Abstract

Let J ∈ C(ℝ), J ≥ 0, ∫ℝ J = 1 and consider the nonlocal diffusion operator M[u] = J * u - u. We study the equation Mu + ∫(x,u) = 0, u ≥ 0, in ℝ, where f is a KPP-type nonlinearity, periodic in x. We show that the principal eigenvalue of the linearization around zero is well defined and that a nontrivial solution of the nonlinear problem exists if and only if this eigenvalue is negative. We prove that if, additionally, J is symmetric, then the nontrivial solution is unique. © 2008 Society for Industrial and Applied Mathematics.