## 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 $(p(t)| y'| ^{\alpha-1}y')' + \lambda q(t)| y| ^{\alpha-1}y = 0, \quad t \geq a ,$ 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 $$[a,\infty)$$.

### MSC:

 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations

### Keywords:

oscillation; quasilinear differential equation
Full Text:

### 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. · Zbl 0511.34006 [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] Hille, E., Non-oscillation theorems, Trans. Amer. Math. Soc., 64, 234-252 (1948) · Zbl 0031.35402 · doi:10.2307/1990500 [4] Kusano, T.; Naito, Y.; Ogata, A., Strong oscillation and nonoscillation of quasilinear differential equations of second order, Differential Equations and Dynamical System, 2, 1-10 (1994) · Zbl 0869.34031 [5] Kusano, T.; Yoshida, N., Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. Anal. Appl., 189, 115-127 (1995) · Zbl 0823.34039 · doi:10.1006/jmaa.1995.1007 [6] Mirzov, D. D., On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems, J. Math. Anal. Appl., 53, 418-425 (1976) · Zbl 0327.34027 · doi:10.1016/0022-247X(76)90120-7 [7] Mirzov, D. D., On the oscillation of solutions of a system of differential equations, Mat. Zametki, 23, 401-404 (1978) · Zbl 0407.34034 [8] Nehari, Z., Oscillation criteria for second-order linear differential equations, Trans. Amer. Math. Soc., 85, 428-445 (1957) · Zbl 0078.07602 · doi:10.2307/1992939
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.