The Existence of Global Solution for a Class of Semilinear Equations on   Heisenberg Group
  Based on the concepts of a generalized critical point and the corresponding generalized P.S. condition introduced by Duong Minh Duc[1], we have proved a new $Z_2$ index theorem and get a result on multiplicity of generalized critical points. Using the result and a quite standard variational method, it is found that the equation $$ -\Delta_{H^n} u=|u|^{p-1} u ~ x\in H^n $$ has infinite positive solutions. Our approach can also be applied to study more general nonlinear problems. 
Zhujun Zheng, Keping Lu, Luo Xuebo