Uniqueness of a boson star at every subcritical mass
For $u \in H^{1/2}(\mathbb{R}^3)$ let $$D(u) = \iint \frac{|u(x)|^2\, |u(y)|^2}{|x - y|}\, dx\, dy, \qquad K(u) = \langle u, \sqrt{-\Delta}\, u \rangle,$$ $$N_* = \inf_{u \ne 0} \frac{2 K(u)\, \|u\|_2^2}{D(u)}, \qquad \mathcal{E}(u) = \langle u, (\sqrt{-\Delta + 1} - 1) u \rangle - \tfrac{1}{2} D(u),$$ where the square roots are Fourier multipliers; this is the pseudo-relativistic Hartree (boson star) functional with rest mass one. Lieb and Yau proved that for every $0 < N < N_*$ the infimum of $\mathcal{E}$ over $\|u\|_2^2 = N$ is attained. Every minimizer is, up to translation and a constant phase, positive, radial and decreasing. Lenzmann proved that the minimizer is unique up to translation and phase for all sufficiently small $N$ outside an at most countable set; Guo and Zeng removed the exceptional set, and Chen and Zeng gave another proof of the small-mass case. Chen, Guo, Wei and Yu (2026 preprint) proved uniqueness for all $N$ sufficiently close to $N_*$.
Problem. For every $0 < N < N_*$, is the minimizer unique up to translation and constant phase, that is, do any two minimizers $u$, $v$ satisfy $v(x) = e^{i\theta} u(x - x_0)$ for some $\theta \in \mathbb{R}$ and $x_0 \in \mathbb{R}^3$?
Source: Lieb and Yau (1987); see Lenzmann (2009) and Guo and Zeng (2020), Section 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, boson stars, ground states, pseudo-relativistic Hartree equations
References
@article{lieb1987,
author = {Lieb, Elliott H. and Yau, Horng-Tzer},
title = {The {C}handrasekhar theory of stellar collapse as the limit of quantum mechanics},
journal = {Comm. Math. Phys.},
volume = {112},
number = {1},
year = {1987},
pages = {147--174},
doi = {10.1007/BF01217684},
}
@article{lenzmann2009,
author = {Lenzmann, Enno},
title = {Uniqueness of ground states for pseudorelativistic {H}artree equations},
journal = {Anal. PDE},
volume = {2},
number = {1},
year = {2009},
pages = {1--27},
doi = {10.2140/apde.2009.2.1},
eprint = {0801.3976},
archivePrefix = {arXiv},
}
@article{guo2020,
author = {Guo, Yujin and Zeng, Xiaoyu},
title = {The {L}ieb--{Y}au conjecture for ground states of pseudo-relativistic {B}oson stars},
journal = {J. Funct. Anal.},
volume = {278},
number = {12},
year = {2020},
pages = {108510},
doi = {10.1016/j.jfa.2020.108510},
eprint = {1904.06957},
archivePrefix = {arXiv},
}
@misc{chen2026a,
author = {Chen, Pan and Zeng, Xiaoyu},
title = {A new proof of uniqueness of ground state for pseudo-relativistic {H}artree equation},
year = {2026},
eprint = {2608.08493},
archivePrefix = {arXiv},
}
@misc{chen2026b,
author = {Chen, Pan and Guo, Qi and Wei, Juncheng and Yu, Yuanyang},
title = {Uniqueness and nondegeneracy of ground state for the {$L^2$}-critical boson star equation},
year = {2026},
eprint = {2609.23771},
archivePrefix = {arXiv},
}
@misc{aim-lieb-yau-boson-star-uniqueness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of a boson star at every subcritical mass}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/201-lieb-yau-boson-star-uniqueness.md},
note = {Entry 201 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00449, "Uniqueness of a boson star at every subcritical mass", clopen.solutions.