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 $C^{\infty}$ regularity of $L^2_2$ solutions to a nonlinear fourth order variational \mbox{equation} in $4$ dimensions. This equation was considered by Chang, Gursky, and Yang in \cite{CGY}, where regularity was obtained only for minimizers using techniques from Morrey~\cite{Morrey} and Schoen-Uhlenbeck~\cite{SU}. The \mbox{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.

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Published 1 January 2000