Regularity of weak solutions to critical exponent variational equations

We present a general method for proving regularity of weak solutions to variational equations with critical exponent nonlinearities. We will focus primarily on the C regularity of L 2 2 solutions to a nonlinear fourth order variational equation in 4 dimensions. This equation was considered by Chang, Gursky, and Yang in CGY99, where regularity was obtained only for minimizers using techniques from Morrey Mor48 and Schoen-Uhlenbeck SU82. The methods in this paper apply to a more general class of critical exponent variational equations in n dimensions with leading term a power of the Laplacian.