Equation for the normal derivative of the solution of Poisson's equation

Question

The solution of Poisson's equation \begin{align*} \Delta u &= 0 ~~~\text{in } B_r \\ u &= g~~~\text{in } \partial B_r, \end{align*} is well known. For some $a\in\Bbb R\setminus \{0\}$ we have $$ u(x) = \frac{r^2-|x|^2}{a} \int_{\partial B_r} \frac{g(y)}{|x-y|^n}dS(y). $$ Does there exist a constant $c\in\mathbb{R}^+$ such that $$\partial_\nu u = c\, u~~~\text{in }\partial B_r\,?$$