\documentclass[reqno]{amsart}
\usepackage{a4wide}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2026 (2026), No. 26, pp. 1--7.\newline
ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2026.26}
\thanks{\copyright 2026. This work is licensed under a CC BY 4.0 license.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2026/26\hfil Solution structure of over-determined problems]
{Solution structure of nonlocal Serrin-type over-determined problems}

\author[K. Sato, F. Takahashi \hfil EJDE-2026/26\hfilneg]
{Kazuki Sato, Futoshi Takahashi}

\address{Kazuki Sato \newline
Department of Mathematics,
Osaka Metropolitan University,
3-3-138, Sumiyoshi-ku,
Sugimoto-cho, Osaka, Japan}
\email{sf22817a@st.omu.ac.jp}

\address{Futoshi Takahashi \newline
Department of Mathematics,
Osaka Metropolitan University,
3-3-138, Sumiyoshi-ku,
Sugimoto-cho, Osaka, Japan}
\email{futoshi@omu.ac.jp}

\thanks{Submitted January 11, 2026. Published April 6, 2026.}
\subjclass[2020]{34C23, 37G99}
\keywords{Over-determined problems; Kirchhoff-type; nonlocal term}

\begin{abstract}
In this article, we study the solution structures of Serrin-type over-determined
problems with Kirchhoff-type nonlocal terms.
We prove that the exact number of solutions is the same as those of some transcendental
equations defined by the nonlocal terms.
We also obtain the explicit form of solutions by using the unique solutions of the
over-determined problems without the nonlocal terms.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega \subset \mathbb{R}^N$ $(N \ge 1)$ be a bounded domain with $C^2$ boundary and
$k \in \{ 1, 2, \dots, N \}$.
In this article, we consider the  fully nonlinear over-determined problem with
Kirchhoff-type nonlocal term,
\begin{equation}\label{OD_HM}
\begin{gathered}
M\Big(\| u \|_{L^{p}(\Omega)}, \| \nabla u \|_{L^{q}(\Omega)} \Big) S_k(D^2 u)
= \lambda \quad \text{in } \Omega, \\
u = 0 \quad \text{on }  \partial\Omega, \\
\frac{\partial u}{\partial \nu} = c > 0 \quad \text{on }  \partial\Omega,
\end{gathered}
\end{equation}
where $0 < p, q \le \infty$, $\lambda > 0$ are given constants,
$c$ is an  unknown positive constant, $\nu$ is the outer unit normal to
$\partial\Omega$, and $M(s, t)$ is a positive function in
$(s, t) \in \mathbb{R}_+ \times \mathbb{R}_+$.
We do not assume any continuity of $M$.
For a real symmetric matrix $A$, let $S_k(A)$ denote the $k$-th elementary symmetric
function of the eigenvalues of $A$ (counted with multiplicity),
\[
 S_k(A) = \sum_{1 \le i_1 < i_2 < \dots < i_k \le N} \lambda_{i_1} \lambda_{i_2}
 \dots \lambda_{i_k}.
\]
Thus $S_1(A) = \sum_{i=1}^N \lambda_i$ and $S_N(A) = \prod_{i=1}^N \lambda_i$.
Finally, let $D^2 u(x) = ( \frac{\partial^2 u}{\partial x_i \partial x_j}(x)
)_{1 \le i, j \le N}$ denote the Hessian matrix of a $C^2$-function $u$.
Then $S_k(D^2 u)$ is called the \emph{$k$-Hessian operator}.
Note that $S_1(D^2 u) = \Delta u$ and $S_N(D^2 u) = {\rm det } (D^2 u)$.

