  We prove some results on the density and multiplicity of positive solutions
to the prescribed Webster scalar curvature problem on the $(2n+1)$-dimensional
standard unit CR sphere $(\mathbb{S} ^{2n+1},\theta_0)$. Specifically, we
construct arbitrarily many multi-bump solutions via the variational gluing
method. In particular, we show the Webster scalar curvature functions of
contact forms conformal to $\theta_0$ are $C^{0}$-dense among bounded functions
which are positive somewhere. Existence results of infinitely many positive
solutions to the related equation $-\Delta_{\mathbb{H}} u=R(\xi) u^{(n+2) /n}$
on the Heisenberg group $\Hn $ with $R(\xi)$ being asymptotically periodic with
respect to left translation are also obtained. Our proofs make use of a refined
analysis of bubbling behavior, gradient flow, Pohozaev identity, as well as
blow up arguments.
