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

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: $${\int }_{Ø}J(\frac{x-y}{g(y)})\frac{\varphi (y)}{g^n(y)}dy+a(x)\varphi =\rho \varphi ,$$ where $Ø\subset {\mathbb{R}}^n$ 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 $({\lambda }_p,{\varphi }_p)$. 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.

References in corpus (2)

• Nonlocal anisotropic dispersal with monostable nonlinearity
• Existence and Uniqueness of Solutions to a Nonlocal Equation with Monostable Nonlinearity

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