Uniqueness of bound states to $\Delta u-u+|u{|}^{p-1}u=0$ in ${\mathbb{R}}^n$, $n\ge 3$

Abstract

We give a positive answer to a conjecture of Berestycki and Lions in 1983 on the uniqueness of bound states to $\Delta u+f(u)=0$ in ${\mathbb{R}}^n$, $u\in H^1({\mathbb{R}}^n)$, $u\not\equiv 0$, $n\ge 3$. For the model nonlinearity $f(u)=-u+|u{|}^{p-1}u$, $1<p<(n+2)/(n-2)$, arisen from finding standing waves of Klein-Gordon equation or nonlinear Schrödinger equation, we show that, for each integer $k\ge 1$, the problem has a unique solution $u=u(|x|)$, $x\in {\mathbb{R}}^n$, up to translation and reflection, that has precisely $k$ zeros for $|x|>0$.