Nonexistence of positive supersolutions for semilinear fractional elliptic equations in exterior domains.

Abstract:

In this talk, our goal is to investigate the nonexistence of positive solutions to nonlinear
fractional elliptic inequalities in exterior domains of Rn, n ≥ 1. Our results extend the classical
Liouville-type theorems of Gidas–Spruck [3] for semilinear elliptic equations, as well as the
framework of Armstrong–Sirakov [1] for supersolutions of elliptic equations, to the nonlocal
setting. They are also closely related to the fundamental solution approach of Felmer–Quaas [2]
for nonlinear integral operators, although our arguments require substantial modifications to
address the interplay between the fractional operator, the geometry of exterior domains, and
the general nonlinearities under consideration. The nonexistence results we obtain hold under
much weaker growth conditions on the nonlinear term f , and they are sharp. Moreover, the
conclusions remain novel not only in exterior domains but also in the whole space.

Posted on Sep 29, 2025 in Differential Equations, Seminars