J. S. W. Wong
abstract:
We are interested in the oscillatory behavior of solutions of Emden-Fowler
equation
$$\ y'' + f(x) |y|^{\gamma - 1} y = 0, \eqno{(1)}$$
where $\gamma > 0$ and $\gamma \ne 1$, $f(x) \in C^1 (0, 1]$ and $f(x) > 0$ for
$x\in (0, 1]$. A solution $y(x)$ is rectifiable oscillatory if the solution
curve $\{(x, y(x)) : x \in (0, 1]\}$ has a finite arc-length. When the
arc-length of the solution curve is infinite, the solution $y(x)$ is said to be
unrectifiable oscillatory. We prove integral criteria in terms of $f(x)$ which
are necessary and sufficient for both rectifiable and unrectifiable oscillations
of all solutions of \eqref{1}. For a discussion on rectifiable oscillation of
the linear differential equation, i.e. the equation (1) when $\gamma = 1$, we
refer to Pa\v{s}i\'{c} [Rectifiable and unrectifiable oscillations for a class
of second-order linear differential equations of Euler type. J. Math. Anal.
Appl. 335 (2007), No. 1, 724-738], Wong [On rectifiable oscillations
of Euler type second order linear differential equations.
Electron. J. Qual. Diff. Equ. 20 (2007), 1-12].
Mathematics Subject Classification: 34C10, 34C15
Key words and phrases: Emden-Fowler equations, oscillation, rectifiable, infinite arc-length