Let \(u\in\overset\circ w^ 1_ n(D)\) and \(\int_{D}| u_ x|^ n dx\leq 1\) where D is an open domain in \(R^ n\), \(n\geq 2\). J. Moser [Indiana Univ. Math. J. 20, 1077-1092 (1971; Zbl 0203.437)] proved that there exists a constant \(C_ n\) which depends only on n such that \(\int_{D}e^{\alpha u^ p} dx\leq C_ n\int_{D}dx\) where \(p=n/(n- 1)\), \(\alpha \leq n\omega_{n-1}^{1/n-1}\) \((\omega_{n-1}\) is the (n- 1)-dimensional surface area of the unit sphere). The main result of this paper is the following: Let D be the unit ball in \(R^ n\). Then there is extremal function u satisfying \(\int_{D}| u_ x|^ n dx\leq 1\) for which (1/\(\int_{D}dx)\int e^{\alpha_ nu^ p}\) is the largest.