Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems

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Multiplicity of solutions for a fourth order equation with power-type nonlinearity (2010, background): “We consider only radial solutions, since all positive smooth solutions of (2) are radial, see Berchio et al. [ 6 ].”
Characterization of balls by Riesz-potentials (2009, methodology): “some important improvements of the moving plane method were achieved by Chang and Yang [4], Berchio et al. [ 2 ], Li [12], Chen et al. [5], Jin and Li [11] and Birkner et al. [3].”
Symmetry of Solutions to Variational Problems for Nonlinear Elliptic Equations via Reflection Methods (2010, methodology): “Finally, in joint work with Gazzola, Berchio and Ferrero [ 15 , 47] we studied symmetry via polarization for higher order semilinear boundary value problems.”
Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS (2010, background): “Another advantage for working on the ball is that all solutions will be radially symmetric and decreasing as was shown recently in [18].”
A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems (2007, unknown): “Using this we find that for every fixed δ > 0 there exists σ(δ) ∈ (0, 1) such that Φk ≤ (1 − σ)k‖∇ku‖Lp(Ω∩Bσ) + δ ≤ (1 − σ)k ( (1 − σ)2m−kǫ‖∇2mu‖Lp(Ω∩Bσ) + Cǫ −k 2m−k (1 − σ)−k‖u‖Lp(Ω∩Bσ) ) + δ = ǫ(1 − σ)2m‖∇2mu‖Lp(Ω∩Bσ) + Cǫ −k 2m−k ‖u‖Lp(Ω∩Bσ) + δ.”