In  \cite{BNST}, the authors considered the Serrin-type over-determined problem for
the $k$-Hessian operator,
\begin{equation} \label{OD_H}
 \begin{gathered}
 S_k(D^2 u) = \binom{N}{k} \quad \text{in }  \Omega, \\
 u = 0 \quad \text{on }  \partial\Omega, \\
 \frac{\partial u}{\partial \nu} = c > 0 \quad \text{on } \partial\Omega,
 \end{gathered}
\end{equation}
where $c > 0$ is unknown, $\binom{N}{k} = \frac{N !}{k! (N-k)!}$ for
$k \in \{1,2,\dots, N \}$.
They prove that if \eqref{OD_H} admits a solution $u \in C^2(\overline{\Omega})$ for some
$k \in \{1,2,\dots, N \}$, then, up to a translation, $\Omega$ must be a ball,
say $\Omega = B_R(x_0)$ for $R > 0$ and $x_0 \in \mathbb{R}^N$, and the solution is of the form
\begin{equation}\label{U_R}
 u(x) = U_{R, x_0}(x) :=  \frac{|x-x_0|^2 - R^2}{2}
\end{equation}
with $c = R$.
Note that the $k$-Hessian operator $S_k(D^2 u)$ is fully nonlinear and in general
is not elliptic.
In spite of these difficulties, the authors in \cite{BNST} give a shorter
alternative proof which does not exploit maximum principles directly
and to extend the famous result by Serrin \cite{Serrin} (for $k=1$) in this setting.
Their method uses a Pohozaev-type identity by Tso \cite{Tso} and reminds us of
an alternative proof by Weinberger \cite{Weinberger}.

Recently, the Serrin-type overdetermined problem with Kirchhof-type nonlocal term,
\[
\begin{gathered}
 M\Big(\| u \|_{L^{p}(\Omega)}, \| \nabla u \|_{L^{q}(\Omega)} \Big) \Delta u = \lambda \ \text{in} \ \Omega, \\
 u = 0 \quad \text{on} \ \partial\Omega, \\
 \frac{\partial u}{\partial \nu} = c \quad \text{on} \ \partial\Omega,
\end{gathered}
\]
where $0 < p, q \le \infty$, $\lambda > 0$ are given constants, $c$ is an unknown constant,
and $M: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ is a positive function,
was considered in \cite{Sato-TF}.
There the exact number of solutions according to the values of the bifurcation parameter
$\lambda > 0$ is determined by a transcendental equation defined by the nonlocal term
for a real unknown variable.

In this article, we extend the main result in \cite{Sato-TF} to  problem \eqref{OD_HM}.
The original argument comes from \cite{ACM}.

\begin{theorem} \label{Theorem:H}
Let $M : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, $0 < p, q \le \infty$, $\lambda > 0$,
and $k \in \{ 1, 2, \dots, N \}$.
Then if \eqref{OD_HM} admits a solution $u \in C^2(\overline{\Omega})$ for an unknown
constant $c > 0$, then $\Omega$ must be a ball and $u$ must be radially symmetric
 with respect to the center of the ball.

Let $\Omega = B_R(x_0)$ with $R > 0$ and $x _0 \in \mathbb{R}^N$. Consider the system
of equations with respect to $(s, t) \in \mathbb{R}_+ \times \mathbb{R}_+$:
\begin{equation}\label{System}
 \begin{gathered}
 s = \Big\{ \frac{M(s, t)}{\lambda} \binom{N}{k} \Big\}^{-1/k}
  \| U_{R, x_0} \|_{L^{p}(B_R(x_0))} \\
 t = \Big\{ \frac{M(s, t)}{\lambda} \binom{N}{k} \Big\}^{-1/k}
  \| \nabla U_{R, x_0} \|_{L^{q}(B_R(x_0))}
 \end{gathered}
\end{equation}
where $U_{R, x_0}$ defined by \eqref{U_R} is the unique solution of \eqref{OD_H}
for $\Omega = B_R(x_0)$.
Then for $\lambda > 0$,  problem \eqref{OD_HM} for $\Omega = B_R(x_0)$ has the same
number of solutions of system  \eqref{System}.
Also  define
\begin{equation}\label{g(s)}
 g(s) =  s^k M\Big(s, \frac{\| \nabla U_{R, x_0} \|_{L^{q}(B_R(x_0))}}
 {\| U_{R, x_0} \|_{L^{p}(B_R(x_0)}} s \Big) \binom{N}{k}
\end{equation}
for $s > 0$.
Then the number of solutions of \eqref{System} is the same as the number of solutions
to the equation
\begin{equation} \label{Single}
 g(s) = \lambda \| U_{R, x_0} \|_{L^{p}(B_R(x_0))}^k
\end{equation}
with respect to $s > 0$.
Moreover, any solution $u_{\lambda}$ of \eqref{OD_HM} has the form
\[
 u_{\lambda}(x) = s_* \frac{U_{R,x_0}(x)}{\| U_{R, x_0} \|_{L^{p}(B_R(x_0))}}
\]
where $s_*$ is any solution of \eqref{Single}.
\end{theorem}

