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: $\int_ØJ(\frac{x-y}{g(y)})\frac{ϕ(y)}{g^n(y)}\, dy +a(x)ϕ=ρϕ,$ where $Ø\subset\R^n$ is an open connected set, $J$ a nonnegative 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 characterize the solutions of some nonlinear nonlocal reaction diffusion equations.

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