Stability and asymptotic behavior 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 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-->1 or p-->+oo, which is of independent interest. It allows to detect the limit profile and other qualitative properties of these solutions.