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

TL;DR

This paper proves the uniqueness of bound state solutions with a specific number of zeros for a class of nonlinear elliptic equations in higher dimensions, confirming a longstanding conjecture.

Contribution

It establishes the uniqueness of radially symmetric bound states with a given number of zeros for a nonlinear PDE, resolving a conjecture from 1983.

Findings

Unique solutions with exactly k zeros for each integer k≥1

Solutions are radially symmetric and unique up to translation and reflection

Confirms the conjecture of Berestycki and Lions from 1983

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 Schr\"odinger 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 $|x|>0$.