## 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 $(a(t)y')'+r(t)| y| ^{\lambda}\text{sgn}y=0,\tag{E}$ 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)].

### 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.)

Zbl 1052.34021
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### References:

 [1] Bartu[sbreve]ek M, The Nonlinear Limit-point/Limit-circle Problem, Birkhäuser (2004) · Zbl 1052.34021 [2] Bartu[sbreve]ek M, J. Math. Anal. Appl. 209 pp 122– (1997) · Zbl 0957.34023 [3] Bartu[sbreve]ek M, Arch. Math. Brno. 34 pp 13– (1998) [4] Bartu[sbreve]ek M, Differential Eqs. Dynamical Syst. 9 pp 49– (2001) [5] Dunford N, Linear Operators; Part II: Spectral Theory, Wiley (1963) [6] DOI: 10.1016/0022-0396(80)90032-7 · Zbl 0441.34024 [7] Graef JR, Pacific J. Math. 104 pp 85– (1983) [8] Graef JR, Publ. Math. Debrecen 36 pp 85– (1989) [9] Graef JR, J. Differential Equations 17 pp 461– (1975) · Zbl 0298.34028 [10] Graef JR, Publ. Math. Debrecen 24 pp 39– (1977) [11] DOI: 10.1016/0362-546X(83)90062-7 · Zbl 0535.34023 [12] Kiguradze I, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer (1993) · Zbl 0782.34002 [13] Titchmarsh EC, Eigenfunction Expansions Associated with Second-Order Differential Equations (1962) [14] DOI: 10.4153/CJM-1949-018-x · Zbl 0031.30801 [15] DOI: 10.1007/BF01474161 · JFM 41.0343.01
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