Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation (Q1914827)

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.