Refinement of asymptotic behavior of the eigenvalues for the linearized Liouville–Gel’fand problem

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 with solutions that blow-up at m points. Our problem is the case with an inhomogeneous coefficient in a 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 reach the conclusion.