Integrating a Poisson kernel in $n$ dimensional unit sphere

Question

Let \begin{equation*} P(x,y)=\frac{1}{\omega_n R} \frac{R^2-|x|^2}{|x-y|^n} \end{equation*} be a Poisson kernel where $x$, $y$ are in $R^n$, $|x|<R$, $|y|=R$, $\omega_n$ is area of n dimensional unit sphere. Then what is $$\int_{\partial B_1(0)}P(x,y)dA_y=?$$

Answer 1

Hint: The function

$$g(x) = \int_{\partial B_1(0)} P(x, y) dA_y = \int_{\partial B_1(0)} 1 \cdot P(x, y) dA_y$$

satisfies $\Delta g = 0$ in $B_1(0)$ and $g(y) = 1$ on $\partial B_1(0)$. What can $g$ be?