Uniqueness of radial bound states for an equation with an odd cubic-like nonlinearity
Abstract
We investigate radial bound states with prescribed nodes for the odd cubic equation Δu+u(absu−b)(c−absu)=0 in Rn, n≥2, 0<b<c. Writing a=b/c, we prove that a radial bound state with exactly k≥1 simple zeros and positive central value exists if and only if 0<a<1/2. For each k, this solution is unique for all 0<a<1/2 in dimension two, for a≥(7−3√5)/2 in dimension three, and for a≥(43−5√(33))/32 in dimension four. For one node, we identify exact ranges obtained from a tangent comparison and a quotient inequality in every dimension n≥3; the threshold is dimension-independent for 3≤ n≤23 and is characterized implicitly for n≥24. These results address the natural odd counterpart of the cubic nonlinearity highlighted in Tang's prescribed-node uniqueness conjecture [M. Tang, \emph{Invent. Math.}, 2026]. The main challenges are the second positive zero of the cubic source, the lack of a global comparison inequality, and supercritical cubic growth in higher dimensions. We overcome them by an amplitude confinement argument, controlled extensions preserving sharp comparison estimates, and a subcritical truncation for existence. A complementary small-a uniqueness range in dimensions three to five follows from a recent massive double-power result.
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BibTeX
@misc{chen2026uniqueness,
title = {{Uniqueness of radial bound states for an equation with an odd cubic-like nonlinearity}},
author = {Wenjing Chen and Zexi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34025},
eprint = {2610.34025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34025},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniqueness of radial bound states for an equation with an odd cubic-like nonlinearity
AU - Wenjing Chen
AU - Zexi Wang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34025
UR - https://arxiv.org/abs/2610.34025
AB - We investigate radial bound states with prescribed nodes for the odd cubic equation Δu+u(absu−b)(c−absu)=0 in R^n, n≥2, 0<b<c. Writing a=b/c, we prove that a radial bound state with exactly k≥1 simple zeros and positive central value exists if and only if 0<a<1/2. For each k, this solution is unique for all 0<a<1/2 in dimension two, for a≥(7−3√5)/2 in dimension three, and for a≥(43−5√(33))/32 in dimension four. For one node, we identify exact ranges obtained from a tangent comparison and a quotient inequality in every dimension n≥3; the threshold is dimension-independent for 3≤ n≤23 and is characterized implicitly for n≥24. These results address the natural odd counterpart of the cubic nonlinearity highlighted in Tang's prescribed-node uniqueness conjecture [M. Tang, \emph{Invent. Math.}, 2026]. The main challenges are the second positive zero of the cubic source, the lack of a global comparison inequality, and supercritical cubic growth in higher dimensions. We overcome them by an amplitude confinement argument, controlled extensions preserving sharp comparison estimates, and a subcritical truncation for existence. A complementary small-a uniqueness range in dimensions three to five follows from a recent massive double-power result.
ER -