The multiplicity of solutions for a class of quasilinear elliptic boundary value problems on the Heisenberg group was concerned.In the whole space,the main coefficients and their derivatives were assumed to be bounded,and the nonlinear term satisfies superlinear growth conditions.Under the above assumptions,the functional is continuous but not differentiable in the whole space.So,the nonsmooth critical point theory should be applied.The concepts about weak slope,critical point,(PS)_c conditions and some fundamental lemmas in the nonsmooth critical point theory were introduced.The properties of the critical point of the functional were analysed.The strong convergence of the (PS)_c sequences was proved by using nonlinear functional theory,Fatou's lemma,Lebesgue's dominated convergence theorem and Brezis-Browder theorem.Moreover,by virtue of the generalized Mountain Pass lemma,the existence of infinite weak solutions of the boundary value problem was confirmed and these solutions are separable from one another.