In the second part, we study an exterior over-determined problem with
Kirchhoff-type nonlocal term as described below.
Let $D \subset \mathbb{R}^N$ $(N \ge 2)$ be a bounded $C^2$-domain containing the origin
as an interior point. We consider
\begin{equation} \label{OD_EM}
 \begin{gathered}
 M\Big(\| u \|_{L^{p}(\Omega)}, \| \nabla u \|_{L^{q}(\Omega)} \Big) \Delta u(x)
 = \lambda |x|^{-N-2}, \quad x \in \Omega, \\
 u(x) = 0, \quad x \in \partial \Omega, \\
 |\nabla u(x)| = c |x|^{-N} > 0, \quad x \in \partial \Omega, \\
 |u(x)| = o(1) \quad (|x| \to \infty), \quad (\text{if }  N \ge 3), \\
 |u(x)| = O(1) \quad (|x| \to \infty), \quad (\text{if }  N = 2),
 \end{gathered}
\end{equation}
where $\Omega = \mathbb{R}^N \setminus \overline{D}$ is an exterior domain,
$0 < p, q \le \infty$, $\lambda > 0$ are given constants, $c$ is an unknown positive
constant, and $M(s, t)$ is a positive function in $(s, t) \in \mathbb{R}_+ \times \mathbb{R}_+$.

Let $B$ be the closed unit ball in $\mathbb{R}^N$ for $N \ge 2$ and define
\begin{equation} \label{U}
 U(x) = \frac{1}{2} \big( |x|^{-N} - |x|^{2-N}\big) \quad \text{for } |x| > 1.
\end{equation}
Then direct computations show that $U$ is a solution of
\[
 \begin{gathered}
 \Delta U = N |x|^{-N-2} \ \text{in }  B^c = \mathbb{R}^N \setminus B, \\
 U = 0 \quad \text{on }  \partial B^c, \\
 |\nabla U| = 1 \quad \text{on }  \partial B^c, \\
 |U(x)| \to 0 \quad (|x| \to \infty), \quad (\text{if }  N \ge 3), \\
 |U(x)| \to \frac{1}{2} \quad (|x| \to \infty), \quad (\text{if }  N = 2).
 \end{gathered}
\]
Also note that $U \in L^p(B^c)$ if and only if $\frac{N}{N-2} < p \le \infty$
if $N \ge 3$, $p = \infty$ if $N = 2$;
and $|\nabla U| \in L^q(B^c)$ if and only if $\frac{N}{N-1} < q \le \infty$
if $N \ge 3$, $\frac{2}{3} < q \le \infty$ if $N = 2$.
In this case, we compute
\begin{gather*}
\| U \|_{L^p(B^c)}^p
 = (\frac{1}{2})^{p+1} |\mathbb{S}^{N-1}|
 B\Big(\frac{p(N-2)-N}{2}, p+1\Big), \quad (\frac{N}{N-2} < p < \infty, \ N \ge 3), \\
\| U \|_{L^\infty(B^c)} = \Big(\frac{1}{N-2}\Big)\Big(\frac{N}{N-1}\Big)^{-N/2},
  \quad (N \ge 3), \\
\| U \|_{L^\infty(B^c)}
 = \frac{1}{2}, \quad (N = 2), \\
\begin{aligned}
\| \nabla U \|_{L^q(B^c)}^q
&= (\frac{1}{2})^{q+1} |\mathbb{S}^{N-1}| N^q
 \Big(\frac{N-2}{N}\Big)^{\frac{q(N+1)-N}{2}} 
 \Big(B \big(\frac{q(N-1)-N}{2}, q+1\big)\\
&\quad  + B_{2/N}\big( q+1, \frac{-(N+1)q + N}{2} \big) \Big),
  \quad (\frac{N}{N-1} < q < \infty, \; N \ge 3),
\end{aligned}\\
 \| \nabla U \|_{L^q(B^c)}^q = \frac{2\pi}{3q-2} \quad (\frac{2}{3} < q < \infty, \; N = 2), \\
 \| \nabla U \|_{L^\infty(B^c)}
 = \big(\frac{N}{N-1}\big)\big(\frac{N(N+1)}{(N-1)(N-2)}\big)^{-(N+1)/2}, \quad (N \ge 3),  \\
 \| \nabla U \|_{L^\infty(B^c)} = 1, \quad (N = 2),
\end{gather*}
where $B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$ denotes the Beta function, and
$B_z(x, y) = \int_0^z t^{x-1} (1-t)^{y-1} dt$ denotes the incomplete Beta function
for $z \in (0,1)$.

