Nonlinear oscillations of fourth order quasilinear ordinary differential equations (Q653936)

The equation \[ (p(t)| u''| ^{\alpha} \text{sgn}\;u'')'' + q(t)| u| ^{\lambda} \text{sgn}\;u = 0 \] is considered with \(\alpha>0\), \(\lambda>0\) and \(p, q\colon [T, \infty)\mapsto(0,\infty)\) continuous, under the assumptions \[ \displaystyle{\int^{\infty}\left({{t}\over{p(t)}}\right)^{1/\alpha}dt<\infty , \;\int^{\infty}{{t}\over{(p(t))^{1/\alpha}}}dt<\infty}. \] The main results are as follows. Theorem 1. If \(\lambda>\alpha\) and \(c_1t^k\leq p(t)\leq c_2t^k\) with \(c_1>0\), \(c_2>0\), \(k\in {\mathbb R}\) constants, then every solution of the equation is oscillatory if and only if \[ \int^{\infty}t^{1+2\lambda-(k\lambda)/\alpha}q(t)dt = \infty. \] Theorem 2. If \(\lambda<\alpha\) and \(c_1t^k\leq p(t)\leq c_2t^k\;;\;c_3t^l\leq q(t)\leq c_4t^l\) where \(c_i>0\), \(i=1, 2, 3, 4\), \(k>0\), \(l\in {\mathbb R}\), then every solution of the equation is oscillatory if and only if \(\alpha+\lambda+l+2\geq k\). These results are further used in the analysis of the solutions of the binary elliptic system \[ \begin{cases}\triangle u=f(x)| v| ^{\sigma} \text{sgn}\;v, \\ \triangle v = -g(x)| u| ^{\tau} \text{sgn} \;u\end{cases},\;x\in\Omega = \mathbb R^N\setminus B(0, R) \] with \(N\geq 3\), \(f,g\in C(\Omega;(0,\infty))\), \(\sigma,\tau\geq 1\).