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arXiv · 2509.09648

Stability and asymptotic behaviour of one-dimensional solutions in cylinders

Francesca De Marchis·Lisa Mazzuoli·Filomena Pacella
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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.

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Francesca De Marchis, Lisa Mazzuoli, Filomena Pacella. 2025-09-11. Stability and asymptotic behaviour of one-dimensional solutions in cylinders. https://arxiv.org/abs/2509.09648

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