Principal spectral theory and variational characterizations for cooperative systems with nonlocal and coupled diffusion

Abstract

We study a general class of cooperative systems with nonlocal diffusion operators that may or may not be coupled. These systems are either “strong” in cooperation or “strong” in the coupling of the nonlocal diffusion operators, and in the former case, diffusion may not occur in some of the components of the system at all. We prove results concerning the existence, uniqueness, multiplicity, variational characterizations of the principal eigenvalues of these systems, the spectral bound, the essential spectrum, and the relationship between the sign of principal eigenvalue and the validity of the maximum principle. We do so using an elementary method, without resorting to Krein-Rutman theorem.