Extremal functions for Moser's inequality.

Let $\Omega$ be a bounded smooth domain in $R^{n}$, and $u(x)$ a $C^{1}$ function with compact support in $\Omega$. Moser's inequality states that there is a constant $c_{o}$, depending only on the dimension $n$, such that \begin{equation*} \frac{1}{|\Omega|} \int_{\Omega} e^{n \omega_{n-1}^{\frac{1}{n-1}} u^{\frac{n}{n-1}}} dx \leq c_{o} , \end{equation*} where $|\Omega|$ is the Lebesgue measure of $\Omega$, and $\omega_{n-1}$ the surface area of the unit ball in $R^{n}$. We prove in this paper that there are extremal functions for this inequality. In other words, we show that the \begin{equation*} \sup \{\frac{1}{|\Omega|} \int_{\Omega} e^{n \omega_{n-1}^{\frac{1}{n-1}} u^{\frac{n}{n-1}}} dx: u \in W_{o}^{1,n}, \|\nabla u\|_{n} \leq 1 } \end{equation*} is attained. Earlier results include Carleson-Chang (1986, $\Omega$ is a ball in any dimension) and Flucher (1992, $\Omega$ is any domain in 2-dimensions). [ABSTRACT FROM AUTHOR]