Existence and Uniqueness of Solutions to a Nonlocal Equation with Monostable Nonlinearity

Let $J\in C(\mathbb{R})$, $J\ge 0$, \int_{\tiny$\mathbb{R}$} J = 1 and consider the nonlocal diffusion operator $\mathcal{M}[u]=J\star u-u$. We study the equation $\mathcal{M}u+f(x,u)=0$, $u\ge 0$, in $\mathbb{R}$, 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.