Refinement of asymptotic behavior of the eigenvalues for the linearized Liouville-Gel'fand Problem

TL;DR

This paper refines the understanding of the asymptotic behavior of eigenvalues and eigenfunctions for the linearized Liouville-Gel'fand problem with inhomogeneous coefficients, extending previous homogeneous case results.

Contribution

It provides the second term in the asymptotic expansion for eigenvalues and eigenfunctions, considering inhomogeneous coefficients and regularity conditions.

Findings

Derived the second asymptotic term for eigenvalues

Extended results to inhomogeneous coefficients

Analyzed regularity conditions for coefficients

Abstract

We determine the second term of the asymptotic expansions for the first m eigenvalues and eigenfunctions of the linearized Liouville-Gel'fand problem associated to solutions which blow-up at m points. Our problem is the case with an inhomogeneous coefficient in two dimensional domain and we extend the previous studies for the problem with a homogeneous coefficient. We also discuss in detail the required regularity of the coefficient necessary to get the conclusion.