Overdetermined problems for negative power integral equations on bounded domain

Abstract

The aim of this paper is to study symmetry and monotonicity of positive solutions for the following overdetermined problem $\{\begin{array}{ll}{\displaystyle u(x)=A{\int }_{\mathrm{\Omega }}|x-y{|}^{\alpha -n}{u}^{-p}(y)\mathrm{d}y+B,}& \mathrm{in}\mathrm{\Omega },\\ u>0,& \mathrm{in}\mathrm{\Omega },\end{array}$ where $\alpha >n$, p>0, A>0, $B\ge 0$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain. We first prove that $u=\mathrm{const}.$ on $\mathrm{\partial }\mathrm{\Omega }$ if and only if Ω is a ball. Next we consider the partially overdetermined problem. If Γ is a proper open set of $\mathrm{\partial }\mathrm{\Omega }$ and u = C in $\mathrm{\Gamma }\subseteq \mathrm{\partial }\mathrm{\Omega }$, we show that under some assumptions on the geometry of Γ, Ω is a ball. Furthermore, we derive that all positive solutions of above equations are radially symmetric and monotone increasing with respect to the radius by using the method of moving planes.

Keywords:

• Method of moving planes
• overdetermined problem
• partially overdetermined problem
• symmetry and monotonicity of solutions

AMS Subject Classifications:

• Primary: 31B10
• 35J91
• Secondary: 35B06
• 35B65

Acknowledgements

The authors thank Prof. Wenxiong Chen for valuable discussions and encouragement on this manuscript. This work was completed when the first author was visiting the Department of Mathematical Science, Yeshiva University. The first author appreciates the hospitality of Prof. Wenxiong Chen.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Z. L. is supported by the NNSF of China (No. 11801237), the Natural Foundation of Jiangxi Province (No. 20202BABL211001), the Educational Committee of Jiangxi Province (No. GJJ211101), and the Fundamental Research Funds for the Central Universities (No. 2020QNBJRC005). Y.H. is supported by the NNSF of China (Grant Number 12101380), China Postdoctoral Science Foundation (Grant Number 2021M700086).