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

TL;DR

This paper establishes a criterion for the existence of a principal eigenfunction for certain nonlocal operators, with applications to nonlinear reaction-diffusion equations in various scientific fields.

Contribution

It introduces a new criterion for the existence of principal eigenpairs of nonlocal operators and explores their spectral properties and applications.

Findings

Established a criterion for principal eigenpair existence.

Linked the spectrum's largest element to a maximum principle.

Applied results to nonlinear nonlocal reaction-diffusion equations.

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: $$\int_{\O}J(\frac{x-y}{g(y)})\frac{\phi(y)}{g^n(y)}\, dy +a(x)\phi =\rho \phi,$$ 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.