We consider positive solutions of a fractional Lane-Emden type problem in a
bounded domain with Dirichlet conditions. We show that uniqueness and
nondegeneracy hold for the asymptotically linear problem in general domains.
Furthermore, we also prove that all the known uniqueness and nondegeneracy
results in the local case extend to the nonlocal regime when the fractional
parameter s is sufficiently close to 1.Comment: 22 page