Monotonicity and Symmetry of Nonnegative Solutions to -Delta u = f(u) in Half-Planes and Strips

Alberto Farina (1) , Berardino Sciunzi

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

We 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.

Domaines: Mathématiques [math]

Alberto Farina, Berardino Sciunzi. Monotonicity and Symmetry of Nonnegative Solutions to -Delta u = f(u) in Half-Planes and Strips. Advanced Nonlinear Studies, 2017, 17 (2, SI), pp.297-310. ⟨10.1515/ans-2017-0010⟩. ⟨hal-03621417⟩