Lisa Mazzuoli | Search arXiv

arXiv2026

Stability and asymptotic behaviour of one-dimensional solutions in cylinders

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.

math.AP↗

arXiv2026

One-dimensional nodal solutions of the Lane--Emden problem in cylinders: asymptotic behaviour and energy stability

We consider the Lane-Emden problem $-Δu=|u|^{p-1}u$, $p>1$, in bounded domains of a half-cylinder $ω\times(0,+\infty)$, with Dirichlet conditions on the relative boundary and Neumann conditions on the boundary of the cylinder. In flat cylinders $ω\times(0,L)$ 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 $p\to1^+$ and $p\to+\infty$, 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.

math.AP↗