Assume $\frac{N}{N-2} < p \le \infty$ if $N \ge 3$, $p = \infty$ if $N = 2$, and
$\frac{N}{N-1} < q \le \infty$ if $N \ge 3$, $2/3 < q \le \infty$ if $N=2$.
Let us consider the system of equations
\begin{equation} \label{System_E}
 \begin{gathered}
 s = \big\{ \frac{M(s, t)}{\lambda} N \big\}^{-1} \| U \|_{L^{p}(B^c)} \\
 t = \big\{ \frac{M(s, t)}{\lambda} N \big\}^{-1} \| \nabla U \|_{L^{q}(B^c)}
 \end{gathered}
\end{equation}
with respect to $(s, t) \in \mathbb{R}_+ \times \mathbb{R}_+$, where $U$ is in \eqref{U}.
Then we have the following:

\begin{theorem} \label{Theorem:E}
Let $N \ge 2$, $M : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, and
\begin{gather*}
 \frac{N}{N-2} < p \le \infty \quad \text{and} \quad \frac{N}{N-1} < q \le \infty,
 \quad \text{if }  N \ge 3, \\
 p = \infty \quad \text{and} \quad \frac{2}{3} < q \le \infty,
 \quad \text{if } \ N = 2.
\end{gather*}
Assume that \eqref{OD_EM} admits a solution $u \in C^2(\overline{\Omega})$ for an unknown
constant $c > 0$,
where $\Omega = \mathbb{R}^N \setminus \overline{D}$ is an exterior domain.
Then $D$ must be a ball and $u$ must be radially symmetric with respect to the center
of the ball.

Assume, without loss of generality, that $\overline{D} = B$.
Then for $\lambda > 0$, the problem \eqref{OD_EM} for $\Omega = B^c$ has the same number
of solutions of the system  \eqref{System_E}.
Also define
\[
 g(s) =  s M\Big(s, \frac{\| \nabla U \|_{L^{q}(B^c)}}{\| U \|_{L^{p}(B^c)}} s \Big) N
\]
for $s > 0$.
Then the number of solutions of \eqref{System_E} is the same as the number of
solutions to the equation
\begin{equation} \label{Single_O}
 g(s) = \lambda \| U \|_{L^{p}(B^c)}
\end{equation}
with respect to $s > 0$.
Moreover, any solution $u_{\lambda}$ of \eqref{OD_EM} has the form
\[
 u_{\lambda}(x) = s_* \frac{U(x)}{\| U \|_{L^{p}(B^c)}}
\]
where $s_*$ is any solution of \eqref{Single_O}.
\end{theorem}

