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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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##### References:
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