  In this manuscript we consider semilinear PDEs, with a convex nonlinearity,
in a sector-like domain. Using cylindrical coordinates $(r, \theta, z)$, we
investigate the shape of solutions whose derivative in $\theta$ vanishes at the
boundary. We prove that any solution with Morse index less than two must be
either independent of $\theta$ or strictly monotone with respect to $\theta$.
In the special case of a planar domain, the result holds in a circular sector
as well as in an annular, and it can also be extended to a rectangular domain.
The corresponding problem in higher dimensions is also considered, as well as
an extension to unbounded domains. The proof is based on a rotating-plane
argument: a convenient manifold is introduced in order to avoid overlapping the
domain with its reflected image in the case when its opening is larger than
$\pi$.
