## 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
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### References:

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