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

The aim of this paper is to prove a sharp subcritical Moser-Trudinger inequality on the whole Heisenberg group. Let H = C-n x R be the n-dimensional Heisenberg group, Q = 2n+ 2 be the homogeneous dimension of H, Q' = Q/Q-1, and rho(xi) = (|z|(4) + t(2))(1/4) be the homogeneous norm of xi = (z, t) is an element of H. Then we establish the following inequality on H(Theorem 1.1): there exists a positive constant alpha(Q) = Q(2 pi(n)Gamma(1/2)Gamma(Q-1/2)Gamma(Q/2)(-1)Gamma(n)(-1)Gamma(n)(-1))(Q'-1) such that for any pair beta, alpha satisfying 0 <= beta < Q, 0 < alpha < alpha(Q) (1 - beta/Q) there exists a constant 0 < C-alpha,C-beta = C (alpha, beta) < infinity such that the following inequality holds sup parallel to del(u)(H)parallel to(Q)(L) (H) <= 1 1/Q-beta integral(H) 1/rho(xi)(beta) {exp (alpha vertical bar u vertical bar(Q/(Q-1))) - (k=0)Sigma(Q-2) alpha k/k! vertical bar u vertical bar(kQ/(Q-1)) } <= C-alpha,C-beta. The above result is the best possible in the sense when alpha = alpha Q (1 - beta/Q), the integral is still finite for any u is an element of W-1,W-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 parallel to del Hu parallel to(LQ(H)) <= 1 of the function u is much weaker than parallel to del(H)u parallel to(LQ(H)) + parallel to u parallel to(LQ) (H) <= 1 which was assumed in Lam and Lu (2012) [16]. As a consequence, our inequality fails at alpha = alpha(Q)(1 - beta/Q) 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: -Delta(Q)u + V(xi) |u|(Q-2) u = f (xi, u)/rho (xi)(beta) in H (0.1) has a nontrivial weak solution, where the nonlinear term f has the critical exponential growth e(alpha vertical bar u vertical bar Q/Q-1) as u -> infinity, but does not satisfy the Ambrosetti-Rabinowitz condition. (C) 2013 Elsevier Ltd. All rights reserved.