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

Abstract

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.