Asymptotic classification of solutionsto the second-order Emden - Fowler type differential equation with negativepotential

Consider the second-order differential equation of Emden - Fowler type with negative potential y′′ - p (x, y, y′) |y|sgn y = 0: The function p (x; y; y′) is assumed positive, continuous, and Lipschitz continuous in y, y′: In the case ofsingular nonlinearity (0 < k < 1) the solutions to above equation can behavein 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.

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