We discuss some recent uniqueness and nondegeneracy results for non-negative solutions of some fractional semilinear problems in bounded domains with Dirichlet exterior condition.

In particular we can consider least energy solutions in balls or in more general symmetric domains, for problems with power nonlinearities. The symmetry properties of the solutions of the associated linearized equation are also investigated. 

The talk is mainly based on the following joint works:

[1] A. Dieb, I. Ianni, A. Saldaña, Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods, Nonlinear Analysis, 236, 2023, https://doi.org/10.1016/j.na.2023.113354

[2] A. Dieb, I. Ianni, A. Saldaña, Uniqueness and nondegeneracy of least-energy solutions to fractional Dirichlet problems, preprint arXiv:2310.01214