Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity

Let J is an element of C(R), J >= 0, integral(R) J = 1 and consider the nonlocal diffusion operator M[u] = J * u- u. We study the equation Mu + f( 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.