Concerning the strategy of the proof of Theorem \ref{Theorem:H},
we cannot use the maximum principle directly for problem \eqref{OD_HM} because of
the nonlocal terms.
Also it seems difficult to carry out the integral approach in \cite{BNST}, again because
of the nonlocal terms.
The key point of the proof is using the homogeneity of the $k$-Hessian operator
\[
 S_k(\gamma D^2u) = \gamma^k S_k(D^2 u), \quad (\gamma > 0)
\]
to eliminate the nonlocal terms in the equation.
Then we can appeal to the uniqueness of the solution for the over-determined problem
for the $k$-Hessian operators \eqref{OD_H} in \cite{BNST}.

The proof of Theorem \ref{Theorem:E} goes also along the the same lines.
In this case, we use a classical Kelvin transform to rewrite the problem from the
exterior to the interior of the domain.
Then we use the Serrin's characterization of the over-determined problem without
nonlocal terms.
Since the Kelvin transform works well only for the linear operator, we have to
restrict ourselves in the linear problem.
Finally, nonlocal terms are treated by the use of the homogeneity of the Laplacian,
as in the proof of Theorem \ref{Theorem:H}.

The structure of this article is as follows:
In \S 2, we prove Theorem \ref{Theorem:H}.
In \S 3, we obtain a characterization of (the outside of) a ball for some
exterior over-determined problem, Proposition \ref{Prop:E},
and prove Theorem \ref{Theorem:E}.

\section{Proof of Theorem \ref{Theorem:H}}

\begin{proof}
First assume that there exists a solution $u \in C^2(\overline{\Omega})$ of \eqref{OD_HM}
and put
\[
 v = \gamma u
\]
where $\gamma > 0$ is chosen so that
\[
 \gamma = \Big( \frac{M(\| u \|_{L^{p}(\Omega)},
 \| \nabla u \|_{L^{q}(\Omega)})}{\lambda} \binom{N}{k} \Big)^{1/k}.
\]
Note that the $k$-Hessian operator is homogeneous of degree $k$.
Then
\begin{align*}
 S_k(D^2 v) &= S_k (\gamma D^2 u) \\
&= \gamma^k S_k(D^2 u) \\
&\overset{\eqref{OD_HM}}{=}
 \gamma^k  \frac{\lambda}{M(\| u \|_{L^p(\Omega)}, \| \nabla u \|_{L^q(\Omega)})}\\
&= \binom{N}{k}
\end{align*}
in $\Omega$.
Also we see that $v = 0$ on $\partial\Omega$ and
$\frac{\partial v}{\partial \nu} = \gamma c =$ const. on $\partial\Omega$.
Thus by the result of \cite{BNST}, we see that $\Omega$ must be a ball,
say $\Omega = B_R(x_0)$ for some $R > 0$ and $x _0 \in \mathbb{R}^N$ and
$v \equiv U_{R,x_0}(x)$.
This implies that
\begin{equation}
\label{u_form}
 u(x) = \gamma^{-1} U_{R,x_0}(x), \quad
 \nabla u(x) = \gamma^{-1} \nabla U_{R,x_0}(x).
\end{equation}
We define
\begin{equation}
\label{st}
 s = \| u \|_{L^{p}(B_R(x_0))} \quad \text{and} \quad
 t = \| \nabla u \|_{L^{q}(B_R(x_0))}.
\end{equation}
Then by \eqref{u_form}, we have
\[
 \begin{gathered}
 s = \gamma^{-1} \| U_{R,x_0} \|_{L^{p}(B_R(x_0))}, \\
 t = \gamma^{-1} \| \nabla U_{R,x_0} \|_{L^{q}(B_R(x_0))},
 \end{gathered}
\]
which is equivalent to \eqref{System}.
This shows that
\[
 (s, t) = (\| u \|_{L^{p}(B_R(x_0))}, \| \nabla u \|_{L^{q}(B_R(x_0))})
\]
is a solution to \eqref{System} and thus
\[
 \# \{ u : \text{solutions of \eqref{OD_HM} for $\Omega = B_R(x_0)$} \}
  \le \# \{ (s, t) \in (\mathbb{R}_+)^2 : \text{solutions of \eqref{System}} \},
\]
where $\# A$ denotes the cardinality of the set $A$.

