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

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) = (vertical bar z vertical bar(4) + r(2))(1/4), be the homogeneous norm of xi = (z, t) is an element of H. Then we prove the following sharp Moser-Trudinger inequality on H (Theorem 1.6): 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))(Q-1) such that for any pair beta, alpha satisfying 0 <= beta < Q, 0 < alpha <= alpha(Q)(1 - beta/Q) there holds (sic) The constant alpha(Q)(1 - beta/Q) is best possible in the sense that the supremum is infinite if alpha > alpha(Q)(1 - beta/Q). Here tau is any positive number, and parallel to u parallel to(1,tau) = [integral(H)vertical bar del(H)u vertical bar(Q) + tau integral(H)vertical bar u vertical bar(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 alpha < alpha(Q)(1 - beta/Q). 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): -div(H)(vertical bar del(H)u vertical bar(Q-2)del(H)u) + V(xi)vertical bar u vertical bar(Q-2) u = f(xi, u)/(rho(xi)(beta) + epsilon h(xi) with nonlinear terms f of maximal exponential growth exp(alpha vertical bar u vertical bar(Q/Q-1)) as vertical bar u vertical bar -> infinity. In particular, when epsilon = 0, the existence of a nontrivial solution is also given. (C) 2012 Elsevier Inc. All rights reserved.