Year: 2025
Type: article
Abstract: We give a positive answer to a conjecture of Berestycki and Lions in 1983 on the uniqueness of bound states to $\Delta u +f(u)=0$ Δ u + f ( u ) = 0 in $\mathbb{R}^{n}$ R n , $u\in H^{1}(\mathbb{R}^{n})$ u ∈ H 1 ( R n ) , $u\not \equiv 0$ u ≢ 0 , $n\ge 3$ n ≥ 3 . For the model nonlinearity $f(u)=-u+|u|^{p-1}u$ f ( u ) = − u + | u | p − 1 u , $1< p<(n+2)/(n-2)$ 1 < p < ( n + 2 ) / ( n − 2 ) , arising from finding standing waves of Klein-Gordon equation or nonlinear Schrödinger equation, we show that, for each integer $k\ge 1$ k ≥ 1 , the problem has a unique solution $u=u(|x|)$ u = u ( | x | ) , $x\in \mathbb{R}^{n}$ x ∈ R n , up to translation and reflection, that has precisely $k$ k zeros for $|x|>0$ | x | > 0 .
Source: Inventiones mathematicae
Author: Moxun Tang
Institution: Michigan State University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4415026708