Interior bounds for stable elliptic states with sign-changing reaction laws
Let $B_r \subset \mathbb{R}^d$ denote the ball of radius $r$ centred at the origin. A solution $u \in C^2(B_1)$ of $-\Delta u = f(u)$ in $B_1$, with $f \in C^1(\mathbb{R})$, is stable if $$\int_{B_1} f'(u) \varphi^2 \le \int_{B_1} |\nabla \varphi|^2 \qquad (\varphi \in C_c^\infty(B_1)).$$ Cabré, Figalli, Ros-Oton and Serra proved that if $f \ge 0$ and $d \le 9$, then $\|u\|_{C^\alpha(B_{1/2})} \le C \|u\|_{L^1(B_1)}$ for stable solutions, with $\alpha > 0$ and $C$ depending only on $d$. Without a sign condition on $f$, an interior $L^\infty$ bound for stable solutions holding for all nonlinearities was proved for $d \le 4$ by Cabré, and for $d = 5$ Peng, Zhang and Zhou proved an interior Hölder estimate controlled by $\|\nabla u\|_{L^2}$. Stable radial $W^{1,2}(B_1)$ solutions are bounded for $d \le 9$ and every nonlinearity (Cabré and Capella).
Problem. For each $d \in \{6, 7, 8, 9\}$, does there exist $C_d < \infty$ such that, for every $f \in C^1(\mathbb{R})$ and every stable solution $u \in C^2(B_1)$ of $-\Delta u = f(u)$ in $B_1$, $$\|u\|_{L^\infty(B_{1/2})} \le C_d \|u\|_{W^{1,2}(B_1)}?$$ No sign assumption is imposed on $f$ or $u$, and the constant must be independent of $f$.
Source: Cabré, Miraglio and Sanchón (2022), §1.4, and Cabré (2023), who ask whether the sign condition f ≥ 0 is needed for interior regularity of stable solutions; Cabré (2026), §3.5.4, for dimensions 6 to 9. Stated by AIM — Open Applied Problems as a uniform bound in terms of the W^{1,2} norm.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, semilinear elliptic equations, stable solutions
References
@inproceedings{cabre2026,
author = {Cabr{\'e}, Xavier},
title = {Stable solutions to reaction-diffusion elliptic problems},
booktitle = {Proceedings of the International Congress of Mathematicians 2026, Volume 5: Invited Lectures (Sections 9--11)},
publisher = {SIAM},
year = {2026},
pages = {234--248},
doi = {10.1137/25M1803978},
eprint = {2603.03161},
archivePrefix = {arXiv},
}
@incollection{cabre2023,
author = {Cabr{\'e}, Xavier},
title = {Regularity of stable solutions to reaction-diffusion elliptic equations},
booktitle = {European Congress of Mathematics},
publisher = {EMS Press},
year = {2023},
pages = {1--8},
doi = {10.4171/8ECM/13},
eprint = {2112.08277},
archivePrefix = {arXiv},
}
@article{cabre2022p,
author = {Cabr{\'e}, Xavier and Miraglio, Pietro and Sanch{\'o}n, Manel},
title = {Optimal regularity of stable solutions to nonlinear equations involving the $p$-{L}aplacian},
journal = {Adv. Calc. Var.},
volume = {15},
number = {4},
year = {2022},
pages = {749--785},
doi = {10.1515/acv-2020-0055},
eprint = {2006.01445},
archivePrefix = {arXiv},
}
@article{cabre2020,
author = {Cabr{\'e}, Xavier and Figalli, Alessio and Ros-Oton, Xavier and Serra, Joaquim},
title = {Stable solutions to semilinear elliptic equations are smooth up to dimension 9},
journal = {Acta Math.},
volume = {224},
number = {2},
year = {2020},
pages = {187--252},
doi = {10.4310/ACTA.2020.v224.n2.a1},
eprint = {1907.09403},
archivePrefix = {arXiv},
}
@article{cabre2010,
author = {Cabr{\'e}, Xavier},
title = {Regularity of minimizers of semilinear elliptic problems up to dimension 4},
journal = {Comm. Pure Appl. Math.},
volume = {63},
number = {10},
year = {2010},
pages = {1362--1380},
doi = {10.1002/cpa.20327},
}
@article{peng2026,
author = {Peng, Fa and Zhang, Yi Ru-Ya and Zhou, Yuan},
title = {Interior {H}{\"o}lder regularity for stable solutions to semilinear elliptic equations up to dimension 5},
journal = {Adv. Math.},
volume = {503},
year = {2026},
pages = {111199},
doi = {10.1016/j.aim.2026.111199},
eprint = {2204.06345},
archivePrefix = {arXiv},
}
@article{cabre2006,
author = {Cabr{\'e}, Xavier and Capella, Antonio},
title = {Regularity of radial minimizers and extremal solutions of semilinear elliptic equations},
journal = {J. Funct. Anal.},
volume = {238},
number = {2},
year = {2006},
pages = {709--733},
doi = {10.1016/j.jfa.2005.12.018},
}
@misc{aim-stable-semilinear-sign-changing-interior-bound,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Interior bounds for stable elliptic states with sign-changing reaction laws}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/439-stable-semilinear-sign-changing-interior-bound.md},
note = {Entry 439 at commit 24edaad, 2026-10-15},
}
Resolution claims
We investigate interior regularity for stable solutions of the semilinear elliptic equation $-Δu=f(u)$ in an open set $Ω\subseteq \mathbb R^n$, where $f$ is locally Lipschitz and may change sign. We establish sharp interior Hölder estimates for stable classical solutions in every dimension $n\le 9$, thereby fully resolving a question raised by Cabré concerning interior regularity without the nonnegativity assumption on …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.21641v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00571, "Interior bounds for stable elliptic states with sign-changing reaction laws", clopen.solutions.