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On existence of oscillatory solutions of second order Emden-Fowler equations. (English) Zbl 1027.34039

The authors study the second-order Emden-Fowler equation

${y}^{\text{'}\text{'}}+a{|y|}^{\gamma }\text{sgn}y=0,\phantom{\rule{1.em}{0ex}}\gamma >0,\phantom{\rule{4pt}{0ex}}\gamma \ne 1,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $a$ is a positive absolutely continuous function on $\left(0,\infty \right)$. Let $\varphi$ be the function defined by $\varphi \left(x\right)=a\left(x\right){x}^{\frac{\gamma +3}{2}}$, and assume that $\varphi$ is bounded away from zero at infinity. Under this condition, the main result of the paper says that if the negative part of ${\varphi }^{\text{'}}$, ${\varphi }_{-}^{\text{'}}\left(x\right)=-min\left({\varphi }^{\text{'}}\left(x\right),0\right)$, belongs to ${L}^{1}\left(0,\infty \right)$, then equation (1) has oscillatory solutions (that is, solutions with arbitrary large zeroes). The authors provide an example that shows that this result, being applicable to nonmonotonous functions $\varphi$, strictly extends previous results of M. Jasny [Cas. Pest. Mat. 85, 78-82 (1960; Zbl 0113.07603)], J. Kurzweil [Cas. Pest. Mat. 85, 357-358 (1960; Zbl 0129.06204)], J. W. Heidel and D. B. Hinton [SIAM J. Math. Anal. 3, 344-351 (1972; Zbl 0243.34062)], L. H. Erbe and J. S. Muldowney [Ann. Mat. Pura Appl., IV. Ser. 109, 23-38 (1976; Zbl 0345.34022)], and K. Chiou [Proc. Am. Math. Soc. 35, 120-122 (1972; Zbl 0262.34026)].

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A34 Nonlinear ODE and systems, general
##### Keywords:
oscillatory solution; Emden-Fowler equation