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 $p\to \mathrm{\infty }$, which is of independent interest. It allows to detect the limit profile and other qualitative properties of these solutions.