On the number of peaks of the eigenfunctions of the linearized Gel'fand problem

TL;DR

This paper provides a second order estimate for eigenvalues and eigenfunctions of the linearized Gel'fand problem near blow-up points, revealing qualitative properties related to concentration points and eigenvalue multiplicity.

Contribution

It introduces a second order estimate for eigenvalues and eigenfunctions of the linearized Gel'fand problem, advancing understanding of their behavior near blow-up points.

Findings

Derived second order estimates for eigenvalues and eigenfunctions.

Identified qualitative properties of eigenfunctions related to concentration points.

Linked eigenvalue multiplicity to eigenfunction behavior.

Abstract

We derive a second order estimate for the first m eigenvalues and eigenfunctions of the linearized Gel'fand problem associated to solutions which blow-up at m points. This allows us to determine, in some suitable situations, some qualitative properties of the first m eigenfunctions as the number of points of concentration or the multiplicity of the eigenvalue .