One-dimensional nodal solutions of the Lane--Emden problem in cylinders: asymptotic behaviour and energy stability
Abstract
We consider the Lane-Emden problem −Δu=|u|p−1u, p>1, in bounded domains of a half-cylinder ω×(0,+∞), with Dirichlet conditions on the relative boundary and Neumann conditions on the boundary of the cylinder. In flat cylinders ω×(0,L) this problem admits, for every p>1, one-dimensional solutions with any prescribed number m of nodal regions. We study the stability of these domains, as critical points of a related energy functional, with respect to volume-preserving deformations of the domain. We prove that, for p large, the flat cylinders are stable as soon as their height exceeds a threshold growing linearly in p, while, for p close to $1$, both stability and instability occur, with explicit thresholds depending on m and on the first nontrivial Neumann eigenvalue of ω. These results rely on a detailed analysis of the one-dimensional profiles: we describe their asymptotic behaviour, and that of the first eigenvalue of the associated linearized operator, as p→1+ and p→+∞, we obtain expansions of the amplitude with explicit constants, and we prove global comparison results with respect to p, some of which are new also for positive solutions.
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BibTeX
@misc{marchis2026onedimensional,
title = {{One-dimensional nodal solutions of the Lane--Emden problem in cylinders: asymptotic behaviour and energy stability}},
author = {Francesca De Marchis and Lisa Mazzuoli},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06339},
eprint = {2611.06339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06339},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - One-dimensional nodal solutions of the Lane--Emden problem in cylinders: asymptotic behaviour and energy stability
AU - Francesca De Marchis
AU - Lisa Mazzuoli
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06339
UR - https://arxiv.org/abs/2611.06339
AB - We consider the Lane-Emden problem −Δu=|u|^p−1u, p>1, in bounded domains of a half-cylinder ω×(0,+∞), with Dirichlet conditions on the relative boundary and Neumann conditions on the boundary of the cylinder. In flat cylinders ω×(0,L) this problem admits, for every p>1, one-dimensional solutions with any prescribed number m of nodal regions. We study the stability of these domains, as critical points of a related energy functional, with respect to volume-preserving deformations of the domain. We prove that, for p large, the flat cylinders are stable as soon as their height exceeds a threshold growing linearly in p, while, for p close to $1$, both stability and instability occur, with explicit thresholds depending on m and on the first nontrivial Neumann eigenvalue of ω. These results rely on a detailed analysis of the one-dimensional profiles: we describe their asymptotic behaviour, and that of the first eigenvalue of the associated linearized operator, as p→1^+ and p→+∞, we obtain expansions of the amplitude with explicit constants, and we prove global comparison results with respect to p, some of which are new also for positive solutions.
ER -