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

Let Hn=R2n×RHn=R2n×R be the nn-dimensional Heisenberg group, ∇Hn∇Hn be its sub-elliptic gradient operator, and ρ(ξ)=(|z|4+t2)1/4ρ(ξ)=(|z|4+t2)1/4 for ξ=(z,t)∈Hnξ=(z,t)∈Hn be the distance function in HnHn. Denote Q=2n+2Q=2n+2 and Q′=Q/(Q−1)Q′=Q/(Q−1). It is proved in this paper that there exists a positive constant α∗α∗ such that for any pair ββ and αα satisfying 0≤β1, the above integral is still finite for any u∈W1,Q(Hn)u∈W1,Q(Hn). Furthermore the supremum is infinite if α/αQ+β/Q>1α/αQ+β/Q>1, where αQ=QσQ1/(Q−1), σQ=∫ρ(z,t)=1|z|QdμσQ=∫ρ(z,t)=1|z|Qdμ. Actually if we replace HnHn and W1,Q(Hn)W1,Q(Hn) by unbounded domain ΩΩ 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.