Consider following PDE:
$r^2u_{rr} + ru_r + u_{\theta\theta} = 0 , \ \ 1\lt r\lt e , \ \ 0 \lt \theta \lt \frac{\pi}{2}$
with these boundary conditions:
$u(1,\theta) = 0 \\ u(e,\theta) = 0 \\ u(r , 0) = 0 \\ u(r,\frac{\pi}{2}) = \sin(4\pi \ln r)$
Using separation of variables:
$\theta'' + K\theta = 0 \\ r^2R'' + rR' - KR = 0$
My main question is about BCs. For solving these problems, we should consider $u(r,\theta)$ be a periodic in $\theta$ but if we apply this condition then $K>0$ and the second equation using the first two BCs lead to $R$ be identically zero. So we shouldn't apply periodic condition here? When it is necessary?