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

Jérôme Coville (1)

1 BioSP - Biostatistique et Processus Spatiaux

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

Mots clés: Nonlocal diffusion operators; Principal eigenvalue; Non-trivial solution; Asymptotic behaviour

Domaines: Equations aux dérivées partielles [math.AP]; Analyse classique [math.CA]

Jérôme Coville. On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators. Journal of Differential Equations, 2010, 249 (11), pp.2921-2953. ⟨10.1016/j.jde.2010.07.003⟩. ⟨hal-00603471⟩