Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations with sub-homogeneity (Q5935589)

The authors investigate the asymptotic behaviour of positive solutions to the quasilinear ordinary differential equation NEWLINE\[NEWLINE(|u'|^{\alpha-1} u')'= p(t)|u|^{\lambda- 1}u,NEWLINE\]NEWLINE subject to the general conditions: (i) \(\alpha\) and \(\lambda\) are positive constants which satisfy \(0< \lambda<\alpha\); (ii) \(p: [t_0,\infty)\to (0,\infty)\) is a continuous function such that \(p(t)\sim t^\alpha\) as \(t\to\infty\). The case \(\alpha= 1\) is the well-known Emden-Fowler equation. The uniqueness of positive decaying solutions is also proved.