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arXiv · 2610.15232

Uniqueness of Bound States for Double-Power Scalar Field Equations

Yujin Guo·Wei Ji·Yong Luo
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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,\ 1 1$ 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 $(*)$.

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Yujin Guo, Wei Ji, Yong Luo. 2026-10-10. Uniqueness of Bound States for Double-Power Scalar Field Equations. https://arxiv.org/abs/2610.15232

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