Resumen de Qualitative properties and classification of nonnegative solutions to −Δu=f(u) in unbounded domains when f(0)<0

Alberto Farina , Berardino Sciunzi

• We consider nonnegative solutions to −Δu=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.