Mathematics

We present recent uniqueness results for non-negative solutions of semilinear equations with a power nonlinearity set in bounded domains with Dirichlet boundary conditions. 

We start discussing the local case where, for the Lane-Emden problem, we can give a positive answer to a uniqueness conjecture in convex domains.

Then we also consider the nonlocal setting, where new difficulties arise, and we are able to show uniqueness  for least energy solutions in balls or in more general symmetric domains.

The problem of the uniqueness is strictly related to the study of the nondegeneracy of the solutions, hence during the talk the properties of the associated linearized equation are also investigated.

 

The talk is mainly based on the following joint works:

[1] F. De Marchis, M. Grossi, I. Ianni, F. Pacella, Morse index and uniqueness of positive solutions of the Lane-Emden problem in planar domains, J. Math. Pures Appl. 128, 2019.

[2] 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

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