Regular MEMS extremal states without an asymptotic curvature condition
Let $3 \le d \le 6$, let $\Omega \subset \mathbb{R}^d$ be a bounded smooth domain, and let $f \in C^2([0,1))$ be positive, nondecreasing and convex, with $f(t) \to \infty$ as $t \uparrow 1$ and $\int_0^1 f(t) \, dt = \infty$. Consider $$-\Delta u = \lambda f(u) \quad \text{in } \Omega, \qquad u = 0 \quad \text{on } \partial\Omega, \qquad 0 < u < 1.$$ Let $\lambda^*$ be the supremum of the parameters $\lambda > 0$ admitting a classical solution, let $u_\lambda$ be the pointwise minimal classical solution for $0 < \lambda < \lambda^*$, and set $u^* = \lim_{\lambda \uparrow \lambda^*} u_\lambda$.
Bruera and Cabré proved that a stable classical solution $u$ of $-\Delta u = f(u)$ in $\Omega$, $u = 0$ on $\partial\Omega$, with $f$ nonnegative, nondecreasing, convex and $\int_0^1 f = \infty$, satisfies $\|u\|_{L^\infty(\Omega)} \le C < 1$ for a constant $C$ depending only on $\Omega$ and $f$, when $d \le 2$, and when $3 \le d \le 6$ under the Crandall–Rabinowitz type condition $$\liminf_{t \uparrow 1} \frac{f(t) f''(t)}{f'(t)^2} > 1,$$ equivalently, $f^{-\theta}$ is concave near $1$ for some $\theta > 0$. For $d \ge 7$ there are singular stable solutions with $f(u)$ a multiple of $(1-u)^{-p}$ for some $p > 1$, such as $1 - |x|^{2/(1+p)}$ on the unit ball. Peng and Villegas proved, without the condition, that for $2 \le d \le 6$ every stable radial solution $0 \le u \le 1$ of $-\Delta u = f(u)$ on the unit ball vanishing on its boundary, with $f \ge 0$ and $\int_0^1 f = \infty$, satisfies $\|u\|_{L^\infty} < 1$.
Problem. Must $\|u^*\|_{L^\infty(\Omega)} < 1$, without assuming $\liminf_{t \uparrow 1} f(t) f''(t)/f'(t)^2 > 1$?
Source: Bruera and Cabré (2025), open problem in §1.1, for stable solutions with singular nonlinearities in general; Cabré (2026), §3.5.3. Stated for the extremal solution of the Dirichlet problem by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, extremal solutions, singular nonlinearities
References
@article{bruera2026,
author = {Bruera, Renzo and Cabr{\'e}, Xavier},
title = {Regularity of stable solutions to the {MEMS} problem up to the optimal dimension 6},
journal = {Calc. Var. Partial Differential Equations},
volume = {65},
number = {9},
year = {2026},
pages = {256},
doi = {10.1007/s00526-026-03419-w},
eprint = {2507.20916},
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{peng2026mems,
author = {Peng, Fa and Villegas, Salvador},
title = {Regularity of stable radial solutions to semilinear elliptic equations in {MEMS} problems},
journal = {J. Differential Equations},
volume = {471},
year = {2026},
pages = {114381},
doi = {10.1016/j.jde.2026.114381},
eprint = {2602.21115},
archivePrefix = {arXiv},
}
@misc{aim-mems-extremal-without-crandall-rabinowitz,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Regular MEMS extremal states without an asymptotic curvature condition}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/438-mems-extremal-without-crandall-rabinowitz.md},
note = {Entry 438 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-00616, "Regular MEMS extremal states without an asymptotic curvature condition", clopen.solutions.