On the other hand, let $\Omega = B_R(x_0)$ for some $R > 0$ and $x_0 \in \mathbb{R}^N$
and let $(s, t) \in (\mathbb{R}_+)^2$ be a solution to \eqref{System}.
Note that by \eqref{System}, we have
\begin{equation}  \label{System2}
 \frac{s}{\| U_{R, x_0} \|_{L^{p}(B_R(x_0))}}
 = \frac{t}{\| \nabla U_{R,x_0} \|_{L^{q}(B_R(x_0))}}
 = \Big( \frac{M(s, t)}{\lambda} \binom{N}{k} \Big)^{-1/k}.
\end{equation}
Thus if we define
\begin{equation} \label{u_def}
 u(x) = s \frac{U_{R, x_0}(x)}{\| U_{R, x_0} \|_{L^{p}(B_R(x_0))}}
 \quad \big(= t \frac{U_{R, x_0}(x)}{\| \nabla U_{R, x_0} \|_{L^{q}(B_R(x_0))}} \big),
\end{equation}
then we have $u = 0$, $\frac{\partial u}{\partial \nu} = $const. on
$\partial B_R(x_0)$ and \eqref{st} holds by \eqref{u_def}.
Moreover, we have
\begin{align*}
 &M\Big(\| u \|_{L^{p}(B_R(x_0))}, \| \nabla u \|_{L^{q}(B_R(x_0))}\Big) S_k(D^2 u) \\
 &\overset{\eqref{st}}{=} M(s, t) S_k(D^2 u) \\
 &\overset{\eqref{u_def}}{=} M(s, t)
  \Big( \frac{s}{\| U_{R, x_0} \|_{L^{p}(B_R(x_0))}} \Big)^k
  \underbrace{S_k(D^2 U_{R, x_0})}_{=\binom{N}{k}} \\
 &\overset{\eqref{System2}}{=} M(s, t)
 \Big( \frac{M(s,t)}{\lambda} \binom{N}{k} \Big)^{-1}
  \binom{N}{k} = \lambda
\end{align*}
on $B_R(x_0)$.
This shows that
\[
 \# \{ u : \text{solutions of \eqref{OD_HM} for $\Omega = B_R(x_0)$} \}
 \ge \# \{ (s, t) \in (\mathbb{R}_+)^2 : \text{solutions of \eqref{System}} \}.
\]
Thus the number of solutions of \eqref{OD_HM} for $\Omega = B_R(x_0)$ and that of
\eqref{System} are the same.

Also by \eqref{System2}, we can rewrite the system of equations \eqref{System} into
a single equation for $s$,
\[
 s = \Big( \frac{M\big(s, \frac{\| \nabla U_{R, x_0} \|_{L^{q}(B_R(x_0))}}
 {\| U_{R, x_0} \|_{L^{p}(B_R(x_0))}}\big) s}
 {\lambda} \binom{N}{k} \Big)^{-1/k}
 \| U_{R, x_0} \|_{L^p(B_R(x_0)},
\]
which is equivalent to \eqref{Single} with $g(s)$ in \eqref{g(s)}.
Thus the number of solutions of \eqref{System} for $\Omega = B_R(x_0)$ and that
of \eqref{Single} are also the same.
\end{proof}



