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

The authors concern the oscillatory (and nonoscillatory) behaviour of quasilinear differential equations of the form

$\left(p\left(t\right)|{y}^{\text{'}}{|}^{\alpha -1}{y}^{\text{'}}{\right)}^{\text{'}}+\lambda q\left(t\right){|y|}^{\alpha -1}y=0,\phantom{\rule{1.em}{0ex}}t\ge a,$

where $\alpha$ and $a$ are positive constants, $p\left(t\right)$ and $q\left(t\right)$ are continuous functions on $\left[a,\infty \right)$ 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

$\sum _{i=1}^{N}{D}_{i}{\left(|Du|}^{m-2}{D}_{i}{u\right)+c\left(|x|\right)|u|}^{m-2}u=0,\phantom{\rule{1.em}{0ex}}x\in {E}_{\alpha },$

with $m>1$, $N\ge 2$, ${D}_{i}=\partial /\partial {x}_{i}$, $i=1,\cdots ,N$, $D=\left({D}_{1},\cdots ,{D}_{N}\right)$, ${E}_{\alpha }=\left\{x\in {ℝ}^{N}:|x|\ge A\right\}$, $a>0$, and $c\left(t\right)$ is a nonnegative function on $\left[a,\infty \right)$.

##### MSC:
 34C15 Nonlinear oscillations, coupled oscillators (ODE)
##### 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. [3] E. Hille, Non-oscillation theorems,Trans. Amer. Math. Soc.,64 (1948), 234–252. · doi:10.1090/S0002-9947-1948-0027925-7 [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. [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 · doi:10.1006/jmaa.1995.1007 [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 · doi:10.1016/0022-247X(76)90120-7 [7] D. D. Mirzov, On the oscillation of solutions of a system of differential equations,Mat. Zametki,23 (1978), 401–404 (Russian). [8] Z. Nehari, Oscillation criteria for second-order linear differential equations,Trans. Amer. Math. Soc.,85 (1957), 428–445. · doi:10.1090/S0002-9947-1957-0087816-8