Exact eigenvalues and eigenfunctinos for a one-dimensional Gel'fand problem

TL;DR

This paper derives explicit formulas for all eigenvalues and eigenfunctions of the linearized problem at solutions of a 1D Gel'fand problem and analyzes their asymptotic behavior as solutions grow large.

Contribution

It provides exact expressions for eigenvalues and eigenfunctions and studies their asymptotics, advancing understanding of the spectral properties of the Gel'fand problem.

Findings

Explicit formulas for eigenvalues and eigenfunctions

Asymptotic analysis as solution norm diverges

Enhanced understanding of spectral behavior in Gel'fand problems

Abstract

It is known that every positive solution of a one-dimensional Gel'fand problem can be written explicitly. In this paper we obtain exact expressions of all the eigenvalues and eigenfunctions of the linearized eigenvalue problem at each solution. We also study asymptotic behaviors of eigenvalues and eigenfunctions as the $L^{\infty}$-norm of the solution goes to the infinity.