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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

(a(t)y ' ) ' +r(t)|y| λ sgny=0,(E)

where λ is a positive real number, λ1, a(t)>0,r(t)>0, and a,rAC loc 1 ( + ). It is said that a solution y(t) of (E) is of nonlinear limit-circle type, if 0 |y(t)| λ+1 dt<; 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 0 du/a(u)< or equal to , 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 δ |y| λ sgny=0 (with λ>0,λ1,δ0) and y '' +e t |y| λ sgny=0 (with λ3). 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)].

34B20Weyl theory and its generalizations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34C15Nonlinear oscillations, coupled oscillators (ODE)
34B30Special ODE (Mathieu, Hill, Bessel, etc.)