Advanced Nonlinear Studies 12 (2012), 659–681 On Nonuniformly Subelliptic Equations of Q−sub-Laplacian Type with Critical Growth in the Heisenberg Group ∗ Nguyen Lam, Guozhen Lu † Department of Mathematics Wayne State University, Detroit, Michigan 48202 e-mail: nguyenlam@wayne.edu, gzlu@math.wayne.edu Hanli Tang School of Mathematical Sciences Beijing Normal University, Beijing, China 100875 e-mail: rainthl@163.com Received in revised form 05 April 2012 Communicated by Susanna Terracini Abstract Let H n = R 2n × R be the n−dimensional Heisenberg group, ∇ H n be its subelliptic gradient operator, and ρ (ξ) = ( |z| 4 + t 2 ) 1/4 for ξ = (z, t) ∈ H n be the distance function in H n . Denote H = H n , Q = 2n + 2 and Q ′ = Q/(Q − 1). Let Ω be a bounded domain with smooth boundary in H. Motivated by the Moser-Trudinger inequalities on the Heisenberg group, we study the existence of solution to a nonuniformly subelliptic equation of the form − div H (a (ξ, ∇ H u (ξ))) = f (ξ,u(ξ)) ρ(ξ) β + εh(ξ) in Ω u ∈ W 1,Q 0 (Ω)\{0} u ≥ 0 in Ω , where f : Ω × R → R behaves like exp ( α |u| Q ′ ) when |u|→∞. In the case of Q−sub- Laplacian − div H ( |∇ H u| Q−2 ∇ H u ) = f (ξ,u) ρ(ξ) β + εh(ξ) in Ω u ∈ W 1,Q 0 (Ω)\{0} u ≥ 0 in Ω , we will apply minimax methods to obtain multiplicity of weak solutions. 2010 Mathematics Subject Classification. 42B37, 35J92, 35J62. Key words. Moser-Trudinger inequality, Heisenberg group, subelliptic equations, Q-subLaplacian, Mountain-Pass theorem, Palais-Smale sequences, existence and multiplicity of solutions. ∗ Research is partly supported by a US NSF grant DMS–0901761. † Corresponding Author: Guozhen Lu at gzlu@math.wayne.edu. 659