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Nonlinear oscillations of fourth order quasilinear ordinary differential equations. (English) Zbl 1249.34111
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 $$N\geq 3$$, $$f,g\in C(\Omega;(0,\infty))$$, $$\sigma,\tau\geq 1$$.

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations
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