Regularity for a fourth-order critical equation with gradient nonlinearity

Given Ω   a smooth bounded domain of RnRn, n⩾3n⩾3, we consider functions u∈H2,02(Ω) that are weak solutions to the equationΔ2u+au=−div(f|x|s|∇u|2⋆−2∇u)in Ω, where 2⋆:=2(n−s)n−2, s∈[0,2)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)s∈[0,2) and the relative position of Ω   with respect to the origin. In particular, the solutions are in C4(Ω¯) when s=0s=0.