Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation

Joel Kübler; Tobias Weth

doi:10.3934/dcds.2020032

Article Contents

2020, Volume 40, Issue 6: 3629-3656. Doi: 10.3934/dcds.2020032 open access

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Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation

Joel Kübler and
Tobias Weth^,

Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, D-60629 Frankfurt a.M., Germany

^* Corresponding author: Tobias Weth
^* Corresponding author: Tobias Weth

Dedicated to Wei-Ming Ni with admiration and appreciation

Received: December 31, 2018

Revised: May 31, 2019

Published: March 2020

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Abstract

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem

$ \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -\Delta u & = |x|^\alpha |u|^{p-2}u&&\qquad \text{in $ {\bf B}$,}\\ u& = 0&&\qquad \text{on $\partial {\bf B}$,} \end{aligned} \right. \end{equation*} $

in the unit ball $ {\bf B} \subset \mathbb{R}^N,N\geq 3 $, $ p>2 $ in the limit $ \alpha \to +\infty $. More precisely, for a given positive integer $ K $, we derive asymptotic $ C^1 $-expansions for the negative eigenvalues of the linearization of the unique radial solution $ u_\alpha $ of $ (H) $ with precisely $ K $ nodal domains and $ u_\alpha(0)>0 $. As an application, we derive the existence of an unbounded sequence of bifurcation points on the radial solution branch $ \alpha \mapsto (\alpha,u_\alpha) $ which all give rise to bifurcation of nonradial solutions whose nodal sets remain homeomorphic to a disjoint union of concentric spheres.

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Mathematics Subject Classification: 35J25, 35B32, 35P15.

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References

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