Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities
The paper deals with the quasilinear second order differential equation with singular nonlinear term \[ (| u'| ^{\alpha-1}u')' = p(t)u^{-\beta}\;,\text{ on }[t_0,\infty),\;\alpha>0,\;\beta>0, p\in C^1\text{ with }p(t)>0. \] A solution is a function \(u\) such that \(u\) and \(| u'| ^{\alpha-1}u'\) are \(C^1\) functions and satisfy the equation for large \(t>0\). The following problems are discussed throughout the paper: asymptotic equivalence for two equations where \(p(t)\equiv a(t),\;b(t)\) respectively, and \[ \lim_{t\rightarrow\infty}\;a(t)/b(t)=1; \] asymptotic forms of asymptotic superlinear solutions, i.e. \[ \lim_{t\rightarrow\infty}\;u'(t)=\lim_{t\rightarrow\infty}\;u(t)/t=+\infty; \] asymptotic forms of decaying solutions, i.e. \[ \lim_{t\rightarrow\infty}\;u'(t)=\lim_{t\rightarrow\infty}\;u(t) = 0; \] uniqueness of decaying solutions.
Quasilinear Ordinary Differential Equations
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