Let $k$ be a non-zero real number. Consider the problem
$$ \nabla^2 \phi = 0 \ \ \ \mbox{for} \ \ \ 1 \leq r \leq 2, \ \ \ \ \alpha\phi + \frac{\partial\phi}{\partial r} = k\cos\theta \ \ \ \mbox{on} \ \ r = 1, \ \ \alpha\phi + \frac{\partial\phi}{\partial r} = 0 \ \ \ \mbox{on} \ \ r = 2 $$
What can you say about existence and uniqueness of the solution to the problem in terms of $\alpha$?
For uniqueness, I am aware e.g. that for $\alpha > 0$ it holds (well-known for Robin conditions) and that for $\alpha = 0$ it holds up to additive constants.
However, now with the Fourier series approach I got a unique solution (respecting the general solution to Laplace's equation) for $\alpha \neq 0, \frac{5\pm \sqrt{97}}{12}, \frac{1}{2\log 2}$, and existence for $\alpha \neq \frac{5\pm \sqrt{97}}{12}$.
My solutions for $\alpha = \frac{1}{2\log 2}$ are $c(2\log 2 - \log r) + (A_1r + B_1/r)\cos\theta$ where $c$ is arbitrary and $A_1, B_1$ satisfy $k=( \alpha + 1)A_1 + (\alpha - 1)B_1$ and $(2\alpha + 1)A_1 + (\frac{\alpha}{2} - \frac{1}{4})B_1 = 0$. How does this happen, as $\frac{1}{2\log 2} > 0$?
Any help appreciated!