Bounded solutions with prescribed numbers of zeros for the Emden-Fowler differential equation
The subject of this comprehensive paper is the Emden-Fowler equation (1) \(y'' + p(t) | y |^ \alpha \text{sgn} y = 0\) with \(p \in C^ 1 [t_ 0,\infty)\), \(p(t)>0\), which is investigated for \(\alpha>1\) and \(0<\alpha<1\) separately. The author derives rather sharp conditions on \(p(t)\) such that there exist solutions \(y(t)\) with any prescribed number of zeros in the unbounded interval \((t_ 0,\infty)\), thus generalizing several previous results on this topic. In the superlinear case \(\alpha>1\), the statement reads as follows: Assume \(\lim_{t \to \infty} \sup tp' (t)p(t)^{-1} < - (\alpha + 3)/2\). Then, for each \(k \in \mathbb{N}\) there exists a solution \(y_ k\) of (1) with precisely \(k-1\) zeros in \((t_ 0,\infty)\), where, moreover, \(y_ k(t_ 0)=0\) and the limits \(\lambda_ k : = \lim_{t \to \infty} y_ k(t)\) exist and satisfy \(0<\lambda_ 1 < \lambda_ 2 < \dots\). The proofs are mostly elementary, using methods and results from previous work by several authors and the present author himself. The results on equation (1) are also applied to the investigation of radial solution \(u(| x |)\) of the elliptic equation \(\Delta u + q (| x |) | u |^ \alpha \text{sgn} u=0\), which are defined in either the interior or exterior domain of the unit ball in \(\mathbb{R}^ n\), or on the whole space \(\mathbb{R}^ n\). Imposing appropriate boundary conditions on \(| x | = 1\) and/or limiting behavior for \(| x | \to \infty\), the author formulates conditions on \(q (| x |)\) (depending on \(\alpha)\) which again imply the existence of solutions \(u(r)\) with any prescribed number of zeros.
scientific article; zbMATH DE number 597301