OP-00836 Solution claimed

Extinction at the critical Dirichlet eigenvalue for strong-KPP reactions

Let $\Omega \subset \mathbb{R}^d$, $d \ge 2$, be a connected unbounded domain that is 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. The domain need not be periodic or geometrically convergent at infinity. Let $$\lambda_1(\Omega) = \inf_{0 \ne \phi \in C_c^\infty(\Omega)} \frac{\int_\Omega |\nabla\phi|^2}{\int_\Omega \phi^2},$$ and 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 $(1,\infty)$, with $f(s)/s$ strictly decreasing on $(0,1]$ (a strong-KPP reaction). Consider the Dirichlet problem $$-\Delta u = f(u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega.$$ Berestycki and Graham proved that it has a positive bounded solution when $\lambda_1(\Omega) < f'(0)$ and none when $\lambda_1(\Omega) > f'(0)$, and that it has none when $\lambda_1(\Omega) = f'(0)$ and $\Omega$ is bounded or periodic.

Problem. If $\lambda_1(\Omega) = f'(0)$, must the Dirichlet problem have no bounded positive classical solution?

Source: Berestycki and Graham (2025), Conjecture 1.7; stated for the Dirichlet problem on connected unbounded domains in 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, generalized principal eigenvalue, reaction–diffusion equations

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},
}
@misc{aim-strong-kpp-critical-extinction,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Extinction at the critical Dirichlet eigenvalue for strong-KPP reactions}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/380-strong-kpp-critical-extinction.md},
  note         = {Entry 380 at commit 24edaad, 2026-10-15},
}

Resolution claims

A positive equilibrium at the critical Dirichlet eigenvalue v1 GitHub ↗

Kaiwen Jin

We construct a connected unbounded uniformly smooth domain in $\R^{21}$ whose bottom Dirichlet eigenvalue is $1$, but on which $-\Delta u=u-u^2$ has a bounded positive solution with zero boundary values. Thus equality between the spectral threshold and the derivative of a strict KPP reaction at zero does not force extinction on all uniformly smooth domains. The domain is a slight …

Counterexample · not verified · fidelity not established · posted on GitHub 15 Oct 2026

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-00836, "Extinction at the critical Dirichlet eigenvalue for strong-KPP reactions", clopen.solutions.