Nonnegative solutions of semilinear elliptic equations in half-spaces

We consider the semilinear elliptic problem {-Delta u = f (u) in R-+(N) (0.1) u = 0 on partial derivative R-+(N) where the nonlinearity f is assumed to be C-1 and globally Lipschitz with f (0) < 0, and R-+(N) = {x is an element of R-N : x(N) > 0} stands for the half-space. Denoting by u(0) the unique solution of the one-dimensional problem -u '' = f(u) with u(0) = u '(0) = 0, we show that nonnegative solutions u of (0.1) which verify u(x) >= u(0)(x(N)) in R-+(N) either are positive and monotone in the x(N) direction or coincide with u(0). As a particular instance, when f (t) = t - 1, we prove that the unique nonnegative (not necessarily bounded) solution of (0.1) is u(x) = 1- cos x(N). This solves in a strengthened form a conjecture posed by Berestycki, Caffarelli and Nirenberg in 1997. (C) 2016 Elsevier Masson SAS. All rights reserved.