Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems

Let B = B 1 ( 0 ) be the unit ball in R n and r = | x | . We study the poly-harmonic Dirichlet problem { ( − Δ ) m u = f ( u ) in B , u = ∂ u ∂ r = ⋯ = ∂ m − 1 u ∂ r m − 1 = 0 on ∂ B . the corresponding integral equation and the method of moving planes in integral forms, we show that the positive solutions are radially symmetric and monotone decreasing about the origin. We also obtain regularity for solutions.