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

Let H-n = R-2n x R be the n-dimensional Heisenberg group, del(n)(H) be its sub-elliptic gradient operator, and rho(xi) = (vertical bar z vertical bar(4) + t(2))(1/4) for xi = (z, t) is an element of H-n be the distance function in H-n. Denote Q = 2n + 2 and Q' = Q/(Q - 1). It is proved in this paper that there exists a positive constant alpha* such that for any pair beta and alpha satisfying 0 <= beta < Q and alpha/alpha* + beta/Q <= 1, (parallel to u parallel to W1,Q) sup((Hn)) (<= 1) integral(Hn) 1/rho(xi)(beta) {e(alpha)vertical bar u vertical bar(Q)' - (Q-2)Sigma(k=0) alpha(k)vertical bar u vertical bar kQ'/k!} d xi < infinity, where W-1,W-Q (H-n) is the Sobolev space on H-n. When alpha/alpha* + beta/Q > 1, the above integral is still finite for any u is an element of W-1,W-Q (H-n). Furthermore the supremum is infinite if alpha/alpha(Q)+ beta/Q > 1, where alpha(Q) = Q(sigma Q)(1)/((Q-1)), sigma(Q) = integral(rho(z,t)=1) vertical bar z vertical bar(Q)d mu. Actually if we replace H-n and W-1,W-Q(H-n) by unbounded domain Omega and W-0(1,Q) (Omega) respectively, the above inequality still holds. As an application of this inequality, a sub-elliptic equation with exponential growth is considered. (C) 2011 Elsevier Ltd. All rights reserved.