A fractional Hopf Lemma for sign-changing solutions

TL;DR

This paper establishes a boundary Hopf Lemma for fractional elliptic solutions without sign restrictions and proves that non-trivial radial solutions cannot have infinitely many zeros near the boundary, with sharpness demonstrated through examples.

Contribution

It introduces a fractional Hopf Lemma applicable to sign-changing solutions and analyzes zero accumulation behavior near boundaries.

Findings

Established a Hopf Lemma for fractional elliptic solutions without sign constraints.

Proved non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary.

Provided examples demonstrating the sharpness of the results.

Abstract

In this paper we prove some results on the boundary behavior of solutions to fractional elliptic problems. Firstly, we establish a Hopf Lemma for solutions to some integro-differential equations. The main novelty of our result is that we do not assume any global condition on the sign of the solutions. Secondly, we show that non-trivial radial solutions cannot have infinitely many zeros accumulating at the boundary. We provide concrete examples to show that the results obtained are sharp.