  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.
