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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2020.tde-19032020-095419
Document
Author
Full name
Wendel Leite da Silva
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2020
Supervisor
Committee
Santos, Ederson Moreira dos (President)
Figueiredo, Giovany de Jesus Malcher
Miyagaki, Olimpio Hiroshi
Soares, Sérgio Henrique Monari
 
Title in English
Qualitative properties of radial solutions of the Hénon equation
Keywords in English
Asymptotic behavior
Hénon equation
Morse index
Nodal radial solutions
Semilinear elliptic equations
Abstract in English
In this work, we study qualitative properties of radial solutions to the Hénon problem { - Δu = ΙxΙαΙuΙp-1 in B; u = 0 on ∂B; where B ⊂ RN is the unit ball centered at the origin, N ≥ 2, α ≥ 0 and p > 1. We obtained results about the computation of the Morse index and the asymptotic profile, as α → ∞, of both positive and sign changing radial solutions. More precisely, we divided this work into two parts. Firstly, considering the case N = 2, we proved that the Morse index of the radial solutions uα, with the same number of nodal sets, is monotone non-decreasing with respect to α. Moreover, we present a lower bound for the Morse indices m(uα), which is better than those that already exist in the literature, showing in particular that m(uα) → ∞ as α → ∞. Secondly, considering N ≥ 3, we show that the two-dimensional Lane-Emden equation can be seen as a limit problem for the Hénon equation. Finally, we used this fact to obtain some qualitative consequences of these solutions.
 
Title in Portuguese
Propriedades qualitativas de soluções radiais da equação de Hénon
Keywords in Portuguese
Comportamento assintótico
Equação de Hénon
Equações elípticas semilineares
Índice de Morse
Soluções radiais nodais
Abstract in Portuguese
Neste trabalho, estudamos propriedades qualitativas de soluções radiais para o problema de Hénon ( { - Δu = ΙxΙαΙuΙp-1 in B; u = 0 on ∂B onde B ⊂ RN é a bola unitária centrada na origem, N ≥ 2, α ≥ 0 e p > 1. Obtivemos resultados sobre o cálculo do índice de Morse e o perfil assintótico, quando α → ∞, das soluções radiais, as positivas e também as que trocam de sinal. Mais precisamente, dividimos este trabalho em duas partes. Primeiramente, considerando o caso N = 2, provamos que o índice de Morse das soluções radiais uα, com o mesmo número de conjuntos nodais, é monótono não decrescente com respeito α. Além disso, apresentamos uma cota inferior para os índices de Morse m(uα), melhor que aquelas já existentes na literatura, o que mostra em particular que m(uα) → ∞ quando α → ∞. Segundamente, considerando N ≥ 3, mostramos que a equação de Lane-Emden bidimensional pode ser vista como um problema limite para a equação de Hénon. Por fim, utilizamos este fato para obter algumas consequências qualitativas destas soluções.
 
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WendelLeitedaSilva.pdf (960.60 Kbytes)
Publishing Date
2020-03-19
 
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