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

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.