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

Bartušek, Miroslav; Graef, John R.

Appl. Anal. 83, No. 5, 461-476 (2004).

The paper deals with qualitative aspects of the solutions of the nonlinear second-order Emden-Fowler equation \[ (a(t)y')'+r(t)| y| ^{\lambda}\text{sgn}y=0,\tag{E} \] where \(\lambda\) is a positive real number, \(\lambda\neq1\), \(a(t)>0, r(t)>0\), and \(a,r \in AC_{\text{loc}}^{1}(\mathbb{R}^{+})\). It is said that a solution \(y(t)\) of (E) is of nonlinear limit-circle type, if \(\int_{0}^{\infty}| y(t)| ^{\lambda+1}dt<\infty\); 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 \(\int_{0}^{\infty}du/a(u)<\infty\) 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 \(y''+t^{\delta}| y| ^{\lambda}\text{sgn}y=0\) (with \(\lambda>0, \lambda\neq1, \delta\geq0)\) and \(y''+e^{t}| y| ^{\lambda}\text{sgn}y=0\) (with \(\lambda\geq3\)). These examples are used to illustrate how the results obtained in this paper improve similar results contained in [M. Bartušek, Z. Došlá and J. R. Graef, The nonlinear limit-point/limit-circle problem.Boston, MA: Birkhäuser (2004; Zbl 1052.34021)].

Reviewer: Antonio Linero Bas (Murcia)

MSC:

34B20	Weyl theory and its generalizations for ordinary differential equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
34B30	Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)

Keywords:

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

Citations:

Zbl 1052.34021

References:

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