Uniqueness of strong-KPP equilibria on general unbounded domains
Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be connected, unbounded and uniformly $C^{2,\gamma}$ for some $0 < \gamma < 1$: its boundary has graph charts of a common positive radius with uniformly bounded $C^{2,\gamma}$ norms. Let $f \in C^{1,\gamma}([0,\infty))$ satisfy $f(0) = f(1) = 0$, $f'(0) > 0$, $f > 0$ on $(0,1)$ and $f < 0$ on $(1,\infty)$, and let $f(s)/s$ be strictly decreasing on $(0,1]$ (a strong-KPP reaction). For $\beta \in [0,1)$ consider $$-\Delta u = f(u) \quad \text{in } \Omega, \qquad \beta\, \partial_n u + (1 - \beta)\, u = 0 \quad \text{on } \partial\Omega,$$ where $n$ is the outward normal; $\beta = 0$ is the Dirichlet condition. Berestycki and Graham proved that this problem has at most one positive bounded solution whenever $f'(0)$ lies outside the closure of the principal limit spectrum of $\Omega$, the set of generalized principal eigenvalues of $-\Delta$, with the same boundary condition, on the connected limits of translates of $\Omega$.
Problem. For each fixed $\beta \in [0,1)$, is there at most one bounded positive classical solution? No periodicity or spectral-gap assumption at spatial infinity is made.
Source: Berestycki and Graham (2025), Conjecture 1.3, repeated as Open Question 3 in Berestycki and Graham (2026); stated for connected unbounded domains in dimension d ≥ 2 and β ∈ [0, 1) by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, reaction–diffusion equations, strong-KPP reactions
References
@article{berestycki2025,
author = {Berestycki, Henri and Graham, Cole},
title = {The steady states of strong-{KPP} reactions in general domains},
journal = {J. Eur. Math. Soc. (JEMS)},
year = {2025},
doi = {10.4171/JEMS/1578},
eprint = {2212.06611},
archivePrefix = {arXiv},
}
@article{berestycki2026,
author = {Berestycki, Henri and Graham, Cole},
title = {A stable-compact method for qualitative properties of semilinear elliptic equations},
journal = {Arch. Ration. Mech. Anal.},
volume = {250},
number = {2},
year = {2026},
pages = {21},
doi = {10.1007/s00205-026-02168-6},
eprint = {2309.16642},
archivePrefix = {arXiv},
}
@misc{aim-strong-kpp-unbounded-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 strong-KPP equilibria on general unbounded domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/379-strong-kpp-unbounded-uniqueness.md},
note = {Entry 379 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct a connected unbounded domain in $\R^{21}$ with uniformly $C^{2,1/2}$ boundary on which the Dirichlet Fisher--KPP equation has two distinct bounded positive equilibria. The boundary consists of two smooth global graphs whose separation tends to the critical slab width. A corrected transverse mode gives a slowly decaying subsolution, while a compact enlargement of the domain permits a second equilibrium …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00837, "Uniqueness of strong-KPP equilibria on general unbounded domains", clopen.solutions.