  In this paper we address some questions about symmetry, radial monotonicity,
and uniqueness for a semilinear fourth-order boundary value problem in the ball
of $\mathbb R^2$ deriving from the Kirchhoff-Love model of deformations of thin
plates. We first show the radial monotonicity for a wide class of biharmonic
problems. The proof of uniqueness is based on ODE techniques and applies to the
whole range of the boundary parameter. For an unbounded subset of this range we
also prove symmetry of the ground states by means of a rearrangement argument
which makes use of Talenti's comparison principle. This paper complements the
analysis in [G. Romani, Anal. PDE 10 (2017), no. 4, 943-982], where existence
and positivity issues have been investigated.
