Uniqueness of bound states to $Δu-u+|u|^{p-1}u= 0$ in $\mathbb{R}^n$, $n\ge 3$
Moxun Tang
TL;DR
The paper resolves the Berestycki–Lions conjecture for the model nonlinearity $f(u)=-u+|u|^{p-1}u$ with $1<p<(n+2)/(n-2)$, $n\ge 3$, by proving that for each $k\ge 1$ there exists a unique radial bound state with exactly $k$ zeros in $\mathbb{R}^n$, up to translations and reflections. The authors implement a shooting method, analyzing the variation $v(r,\alpha)$ and employing a phase-decomposed framework that tracks zeros and critical points across phases, supported by energy and Pohozaev-type identities, positivity lemmas, and bridging constructions. Key ingredients include the phase-transition lemma, the positivity of $Q$, $M$, $T_2$, and the careful handling of delicate regimes (notably $n=3$, $p\in(1,2)$). This approach yields a robust, inductive mechanism establishing uniqueness of $k$-node bound states and provides a structured toolkit applicable to related nonlinear elliptic problems and standing-wave analyses in Klein–Gordon and Schrödinger equations.
Abstract
We give a positive answer to a conjecture of Berestycki and Lions in 1983 on the uniqueness of bound states to $Δu +f(u)=0$ 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 $|x|>0$.