Remarks on the Extremal Functions for the Moser-Trudinger Inequalities

Yuxiang Li

We will show in this paper that if $\lambda$ is very close to 1, then $$I(M,λ,m)= \sup_{u\in H^{1,n}_0(M) ,\int_M|\nabla u|^ndV=1}\int_Ω(e^{α_n |u|^\frac{n}{n-1}}-λ\sum\limits_{k=1}^m\fr…