Emden-Fowler equations with infinitely extendible unbounded solutions
The author proves the following two theorems: Theorem 1. For any piecewise continuous and monotonous function \(\phi\) (t)\(\geq 1\) on \(t\geq 0\), \(\phi (t)\to +\infty\) when \(t\to +\infty\), and any number \(\mu >1\), there exists a piecewise continuous function p(t)\(\geq 0\), satisfying (1) \(\int^{+\infty}_{0}(p(\tau)/\phi (\tau))\tau^{\lambda}d\tau =+\infty\) such that the Emden-Fowler equation (2) \(U''=p(t)| U|^{\lambda}sgn U\), \(\lambda >1\), \(t\geq 0\) has a two parameter family of regular solutions u(t), with (3) \(0<\dot U(t)/u(t)<t^{- 1}\phi^{\mu}(t)\) when \(t\geq t_ 1>0\). On the other hand, if \(\mu <1\), then for any piecewise continuous function p(t)\(\geq 0\) satisfying (1), (2) has no solution, which can satisfy (3) for all \(t\geq t_ u>0.\) This theorem answers a question of I. T. Kiguradze, as to whether the condition (4) \(\int^{\infty}_{0}p(\tau)\tau^{\lambda}d\tau <+\infty\) is necessary or not for (2) to have unbounded regular solution with \(p(t)\geq t^{1-\lambda}\). const., which obviously satisfies (5) \(\int^{\infty}_{1}p(\tau)\tau^{\lambda}d\tau =+\infty\) ((4) was proved to be a sufficient condition by Kiguradze in 1965), and also establishes a relation between u(t) and u'(t). Theorem 2. Equation (2) with any piecewise continuous function p(t)\(\geq 0\) satisfying (6) \(\int^{+\infty}_{1}\tau^{- 2}[p(\tau)\tau^{\lambda +3}]^{\mu}d\tau =+\infty\), \(\mu >0\) has no unbounded regular solution, if \(\mu <\). On the other hand, for any \(\mu >\), there exists a piecewise continuous function p(t)\(\geq 0\), satisfying (6) and (5), for which equation (2) has a two parameter family of unbounded regular solutions. In case \(\mu =\), there exists p(t)\(\geq 0\) satisfying (6), for which equation (2) has no unbounded regular solution, but there also exists p(t)\(\geq 0\) satisfying (6) and (4), for which equation (2) has a two parameter family of unbounded regular solutions.
scientific article; zbMATH DE number 3881080