Assume the 2D Laplace Equation $$\nabla^2 u =0 $$ in polar coordinates $(r, \theta)$. The general solution is $$u(r,\theta) = \frac{A_0}{2} + \sum^{\infty}_{n=1}r^n[A_n\cos(n\theta) + B_n\sin(n\theta)].$$ Now assume Neumann boundary conditions: $$\frac{\partial u}{\partial \theta}(r,\pm\frac{\pi}{2}) = 0$$ and $$\frac{\partial u}{\partial \theta}(1,\theta) = u(1,\theta) - 1.$$
Does a non-trivial solution exist? If yes, then what is the solution?