Morse index computation for radial solutions of the H 'enon problem in\n the disk

2021/02/26 by Annalisa Amadori, Amadori, Annalisa, Francesca De Marchis +3
Mathematics · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2102.13553

openalex publication_date 2021/02/26 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We compute the Morse index textsfm(u_p) of any radial solution u_p\nof the semilinear problem: \
labelproblemaAbstract
tagP\n
left
\-
Delta u=|x|
^alpha|u|^p-1u · amp;
mboxin B

u=0\n · amp;
mbox on
partial B \
right. where B is the\nunit ball of mathbb R² centered at the origin, \α\≥ 0 is fixed\nand p>1 is sufficiently large. In the case \α=0, i.e. for the\n\Lane-Emden problem, this leads to the following Morse index formula\n\
textsfm(u_p) = 4m²-m-2, for p large enough, where m is the\nnumber of nodal domains of u.\n

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