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Zbl 0906.34024
Kusano, T.; Naito, Y.
Oscillation and nonoscillation criteria for second order quasilinear differential equations.
(English)
[J] Acta Math. Hung. 76, No.1-2, 81-99 (1997). ISSN 0236-5294; ISSN 1588-2632/e

The authors concern the oscillatory (and nonoscillatory) behaviour of quasilinear differential equations of the form $$(p(t)| y'| ^{\alpha-1}y')' + \lambda q(t)| y| ^{\alpha-1}y = 0, \quad t \geq a ,$$ where $\alpha$ and $a$ are positive constants, $p(t)$ and $q(t)$ are continuous functions on $[a,\infty)$ and $\lambda > 0$ is a parameter. For a fixed $\lambda$ all solutions are either oscillatory or else nonoscillatory. Here, oscillation and nonoscillation criteria are given in terms of $p,q$ and $\lambda$. The results find applications to quasilinear degenerate elliptic partial differential equations of the type $$\sum_{i=1}^N D_i (| Du| ^{m-2} D_i u) + c(| x|) | u| ^{m-2}u = 0, \quad x \in E_\alpha,$$ with $m>1$, $N \geq 2$, $D_i = \partial/\partial x_i$, $i = 1,\dots,N$, $D=(D_1,\dots,D_N)$, $E_\alpha = \{ x \in {\bbfR^N} : | x| \geq A \}$, $a > 0$, and $c(t)$ is a nonnegative function on $[a,\infty)$.
[Stefan Siegmund (Augsburg)]
MSC 2000:
*34C15 Nonlinear oscillations of solutions of ODE

Keywords: oscillation; quasilinear differential equation

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