\section{Exterior over-determined problem}

Let $D \subset \mathbb{R}^N$, $N \ge 2$ be a bounded $C^2$-domain containing the origin as
an interior point,
and put $\Omega = \mathbb{R}^N \setminus \overline{D}$.
First, we consider an exterior over-determined problem without Kirchhoff term:
\begin{equation}\label{OD_E}
 \begin{gathered}
 \Delta u = N |x|^{-N-2} \ \text{in }  \Omega, \\
 u = 0 \quad \text{on }  \partial\Omega, \\
 |\nabla u| = c|x|^{-N} \quad \text{on }  \partial\Omega, \\
 |u(x)| = o(1) \quad (|x| \to \infty), \quad (\text{if } N \ge 3), \\
 |u(x)| = O(1) \quad (|x| \to \infty), \quad (\text{if }  N = 2),
 \end{gathered}
\end{equation}
where $c$ is an unknown positive constant.
Direct computation shows that $U(x)$ in \eqref{U} is the exact solution of
\eqref{OD_E} for $\Omega = B^c$ with $c = 1$.
On the uniqueness, we have the following proposition.

\begin{proposition} \label{Prop:E}
Let $N \ge 2$ and $\Omega = \mathbb{R}^N \setminus \overline{D}$ be an exterior domain.
If problem \eqref{OD_E} admits a solution $u \in C^2(\overline{\Omega})$,
then $D$ must be a ball and $u$ must be radially symmetric with respect to the
center of the ball.
Let $\Omega = B^c = \mathbb{R}^N \setminus B$. Then $U(x)$ in \eqref{U} is the unique
solution of \eqref{OD_E} and $c$ must satisfy $c = 1$.
\end{proposition}

