Monotonicity of the Morse index of radial solutions of the Hénon equation in dimension two

2018/12/07 by Wendel Leite da Silva, da Silva, Wendel Leite, Ederson Moreira dos Santos +1
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1812.03098

openalex publication_date 2018/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the equation -Δu = |x|^α |u|^p-1u, x ∈ B, u=0 on ∂ B, where B ⊂ \mathbb R² is the unit ball centered at the origin, α≥0, p>1, and we prove some results on the Morse index of radial solutions. The contribution of this paper is twofold. Firstly, fixed the number of nodal sets n≥1 of the solution u_α,n, we prove that the Morse index m(u_α,n) is monotone non-decreasing with respect to α. Secondly, we provide a lower bound for the Morse indices m(u_{α, n}), which shows that m(u_{α, n}) → +∞ as α→ + ∞.

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