Extremal Functions for Trudinger-Moser Type Inequalities in ℝ^N

Let N≥2, αN =Nω 1/(N−1) N−1 , where ωN−1 denotes the area of the unit sphere in RN . In this note, we prove that for any 0<α<αN and any β>0, the supremum sup u∈W1,N(RN),‖u‖ W1,N(RN ) ≤1 ∫ RN |u| ( e N N−1 − N−2 ∑ j=0 αj j! |u| Nj N−1 ) dx can be attained by some function u∈W1,N(RN) with ‖u‖W1,N(RN)=1. Moreover, when α≥αN , the above supremum is infinity. AMS Subject Classifications: 46E35 Chinese Library Classifications: O175.29