A fractional Hopf Lemma for sign-changing solutions

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.

KEYWORDS:

• Hopf Lemma
• qualitative behavior
• nonlocal equations
• zeros of solutions

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 (1.1)(1.1) $$\mathit{\text{Lw}}(x)=\mathit{\text{PV}}{\int }_{{\mathbb{R}}^n}\frac{w(x)-w(x+y)}{|y{|}^{n+2s}}a\left(\frac{y}{\left|y\right|}\right)\mathit{\text{dy}},$$(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 (5.13)(5.13) $$\begin{array}{cc}0\hfill & ⩾\frac{c_1{\rho }_0^{n-1}}{2}{\int }_{-1}^1\varphi (\tau )\mathit{d\tau }{\int }_{\partial B_1}G({\rho }_{0}e,{\rho }_0\omega )d{H}_{\omega }^{n-1}-C\hfill \\ \hfill & ⩾\frac{c c_1}{2{\rho }_0^{1-2s}}{\int }_{-1}^1\varphi (\tau )\mathit{d\tau }{\int }_{\partial B_1\cap \left\{\right|e-\omega |<\eta \}} \frac{d{H}_{\omega }^{n-1}}{|e-\omega {|}^{n-2s}}-C\hfill \\ \hfill & =+\infty .\hfill \end{array}$$(5.13) when $n⩾2$. Notice that we have used there that $n-2s⩾n-1$. When n = 1, we just obtain $\frac{c c_1}{2{\rho }_0^{1-2s}}{\int }_{-1}^1\varphi (\tau )\mathit{d\tau }|\hspace{0.17em}\mathrm{\text{log}}\hspace{0.17em}0|-C=+\infty .$

Additional information

Funding

Nicola Soave is partially supported by the project no. 2022R537CS “Nodal Optimization, NOnlinear elliptic equations, NOnlocal geometric problems, with a focus on regularity (NO ³)” - funded by European Union - Next Generation EU within the PRIN 2022 program (D.D. 104 - 02/02/2022 Ministero dell’Università e della Ricerca, Italy), and by the INdAM - GNAMPA Project, cod. CUP_ E53C22001930001 “Regolarità e singolarità in problemi con frontiere libere”. Enrico Valdinoci is supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations”.