Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications

Abstract

Let H=Cn×R be the n-dimensional Heisenberg group, Q=2n+2 be the homogeneous dimension of H, Q′=QQ−1, and ρ(ξ)=(|z|4+t2)14 be the homogeneous norm of ξ=(z,t)∈H. Then we prove the following sharp Moser–Trudinger inequality on H (Theorem 1.6): there exists a positive constant αQ=Q(2πnΓ(12)Γ(Q−12)Γ(Q2)−1Γ(n)−1)Q′−1 such that for any pair β,α satisfying 0≤β<Q, 0<α≤αQ(1−βQ) there holds sup‖u‖1,τ≤1∫H1ρ(ξ)β{exp(α|u|Q/(Q−1))−∑k=0Q−2αkk!|u|kQ/(Q−1)}≤C(Q,β,τ)<∞. The constant αQ(1−βQ) is best possible in the sense that the supremum is infinite if α>αQ(1−βQ). Here τ is any positive number, and ‖u‖1,τ=[∫H|∇Hu|Q+τ∫H|u|Q]1/Q.Our result extends the sharp Moser–Trudinger inequality by Cohn and Lu (2001) [19] on domains of finite measure on H and sharpens the recent result of Cohn et al. (2012) [18] where such an inequality was studied for the subcritical case α<αQ(1−βQ). We carry out a completely different and much simpler argument than that in Cohn et al. (2012) [18] to conclude the critical case. Our method avoids using the rearrangement argument which is not available in an optimal way on the Heisenberg group and can be used in more general settings such as Riemanian manifolds, appropriate metric spaces, etc. As applications, we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of Q-Laplacian type on the Heisenberg group (Theorems 1.8, 1.9, 1.10 and 1.11): −divH(|∇Hu|Q−2∇Hu)+V(ξ)|u|Q−2u=f(ξ,u)ρ(ξ)β+εh(ξ) with nonlinear terms f of maximal exponential growth exp(α|u|QQ−1) as |u|→∞. In particular, when ε=0, the existence of a nontrivial solution is also given.