Second radial eigenfunctions to a fractional Dirichlet problem and uniqueness for a semilinear equation

We analyze the shape of radial second Dirichlet eigenfunctions of fractional Schr\"odinger type operators of the form (-Δ)ˢ +V in the unit ball B in RN with a nondecreasing radial potential V. Specifically, we show that the eigenspace corresponding to the second radial eigenvalue is simple and spanned by an eigenfunction u which changes sign precisely once in the radial variable and does not have zeroes anywhere else in B. Moreover, by a new Hopf type lemma for supersolutions to a class of degenerate mixed boundary value problems, we show that u has a nonvanishing fractional boundary derivative on ∂ B. We apply this result to prove uniqueness and nondegeneracy of positive ground state solutions to the problem (-Δ)ˢ u+λ u=uᵖ on B, $\; u=0$ on RN B. Here s∈ (0,1), λ≥ 0 and $p>1$ is strictly smaller than the critical Sobolev exponent.