In this paper, we address questions related to symmetry, radial monotonicity, and uniqueness for a semilinear fourth-order boundary value problem in the ball of \(\mathbb {{R}}^2\) originating from the Kirchhoff–Love model of deformations of thin plates. We first show the radial monotonicity for a broad 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 work complements the results in Romani (Anal PDE 10(4):943–982, 2017), where existence and positivity of solutions were previously analyzed.
