I am given the following nonlinear Poisson-Boltzmann equation and asked to derive a linearized form for a current approximation of $u$: $$ (-\Delta u)\,+k^2\sinh(u) = \rho(x) $$ $$ x \in \Omega, \quad u_{\partial\mkern2mu \Omega} = g $$
For the 1D case I am given $ k = 1, \rho = 0 $ and the exact solution is: $$ \bar u(s) = \frac{\ln(1+\cos(s))}{\ln(1-\cos(s))} $$
Now for the 2D problem I am told $ k =1, \quad \Omega(0,1)^2 $
The exact solution is:
$$ u(x) = \bar u(0.1 + (x,a)) $$
Where: $$ a = \frac{(1.0,2.0)}{\sqrt5} $$
I need to find $ g, \rho$ such that the exact solution of $ u(x) $ is satisfied.
This is pretty foreign to me considering I am an engineer and don't deal with nonlinear equations of this type. Any help would be greatly appreciated as I need this linearization for a project.