Oscillation and nonoscillation criteria for second order quasilinear differential equations. (English) Zbl 0906.34024

Kusano, T.; Naito, Y.

Acta Math. Hung. 76, No. 1-2, 81-99 (1997).

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 \(m>1\), \(N \geq 2\), \(D_i = \partial/\partial x_i\), \(i = 1,\dots,N\), \(D=(D_1,\dots,D_N)\), \(E_\alpha = \{ x \in {\mathbb{R}^N} : | x| \geq A \}\), \(a > 0\), and \(c(t)\) is a nonnegative function on \([a,\infty)\).

Reviewer: Stefan Siegmund (Augsburg)

MSC:

34C15

Nonlinear oscillations and coupled oscillators for ordinary differential equations

Keywords:

oscillation; quasilinear differential equation

References:

[1]	Á. Elbert, A half-linear second order differential equation, Colloquia Math. Soc. János Bolyai 30:Qualitative Theory of Differential Equations (Szeged, 1979), 153–180.
[2]	Á. Elbert, Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations, Lecture Notes in Mathematics, Vol. 964:Ordinary and Partial Differential Equations (1982), pp. 187–212. · Zbl 0528.34034
[3]	E. Hille, Non-oscillation theorems,Trans. Amer. Math. Soc.,64 (1948), 234–252. · Zbl 0031.35402
[4]	T. Kusano, Y. Naito, and A. Ogata, Strong oscillation and nonoscillation of quasilinear differential equations of second order,Differential Equations and Dynamical System,2 (1994), 1–10. · Zbl 0869.34031
[5]	T. Kusano, and N. Yoshida, Nonoscillation theorems for a class of quasilinear differential equations of second order,J. Math. Anal. Appl.,189 (1995), 115–127. · Zbl 0823.34039
[6]	D. D. Mirzov, On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems,J. Math. Anal. Appl.,53 (1976), 418–425. · Zbl 0327.34027
[7]	D. D. Mirzov, On the oscillation of solutions of a system of differential equations,Mat. Zametki,23 (1978), 401–404 (Russian). · Zbl 0423.34047
[8]	Z. Nehari, Oscillation criteria for second-order linear differential equations,Trans. Amer. Math. Soc.,85 (1957), 428–445. · Zbl 0078.07602