Remarks on the Extremal Functions for the Moser-Trudinger Inequalities

We will show in this paper that if $\lambda$ is very close to 1, then $$I(M,\lambda,m)= \sup_{u\in H^{1,n}_0(M) ,\int_M|\nabla u|^ndV=1}\int_\Omega (e^{\alpha_n |u|^\frac{n}{n-1}}-\lambda\sum\limits_{k=1}^m\frac{|\alpha_nu^\frac{n}{n-1}|^k} {k!})dV,$$ can be attained, where $M$ is a compact manifold with boundary. This result gives a counter example to the conjecture of de Figueiredo, do \'o, and Ruf in their paper titled "On a inequality by N.Trudinger and J.Moser and related elliptic equations" (Comm. Pure. Appl. Math.,{\bf 55}:135-152, 2002).