On asymptotic behaviour of solutions with infinite derivative for regular second-order Emden–Fowler type differential equations with negative potential
Abstract:
In this paper we consider the second-order Emden–Fowler type differential equation with negative potential $y''-p(x,\, y,\, y') |y|^k \text{ sgn } y=0$ in case of regular nonlinearity $k>1.$ We assume that the function $p(x,\, u,\, v)$ is continuous in $x$ and Lipschitz continuous in two last variables. We investigate asymptotic behaviour of non-extensible solutions to the equation above. We consider the case of a positive function $p(x,\, u,\, v)$ unbounded from above and bounded away from 0 from below. The conditions guaranteeing an existence of a vertical asymptote of all nontrivial non-extensible solutions to the equation are obtained. Also the sufficient conditions providing the following solutions' properties $\lim\limits_{x \to a} |y'(x)| = +\infty$, $\lim\limits_{x \to a} |y(x)| <+ \infty,$ where $a < \infty$ is a boundary point, are obtained.
Keywords:second-order Emden–Fowler type differential equations, regular nonlinearity, asymptotic behaviour.
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