On the CR Nirenberg problem: Density and multiplicity of solutions

We prove some results on the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the [Formula: see text]-dimensional standard unit CR sphere [Formula: see text]. 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 [Formula: see text] are [Formula: see text]-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation [Formula: see text] on the Heisenberg group [Formula: see text] with [Formula: see text] 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.