https://u-picardie.hal.science/hal-03621424Farina, AlbertoAlbertoFarinaLAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 - UPJV - Université de Picardie Jules Verne - CNRS - Centre National de la Recherche ScientifiqueSciunzi, BerardinoBerardinoSciunziQualitative properties and classification of nonnegative solutions to -Delta u = f (u) in unbounded domains when f (0) < 0HAL CCSD2016[MATH] Mathematics [math]DESSAIVRE, Louise2022-03-28 10:57:132023-03-24 14:53:262022-03-28 10:57:13enJournal articles10.4171/RMI/9181We 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.