The Moser-Trudinger inequality in unbounded domains of Heisenberg group and sub-elliptic equations

摘要

Let ~(Hn)=R2 ~n×R be the n-dimensional Heisenberg group, ~(?Hn) be its sub-elliptic gradient operator, and ρ(ξ)=(|z|4+ ~(t2))14 for ξ=(z,t)∈ ~(Hn) be the distance function in ~(Hn). Denote Q=2n+2 and ~Q′=Q(Q-1). It is proved in this paper that there exists a positive constant ~α* such that for any pair β and α satisfying 0≤β1, the above integral is still finite for any u∈W1,~Q(~(Hn)). Furthermore the supremum is infinite if α ~(αQ)+βQ>1, where ~(αQ)=QσQ1(Q-1), ~(σQ)=∫ _(ρ(z,t)=1)| ~(z|Q)dμ. Actually if we replace ~(Hn) and W1,~Q(~(Hn)) by unbounded domain Ω and W01,Q(Ω) respectively, the above inequality still holds. As an application of this inequality, a sub-elliptic equation with exponential growth is considered.