On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators

Abstract

In this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro-magnetism. More precisely, we study the following eigenvalue problem: ∫ Ω J ( x − y g ( y ) ) ϕ ( y ) g n ( y ) d y + a ( x ) ϕ = ρ ϕ , where Ω ⊂ R n is an open connected set, J a non-negative kernel and g a positive function. First, we establish a criterion for the existence of a principal eigenpair ( λ p , ϕ p ) . We also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator. As an application of these results we construct and characterise the solutions of some nonlinear nonlocal reaction diffusion equations.