Li, Yu Xiang, Remarks on the Extremal Functions for the MoserTrudinger Inequality, Acta mathematica Sinica. English series (Print) , 22(2), 2006, pp. 545-550
We will show in this paper that if . is very close to 1, then I(M,.,m)=supu.H1,n0(M),.M|.u|ndV=1....e.n|u|nn.1...k=1m...nunn.1 ..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 and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" (Comm. Pure. Appl. Math., 55, 135.152, 2002).