Optimal dimension for bounded subquadratic p-Laplace extremal states
Let $1 < p < 2$ and let $d$ be an integer with $2 \le d < p + 4p/(p-1)$. Let $\Omega \subset \mathbb{R}^d$ be bounded, smooth and strictly convex. Suppose $f \in C^2([0,\infty))$ is positive, strictly increasing and convex, and $f(t)/t^{p-1} \to \infty$ as $t \to \infty$. For $$-\operatorname{div}(|\nabla u|^{p-2} \nabla u) = \lambda f(u), \qquad u \in W^{1,p}_0(\Omega), \quad u > 0,$$ with the weak equation tested against $C_c^\infty(\Omega)$, let $u_\lambda$ be the minimal bounded weak solution for $0 < \lambda < \lambda^*$, where $\lambda^*$ is the supremum of the parameters with such a solution.
For the exponential nonlinearity, García Azorero, Peral and Puel proved that stable energy solutions are bounded when $d < p + 4p/(p-1)$, and that this is optimal: for $d \ge p + 4p/(p-1)$, $u(x) = \log(|x|^{-p})$ is a singular stable energy solution of $-\operatorname{div}(|\nabla u|^{p-2} \nabla u) = p^{p-1}(d-p) e^u$ on the unit ball, vanishing on its boundary. In the radial case, boundedness of stable solutions in the range $d < p + 4p/(p-1)$ is due to Cabré, Capella and Sanchón. Cabré, Miraglio and Sanchón proved that $u^* = \lim_{\lambda \uparrow \lambda^*} u_\lambda$ is bounded on smooth strictly convex domains when $1 < p < 2$ and $d < 5p$, for $f$ as above; for $p \ge 2$ their result reaches the full range $d < p + 4p/(p-1)$.
Problem. Must the monotone extremal limit $u^* = \lim_{\lambda \uparrow \lambda^*} u_\lambda$ belong to $L^\infty(\Omega)$?
Source: Cabré, Miraglio and Sanchón (2022), §1.4; also Cabré (2023) and Cabré (2026), §3.5.2. Stated for strictly convex domains by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, extremal solutions, p-Laplace equations
References
@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},
}
@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},
}
@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},
}
@article{garciaazorero1994,
author = {Garc{\'i}a Azorero, J. and Peral Alonso, I. and Puel, J.-P.},
title = {Quasilinear problems with exponential growth in the reaction term},
journal = {Nonlinear Anal.},
volume = {22},
number = {4},
year = {1994},
pages = {481--498},
doi = {10.1016/0362-546X(94)90169-4},
}
@article{cabre2009,
author = {Cabr{\'e}, Xavier and Capella, Antonio and Sanch{\'o}n, Manel},
title = {Regularity of radial minimizers of reaction equations involving the $p$-{L}aplacian},
journal = {Calc. Var. Partial Differential Equations},
volume = {34},
number = {4},
year = {2009},
pages = {475--494},
doi = {10.1007/s00526-008-0192-3},
}
@misc{aim-subquadratic-p-laplace-extremal-critical-dimension,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Optimal dimension for bounded subquadratic p-Laplace extremal states}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/444-subquadratic-p-laplace-extremal-critical-dimension.md},
note = {Entry 444 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-00645, "Optimal dimension for bounded subquadratic p-Laplace extremal states", clopen.solutions.