Regularity for a fourth-order critical equation with gradient nonlinearity

Abstract

Given Ω a smooth bounded domain of R n , n 3, we consider functions u ∈ H 2 2,0 (Ω) that are weak solutions to the equationwhere 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.