Remarks on the Extremal Functions for the Moser-Trudinger Inequalities

Abstract

We will show in this paper that if $\lambda$ is very close to 1, then $$I(M,\lambda ,m)=\underset{u\in H_0^{1,n}(M),{\int }_M|\nabla u{|}^{n}dV=1}{\sup }{\int }_{\Omega }(e^{{\alpha }_n|u{|}^{\frac{n}{n-1}}}-\lambda \sum _{k=1}^m\frac{|{\alpha }_n{u}^{\frac{n}{n-1}}{|}^k}{k!})dV,$$ can be attained, where $M$ is a compact manifold with boundary. This result gives a counter example to the conjecture of de Figueiredo, do ó, 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).

9 pages