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

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

2011-06-25

arxiv.org/abs/1106.5137v1

J. Differential Equations 249 (2010) 2921-2953

Mathematics

Analysis of PDEs

Scientific paper

10.1016/j.jde.2010.07.003

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: $$\int_{\O}J(\frac{x-y}{g(y)})\frac{\phi(y)}{g^n(y)}\, dy +a(x)\phi =\rho \phi,$$ where $\O\subset\R^n$ 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 $(\lambda_p,\phi_p)$. 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 characterize the solutions of some nonlinear nonlocal reaction diffusion equations.

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