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 \([a,\infty)\).


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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