  We prove the uniqueness and nondegeneracy of least-energy solutions of a
fractional Dirichlet semilinear problem in sufficiently large balls and in more
general symmetric domains. Our proofs rely on uniform estimates on growing
domains, on the uniqueness and nondegeneracy of the ground state of the problem
in RN , and on a new symmetry characterization of the eigenfunctions of the
linearized eigenvalue problem in domains which are convex in the x1 - direction
and symmetric with respect to a hyperplane reflection.
