Stability and asymptotic behaviour of one-dimensional solutions in cylinders
Francesca De Marchis, Lisa Mazzuoli, Filomena Pacella
TL;DR
This work examines positive one-dimensional solutions to Lane-Emden type problems in cylindrical domains and their stability under small, volume-preserving domain perturbations. The authors perform detailed nondegeneracy and sharp asymptotic analyses as the nonlinearity exponent $p$ tends to $\infty$ and to $1$, obtaining explicit limit profiles tied to the Green function and a Liouville-type equation, as well as precise eigenvalue scaling. Leveraging these asymptotics, they establish rigorous stability/instability criteria for energy-stationary pairs in hypograph domains, revealing sharp thresholds that separate stable energy-minimizing cylinders from unstable configurations and clarifying the role of domain height $L$ and cross-sectional geometry via $\lambda_1(\omega)$. The results extend prior work by providing explicit asymptotics for the Lane-Emden nonlinearity and have potential implications for domain bifurcation and overdetermined problems. The methods combine variational structure, nondegeneracy, and meticulous rescaling to connect one-dimensional limits with higher-dimensional stability questions.
Abstract
We consider positive one-dimensional solutions of a Lane-Emden relative Dirichlet problem in a cylinder and study their stability/instability properties as the energy varies with respect to domain perturbations. This depends on the exponent $p >1$ of the nonlinearity and we obtain results for $p$ close to 1 and for $p$ large. This is achieved by a careful asymptotic analysis of the one-dimensional solution as $p \to 1$ or $p \to \infty$, which is of independent interest. It allows to detect the limit profile and other qualitative properties of these solutions.