Asymptotic forms of solutions of half-linear ordinary differential equations with integrable perturbations (Q6137335)

In this paper, the quasilinear ordinary differential equations of the form \[ (|u'|^{\alpha-1}u')' = \alpha(1+b(t))|u|^{\alpha-1}u,\quad t\geq t_0 \geq 0 \] is considered, where \(a>0\) is a constant and \(b(t)\) is a continuous function defined on \([t_0, \infty)\). The authors' previous result says that every nontrivial solution of the above equation has the asymptotic form \( u(t)\sim ce^t\) or \(u(t)\sim ce^{-t}\) as \(t\to\infty\) if \(b(t)\) satisfies \(\lim_{t\to\infty}b(t) =0\) and \(\int^{\infty}|b(t)| dt <\infty\). In this paper, under the weaker condition \(b\in L^p[t_0, \infty)\) for some \(p>1\), it is proved that every nontrivial solution of the above equation is of constant sign near \(\infty\). Moreover, every positive solution satisfies either \(u'\sim u\) or \(u'\sim -u\) as \(t\to \infty\). In particular, if \(1<p\leq 2\), it is also proved that every positive solution has the asymptotic form \[ u(t) = c\exp\left(t+\frac{1}{\alpha+1}\int^{\infty}b(s)ds +o(1)\right)\quad (t\to\infty) \] or \[ u(t) = c\exp\left(-t-\frac{1}{\alpha+1}\int^{\infty}b(s)ds +o(1)\right)\quad (t\to\infty). \]