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)(s) u = integral(u) in B, u = 0 in R-N\B, and the nonlinearity fis of class C-1. We prove that for s is an element of (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 is an element of (0, 1/2], the same conclusion holds under an additional assumption on f. In particular, our results apply to the Dirichlet eigenvalue problem for the operator (-Delta)(s) in B for all s.(0, 1), and imply that eigenfunctions corresponding to the second Dirichlet eigenvalue in Bare antisymmetric. This resolves a conjecture of Banuelos and Kulczycki. (C) 2021 Elsevier Inc. All rights reserved.