Asymptotic profile and Morse index of the radial solutions of the Henon equation

Author(s)

da Silva, Wendel Leite ; dos Santos, Ederson Moreira

• Document type: Journal article
• Source: Journal of Differential Equations; v. 287, p. 24-pg., 2021-04-02.

Abstract

We consider the Henon equation -Delta u = vertical bar x vertical bar(alpha)vertical bar u vertical bar p(-1)u in B-N, u = 0 on partial derivative B-N, (P-alpha) where B-N C R-N is the open unit ball centered at the origin, N >= 3, p > 1 and alpha > 0 is a parameter. We show that, after a suitable rescaling, the two-dimensional Lane-Emden equation -Delta w = vertical bar w vertical bar(p-1)w in B-2, w= 0 on partial derivative B-2, where B-2 C R-2 is the open unit ball, is the limit problem of (P-alpha), as alpha -> infinity, in the framework of radial solutions. We exploit this fact to prove several qualitative results on the radial solutions of (P-alpha) with any fixed number of nodal sets: asymptotic estimates on the Morse indices along with their monotonicity with respect to alpha; asymptotic convergence of their zeros; blow up of the local extrema and on compact sets of B-N. All these results are proved for both positive and nodal solutions. (C) 2021 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 15/17096-6 - Problems on Elliptic PDEs: systems and equations
• Grantee: Ederson Moreira dos Santos
• Support Opportunities: Regular Research Grants