Spectral asymptotics of radial solutions and nonradial bifurcation for the H\'enon equation

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.