This article is cited in 2 papers

Mathematics

Asymptotic classification of solutions to the second-order Emden–Fowler type differential equation with negative potential

K. M. Dulina, T. A. Korchemkina

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russian Federation

Abstract: 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 ($0<k<1$) 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 ($0<k<1$) are obtained.

Keywords: second-order ordinary differential equations, equations of Emden–Fowler type, non-extensible solutions, maximally uniquely extended solutions, asymptotic classification, regular nonlinearity, singular nonlinearity.

UDC: 517.925.44

Received: 08.07.2015

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