Uniqueness of prescribed-node radial bound states for a zero-mass double-power equation
Abstract
We study radial bound states of Δu−absuq−1u+absup−1u=0 in Rn, n≥3, 1<q<p<(n+2)/(n−2). For every integer k≥1, we prove that there is exactly one radial bound state with k simple zeros and u(0)>0. Its absolute maximum is attained at the origin. This confirms the zero-mass, Sobolev-subcritical case of Tang's double-power uniqueness conjecture. The proof adapts Tang's phase-transition analysis and combines it with a finite-level separation argument inspired by Cortázar--García-Huidobro--Yarur. Three main difficulties arise in the zero-mass double-power setting: the sign changes of key comparison quantities in the low-amplitude region, the possible failure of pointwise positivity of an auxiliary quantity when n=3 and p+q<3, and the absence of exponential decay at infinity. We overcome these difficulties by introducing bridge functionals to propagate the required sign information through low amplitudes, establishing a weighted integral positivity estimate in the exceptional three-dimensional range, and combining a radial-flux tail estimate with finite-level inverse-branch comparison. We also establish the strict ordering of the bound-state shooting heights and show that the shooting variation along the k-node state has exactly k+1 simple zeros.
Related
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BibTeX
@misc{chen2026uniqueness,
title = {{Uniqueness of prescribed-node radial bound states for a zero-mass double-power equation}},
author = {Wenjing Chen and Zexi Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15405},
eprint = {2610.15405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15405},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniqueness of prescribed-node radial bound states for a zero-mass double-power equation
AU - Wenjing Chen
AU - Zexi Wang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15405
UR - https://arxiv.org/abs/2610.15405
AB - We study radial bound states of Δu−absu^q−1u+absu^p−1u=0 in R^n, n≥3, 1<q<p<(n+2)/(n−2). For every integer k≥1, we prove that there is exactly one radial bound state with k simple zeros and u(0)>0. Its absolute maximum is attained at the origin. This confirms the zero-mass, Sobolev-subcritical case of Tang's double-power uniqueness conjecture. The proof adapts Tang's phase-transition analysis and combines it with a finite-level separation argument inspired by Cortázar--García-Huidobro--Yarur. Three main difficulties arise in the zero-mass double-power setting: the sign changes of key comparison quantities in the low-amplitude region, the possible failure of pointwise positivity of an auxiliary quantity when n=3 and p+q<3, and the absence of exponential decay at infinity. We overcome these difficulties by introducing bridge functionals to propagate the required sign information through low amplitudes, establishing a weighted integral positivity estimate in the exceptional three-dimensional range, and combining a radial-flux tail estimate with finite-level inverse-branch comparison. We also establish the strict ordering of the bound-state shooting heights and show that the shooting variation along the k-node state has exactly k+1 simple zeros.
ER -