Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods

TL;DR

This paper proves the uniqueness and nondegeneracy of positive solutions for fractional Lane-Emden problems in bounded domains, extending local case results to nonlocal regimes near s=1 using asymptotic methods.

Contribution

It establishes the extension of uniqueness and nondegeneracy results from local to nonlocal fractional problems near s=1, using asymptotic analysis.

Findings

Uniqueness and nondegeneracy hold for fractional problems in general domains.

Results extend known local case properties to nonlocal fractional regimes near s=1.

Asymptotic methods are effective in analyzing fractional PDEs.

Abstract

We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.