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

Abstract

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.

25 pages. In this revised version, we have improved our symmetry result Theorem 1.2, which now holds for general non-negative solutions. Due to a mistake found in our previous version (see Remark 3.6), Theorem 1.1 is now stated only for large balls