Geodesic
Asymptotic classification of solutions to the second-order Emden–Fowler type differential equation with negative potential
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 6 (2015), pp. 50-56 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Consider the second-order differential equation of Emden–Fowler type with negative potential $y'' - p\left(x, \, y,\, y'\right) |y|^k \, \mathrm{sgn} \, y = 0$. The function $p\left(x, \, y, \, y'\right)$ is assumed positive, continuous, and Lipschitz continuous in $y$, $y'.$ In the case of singular nonlinearity ($0$) the solutions to above equation can behave in a special way not only near the boundaries of their domains but also near internal points of the domains. This is why a notion of maximally uniquely extended solutions is introduced. Asymptotic classification of non-extensible solutions to above equation in case of regular nonlinearity ($k>1$) and classification of maximally uniquely extended solutions to above equation in case of singular nonlinearity ($0$) are obtained.
Keywords: second-order ordinary differential equations, equations of Emden–Fowler type, maximally uniquely extended solutions, asymptotic classification, regular nonlinearity, singular nonlinearity.
Mots-clés : non-extensible solutions 
@article{VSGU_2015_6_a5,
     author = {K. M. Dulina and T. A. Korchemkina},
     title = {Asymptotic classification of solutions to the second-order {Emden{\textendash}Fowler} type differential equation with negative potential},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {50--56},
     year = {2015},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2015_6_a5/}
}
TY  - JOUR
AU  - K. M. Dulina
AU  - T. A. Korchemkina
TI  - Asymptotic classification of solutions to the second-order Emden–Fowler type differential equation with negative potential
JO  - Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
PY  - 2015
SP  - 50
EP  - 56
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/VSGU_2015_6_a5/
LA  - ru
ID  - VSGU_2015_6_a5
ER  - 
%0 Journal Article
%A K. M. Dulina
%A T. A. Korchemkina
%T Asymptotic classification of solutions to the second-order Emden–Fowler type differential equation with negative potential
%J Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
%D 2015
%P 50-56
%N 6
%U http://geodesic.mathdoc.fr/item/VSGU_2015_6_a5/
%G ru
%F VSGU_2015_6_a5
K. M. Dulina; T. A. Korchemkina. Asymptotic classification of solutions to the second-order Emden–Fowler type differential equation with negative potential. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 6 (2015), pp. 50-56. http://geodesic.mathdoc.fr/item/VSGU_2015_6_a5/

[1] Kiguradze I. T., Chanturia T. A., Asymptotic properties of solutions of nonautonomous ordinary differential equations, Nauka, M., 1990, 432 pp. (in Russian)

[2] Kondrat'ev V. A., Nikishkin V. A., “On the positive solutions of the equation $y''=p(x)y^k$”, Some problems of qualitative theory of differential equations and the theory of motion control, Saransk, 1980, 134–141 (in Russian)

[3] Astashova I. V., “Qualitative properties of solutions to quasilinear ordinary differential equations”, Qualitative properties of solutions to the differential equations and related topics of spectral analysis, scientific edition, ed. I. V. Astashova, IuNITI-DANA, M., 2012, 22–288 (in Russian)

[4] Astashova I. V., “On asymptotic classification of solutions to the singular third- and fourth-order Emden–Fowler Equations”, Czech-Georgian Workshop on Boundary Value Problems, Institute of Mathematics, Academy of Sciences of the Czech Republic, 2015 (in English) http://users.math.cas.cz/s̃remr/wbvp2015/abstracts/astashova.pdf

[5] Astashova I. V., “On existence of quasi-periodic solutions to a nonlinear singular higher-order differential equation and asymptotic classifcation of its solutions for the forth order”, International Workshop on the Qualitative Theory of Differential Equations, QUALITDE-2014, 2014 (in English) http://rmi.tsu.ge/eng/QUALITDE-2014/workshop_2014.htm

[6] Astashova I. V., “On the asymptotic behavior of solutions of differential equations with a singular nonlinearity”, Differential equations, 50:11 (2014), 1551–1552 (in Russian) | DOI | Zbl

[7] Dulina K. M., Korchemkina T. A., “On existence of solutions to second-order Emden–Fowler type differential equations with prescribed domain”, Proceedings of International miniconference “Qualitative theory of differential equations and applications”, MESI, M., 2014, 19–27 (in Russian)