Neumann problem, stuck on a boundary condition. I am stuck on a problem that I am trying for exam practice and I would very much appreciate a hint to help me out, here is the section where I am stuck:

A solution is sought to the Neumann problem for $\nabla^2 u = 0$ in the half plane $z > 0$:
  $u = O(|x|^{−a}),
\frac{\partial u}{\partial r}
= O(|x|^{−a−1}) ~~ \mathrm{as} ~~  |x| → ∞,~~ \frac{\partial u}{\partial z} ~
= p(x, y) ~ on ~ z = 0,
\mathrm{where}~ a > 0$. It is assumed that
  $\int_{\infty}^{\infty}\int_{-\infty}^{\infty}
p(x, y) dx dy = 0$. Explain why this condition is
  necessary.

My feeling is that this is to do with Green's third identity and that we need the normal derivative in the $x-y$ plane to be integrable in order to find out solution with a Green's function, am I correct?

EDIT:
the divergence theorem sorts this out. 

Let $f(z)=\int\int u_z$. Then $f_z=\int\int u_{zz} =\int\int (-u_{xx}-u_{yy})$. Consider this quantity and also $\lim _{z\to\infty} f(z)$. That should give you the result you want.
