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Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation. (English) Zbl 0856.34058
The author uses the Haraux-Weissler equation $$\Delta U+ {1\over 2} \langle x, \text{grad } U\rangle+ {1\over p- 1} U+ |U|^{p- 1} U= 0$$ in $$\mathbb{R}^n$$ as a motivation (via radial solutions) to study the ODE problem $$u''+ ({n- 1\over r}+ {r\over 2}) u'+ {1\over p- 1} u+ |u|^{p- 1} u= 0$$ in $$\mathbb{R}^+$$ with initial conditions $$u(0)= \alpha> 0$$, $$u'(0)= 0$$. He looks for solutions $$u$$ with a prescribed number $$i$$ of zeroes and a certain exponential decay at infinity. The main result is a uniqueness theorem (for $$(n -2) p\leq n$$) saying that given $$i$$ there is at most one $$\alpha$$ such that $$u$$ meets these conditions. Its proof is based on two propositions: i) The number of zeroes of $$u$$ increases with $$\alpha$$ and ii) in fact strictly when $$\alpha$$ passes $$\varphi(0)$$, where $$\varphi$$ is a decaying solution.
Reviewer: N.Weck (Essen)

##### MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 35B40 Asymptotic behavior of solutions to PDEs 35J60 Nonlinear elliptic equations
##### Keywords:
Haraux-Weissler equation; exponential decay; uniqueness
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