Year: 2026
Type: preprint
Abstract: We study the uniqueness of bound states for the following scalar field equation $$ -Δu+u=-\varepsilon |u|^{q-1}u+|u|^{p-1}u \,\ \ \mbox{in}\,\ \mathbb R^N, \,\ \varepsilon>0,\tag*{$(*)$} $$ where $ N\ge3,\ 11$ satisfies $q \neq p$. For each integer $k \geq 1$, we prove that the equation $(*)$ has, up to translation and reflection, a unique bound state with precisely $k$ nodes in a natural coupling regime, where $\varepsilon>0$ is sufficiently small. Our results address a conjecture of [H. Berestycki and P. L. Lions, ARMA, 1983][M. X. Tang, Invent. Math., 2026] on the uniqueness of fixed-node bound states for the equation $(*)$.
Source: arXiv (Cornell University)
Authors: Guo, Yujin, Ji, Wei, Luo, Yong
Institution: Central China Normal University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G3933406863, G7342254039, G8224396934
OpenAlex ID: W7221766889