https://u-picardie.hal.science/hal-03621417Farina, AlbertoAlbertoFarinaLAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 - UPJV - Université de Picardie Jules Verne - CNRS - Centre National de la Recherche ScientifiqueSciunzi, BerardinoBerardinoSciunziMonotonicity and Symmetry of Nonnegative Solutions to -Delta u = f(u) in Half-Planes and StripsHAL CCSD2017[MATH] Mathematics [math]DESSAIVRE, Louise2022-03-28 10:57:062023-03-24 14:53:262022-03-28 10:57:06enJournal articles10.1515/ans-2017-00101We consider nonnegative solutions to -Delta u = f(u) in half-planes and strips, under zero Dirichlet boundary condition. Exploiting a rotating and sliding line technique, we prove symmetry and monotonicity properties of the solutions, under very general assumptions on the nonlinearity f. In fact, we provide a unified approach that works in all cases: f(0)< 0, f(0)= 0 or f(0)> 0. Furthermore, we make the effort to deal with nonlinearities f that may be not locally-Lipschitz continuous. We also provide explicit examples showing the sharpness of our assumptions on the nonlinear function f.