Principal eigenvalues of a class of nonlinear integro-differential operators

TL;DR

This paper investigates principal eigenvalues of nonlinear integro-differential operators, establishing their existence, uniqueness, and related properties, and explores applications to nonlinear problems using bifurcation theory.

Contribution

It introduces new results on the existence, simplicity, and properties of principal eigenvalues for a class of nonlinear integro-differential operators, including anti-maximum principles.

Findings

Existence of two principal eigenvalues in bounded smooth domains.

Proven simplicity of principal eigenfunctions in viscosity sense.

Established maximum principles and continuity of eigenvalues with domain changes.

Abstract

We consider a class of nonlinear integro-differential operators and prove existence of two principal (half) eigenvalues in bounded smooth domains with exterior Dirichlet condition. We then establish simplicity of the principal eigenfunctions in viscosity sense, maximum principles, continuity property of the principal eigenvalues with respect to domains etc. We also prove an anti-maximum principle and study existence result for some nonlinear problem via Rabinowitz bifurcation-type results.