%0 Journal Article %T Qualitative properties and classification of nonnegative solutions to -Delta u = f (u) in unbounded domains when f (0) < 0 %+ Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA) %A Farina, Alberto %A Sciunzi, Berardino %< avec comité de lecture %@ 0213-2230 %J Revista Matemática Iberoamericana %I European Mathematical Society %V 32 %N 4 %P 1311-1330 %8 2016 %D 2016 %Z 1405.3428 %R 10.4171/RMI/918 %Z Mathematics [math]Journal articles %X We consider nonnegative solutions to -Delta u = f (u) in unbounded Euclidean domains under zero Dirichlet boundary conditions, where f is merely locally Lipschitz continuous and satisfies f (0) < 0. In the half-plane, and without any other assumption on u, we prove that u is either one-dimensional and periodic or positive and strictly monotone increasing in the direction orthogonal to the boundary. Analogous results are obtained if the domain is a strip. As a consequence of our main results, we answer affirmatively to a conjecture and to an open question posed by Berestycki, Caffarelli and Nirenberg. We also obtain some symmetry and monotonicity results in the higher-dimensional case. %G English %L hal-03621424 %U https://u-picardie.hal.science/hal-03621424 %~ CNRS %~ UNIV-PICARDIE %~ INSMI %~ U-PICARDIE %~ LAMFA