004; Zbl 1052.34021)].
[Antonio Linero Bas (Murcia)]
MSC 2000:
*34B20 Weyl theory and its generalizations
34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34C15 Nonlinear oscillations of solutions of ODE
34B30 Special ODE

Keywords: Emden-Fowler equation; nonlinear limit-circle solution; nonlinear limit-point solution

Citations: Zbl 1052.34021

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