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 eigen-value problem: integral(Omega) J(x - y/g(y))phi(y)/g(n)(y)dy + a(x)phi = rho phi, where Omega subset of 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 (lambda(p), phi(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. (C) 2010 Published by Elsevier Inc.