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

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.
Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Journal of Mathematical Physics, 60 (2019), 021506 and may be found at https://doi.org/10.1063/1.5021825