A single blow-up point for singular extremal states on convex domains
Let $d \ge 10$, let $\Omega \subset \mathbb{R}^d$ be bounded, smooth and convex, and let $f \in C^\infty([0,\infty))$ be positive, increasing and convex with $f(t)/t \to \infty$ as $t \to \infty$. For the Dirichlet problem $$-\Delta u = \lambda f(u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega,$$ let $u_\lambda$ be the minimal positive classical solution for $0 < \lambda < \lambda^*$, where $\lambda^*$ is the maximal parameter of this branch, and let $u^* = \lim_{\lambda \uparrow \lambda^*} u_\lambda$, the extremal solution. Cabré, Figalli, Ros-Oton and Serra proved that $u^*$ is bounded when $d \le 9$. For $d \ge 10$, on the unit ball with $f(u) = e^u$, one has $\lambda^* = 2(d-2)$ and $u^* = \log(1/|x|^2)$ (Joseph and Lundgren). When $\Omega$ is a ball, $u^*$ is radially decreasing, so if it is unbounded its only singular point is the centre (Brezis and Vázquez).
Problem. Suppose $u^* \notin L^\infty(\Omega)$. Must the set $$\Sigma = \Bigl\{x \in \overline{\Omega} : \mathop{\mathrm{ess\,sup}}_{\Omega \cap B_r(x)} u^* = \infty \text{ for every } r > 0\Bigr\}$$ consist of exactly one point?
Source: Brezis and Vázquez (1997), Problem 5; restated by Brezis (2023), Open Problem 6.2. Stated by AIM — Open Applied Problems with an explicit definition of the blow-up set.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, extremal solutions, singular sets
References
@article{brezis1997,
author = {Brezis, Ha{\"i}m and V{\'a}zquez, Juan Luis},
title = {Blow-up solutions of some nonlinear elliptic problems},
journal = {Rev. Mat. Univ. Complut. Madrid},
volume = {10},
number = {2},
year = {1997},
pages = {443--469},
}
@article{brezis2023,
author = {Brezis, Ha{\"i}m},
title = {Some of my favorite open problems},
journal = {Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.},
volume = {34},
number = {2},
year = {2023},
pages = {307--335},
doi = {10.4171/RLM/1008},
}
@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{joseph1973,
author = {Joseph, D. D. and Lundgren, T. S.},
title = {Quasilinear {D}irichlet problems driven by positive sources},
journal = {Arch. Rational Mech. Anal.},
volume = {49},
number = {4},
year = {1973},
pages = {241--269},
doi = {10.1007/BF00250508},
}
@misc{aim-convex-domain-extremal-single-singularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A single blow-up point for singular extremal states on convex domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/440-convex-domain-extremal-single-singularity.md},
note = {Entry 440 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00584, "A single blow-up point for singular extremal states on convex domains", clopen.solutions.