Extremal Functions for Trudinger-Moser Type Inequalities in IRN

摘要

Let N≥ 2,αN =NωN 1/(N-1)N-1,where ωN-1 denotes the area of the unit sphere in IRN.In this note,we prove that for any 0 ＜ α ＜ αN and anyβ ＞ 0,the supremum u∈W1,N(IRN),sup||u||W1,N(IRN)≤1∫IRN|u|β(eα|u|N/N-1-N-2Σj=0αj/j!|u|Nj/N-1)dxcan be attained by some function u ∈ W1'N (RN) with || u|| w1,N (IRN) =1.Moreover,when α ≥ αN,the above supremum is infinity.