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

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:∫ΩJ(x−yg(y))ϕ(y)gn(y)dy+a(x)ϕ=ρϕ, where Ω⊂RnΩ⊂Rn 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 (λp,ϕp)(λp,ϕp). 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.