Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation
Joel Kübler, Tobias Weth · Discrete and Continuous Dynamical Systems · 2019
We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{document}$ \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*} $\end{document} 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.