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.
Acknowledgments
It is a pleasure to thank Ovidiu Savin for inspiring discussions.
Notes
1 Here we skate around the minor regularity requirements on w in order to write Equation(1.1)(1.1) (1.1) pointwise: at this level, we are implicitly assuming w to be “regular enough,” but a more precise setting will be discussed in the forthcoming Remark 1.10.
2 For simplicity, we wrote Equation(5.13)(5.13) (5.13) when . Notice that we have used there that . When n = 1, we just obtain