\begin{proof}
First we consider $N \ge 3$ and assume that $u \in C^2(\overline{\Omega})$ is a solution
of \eqref{OD_E}.
Define a function $w: \Omega^* \setminus \{ 0 \} \to \mathbb{R}$ such that
\begin{equation}
\label{Kelvin}
 w(y) = |y|^{2-N} u\Big( \frac{y}{|y|^2}\Big), \quad (y \in \Omega^* \setminus \{ 0 \}),
\end{equation}
where
\[
 \Omega^* = \{ y = \frac{x}{|x|^2}  :  x \in \Omega \} \cup \{ 0 \},
\]
that is, $w$ is the Kelvin transform of $u$.
Note that $x = \frac{y}{|y|^2} \in \Omega$ is equivalent to
$y = \frac{x}{|x|^2} \in \Omega^* \setminus \{ 0 \}$.
By a well-known formula for the Kelvin transform and the equation in \eqref{OD_E},
we have
\begin{align*}
 \Delta_y w(y) = |y|^{-2-N} (\Delta_x u) \big(\frac{y}{|y|^2}\big)
 = |x|^{2+N} \Delta_x u(x) = N
\end{align*}
for $y \in \Omega^* \setminus \{ 0 \}$.
Since $y \in \partial \Omega^*$ is equivalent to $x \in \partial \Omega$,
$w(y) \big|_{y \in \partial \Omega^*} = u(x) \big|_{x \in \partial \Omega} = 0$.
Furthermore,  direct computations give that
\begin{align*}
 \nabla_y w(y) = |y|^{-N} \left\{ (2-N) u(x) y - 2 (\nabla_x u(x) \cdot x) y + \nabla_x u(x) \right\}
\end{align*}
where $x = \frac{y}{|y|^2}$, $y \in \Omega^* \setminus \{ 0 \}$.
Thus
\begin{align*}
 \nabla_y w(y) \big|_{y \in \partial\Omega^*}
 &= \Big[ |x|^N \Big\{ (2-N) \underbrace{u(x)}_{=0} \frac{x}{|x|^2}
  - 2 (\nabla_x u(x) \cdot x) \frac{x}{|x|^2} + \nabla_x u(x) \Big\}
  \Big]_{x \in \partial\Omega} \\
 &= |x|^N \Big( - 2 (\nabla_x u(x) \cdot \frac{x}{|x|}) \frac{x}{|x|}
  + \nabla_x u(x) \Big)_{x \in \partial\Omega} \\
 & = |x|^N \Big( \vec{a}(x) - \vec{b}(x) \Big),
\end{align*}
where
\[
 \vec{a}(x) = \nabla_x u(x) - (\nabla_x u(x) \cdot \frac{x}{|x|}) \frac{x}{|x|},
 \quad \vec{b}(x) = (\nabla_x u(x) \cdot \frac{x}{|x|}) \frac{x}{|x|}.
\]
Note that
\[
 \vec{a}(x) + \vec{b}(x) = \nabla_x u(x),  \quad \vec{a}(x) \perp \vec{b}(x)
 \quad \text{for } x \in \partial\Omega,
\]
thus $|\vec{a}(x)|^2 + |\vec{b}(x)|^2 = |\nabla_x u(x)|^2$. %by Pythagorean theorem.
Therefore,
\begin{align*}
 |\nabla_y w(y)|^2 \big|_{y \in \partial\Omega^*}
= |x|^{2N} |\nabla_x u(x)|^2 \big|_{x \in \partial\Omega}.
\end{align*}
Since $u$ is a solution of \eqref{OD_E}, we see that $w$ in \eqref{Kelvin} satisfies
\begin{equation} \label{OD_Serrin}
 \begin{gathered}
 \Delta w = N \quad \text{in }  \Omega^* \setminus \{ 0 \}, \\
 w = 0 \quad \text{on }  \partial \Omega^*, \\
 |\nabla w| = c > 0 \quad \text{on }  \partial \Omega^*.
 \end{gathered}
\end{equation}
Furthermore, since $u$ satisfies $\lim_{|x| \to \infty} u(x) = 0$, its Kelven
transform $w$ satisfies
\[
 \lim_{|y| \to 0} \frac{|w(y)|}{|y|^{2-N}} = 0.
\]
Thus $y = 0$ is a removable singularity and $w$ satisfies the equation on the whole
$\Omega^*$. Then Serrin's result assures that $\Omega^*$ must be a ball and $w$ must
be radially symmetric with respect to the center of the ball.
This implies that $\Omega$ must be the compliment of the ball and $u$ is radially
symmetric with respect to the same point.
If $\Omega = B^c$, then $\Omega^* = {\rm int }(B)$ and
\[
 w(y) = \frac{|y|^2-1}{2} \quad (y \in {\rm int }(B))
\]
is the unique solution of \eqref{OD_Serrin} with $c = 1$.
Then by \eqref{Kelvin}, we have
\[
 u(x) = |y|^{N-2} w(y) = \frac{|y|^N - |y|^{N-2}}{2}
 = \frac{|x|^{-N} - |x|^{2-N}}{2}.
\]

When $N = 2$, the same computation also holds for the Kelvin transform
$w(y) = u(\frac{y}{|y|^2})$.
Moreover, we can claim that $y=0$ is again removable since in this case
\[
 \lim_{|y| \to 0} \frac{|w(y)|}{\log (1/|y|)} = 0
\]
follows
from the weaker condition at $\infty$: $u(x) = O(1)$ as $|x| \to \infty$.
We have finished the proof of Proposition \ref{Prop:E}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{Theorem:E}]
Once the uniqueness of the explicit solution $U$ in \eqref{U} to  problem \eqref{OD_E}
is proven in Proposition \ref{Prop:E},
the proof of Theorem \ref{Theorem:E} can be done in exactly the same way as
the proof of Theorem \ref{Theorem:H}.
\end{proof}

\subsection*{Acknowledgements}
F. Takahashi was supported by the JSPS Grant-in-Aid for Scientific Research (B)
No. 23K25781, and  by the Osaka Central University Advanced Mathematical
Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).

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\end{document}





