Asymptotic forms of positive solutions of third-order Emden-Fowler equations (Q1849157)

The study of qualitative behavior of third-order ordinary differential equations has been of interest from the seventies. In this setting, the authors consider equations of Emden-Fowler type, that is, \[ u'''(t)=p(t) | u(t)|^{\lambda-1}u(t), \] where \(\lambda>1\) and \(p(t)\) behaves like a positive \(C^{1}\) function near \(+\infty\) (that is \(lim_{t\rightarrow\infty} \frac{p(t)}{t^{\sigma}}\) for some \(\sigma\in\mathbb{R}\)). The main interest of such equations is to understand the asymptotic behavior of their solutions near \(+\infty\) and if it is possible to give their form. In fact \(u\in C^{3}\) is a solution of the former equation if \(u\) does not vanishes and satisfies the equation near \(+\infty\). \textit{I. T. Kiguradze} and \textit{T. A. Chanturiya} [Asymptotic properties of solutions of nonautonomous ordinary differential equations, Dordrecht: Kluwer Academic Publisher, (1992; Zbl 0782.34002)] gave five forms of asymptotic behavior (asymptotically constant solutions (AC), intermediate order solutions (I), asymptotically linear solutions (AL), asymptotically quadratic solutions (AQ) and asymptotically superquadratic solutions (ASQ)) and characterized them in terms of some integrals of \textit{p(t)} in the four first cases. Using these characterizations, the authors obtain the following results: The differential equation has: (1) an AC-solution if and only if \(\sigma+3<0\),(2) an AL-solution if and only if \(\sigma+\lambda+2<0\),(3) an AQ-solution if and only if \(\sigma+2\lambda+1<0\) and (4) an ASQ-solution if \(\sigma+2\lambda+1<0.\) Putting \(k=-\frac{\sigma+3}{\lambda-1}\), the authors improve also some results from \textit{P. A. Ohme} [Ann. Mat. Pura Appl. (4) 104, 43--65 (1975; Zbl 0307.34046)]: (a) If \(k>2\), then (1) has one of the following asymptotic forms: \(c_{1}\), \(c_{2}t\), \(c_{3}t^{2}\) or \(at^{k}\), where \(a^{\lambda-1}=k(k-1)(k-2)\) (\(c_{1}, c_{2}\) and \(c_{3}\) are positive constants). (b) If \(k=2\) the forms are: \(c_{1}\) and \(c_{2}t\). (c) For \(1<k<2\), we have the forms \(c_{1}\) and \(c_{2}t\). (d) For \(k=1\), we have \(c_{2}\) and \(\infty\). (e) For \(0<k<1\), we have \(c_{1}\) or \(at^{k}\), but in this case the description is not complete.