Asymptotic forms of positive solutions of second-order quasilinear ordinary differential equations. (English) Zbl 0982.34049

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 \[ (|u'|^{\alpha- 1}u')'= p(t)|u|^{\lambda- 1}u\tag{1} \] 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).


34E05 Asymptotic expansions of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J60 Nonlinear elliptic equations