Morse index computation for radial solutions of the Hénon problem in the disk. (February 2022)

• Record Type: Journal Article
• Authors: Amadori, Anna Lisa; De Marchis, Francesca; Ianni, Isabella
• Is Part Of: Nonlinear analysis. Volume 215(2022)
• Publication Date: 2022-02
• Subjects: 35B05 -- 35B06 -- 35J91; Superlinear elliptic boundary value problem -- Lane–Emden problem -- Hénon problem -- Sign-changing radial solution -- Asymptotic analysis -- Morse index; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112645
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We compute the Morse index m ( u p ) of any radial solution u p of the semilinear problem: (P) − Δ u = | x | α | u | p − 1 u in B u = 0 on ∂ B where B is the unit ball of R 2 centered at the origin, α ≥ 0 is fixed and p > 1 is sufficiently large. In the case α = 0, i.e. for the Lane–Emden problem, this leads to the following Morse index formula m ( u p ) = 4 m 2 − m − 2, for p large enough, where m is the number of nodal domains of u .