Sharp subcritical Moser–Trudinger inequalities on Heisenberg groups and subelliptic PDEs

Abstract

The aim of this paper is to prove a sharp subcritical Moser–Trudinger inequality on the whole Heisenberg group. 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 establish the following inequality on H (Theorem 1.1): 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 exists a constant 0<Cα,β=C(α,β)<∞ such that the following inequality holds sup‖∇Hu‖LQ(H)≤11‖u‖LQ(H)Q−β∫H1ρ(ξ)β{exp(α|u|Q/(Q−1))−∑k=0Q−2αkk!|u|kQ/(Q−1)}≤Cα,β. The above result is the best possible in the sense when α≥αQ(1−βQ), the integral is still finite for any u∈W1,Q(H), but the supremum is infinite.In contrast to the analogous inequality in Euclidean spaces proved in Adachi and Tanaka (1999) [6] using symmetrization, our argument is completely different and avoids the symmetrization method which is not available on the Heisenberg group in an optimal way. Moreover, our restriction on the norm ‖∇Hu‖LQ(H)≤1 of the function u is much weaker than ‖∇Hu‖LQ(H)+‖u‖LQ(H)≤1 which was assumed in Lam and Lu (2012) [16]. As a consequence, our inequality fails at α=αQ(1−βQ) in contrast to the one in [16].As an application of this inequality, we will prove that the following nonlinear subelliptic equation of Q-Laplacian type without perturbation: (0.1)−ΔQu+V(ξ)|u|Q−2u=f(ξ,u)ρ(ξ)β in H has a nontrivial weak solution, where the nonlinear term f has the critical exponential growth eα|u|QQ−1 as u→∞, but does not satisfy the Ambrosetti–Rabinowitz condition.