Morse index versus radial symmetry for fractional Dirichlet problems

In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$(-\Delta)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,}$$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet eigenvalue problem for the operator $(-\Delta)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Ba\~{n}uelos and Kulczycki.