A Note on a Fourth Order Pde with Critical Nonlinearity

(1.3) 2(1 + α)∆w + 2β div ( αD(|Dw|2)− (α∆w + β|Dw|2)Dw) = div(T (Dw, ·))− (E′(w − w)− E′(w − w)) where E′(w − w) = 1 volM ∫ E′(w − w). Chang, Gursky and Yang [CGY] proved that any F -minimizing solution u ∈ W 2,2(M) to (1.3) is actually smooth. It was asked in [CGY] whether any weak solution u ∈ W 2,2(M) is smooth. Indeed, Uhlenbeck and Viaclovsky [UV] confirmed this recently and proved the smoothness for any weak solution u ∈ W 2,2(M) to (1.3). The proof in [CGY] relied on F -minimality. The idea in [UV] is based on some uniqueness properties for small perturbations of ∆2 in various Sobolev spaces and seems to be an indirect argument. Here we provide an alternative and direct proof of the smoothness for weak solutions to (1.3); namely, we show that under a smallness assumption on the W 2,2 norm, the normalized Lpnorm of the gradient of u on a ball decays like a positive power of the radius