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Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem

  • Anna Lisa Amadori
  • Francesca GladialiEmail author
Article

Abstract

We compute the Morse index of nodal radial solutions to the Hénon problem
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u = |x|^{\alpha }|u|^{p-1} u &{} \quad \text {in}\, B, \\ u= 0 &{} \quad \text {on}\, \partial B, \end{array} \right. \end{aligned}$$
where B stands for the unit ball in \({{\mathbb {R}}}^N\) in dimension \(N\ge 3\), \({\alpha }>0\) and p is close to the threshold exponent for existence of solutions \(p_{\alpha }=\frac{N+2+2\alpha }{N-2}\), obtaining that either
$$\begin{aligned} \begin{array}{ll} m(u_p) = m \sum \limits _{j=0}^{1+\left[ {\alpha }/{2}\right] } N_j&{} \quad \text{ if } \alpha \text{ is } \text{ not } \text{ an } \text{ even } \text{ integer, } \text{ or } \\ m(u_p) = m\sum \limits _{j=0}^{ \alpha /2} N_j + (m-1) N_{1+\alpha / 2} &{} \quad \text{ if } \alpha \text{ is } \text{ an } \text{ even } \text{ number. } \end{array} \end{aligned}$$
Here \(N_j\) denotes the multiplicity of the spherical harmonics of order j, and m stands for the number of nodal zones of u. The computation builds on a characterization of the Morse index by means of a one dimensional singular eigenvalue problem, and is carried out by a detailed picture of the asymptotic behavior of both the solution and the singular eigenvalues and eigenfunctions. In particular it is shown that nodal radial solutions have multiple blow-up at the origin, and converge (up to a suitable rescaling) to the bubble shaped solution of a limit problem. As side outcome we see that solutions are nondegenerate for p near \(p_{\alpha }\), and we give an existence result in perturbed balls.

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze e TecnologieUniversità di Napoli “Parthenope”NaplesItaly
  2. 2.Dipartimento di Chimica e FarmaciaUniversità di SassariSassariItaly

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