Uniqueness and nondegeneracy of least-energy solutions to fractional Dirichlet problems

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 \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ , and on a new symmetry characterization of the eigenfunctions of the linearized eigenvalue problem in domains which are convex in the \(x_1\) $x_1$ -direction and symmetric with respect to a hyperplane reflection.