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

Symmetry and uniqueness for a hinged plate problem in a ball