Qualitative analysis to an eigenvalue problem of the Hénon equation

Abstract

In this paper we study the following eigenvalue problem{−Δv=λC(α)(pα−ε)|x|αuεpα−ε−1vinΩ,u=0on∂Ω, where Ω⊂RN is a smooth bounded domain containing the origin, C(α)=(N+α)(N−2), N≥3, pα=N+2+2αN−2, α>0, ε>0 is a small parameter and uε is a single peaked solution of Hénon equation{−Δu=C(α)|x|αupα−εinΩ,u>0inΩ,u=0on∂Ω, which established by Gladiali and Grossi (2012) [21]. By using various local Pohozaev identities and blow-up analysis, we prove some asymptotic behavior of the eigenvalues λε,i and corresponding eigenfunctionsvε,i,i=2,⋯,∑1≤k<2+α2(N+2k−2)(N+k−3)!(N−2)!k!+2 when α is not an even integer. As a consequence, if 0<α<2, we have that the Morse index of the single peaked solutions is N+1, which gives an affirmative answer to a conjecture raised by Gladiali and Grossi.