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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=−ε |u|q−1u+|u|p−1u in ℝN, ε>0,tag*$(*)$ where $ N\ge3,\ 1<p<(N+2)/(N-2)$ and q>1 satisfies q ≠ p. For each integer k ≥ 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 ε>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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Cite

BibTeXRIS
BibTeX
@misc{guo2026uniqueness,
  title = {{Uniqueness of Bound States for Double-Power Scalar Field Equations}},
  author = {Yujin Guo and Wei Ji and Yong Luo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15232},
  eprint = {2610.15232},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15232},
}
RIS
TY  - UNPB
TI  - Uniqueness of Bound States for Double-Power Scalar Field Equations
AU  - Yujin Guo
AU  - Wei Ji
AU  - Yong Luo
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15232
UR  - https://arxiv.org/abs/2610.15232
AB  - We study the uniqueness of bound states for the following scalar field equation −Δu+u=−ε |u|^q−1u+|u|^p−1u in ℝ^N, ε>0,tag*$(*)$ where $ N\ge3,\ 1<p<(N+2)/(N-2)$ and q>1 satisfies q ≠ p. For each integer k ≥ 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 ε>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 (*).
ER  -