Hi 


Consider Poisson equation with Neumann boundary condition but the right hand side of boundary condition is in term of the unknown function $u$.
How we can solve it?


$\Delta u(x) = f(x)\quad in~ \Omega$


$\frac{\partial u(x)}{\partial n }=g(u(x))\quad on~\partial \Omega$


where n is outward normal vector.


For special case let $g=\sqrt u$.



Poisson equation with special Neumann BC 
Set $G$ a primitive of $g$. Then the solution is a critical point of the functional $E:H^1(\Omega)\rightarrow{\mathbb R}$ defined by
$$E[u]:=\int\_\Omega \left(\frac12|\nabla u|^2+fu\right)dx-\int\_{\partial\Omega}G(u)ds.$$
If $G(\pm\infty)=-\infty$, you may look for a minimum of $E$ over $H^1(\Omega)$. When $g$ is a decreasing function, $G$ is concave and $E$ is strictly convex: your solution is unique.


