Qualitative properties of radial solutions of the Hénon equation

• Ano de defesa: 2020
• Autor(a) principal: Silva, Wendel Leite da
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Asymptotic behavior; Comportamento assintótico; Equação de Hénon; Equações elípticas semilineares; Hénon equation; Índice de Morse; Morse index; Nodal radial solutions; Semilinear elliptic equations; Soluções radiais nodais
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-19032020-095419/

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.