How to evaluate $C$? 
Let B be the unit ball in the plane and let $u$ be a solution of the boundary
  value problem:
$∆u = C$ in $B$
$\frac{∂u}{∂n} = 1 $ on $ ∂B$
where $∆ $denotes the Laplace operator, ∂B denotes the boundary of $B$ and
  $\frac{∂u}{∂n}$ denotes the outer normal derivative on the boundary. Evaluate $C$, given
  that it is a constant.

My attempt : actually  im thinking about  Green's identity
$$\int_\Omega \Delta u \, dx = \int_{\partial \Omega} \frac{\partial u}{\partial n} \, dS.$$
But here i don't  know  how to  colloborate   this formula with my  given problem

Very close, to wit:
The divergence theorem--AKA Green's identity--states
$\displaystyle \int_B \nabla \cdot \nabla u \; dA = \int_{\partial B} \dfrac{\partial u}{\partial n} \; dS, \tag 1$
$dA$ and $dS$ being the area and length elements on $B$ and $\partial B$, respectively.  Given that
$ \dfrac{\partial u}{\partial n} = 1, \tag 2$
we find
$\displaystyle \int_{\partial B} \dfrac{\partial u}{\partial n} \; dS = \int_{\partial B} 1\; dS = 2\pi(1) = 2\pi; \tag 3$
on the other hand, 
$\displaystyle \int_B \nabla \cdot \nabla u \; dA = \int_B \nabla^2 u \; dA = \int_B C \; dA = C \pi (1)^2 = C \pi; \tag 4$
equating these two yields
$C \pi = 2\pi, \tag 5$
that is,
$C = 2. \tag 6$
