Morse index versus radial symmetry for fractional Dirichlet problems

Abstract

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 $$(-\Delta {)}^{s}u=f(u)\text{ in B},u=0\text{in RN∖B,}$$ 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 Bañuelos and Kulczycki.

18 pages