Singular Kneser solutions of higher-order quasilinear ordinary differential equations

Abstract

In this paper we give a new sufficient condition in order that all nontrivial Kneser solutions of the quasilinear ordinary differential equation \[ D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = (-1)^{n}p(t)|x|^{\beta}\mathrm{sgn}\,x, \quad t \geq a, \tag{1.1} \label{abseq} \] are singular. Here, $D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)$ is the $n$th-order iterated differential operator such that \begin{equation*} D(\alpha_n, \alpha_{n-1}, \dots, \alpha_1)x = D(\alpha_n)D(\alpha_{n-1})\cdots D(\alpha_1)x \end{equation*} and, in general, $D(\alpha)$ is the first-order differential operator defined by $D(\alpha)x = (d/dt)\left(|x|^{\alpha}\mathrm{sgn}\,x\right)$ for $\alpha > 0$. In the equation \eqref{abseq}, the condition $\alpha_1\alpha_2\cdots\alpha_n > \beta$ is assumed. If $\alpha_1 = \alpha_2 = \cdots = \alpha_n = 1$, then one of the results of this paper yields a well-known theorem of Kiguradze and Chanturia.

Keywords

• kneser solutions
• singular solutions
• quasilinear equations