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Structure of positive radial solutions to the Haraux-Weissler equation. (English) Zbl 0934.34028
The authors consider the system \[ u_{rr}+{n-1\over r} u_r+{r \over 2}u_r+ \lambda u+|u|^{p-1}u=0,\;r>0,\;u(0)=\alpha>0, \tag{1} \] with \(n\geq 3\), \(p>1\), \(\lambda>0\). The main result is the following: Suppose \(n\geq 3\), \(1<p<(n+2)/(n-2)\). If \(0<\lambda\leq(n-2)/2\), then there exists a unique positive number \(\alpha_\lambda\) such that \(u(r;\alpha_\lambda)\) is a rapidly decaying solution. Morover, \(u(r,\alpha)\) is a crossing solution for every \(\alpha \in(\alpha_\lambda,\infty)\) and a slowly decaying solution for every \(\alpha\in (0,\alpha_\lambda)\).
Reviewer: S.Mazanik (Minsk)

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
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