Regularity for a fourth-order critical equation with gradient nonlinearity

Given Ω a smooth bounded domain of R n , n ⩾ 3 , we consider functions u ∈ H 2 , 0 2 ( Ω ) that are weak solutions to the equation Δ 2 u + a u = − div ( f | x | s | ∇ u | 2 ⋆ − 2 ∇ u ) in Ω , where 2 ⋆ : = 2 ( n − s ) n − 2 , s ∈ [ 0 , 2 ) and a , f ∈ C ∞ ( Ω ¯ ) . In this article, we prove the maximal regularity of solutions to the above equation, depending on the value of s ∈ [ 0 , 2 ) and the relative position of Ω with respect to the origin. In particular, the solutions are in C 4 ( Ω ¯ ) when s = 0 .