Asymptotic behavior of increasing solutions to a system of [formula omitted] nonlinear differential equations

Abstract

We consider the system xi′=ai(t)|xi+1|αisgnxi+1, i=1,…,n, n≥2, where ai, i=1,…,n, are positive continuous functions on [a,∞), αi∈(0,∞), i=1,…,n, with α1⋯αn<1, and xn+1 means x1. We analyze the asymptotic behavior of the solutions to this system whose components are eventually positive increasing. In particular, we derive exact asymptotic formulas for the extreme case, where all the solution components tend to infinity (the so-called strongly increasing solutions). We offer two approaches: one is based on the asymptotic equivalence theorem, and the other utilizes the theory of regular variation. The above-mentioned system includes, as special cases, two-term nonlinear scalar differential equations of arbitrary order n and systems of n/2 second-order nonlinear equations (provided n is even), which are related to elliptic partial differential systems. Applications to these objects are presented and a comparison with existing results is made. It turns out that some of our results yield new information even in the simplest case, a second-order Emden–Fowler differential equation.