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

TL;DR

This paper establishes the uniqueness and nondegeneracy of least-energy solutions for fractional Dirichlet problems in large symmetric domains, using uniform estimates, ground state properties, and symmetry characterizations.

Contribution

It introduces new methods to prove uniqueness and nondegeneracy of solutions in fractional problems within symmetric domains, extending previous results.

Findings

Proves uniqueness of least-energy solutions in large symmetric domains.

Demonstrates nondegeneracy of solutions under certain domain conditions.

Provides a new symmetry characterization of eigenfunctions in convex symmetric domains.

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.