Nonnegative solutions of semilinear elliptic equations in half-spaces

Abstract

We consider the semilinear elliptic problem(0.1){−Δu=f(u)in R+Nu=0on ∂R+N where the nonlinearity f is assumed to be C1 and globally Lipschitz with f(0)<0, and R+N={x∈RN:xN>0} stands for the half-space. Denoting by u0 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)≥u0(xN) in R+N either are positive and monotone in the xN direction or coincide with u0. 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⁡xN. This solves in a strengthened form a conjecture posed by Berestycki, Caffarelli and Nirenberg in 1997.