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 $$I(M,\lambda ,m)=\underset{u\in H_0^{1,n}(M),{\int }_M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{n}dV=1}{\sup }{\int }_{\mathrm{\Omega }}(e^{{\alpha }_n\mathrm{\mid }u{\mathrm{\mid }}^{\frac{n}{n-1}}}-\lambda \sum _{k=1}^m\frac{\mathrm{\mid }{\alpha }_n{u}^{\frac{n}{n-1}}{\mathrm{\mid }}^k}{k!})dV,$$ 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).