I need to show that the following Neumann problem has no solution:
$u\, \, \epsilon \, \, C^{2}(\overline{B})) ,\, \, \, \, B=B_{R}(0)),\,\, \gamma\,\,\, $is outside vector normal at the border line$ \\\ \\ \begin{Bmatrix}_{} \Delta u=1 , \, \, x\, \epsilon\, \, B_{R}(0)) \\ \frac{\partial u}{\partial \gamma }=0,\: \: x\, \epsilon \: \partial B_{R}(0)) \end{Bmatrix}$
I know that the Green's first identity should be used but I do not understand how can I use it to show that a solution $u$ does not exist.
How can I reach this conclusion?
If such a solution $u$ exists, then
$\displaystyle \int_B \nabla \cdot \nabla u \; dV = \int_B \nabla^2 u \; dV = \int_B 1 \; dV = \text{vol}(B) > 0, \tag 1$
where $dV$ is the volume form on $B$ and $\text{vol}(B)$ is the volume of $B$.
On the other hand, by the divergence theorem,
$\displaystyle \int_B \nabla \cdot \nabla u \; dV = \int_{\partial B} \nabla u \cdot \gamma \; dS = \int_{\partial B} \dfrac{\partial u}{\partial \gamma} \; dS = \int_{\partial B} 0 \; dS = 0, \tag 2$
where $dS$ is the surface area element on $\partial B$.
We see that (1) and (2) are in contradiction; hence, there is no such $u$.