Partially overdetermined problem in some integral equations

Abstract:

In this paper, we consider the partially overdetermined problem in integral equations as follows: \begin{align} \begin {cases} u(x)=A \int _{\Omega }\frac {1}{|x-y|^{n-\alpha }}u^p(y)dy+B,~~&x\in \Omega ,\\ u>0, ~~&x\in \Omega , \\ u=C,~~&x\in \Gamma \subseteq \partial \Omega ,\notag \end{cases} \end{align} where $0<\alpha <n, ~p>\frac {n}{n-\alpha }, ~A,~ B,~ C$ are positive constants, $\Omega \subset R^n\ (n\geq 2)$ is a bounded domain with $\partial \Omega \in C^1$, and $\Gamma$ is a proper open set of $\partial \Omega$. Under some assumptions on the geometry of $\Gamma$, we prove that $\Omega$ must be a ball and $u$ is radially symmetric and monotone decreasing with respect to the radius.