Uniqueness of positive reaction–diffusion equilibria outside two obstacles
For $d \ge 2$, radii $r_a, r_b > 0$ and centres $a, b \in \mathbb{R}^d$ with $|a - b| > r_a + r_b$, put $$\Omega = \mathbb{R}^d \setminus \bigl(\overline{B_{r_a}(a)} \cup \overline{B_{r_b}(b)}\bigr).$$ Let $f \in C^{1,\gamma}([0,\infty))$, $0 < \gamma < 1$, satisfy $f(0) = f(1) = 0$, $f'(0) > 0$, $f'(1) < 0$, $f > 0$ on $(0,1)$ and $f < 0$ on $(1,\infty)$. The per-capita growth rate $f(s)/s$ may increase, so such reactions need not be of strong-KPP type. Berestycki and Graham proved that for every such $f$ the Dirichlet problem below has a unique positive bounded solution outside any compact obstacle with uniformly $C^{2,\gamma}$ boundary that is star-shaped about a nonempty open set of points, in particular outside a single ball, and noted that their method also gives uniqueness outside two balls that are sufficiently far apart. On every bounded domain, by contrast, some such reaction admits several positive solutions.
Problem. Must $$-\Delta u = f(u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega$$ have exactly one bounded positive solution $u \in C^2(\Omega) \cap C(\overline{\Omega})$, for every such $d$, pair of balls and reaction $f$? The separation of the obstacles is arbitrary subject to disjointness.
Source: Berestycki and Graham (2026), Open Question 7(ii), raised by Bassam Fayad; stated for all disjoint pairs of balls in every dimension d ≥ 2 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, exterior domains, monostable reactions, reaction–diffusion equations
References
@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-reaction-two-obstacle-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 positive reaction–diffusion equilibria outside two obstacles}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/384-reaction-two-obstacle-uniqueness.md},
note = {Entry 384 at commit 24edaad, 2026-10-15},
}
Resolution claims
We construct a positive monostable reaction and two disjoint disks for which an exterior reaction--diffusion equation with zero Dirichlet boundary data has two distinct bounded positive equilibria. This gives a negative answer to the universal two-obstacle uniqueness question posed by Berestycki and Graham. One equilibrium lies above a positive profile on the infinite strip between the disks. A second is …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | 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-00829, "Uniqueness of positive reaction–diffusion equilibria outside two obstacles", clopen.solutions.