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.
Categories: math.AP