Nonlinear and Nonlocal Eigenvalue Problems

TL;DR

This paper investigates a class of nonlinear eigenvalue problems involving convolution operators and superlinear nonlinearities, providing existence proofs, decay properties, numerical methods, and asymptotic analysis of eigenvalues and eigenfunctions.

Contribution

It introduces a variational approach to establish solutions for nonlinear eigenvalue problems with convolution and superlinear terms, including numerical and asymptotic insights.

Findings

Existence of a one-parameter family of solutions with positive eigenvalues

Eigenfunctions are unimodal and exhibit specific decay properties

Asymptotic behavior characterized for small and large eigenvalues

Abstract

We study a class of nonlinear eigenvalue problems which involves a convolution operator as well as a superlinear nonlinearity. Our variational existence proof is based on constrained optimization and provides a one-parameter family of solutions with positive eigenvalues and unimodal eigenfunctions. We also discuss the decay properties and the numerical computations of those eigenfunctions, and conclude with two asymptotic results concerning small and large eigenvalues.