A non-local semilinear eigenvalue problem

TL;DR

This paper investigates a non-local semilinear eigenvalue problem, establishing the uniqueness and isolation of the first eigenvalue under certain boundary conditions and embedding assumptions.

Contribution

It proves the simplicity and isolation of the first eigenvalue for a class of non-local semilinear problems, extending spectral theory results.

Findings

First eigenvalue is simple and isolated

Results hold under specific boundary conditions

Applicable to problems with compact Sobolev embeddings

Abstract

For a non-local semilinear eigenvalue problem, we prove simplicity and isolation of the first eigenvalue with homogeneous Dirichlet boundary conditions on open sets supporting a suitable compact Sobolev embedding.