Hello, I do not know how to apply the nonlinear boundary conditions with this problem:
The answer is the sum of the two different solutions.
This link may be helpful. Look at the very last section on the web page. The last two paragraphs will be useful.
Laplace Equation - Wikiversity
Yes so explicitly you need to solve these two BVP's
Since you are forcing the other boundary conditions to be zero, they will not mess up the other boundary conditions when you sum of the solutions.