Fine Behaviors of Eigenvalues and Eigenfunctions of the Lane-Emden Problem in Dimension Two

Abstract

Recently, qualitative analysis of peaked solutions of Lane-Emden problem in dimension two has been widely considered. In particular, the Morse index of concentrated solutions with a single peak or multi peaks has been computed in [15,16] separately. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions of the linearized Lane-Emden problem associated to peak solutions. Here we establish the fine behaviors of the first $m$ eigenvalues and eigenfunctions of the linearized Lane-Emden problem in dimension two, and correspondingly the number of concentrated points of the first $m$ eigenfunctions are given.