Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations (Q2733873)

The main objective of this paper is to determine the asymptotic forms of all positive solutions to the second-order quasilinear ordinary differential equation of the form NEWLINE\[NEWLINE(|u'|^{\alpha- 1}u')'= p(t)|u|^{\lambda- 1}u\tag{1}NEWLINE\]NEWLINE in terms of \(\alpha\), \(\lambda\), and \(\delta\), where \(\alpha\) and \(\lambda\) are constants satisfying \(0< \alpha< \lambda\); \(p\) is a continuous function defined near \(+\infty\) satisfying \(p(t)\sim t^\delta\) as \(t\to\infty\) for some \(\delta\in \mathbb{R}\). In the special case that \(\alpha= 1\) and \(p(t)\equiv t^\delta\), this problem was fully discussed by Bellman. Very little is known, however, about the asymptotic forms of positive solutions to (1) when \(\alpha\neq 1\). Motivated by these facts, the authors try to extend such results to the more general quasilinear equation (1).