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⁻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.